nontradable market index and its derivatives · therefore, the spot-futures parity and the put-call...

NONTRADABLE MARKET INDEX AND ITS

DERIVATIVES

by

Peng Xu

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Graduate Department of Economics

University of Toronto

c©Copyright by Peng Xu (2009)

Nontradable Market Index and Its Derivatives

Peng Xu

Ph.D. 2009

Graduate Department of Economics

University of Toronto

Abstract

The S&P 500 Index is a leading indicator of U.S. equities and is meant to reflect the

risk and return on the U.S. stock market. Many derivatives based on the S&P 500 are

available to investors. The S&P 500 Futures of the Chicago Mercantile Exchange and

the S&P 500 Index Options of the Chicago Board Options Exchange are both actively

traded.

This thesis argues that the S&P 500 Index is only a summary statistic designed to

reflect the evolution of the stock market. It is not the value of a self-financed tradable

portfolio, and its modifications do not coincide with changes of the value of any mimicking

portfolio, due to the particular way the S&P 500 Index is computed and maintained.

Therefore, the Spot-Futures Parity and the Put-Call Parity do not hold for the S&P 500

Index and its derivatives. Furthermore, its derivatives cannot be priced by using the

standard option pricing models, which assume that the underlying asset is tradable.

ii

Chapter One analyzes why the S&P 500 Index does not represent the value of a self-

financed tradable portfolio and why it cannot be replaced by the value of a tracker such

as the SPDR. In particular, we show that the nonlinear and extreme risk dynamics of

the SPDR and of the S&P 500 Index are very different.

Chapter Two provides empirical evidence that the non-tradability of the S&P 500

Index can explain the Put-Call Parity deviations. Even after controlling for the liquidity

risk of the options, we find that the Put-Call Parity implied dividends depend significantly

on the option strike.

In Chapter Three, we develop an affine multi-factor model to price coherently various

derivatives such as forwards and futures written on the S&P 500 Index, and European put

and call options written on the S&P 500 Index and on the S&P 500 futures. We consider

the cases when the underlying asset is self-financed and tradable and when it is not,

and show the difference between them. When the underlying asset is self-financed and

tradable, an additional arbitrage condition has to be introduced and implies additional

parameter restrictions.

iii

Acknowledgements

This dissertation could not have been accomplished without Professor Christian Gourier-

oux, who not only served as my supervisor but also encouraged and challenged me

throughout my academic program. I sincerely appreciate his guidance and patience.

I would also like to thank Professor Alan White, who gave me many useful suggestions

throughout the dissertation process.

I am grateful to Professor Varouj Aivazian, Professor Frank Mathewson, Professor

Angelo Melino and Professor James Pesando, who encouraged me to enter the Ph.D.

program and helped me in various aspects.

A special thank you goes to Professor Joann Jasiak for her enormous encouragements

and her kindness in reading through the entire thesis and having many helpful discussions

with me.

Finally, I want to thank my parents, Chunhong Wang and Xingli Xu, and my husband,

Yu Chen, for their infinite patience and moral support.

iv

Contents

1 Is the S&P 500 Index tradable? 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The S&P 500 Index, the Trackers and the Index Derivatives . . . . . . . 7

1.2.1 Description of the S&P 500 Index . . . . . . . . . . . . . . . . . . 7

1.2.2 Mimicking the S&P 500 . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.2.1 S&P 500 Index Funds . . . . . . . . . . . . . . . . . . . 16

1.2.2.2 Exchange Traded Funds . . . . . . . . . . . . . . . . . . 17

1.2.2.3 Static Comparison of the Relative Price Changes of the

SPDR and of the S&P 500 Index . . . . . . . . . . . . . 22

1.2.2.4 Dynamic Comparison of the SPDR and the S&P 500 Index 26

1.3 The Effects of the Non-Tradability of the Index . . . . . . . . . . . . . . 29

1.3.1 Derivative Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3.2 Spot-Futures Parity and Put-Call Parity . . . . . . . . . . . . . . 30

1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2 The S&P 500 Index Options and the Put-Call Parity 37

v

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2 S&P 500 Index Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2.1 The Characteristics of Traded Options . . . . . . . . . . . . . . . 40

2.2.2 The Activity on the Option Market . . . . . . . . . . . . . . . . . 42

2.3 The Put-Call Parity Implied Dividends of the S&P 500 Index . . . . . . 49

2.3.1 Calculation of the Implied Dividends . . . . . . . . . . . . . . . . 49

2.3.2 Data and Empirical Results . . . . . . . . . . . . . . . . . . . . . 51

2.3.3 Discussions of the Empirical Results . . . . . . . . . . . . . . . . 55

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3 Non-tradable S&P 500 Index and the Pricing of Its Traded Derivatives 58

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 The Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2.1.1 Historical Dynamics of the Index . . . . . . . . . . . . . 61

3.2.1.2 Specification of the Stochastic Discount Factor . . . . . 64

3.2.2 Pricing Formulas for European Derivatives Written on the Index . 66

3.2.2.1 Power Derivatives Written on the Index . . . . . . . . . 67

3.2.2.2 The Risk-free Term Structure . . . . . . . . . . . . . . . 67

3.2.2.3 Forward Prices for the S&P 500 Index . . . . . . . . . . 69

3.2.2.4 Futures Prices . . . . . . . . . . . . . . . . . . . . . . . 70

3.2.2.5 European Call and Put Options Written on the Index . 71

vi

3.2.3 Pricing Formulas for European Derivatives Written on Futures . . 72

3.2.3.1 Derivatives Written on Futures . . . . . . . . . . . . . . 72

3.2.3.2 European Call Options Written on Futures . . . . . . . . 74

3.3 Parameter Restrictions for a Tradable Index . . . . . . . . . . . . . . . . 74

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A Official Description of the Index 78

B Proofs of Propositions 85

B.1 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 85








C Tables and Figures 98

vii

List of Tables

C.1 Summary Statistics of the SPDR Traded from Jan 2, 2001 to Dec 30, 2005. 99

C.2 Ljung-Box Statistics for Daily Relative Price Changes of the S&P 500

Index and the SPDR ( XR and DRR) from Jan 2, 2001 to Dec 30, 2005. . 100

C.3 Ljung-Box Statistics for Daily Holding Period Returns of the S&P 500

Index and the SPDR ( XR and DRRd) from Jan 2, 2001 to Dec 30, 2005. 100

C.4 Ljung-Box Statistics for Squared Daily Relative Price Changes of the

S&P 500 Index and the SPDR ( XR2 and DRR

2) from Jan 2, 2001 to

Dec 30, 2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

C.5 Ljung-Box Statistics for Squared Daily Holding Period Returns of the

S&P 500 Index and the SPDR ( XR2 and DRR

2) from Jan 2, 2001 to

Dec 30, 2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

C.6 Times-to-Maturity of the S&P 500 Index Options . . . . . . . . . . . . . 102

C.7 Summary Statistics of the Options Traded from Jan 2, 2003 to Dec 31,

2003 (252 days). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

viii

C.8 Summary Statistics of the Actively Traded Options (with traded volume

≥ 2,000 contracts) from Jan 2, 2003 to Dec 31, 2003 (252 days). . . . . . 105

C.9 Summary Statistics of the Actively Traded Call and Put Options with the

Same Strike Prices and Times-to-Maturity from Jan 2, 2003 to Dec 31,

2003 (252 days). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

C.10 Summary Statistics of Implied Dividends, Q(t,K,T-t), with Treasury Bill

Rates as Proxies for the Risk-free Rates . . . . . . . . . . . . . . . . . . . 106

C.11 Summary Statistics of Implied Dividends, Q(t,K,T-t), with Zero Rates as

Proxies for the Risk-free Rates . . . . . . . . . . . . . . . . . . . . . . . . 107

ix

List of Figures

C.1 Bid-Ask Spread and Trading Volume (shares) of the SPDR from Jan 2,

2001 to Dec 30, 2005 (1,256 observations) . . . . . . . . . . . . . . . . . . 108

C.2 Histograms of Bid-Ask Spread and Trading Volume (shares) of the SPDR

from Jan 2, 2001 to Dec 30, 2005 . . . . . . . . . . . . . . . . . . . . . . 109

C.3 The S&P 500 Index and the SPDR×10 from Jan 2, 2001 to Dec 30, 2005

(1,256 observations). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

C.4 Level Difference between the S&P 500 Index and the SPDR × 10 from

Jan 2, 2001 to Dec 30, 2005. . . . . . . . . . . . . . . . . . . . . . . . . . 111

C.5 Level Difference between the S&P 500 Index and the SPDR × 10 from

Feb 1, 1993 to Dec 30, 2005. . . . . . . . . . . . . . . . . . . . . . . . . . 111

C.6 Daily Relative Price Changes (log xt−log xt−1) of the S&P 500 Index (XRt)

and the SPDR (DRRt) from Jan 2, 2001 to Dec 30, 2005 (1,255 observations).112

C.7 Daily Holding Period Returns (log(xt + dt) − log xt−1) of the S&P 500

Index (XRdt ) and the SPDR (DRR

dt ) from Jan 2, 2001 to Dec 30, 2005

(1,255 observations). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

x

C.8 Daily Relative Price Change Difference between the S&P 500 Index and

the SPDR from Jan 2, 2001 to Dec 30, 2005. . . . . . . . . . . . . . . . . 114

C.9 Daily Holding Period Return Difference between the S&P 500 Index and

the SPDR from Jan 2, 2001 to Dec 30, 2005. . . . . . . . . . . . . . . . . 114

C.10 Histogram of Daily Relative Price Changes of the S&P 500 Index (XR)and

the SPDR (DRR) and Daily Holding Period Returns of the SPDR (DRRd)from

Jan 2, 2001 to Dec 30, 2005. . . . . . . . . . . . . . . . . . . . . . . . . . 115

C.11 Histogram of Relative Price Change Difference between the S&P 500 Index

and the SPDR and Histogram of Daily Holding Period Return Difference

between the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30,

2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

C.12 Squared Relative Price Changes of the S&P 500 Index (XR2t ) and the

SPDR (DRR2t ) from Jan 2, 2001 to Dec 30, 2005. . . . . . . . . . . . . . . 117

C.13 Squared Daily Holding Period Returns of the S&P 500 Index and the

SPDR from Jan 2, 2001 to Dec 30, 2005. . . . . . . . . . . . . . . . . . . 117

C.14 Autocorrelation Function for Squared Daily Relative Price Changes of the

S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005. . . . . . 118

C.15 Autocorrelation Function for Squared Daily Holding Period Returns of the

S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005. . . . . . 118

xi

C.16 Cross-Correlation Function for Squared Relative Price Changes of the

S&P 500 Index and the SPDR and Autocorrelation Function for Squared

Relative Price Changes of the S&P 500 Index from Jan 2, 2001 to Dec 30,

2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

C.17 Cross-Correlation Function for Squared Daily Holding Period Returns of

the S&P 500 Index and the SPDR and Autocorrelation Function for Squared

Daily Holding Period Returns of the S&P 500 Index from Jan 2, 2001 to

Dec 30, 2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

C.18 Autocorrelation Function of Max(exp(Rt)− k, 0) for Daily Relative Price

Changes of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30,

2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

C.19 Autocorrelation Function of Max(exp(Rdt )− k, 0) for Holding Period Re-

turns of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30,

2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

C.20 Autocorrelation Function of Max(k− exp(Rt), 0) for Daily Relative Price

Changes of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30,

2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

C.21 Autocorrelation Function of Max(k − exp(Rdt ), 0) for Holding Period Re-

turns of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30,

2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

C.22 Number of Times-to-Maturity on Each Trading Day in 2003 . . . . . . . 125

C.23 Times-to-Maturity on Each Trading Day in 2003 . . . . . . . . . . . . . . 125

xii

C.24 Times-to-Maturity of Options with High Volume on Each Trading Day in

2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

C.25 Strike Prices of Options with High Volume on Each Trading Day in 2003 127

C.26 Moneyness of Options with High Volume on Each Trading Day in 2003 . 127

C.27 Prices of Traded Options with Traded Volume ≥ 2,000 Contracts in 2003 127

C.28 Number and Proportion of Highly Traded Options on Each Trading Day

in 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

C.29 Number of Highly Traded Put Options and Its Proportion in Highly Traded

Options on Each Trading Day in 2003 . . . . . . . . . . . . . . . . . . . . 129

C.30 Number of Actively Traded Options with Time-to-Maturity T1 and Its

Proportion in the Actively Traded Options on Each Trading Day in 2003 130







C.34 Moneyness, Total Volume and Total Value of Actively Traded Options

with Time-to-Maturity T1 on Each Trading Day in 2003 . . . . . . . . . 134



xiii



C.37 Moneyness of Actively Traded Call and Put Options with the Same Strike

Prices and Times-to-Maturity on Each Trading Day in 2003 . . . . . . . 137

C.38 Time-to-Maturity of Actively Traded Call and Put Options with the Same

Strike Prices and Times-to-Maturity on Each Trading Day in 2003 . . . . 137

C.39 Implied Dividends, Q(t,K,T-t), with Treasury Bill Rates as Proxies for the

Risk-free Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

C.40 Implied Dividends, Q(t,K,T-t), with the same t and T-t, but different K,

and with Treasury Bill Rates as Proxies for the Risk-free Rates . . . . . 138

C.41 Difference between the maximum and minimum Q(t,K,T-t), with the same

t and T-t, but different K, and with Treasury Bill Rates as Proxies for the

Risk-free Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

C.42 Implied Dividends, Q(t,K,T-t), with the same t and K, but different T-t,

and with Treasury Bill Rates as Proxies for the Risk-free Rates . . . . . 139


t and K, but different T-t, and with Treasury Bill Rates as Proxies for the

Risk-free Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

C.44 Implied Dividend, Q(t,K,T-t), with Zero Rates as Proxies for the Risk-free

Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

C.45 Implied Dividends, Q(t,K,T-t), with the same t and T-t, but different K,

and with Zero Rates as Proxies for the Risk-free Rates . . . . . . . . . . 141

xiv


t and T-t, but different K, and with Zero Rates as Proxies for the Risk-free

Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

C.47 Implied Dividends, Q(t,K,T-t), with the same t and K, but different T-t,

and with Zero Rates as Proxies for the Riskfree Rates . . . . . . . . . . 142


t and K, but different T-t, and with Zero Rates as Proxies for the Risk-free

Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

xv

Chapter 1

Is the S&P 500 Index tradable?

1

1.1 Introduction

The S&P 500 Index is one of the most commonly used benchmarks for the overall U.S.

stock market. It is a leading indicator of U.S. equities and is meant to reflect the risk and

return of the whole market. A number of derivatives based on the S&P 500 are available

to investors. In particular, the S&P 500 futures traded in the Chicago Mercantile Ex-

change (CME) and the S&P 500 options traded in the Chicago Board Options Exchange

are actively traded because they “extend the range of investment and risk management

strategies available to investors by offering them the possibility of unbundling the market

and nonmarket components of risk and return in their portfolios” [Figlewski (1984a)].

The success of these derivatives has not only attracted the investors, but also drawn

a lot of attention from researchers who study their pricing. For example, the S&P 500

Index and its derivatives have been widely used to test the Spot-Futures Parity, the

Put-Call Parity and different kinds of pricing models.

The Spot-Futures Parity is based on the argument that the spot asset and the for-

ward contract written on it can form a risk-free portfolio in a frictionless market. This

portfolio must earn risk-free interest because otherwise the arbitrageurs will make profit

from this opportunity and close the gap quickly. The futures contract is generally dif-

ferent from the forward contract because of the mark-to-market rule. However, when

the interest rates are non-stochastic, the futures and forward prices are the same [see

Cox, Ingersoll and Ross (1981) and French (1983)], and the parity relationship holds also

for the spot asset and the futures written on it. Figlewski wrote a series of papers on

2

the hedging performance and basis risk in stock index futures and on the index-futures

arbitrage. One key result of these papers is that the Spot-Futures Parity implied by the

No-Arbitrage condition cannot be verified using the S&P 500 Index and the index futures

data. Mackinlay and Ramaswamy (1988) confirmed this result using intraday transaction

data for the S&P 500 stock index futures prices and intraday quotes for the underlying

index. Five explanations have been proposed for the deviation between the futures prices

and index values. First, the index is not self-financed in the sense that the component

stocks of the index pay dividends, while the S&P 500 Index reflects the evolution of the

market value of the 500 listed firms without reflecting the dividends paid by those stocks.

The dividends paid by the index component stocks are not known ex-ante, so arbitrageurs

have to bear additional carrying cost and risk of the dividends[See Figlewski (1984a,b)].

This implies that the arbitrage portfolio is not risk-free and hence does not necessarily

earn risk-free interest. Second, in the real world due to transaction costs such as commis-

sions and bid-ask spreads, the arbitrage trading required to reach the Spot-Futures Parity

cannot be easily completed. Spot and futures prices can deviate from their theoretical

prices without inducing an arbitrage opportunity [See Figlewski (1984a,b), Mackinlay

and Ramaswamy (1988) and reference therein]. Third, the mark-to-market rule of the

futures differentiates it from the forward contract [See Cox, Ingersoll and Ross (1981)

and French (1983)]. Daily cash flows from the futures position may include unanticipated

interest earnings or costs. The covariance between the interest rates and the futures is

almost always nonzero. As a consequence, in general the price of a forward contract with

a certain delivery date is not the same as the price of a futures with the same delivery

3

date. Fourth, there are different taxes for spot and futures earnings, and a tax-timing

option exists for a spot position, but not for a futures position1. This may reduce the

futures price below its predicted level. And fifth, a risk-free arbitrage between a large

portfolio of 500 stocks and a futures contract is impossible, since it is impossible to buy or

sell all of the stocks simultaneously at the desired size to take advantage of the short-run

deviation of the futures price from its implied level [See Figlewski (1984a)]. The first four

explanations have been carefully studied either theoretically and/or empirically. There

was no conclusion about whether they are sufficient to explain the mispricing of futures

contracts relative to the spot index. By contrast, little is known about the fifth explana-

tion. The issue here is that the S&P 500 is a pure index of stocks, that is, an artificial

number, and is non-tradable. Specifically, it is not the price of a portfolio traded in the

financial market, because the investors cannot buy a portfolio at the price of the index

one day and be guaranteed that they can sell it anytime in the future at the future price

of the index. This is because no portfolio can replicate the index perfectly due to the

1Cornell and French (1983) suggested that the discrepancy between the actual and theoretical futuresprices is caused by taxes. Capital gains and losses are only taxed when they are realized. If stock holdershave capital losses, they can transfer part of their losses to the government by selling the stocks anddeduct losses at the ordinary short term rate. If there are capital gains, the investors can delay the sellingand take advantage of the long-term capital gain rate. This so-called “timing option” is not available forfutures traders. All capital gains and losses must be realized either at the end of the fiscal year, or at theexpiry of the futures. Including the tax option in the Cornell and French model reduces the predictedfutures prices. Cornell (1985) tested this conjecture empirically and his results showed that the patternof discrepancy between the actual and predicted futures prices is not consistent with the prediction ofthe timing option model. Figlewski (1984b) and Cornell (1985) discussed the reasons why it was difficultto judge the importance of the timing option in explaining the discrepancy. The value of the timingoption is dependent on unknown parameters such as the average holding period and taxable basis forthe market stock portfolio and the average investor’s marginal tax rate. A huge number of stocks areheld by tax-exempt institutions or by taxable investors whose holding periods are already greater thanone year. If the timing option is the main reason for the futures discount, these investors should takeadvantage of it by selling the stocks and buying the futures and risk-free securities. Furthermore, thetransaction costs or tax-related constraints may also reduce the value of the timing option.

4

composition and management of the index as well as because of the frictions and the lack

of perfect liquidity of the market. In Section 1.2.1, we elaborate on the reasons why the

S&P 500 is non-tradable. We also show that some trackers such as the SPDR can mimic

the linear dynamics of the index. However, the nonlinear and extreme risk dynamics

of the SPDR are very different from those of the index. If the index is non-tradable,

investors cannot form a risk-free portfolio including the index and the futures contract,

so the Spot-Futures Parity does not hold.

The non-tradability of the index can also help explain why the Put-Call Parity does

not hold for the S&P 500 Index and the options written on it. The Put-Call Parity

implied by the No-Arbitrage condition was first formalized by Stoll (1969) and Merton

(1973a). It is based on the argument that the spot asset and its put and call options can

form a risk-free portfolio in a frictionless market. The existence of arbitrageurs ensures

that there is no arbitrage opportunity remaining in the market and the portfolio will earn

a risk-free rate. Using daily and intradaily prices of the S&P 500 and its options from

1986-1989, Kamara and Miller (1995) observed parity deviations even after controlling

for the effects of dividends and transaction costs. Ackert and Tian (2001) found a similar

price deviation using data from 1992 to 1994. They both tested whether this was due

to the “liquidity risk”, that is, the difficulty of buying or selling a portfolio of 500 stocks

simultaneously before adverse price movements occurred. They both observed supporting

evidence that the “liquidity risk” can help explain the parity deviation. However, they

both treated the S&P 500 Index as a tradable asset, which is not very liquid. However,

the index is not just illiquid, but instead completely non-tradable in the market. As

5

for the Spot-Futures Parity, if the index is not tradable in the market, investors cannot

make risk-free arbitrage transactions using the index and the options. Therefore, the

No-Arbitrage condition does not imply the Put-Call Parity for the index and its options.

Let us also consider option pricing. The S&P 500 Index options are frequently used

for testing the option pricing models for the following reasons: First, the S&P options

are among the most liquid options. Second, they are European options. And third, the

underlying asset is an index whose values are less likely to jump due to the effect of

diversification. Therefore, the index satisfies several assumptions underlying the Black-

Scholes model. However, the options are written on an index which is not traded on the

market, and thus the index itself cannot be used in the arbitrage strategy. For example,

the well-known Black-Scholes model [See Black and Scholes (1973) and Merton (1973b)]

assumes that the underlying asset price follows a geometric Brownian Motion process.

The underlying asset and the option can be used to construct an instantaneous risk-

free portfolio, which should earn an instantaneous risk-free return by the No-Arbitrage

argument. The No-Arbitrage argument along with the terminal condition of the option

determines the option price. However, if the underlying asset is not traded, the Black-

Scholes formula could be violated without creating arbitrage opportunities. This is also

true for the option pricing formulas derived from the general equilibrium models. If the

investors account for the non-tradability of the asset, consumption, investment decisions

and equilibrium prices differ from their model prediction.

The rest of this paper is organized as follows: In Section 1.2, we introduce the S&P 500

Index and its derivatives, and explain why the S&P 500 Index is non-tradable and cannot

6

be perfectly mimicked. We analyze the effect of non-tradability on the Spot-Futures

Parity, the Put-Call Parity and the option pricing in Section 1.3. We conclude in Section

1.4. Technical results and details are gathered in the Appendices.

1.2 The S&P 500 Index, the Trackers and the Index

Derivatives

1.2.1 Description of the S&P 500 Index

The S&P 500 Index is a recognized barometer of the U.S. stock market, which has been

widely used in asset pricing studies and is a benchmark for most professional investors.

The S&P 500 Index is based on the stock prices of 500 different companies – including

about 80% industrials, 3% utilities, 1% transportation companies, and 15% financial

institutions. The market value of the 500 firms of the index represents approximately

80% of the value of all stocks traded on the New York Stock Exchange.

The S&P 500 Composite Index is calculated as follows2:

indext+1

indext=

∑i pi(t+1),t+1qi(t+1),t+1∑i pi(t+1),tqi(t+1),t+1

(1.1)

=∑i

pi(t+1),tqi(t+1),t+1∑i pi(t+1),tqi(t+1),t+1

pi(t+1),t+1

pi(t+1),t

, (1.2)

where pi(t+1),t+1 is the last trading price of security i(t + 1) at time t+1, qi(t+1),t+1 is its

number of shares available to public at time t+1, and pi(t+1),t is the last trading price of

security i(t+ 1) at time t. The S&P 500 Index is updated every 30 seconds on a trading

2Appendix A discusses S&P’s presentation of the index, where a so-called ”divisor” is directly intro-duced in the calculation. It is shown that their method and the method presented here are identical.

7

day, so t+1 is generally 30 seconds later than t, except that, if t is the closing time of one

day, then t+1 is the opening time of the next day. Since September 19, 2005, the S&P 500

Index has been moving to float adjustment, that is, the index only includes shares that

are easily available to investors. The shares that are not in the float include those closely

held by other publicly traded companies, control groups, or government agencies, if they

total more than 10%. The Standard and Poor’s Agency determines what percentage

of its shares is available to the public for each company in the index. This percentage

is called the investable weight factor or IWF. The quantity qi(t),t is equal to the Index

Shares, that is, the share count that S&P uses in its index calculations, multiplied by

the IWF. The composition of the index can change over time due to the deletion and

inclusion of stock(s). Hence the set of index component stocks at t, {i(t)}500i=1, may not

be the same as that at t+1, {i(t + 1)}500i=1

3. The index ratio is computed as the sum of

the products of prices and quantities at t + 1 of all stocks, i = 1, · · · , 500, from the set

{i(t+ 1)}500i=1, divided by the sum of the products of prices at t and quantities at t+ 1 of

all stocks, i = 1, · · · , 500, from the set {i(t+ 1)}500i=1.

Equation (1.1) shows that the S&P 500 Index is essentially a Paasche chain index for

prices, where the quantities of the index component stocks at t+ 1 are used, that is,

indextindex0

=t−1∏τ=0

∑i pi(τ+1),τ+1qi(τ+1),τ+1∑i pi(τ+1),τqi(τ+1),τ+1

,

3The changes to the index composition only happen after the market closes. If time t and time t+ 1are within the same day, {i(t)}500i=1 is the same as {i(t + 1)}500i=1. If t is the closing time of one day andt+ 1 is the opening time of the next day, then {i(t)}500i=1 may be different from {i(t+ 1)}500i=1.

8

with 4index0 = 10. The change in the index is defined as a percentage change in the

total market value from one point in time to the next.

Equation (1.2) implies that the rate of return of the index equals the rate of return

that would be earned by an investor holding a portfolio that consists of all 500 stocks

in the index and is weighted in proportion to each stock’s float-adjusted market value

(which is defined here as the product of each component stock’s price and the number of

shares available to public for that company), except that the index does not reflect cash

dividends paid out by those stocks5. Indeed, the S&P 500 is not self-financed and is a

float-adjusted-capitalization-weighted (FACap-weighted) index reflecting the evolution of

the (float-adjusted) market value of the 500 listed firms. Because of the manner in which

the index is weighted, a price change in any stock will affect the index in proportion to

the stock’s relative (float-adjusted) market value.

The S&P 500 is maintained by the S&P Index Committee, which is a team of Standard

& Poor’s economists and index analysts who meet on a regular basis. The identities of

index component stocks and their share numbers used in the index computation may be

adjusted on each trading day after the market closes. In other words, {qi(t),t}500i=1 may be

different from {qi(t+1),t+1}500i=1, if t is the closing time of one day and t+1 is the opening

time of the next day. The adjustment can lead to the following effects: First, after

the market closes at t, {i(t + 1)} is different from {i(t)} and hence {qi(t+1),t+1} differs

from {qi(t),t}. This could be caused by the identity change of index components due to

4The S&P 500 index has a Base Value of 10 and a Base Period of 1941-1943.5In the S&P 500 Index, about 380 stocks pay dividends.

9

the deletion and inclusion of stock(s). For example, 176 index identity changes have

been observed from Jan 5, 2000 to March 31, 2006. Second, after the market closes at

t, {i(t + 1)} can be the same as {i(t)}, but {qi(t+1),t+1} can be different from {qi(t),t}.

That effect occurs when the stock composition of the index remains the same, but the

numbers of shares available to the public change. This happens when the total number

of outstanding shares available to the public of one or more component index securities

changes due to secondary offering, repurchases, conversions, or other corporate actions.

It is possible that both effects occur simultaneously. Changes to the S&P 500 Index are

made whenever they are needed. There is no annual or semi-annual updating frequency.

Instead, changes in response to corporate actions and market developments can be made

at any time. These changes are typically announced after the closure of a trading day,

which is two to five days before the changes are scheduled to be implemented. The Index

Committee also lays down the policies about share changes6.

The above discussion shows that the S&P 500 index is an artificial number constructed

to reflect the evolution of the market. In the remaining part of this subsection, we explain

why no self-financed portfolio can be constructed to replicate the index perfectly due to

the particular way the S&P 500 Index is calculated and maintained.

Let us consider two consecutive trading days. We use t to denote a point in time on

the first day, and t* and t’ the closing time on the first and second day, respectively.

Let us first suppose that there is no transaction cost in trading the component stocks

of the S&P 500, and that none of the stocks pays dividends. Consider an investor who

6See Appendix A for details.

10

buys a portfolio withqi(t),t∑

i pi(t),t−1qi(t),tindext−1

of each stock i(t) at time t. If the investor can buy

each stock at exactly pi(t),t, then the cost of the portfolio is equal to∑

i pi(t),tqi(t),t∑i pi(t),t−1qi(t),t

indext−1

or

indext by Equation (1.1). Suppose there is no adjustment to the index after the market

closes at t*, that is, {i(t′)} is the same as {i(t)} and {i(t∗)}, and {qi(t′),t′} is the same as

{qi(t),t} and {qi(t∗),t∗}. If the investor can sell the portfolio at exactly {pi(t′),t′}500i=1, then at

t′ s/he will get

∑i pi(t),t′qi(t),t∑i pi(t),t−1qi(t),tindext−1

=indext−1

∑i pi(t∗),t∗qi(t∗),t∗∑i pi(t),t−1qi(t),t

∑i pi(t∗+1),t∗qi(t∗+1),t∗+1∑

i pi(t∗),t∗qi(t∗),t∗

×∑

i pi(t′),t′qi(t′),t′∑i pi(t∗+1),t∗qi(t∗+1),t∗+1

=indext−1indext∗

indext−1

∑i pi(t∗+1),t∗qi(t∗+1),t∗+1∑


indext′

indext∗

=indext′ .

The holding period return of the portfolio isindext′indext

. Thus, without any adjustment to

the index composition, the S&P 500 can be replicated using the buy-and-hold portfolio

strategy if all the 500 stocks are perfectly liquid and the investor can buy and sell all

of them at the last trading prices at the desired time. However, not all of the stocks

in the S&P 500 are very liquid, and it is almost impossible for all of the 500 stocks’

transactions to occur at exactly the last trading prices7. Therefore, replicating the index

with the buy-and-hold strategy is almost impossible.

Suppose now that the index composition is adjusted after the stock market closes on

the first day, that is, {qi(t′),t′} is different from {qi(t),t} and {qi(t∗),t∗}, and/or {i(t + 1)}7Indeed, the smallest stocks included in the index are not always very liquid, which explains why the

time interval between two trading can be quite large.

11

is different from {i(t)} and {i(t∗)}. Let us consider again the investor who buys at time

t an index replicating portfolio ofqi(t),t∑


of each stock i(t) with portfolio value

indext by Equation (1.1). The value of the portfolio at t′ is

∑i pi(t),t′qi(t),t∑i pi(t),t−1qi(t),tindext−1

=indext−1

∑i pi(t∗),t∗qi(t∗),t∗∑i pi(t),t−1qi(t),t

∑i pi(t∗+1),t∗qi(t∗+1),t∗+1∑


×∑

i pi(t),t′qi(t),t∑i pi(t∗+1),t∗qi(t∗+1),t∗+1

=indext∗

∑i pi(t∗+1),t∗qi(t∗+1),t∗+1∑


∑i pi(t),t′qi(t),t∑

i pi(t∗+1),t∗qi(t∗+1),t∗+1

6=indext∗indext′

indext∗

=indext′ .

After the adjustment, the portfolio that replicates the index at time t no longer replicates

the index at time t′. In order for the portfolio to replicate the index both before and

after the adjustment, the investor has to adjust his/her portfolio allocation and hold

qi(t′),t′∑i pi(t′),t∗qi(t′),t′

indext∗

of each stock i(t′) at t*. The sales and purchases of the stocks have to be

made at the closing prices of the first day, {pi(t),t∗} and {pi(t′),t∗}, so that this portfolio can

always have the same value as the index. To see this, we have to show that the value of

the portfolio withqi(t),t∑


of each stock i(t) at time t is equal to indext by Equation

(1.1). The value of the same portfolio at t* is∑

i pi(t),t∗qi(t),t∑i pi(t),t−1qi(t),t

indext−1

, which is equal to indext∗ .

The value of the portfolio withqi(t′),t′∑

i pi(t′),t∗qi(t′),t′indext∗

of each stock i(t′) at t* is∑

i pi(t′),t∗qi(t′),t′∑i pi(t′),t∗qi(t′),t′

indext∗

,

which is still equal to indext∗ . The value of the portfolio withqi(t′),t′∑


of each stock

i(t′) at t’ is∑

i pi(t′),t′qi(t′),t′∑i pi(t′),t∗qi(t′),t′

indext∗

, which is equal to indext′ . If there is an adjustment to the

index composition and the investor can make the corresponding adjustment to his/her

12

replicating portfolio using the above mentioned self-financing strategy, the index can still

be replicated. If there is one adjustment to the index, the adjustment to the portfolio has

to involve 500 stocks or more. The new quantity,qi(t′),t′∑


, of each stock that the

investor has to hold depends on {pi(t′),t∗}, {qi(t′),t′}, {pi(t∗),t∗} and {qi(t∗),t∗}. Quantities

{qi(t′),t′} and {qi(t∗),t∗} are usually known before the market is closed. Prices {pi(t′),t∗}

and {pi(t∗),t∗} are the closing prices of {i(t′)} and {i(t∗)} at t*, and are not known until

the market is closed. Therefore, if the investor decides to adjust the portfolio before t∗8,

s/he has to estimate the closing prices pi(t′),t∗ and pi(t∗),t∗ for about 500 stocks, determine

the quantity of each stock to be held based on his/her estimations, and trade the stocks

at prices as close as possible to the closing prices. To minimize the hedging error, the

estimation and the trading should be made as near as possible to the end of the trading

day. Because the market is not perfectly liquid, it may not be possible to trade the stocks

instantaneously and it may take some time to complete the transactions for all stocks.

The lack of perfect foresight and perfect liquidity may lead both the quantities and prices

away from those desired, and this divergence could involve all the relevant stocks. This

can cause large tracking errors. If the investor instead decides to adjust the portfolio at

the beginning of the second day, the opening prices of the second day are generally not

the same as the closing prices of the first day due to unexpected overnight information

flow and market adjustment9, so the investors still cannot obtain the desired quantity

8Empirical evidence shows that in order to minimize tracking errors most institutional investors suchas index funds choose to adjust the portfolio on the effective day, that is, the day after the closure atwhich the change to the index becomes effective [See e.g. Beneish and Whaley (1996) and Cusick (2002)].

9As shown in Beneish and Whaley (1996), Lynch and Mendenhall (1997) and Cusick (2002), after theannouncement of the index addition and/or deletion, there is a significant positive (resp. negative) meanabnormal return until the effective day for the stock to be added (resp. deleted). The overnight and

13

of each stock i(t′) at the desired price. In summary, without perfect foresight and a

perfectly liquid market, it is impossible for the investor to obtain the desired quantity of

each stock i(t′) at the closing price of t*. Since the investor’s portfolio allocation on the

second day is not equal to the index allocation, we still conclude that the S&P 500 index

cannot be perfectly replicated.

Moreover, because the adjustment to the index occurs whenever it is needed, investors

cannot perfectly anticipate how many changes there will be and what changes will take

place during their holding period of the portfolio. This adds more uncertainty to the

future value of the replicating portfolio. Investors cannot buy a portfolio at the price

equal to the index value today and expect to sell it at the price equal to the future index

value later10.

Now let us also take into account the transaction costs and dividends paid by the

component stocks. When an investor holds a portfolio that replicates the index, s/he not

only has to pay the transaction costs when purchasing and selling the portfolio of about

500 stocks, but s/he also needs to pay for the transaction costs at each time the index and

hence the portfolio are adjusted. This makes the replicating strategy described above no

daily mean abnormal returns after the effective day are negative (resp. positive). The trading volume ofthe stock starts rising after the announcement day and peaks on the effective day. What could happen isthat, after the announcement of the change, the risk arbitrageurs know that there will be a huge demand(resp. supply) for the stock to be added (resp. deleted) on the effective day. Then, they start to buy(resp. sell) the stock, driving the price up (resp. down). The index fund managers do not want to buy(resp. sell) the stock earlier because they may not be rewarded for creating “tracking errors”. On theeffective day, the fund managers buy (resp. sell) the stock. The next day, the risk arbitrageurs sell (resp.buy) the extra stocks due to their overestimation of the demand (resp. supply) of the fund manager.

10This is consistent with the empirical results in Mackinlay and Ramaswamy (1988), where the mis-pricing of the futures relative to the spot is positively related to time-to-maturity. For a very shorttime-to-maturity, there is little chance that an adjustment to the index will occur. So investors canuse the buy-and-hold strategy to replicate the index and take advantage of the arbitrage opportunity, ifthere is one after considering the transaction cost, dividend, etc.

14

longer self-financing. Indeed, if most of the index mimicking fund managers adjust their

portfolio about the same time, the transaction cost could be very high. Moreover, the

number of changes is uncertain, which increases the risk on the portfolio and makes it

even more difficult for an investor to replicate the index.

There are about 380 stocks in the S&P 500 Index that pay dividends. The payment

dates and amounts of dividends are generally not perfectly predictable. This by itself

does not cause the non-tradability of the index since a lot of securities traded in the

market pay dividends which are not known ex-ante11. If the market is frictionless, with

perfect liquidity and there is no adjustment to the index, the portfolio with quantity

qi(t),t∑i pi(t),t−1qi(t),t

indext−1

of each stock i(t) purchased at price qi(t),t at time t replicates the index

at time t with value indext. Whenever there is an ex-dividend of the index component

stocks, the relevant stock prices and the index level may adjust accordingly. If the

replicating portfolio pays exactly the same amount of dividends as the index component

stocks, the value of the portfolio will always mimic the index exactly. That is, the index

can be seen as the price of the replicating portfolio which pays dividends and is not

self-financing itself. But as discussed above, even without dividends, the index cannot

be replicated perfectly in a market with friction, lack of perfect liquidity, and periodical

adjustments to the index. The presence of dividends introduces additional risks on the

replicating portfolio. When an investor buys a replicating portfolio at t, s/he not only is

uncertain about the dates and sizes of the future dividends of each component stock, but

11Although the uncertainty of the dividends does not necessarily cause the non-tradability of the index,it may cause Spot-Futures Parity and Put-Call Parity deviations.

15

also about the identities and quantities of the stocks that will be held and pay him/her

dividends during the holding period. Because investors do not know ex-ante what change

will occur to the index, how many changes there will be and when they will occur, they

do not know what stocks and how many shares of these stocks they will have to hold

later on to replicate the index. This aggravates the uncertainty about the dividends of

the replicating portfolio.

Since the S&P 500 Index is non-tradable, many financial products have been intro-

duced on the market to mimic the index. We discuss below the so-called S&P 500 Index

Funds and Exchange Traded Funds12.

1.2.2 Mimicking the S&P 500

1.2.2.1 S&P 500 Index Funds

S&P 500 Index Funds are mutual funds seeking to replicate and track the performance

of the S&P 500 Index. This is accomplished by holding either all of the securities in the

index in the appropriate proportions, or by holding a selected sample of securities that

closely track the desired index. Like any other fund, the index funds entail operating

expenses and transaction costs. Typically, an index fund distributes to shareholders

its net income (interest and dividends, less expenses) as well as any net capital gains

realized from the sales of its holdings. For example, the largest index fund, Vanguard

Index Fund, distributes its income dividends in March, June, September and December,

and its capital gains in December. In addition, the fund may occasionally be required

12Investors also use index futures and risk-free securities to mimic the spot index, although there areviolations of the Spot-Futures Parity.

16

to make supplemental distributions at some other time during the year. Like any open-

end fund, index funds can only be purchased or sold at their end-of-day net asset value.

This implies that the index funds are illiquid within the day and cannot mimic the

S&P 500 Index continuously, while the S&P 500 Index value changes continuously within

the trading day. Therefore, the existence of index funds does not solve the problem of

non-tradability of the S&P 500 Index. Furthermore, in this paper we are interested in

how the non-tradability of the index affect its derivative pricing. Both the futures and

options written on the index are actively traded in the markets and have the opening

prices on the maturity dates as the settle prices. Therefore, we will now focus our analysis

on the mimicking funds which can be traded any time within the trading day.

1.2.2.2 Exchange Traded Funds

The Standard and Poor’s Depositary Receipt (SPDR), often referred to as the “spider”,

is an exchange traded fund which, as stated in the prospectus, holds all of the S&P 500

Index stocks and is designed to reflect the price and yield performance of the S&P 500

Index. As of September 30, 2005, SPDR had $47 billion in total assets. The SPDR Trust

issues and redeems SPDRs only in multiples of 50,000 SPDRs (referred to as “Creation

Units”) in exchange for S&P 500 Index stocks and cash. For example, to create (resp.

redeem) 50,000 SPDRs, the investor will deposit with (resp. be delivered by) the Trustee

a specified portfolio of Index Securities and a cash payment generally equal to dividends

(net of expenses) accumulated up to the time of deposit. The first creation units were

deposited on January 22, 1993. The Trust is scheduled to terminate no later than January

17

22, 2118, but may terminate earlier under certain circumstances. The Creation Units are

redeemable only in kind and not redeemable for cash. The creation and redemption

take place after the closure of the trading day. Regardless of the number of Creation

Units created or redeemed, the investor has to pay $3,000 for each transaction. Each

SPDR represents an undivided ownership interest in the SPDR Trust and has a price

approximately equal to one-tenth of the Index level. It is traded on the American Stock

Exchange (AMEX) like any other equity security at any time within the trading day.

Thus the purchases and sales of the SPDRs are subject to transaction costs such as

bid-ask spread and ordinary brokerage commissions and charges. The minimum trading

unit is one SPDR. The SPDR is an actively traded security. As shown in Table C.1, the

average daily trading volume is over 38 million shares and the average trading value per

day is over $4 billion. About 11.2% of the total shares are traded every day. The Trust

charges a very low expense fee (18.45 basis point in the past13) to the SPDR holders.

The SPDR Trust pays dividends. The ex-dividend date is the third Friday in March,

June, September and December. Beneficial owners are entitled to receive the dividends

accumulated through the quarterly dividend period which ends on the business day before

the ex-dividend day. The dividend is the amount of any cash dividends declared on the

SPDR portfolio during the corresponding period14, net of fees and expenses associated

with the operations of the Trust, and taxes, if applicable. It is paid on the last business

day of April, June, October and January. Due to the fees and expenses, the dividend

13See Elton, Gruber, Comer and Li (2002).14The dividends that the trust receives from the SPDR portfolio are held in an nonearning account

before being distributed to the investors.

18

yield for SPDRs is usually less than that of the stocks in the S&P 500 Index. The trustees

may also distribute the capital gains in January and require an additional distribution

shortly after the end of the year.

To match the composition and weights of stocks held by the Trust with component

stocks of the S&P 500 Index, the trustee adjusts the composition and weights of stocks

held by the Trust periodically to respond to changes in the identity and/or weighting of

the Index. The trustee aggregates certain of these adjustments and makes changes at least

monthly or even more frequently in the case of significant changes to the S&P 500 Index.

Any change in the identity or weighting of an Index Security will cause an adjustment to

the portfolio held by the Trust effective on any day that the New York Stock Exchange

(NYSE) is open for business following the day on which after the close of the market the

change to the S&P 500 Index takes effect.

There are several reasons why the SPDR may not mimic the index perfectly. First,

the SPDR pays dividends later than the index component stocks and holds the cash

dividends in a non-earning account, that is, the cash dividends are not reinvested in the

SPDR. The SPDR also charges management expenses, which lower the amount of the

dividends to be paid. So the timing and the amount of the cash flows from the SPDR are

different from those of the index securities, which may cause the value of the SPDR to

diverge from the index level. Second, the SPDR needs to make occasional adjustments

to its portfolio to respond to the adjustment to the index. As discussed in the last

subsection, this kind of adjustments may be expensive and lead to differences between

the SPDR portfolio and the index portfolio. Third, the index is not traded on the market

19

while the SPDR is, which can induce a liquidity premium for the SPDR. In summary,

due to the different timing and amount of dividends paid by the Index securities and the

SPDR, the way the SPDR deals with the dividends, the expenses of the SPDR Trust,

the maintenance of the index and SPDR portfolio, and the different forces that drive the

price of the SPDR and the prices of the securities components of the index, the dynamics

of the S&P 500 Index and the SPDR can diverge.

In the remainder of this subsection, we examine how closely the SPDR mimics the

S&P 500 Index using daily data from January 2, 2001 to December 30, 200515.

We will check the mimicking performance of the SPDR in two ways. First, we examine

how the relative price change of the SPDR mimics the relative change of the index.

Second, we study how the holding period return of the SPDR mimics the return on the

index. In the first case, the relative price change of the SPDR is defined as

Rt+1 = log(SPDRt+1)− log(SPDRt). (1.3)

In the second case, the dividend from the SPDR is reinvested on the ex-dividend day and

the holding period return over one day is defined as

Rdt+1 = log(SPDRt+1 + dt+1)− log(SPDRt), (1.4)

15Beaulieu and Morgan (2000) studied the high-frequency relationships between the S&P 500 Indexand the SPDR by using minute-by-minute data for November 1997 through February 1998. The authorsused both the covariance estimator of De Jong and Nijman (1997) and the GMM estimation of systems ofsimultaneous equations using imputed data (both filtered and unfiltered). Many one minute intervals didnot contain a SPDR transaction and the two methods dealt with the missing observations in differentways, so the conclusion from the two methods were slightly different. Specifically, they found thatthe SPDR led the index by one minute from the De Jong and Nijman estimator and the filtered dataindicated that the index led the SPDR by one minute with the GMM. In either case, the SPDR did nottrack the index perfectly.

20

where SPDRt is the price of the SPDR at t and dt+1 is the dividend with t + 1 as the

ex-dividend day. Since the index itself pays no dividend, its holding period return is the

same as the relative change defined by Equation (1.1). If the holding period return can

replicate the return of the index, the index can be replicated by the SPDR using the

self-financing strategy.

The data source is the Center for Research in Security Prices (CRSP).

For the S&P 500 Index, we use the level of the Standard & Poor’s 500 Composite

Index at the end of the trading day. These data were collected from publicly available

sources such as the Dow Jones News Service, The Wall Street Journal or the Standard

& Poor’s Statistical Service. While the index does not include dividends, it indicates the

change in price of the component securities.

For the SPDR price, we use the average of the closing bid and ask prices16, which is

the average of the bid and ask prices from the last representative quote before the market

closes. The relative price change is calculated using this data. Figure C.1 presents the

bid-ask spread and the trading volume of the SPDR from January 1, 2001 to December

30, 2005. More than fifteen percent of observations had a negative bid-ask spread. We

do not know whether this is due to data recording errors or whether there are arbitrage

opportunities to be taken advantage of in the market. We do not remove these observa-

tions because their averages do not seem to be outliers. The bid-ask spread of the earlier

sample period is very volatile. This could be caused by the so-called “dotcom bubble”,

16There are 17 days which do not have quotation prices and we use closing prices to replace the averageof bid and ask prices. We set the bid-ask spreads of these days equal to 0.

21

when the stock prices, the index and the SPDR were very volatile. As we can see from

the second graph of Figure C.1, which covers June 1, 2001 to December 30, 2005, most

bid-ask spreads are within the interval (-0.1, 0.1). The mean of the spread is 0.13 and

the variance is 0.27. There are 379 out of 1,256 observations with a bid-ask spread equal

to 0.01. The result is confirmed in Figure C.2 where the historical distributions of the

bid-ask spread and trading volume of SPDR as well as their joint distribution are pre-

sented. The average daily trading volume of the SPDR is about 38 million shares. The

correlation coefficient between the spread and daily volume is -0.31, which is consistent

with the stylized fact that the higher the trading volume, the lower the bid-ask spread.

For the daily holding return of the SPDR, we use the log of ”holding period return”

data from CRSP which is calculated as shown in Equation (1.4) except that SPDRt is

the last sale price of the SPDR at time t.

1.2.2.3 Static Comparison of the Relative Price Changes of the SPDR andof the S&P 500 Index

Figure C.3 shows the SPDR price and the S&P 500 Index level. The SPDR price is

multiplied by 10 and denoted SPDR× 10. As expected, the historical means of the two

time series are very close, as are the historical variances, skewnesses and kurtosis, since

the SPDR is intended to mimic the value of the S&P 500 Index. Figure C.4 shows the

time series of the difference between the SPDR× 10 and the S&P 500. We observe that

the difference diminishes regularly on the ex-dividend days of the SPDR, which creates

a periodic non-stationary feature. In Figure C.4, we see that the difference between the

22

SPDR× 10 and the S&P 500 can be as large as 19.76 index points17. On February 6th,

2001, the SPDR×10 was 19.76 index points below the S&P 500 Index. On that day, the

Standard and Poor’s added (and correspondingly deleted) one stock component. This

change could explain a spike in the series of difference. Another cause of the spike in the

difference between the SPDR × 10 and the S&P 500 on that day can be the so-called

“dotcom bubble”. Indeed during this bubble, the stock prices were very volatile and so

were the index and the SPDR. If the index and the SPDR do not move simultaneously,

this can create a large difference. To highlight this phenomenon, we also plot the dif-

ference series using a longer period from February 1, 1993 through December 30,2005

in Figure C.5. During the “dotcom bubble” period, i.e., roughly between 1997 to 2001,

the difference series was much more volatile than during the other period. The index

dropped from 1354.31 on February 5, 2001 to 1352.26 on February 6, 2001 and continued

to decrease to 1340.89 on February 7, 2001. The SPDR × 10 dropped from 1358.50 on

February 5, 2001 to 1332.50 on February 6, 2001 and increased to 1347.05 on February 7,

2001. It could be that on February 6, 2001, investors who held the SPDR overreacted to

the negative information and underpriced the SPDR. The value of the SPDR rebounded

on February 7, 2001. In conclusion, Figures C.4 and C.5 show that the SPDR does not

mimic perfectly the index.

So far we have examined the daily closing prices. Figures C.6 and C.8 provide the daily

relative price changes of the SPDR and the S&P 500 Index, as defined in Equations (1.3)

17There are 7 observations in which the difference between the two time series is larger than 10 indexpoints. There are 5 cases in which the SPDR× 10 is larger and 2 cases in which the S&P 500 Index islarger.

23

and (1.1), and their difference series. The sample mean of the SPDR return is 0.05 basis

points lower than the sample mean of the index return, which accounts for 24% of the

average daily return of the S&P 500 Index. The historical correlation coefficient between

the two return series is 0.98, suggesting that the SPDR mimics the index rather well as

far as a linear static analysis is concerned. When we consider the higher order moments,

we see that the sample skewnesses of the two time series are different and take values

0.17 for the index and 0.13 for the SPDR, respectively. Again, the differences shown

in Figure C.8 decrease regularly on the ex-dividend days of the SPDR. The difference

series of daily returns can be as large as 223 basis points per day, as occurred on April

4, 2001. From April 3 to April 5, 2001, the index jumped from 1106.46 to 1151.44 and

the SPDR× 10 jumped from 1100 to 1150.45. However, the two series did not increase

simultaneously in that period. On April 4, the index did not change much and closed

at 1103.25 while the SPDR moved to 1121.55. Hence, the different evolutions of the

two processes caused the huge daily return difference. Figures C.7 and C.9 provide the

daily holding period returns, as defined in Equations (1.4) and (1.1), of the SPDR and

the S&P 500 Index, respectively, and their daily difference. The differences between the

holding period returns are bigger than the relative price changes.

Figure C.10 shows the histograms of daily relative price changes of the index and

the SPDR, and the histogram of daily holding period returns of the SPDR. From the

first two histograms, we see that the relative price changes of the index and the SPDR

have different unconditional distributions. The 5% and 95% quantiles corresponding to

the left and right tails of the distribution differ by 11 basis points, which account for

24

6.04% and 6.32% of the corresponding daily index returns, respectively. The medians

themselves differ by about 10% of the index return. The first and third histograms show

the differences between the holding period returns of the index and of the SPDR. The fact

that the quantiles of the SPDR and the S&P 500 distributions do not match can have

important effects on European call prices for which the price thresholds (strikes) play

an important role. Figure C.11 shows the sample distribution of relative price change

differences and daily holding period return differences between the S&P 500 Index and

the SPDR. The relative price change differences are between -0.018 and 0.022, compared

to the relative price changes which are, as shown in Figure C.10, between -0.051 and

0.056. This confirms the error on relative price changes. There are 9.8% of observations

with absolute daily geometric return difference greater than 25 basis point, 28.21% of

observations with absolute daily geometric return differences greater than 10 basis point,

and 59.12% of observations with absolute daily geometric return differences greater than

4 basis points. This is a huge difference given that the medians of the daily geometric

returns of the index and the SPDR are less than 4 basis points. The second histogram

shows that the daily holding period return differences are even more frequently away

from zero and have a relatively larger range.

Figures C.12 and C.13 show the squared daily geometric returns. It is not surprising

that the SPDR and the S&P 500 plots are different, given that they have different daily

relative price changes and holding period returns. We will compare the squares of daily

relative price changes and holding period returns in greater detail in the next subsection

on dynamic analysis.

25

1.2.2.4 Dynamic Comparison of the SPDR and the S&P 500 Index

The analysis above is a first step in a comparison of the properties of the SPDR and the

S&P 500 Index, since it is based only on historical summary statistics and thus neglects

the serial dependence. It is necessary to compare the serial dependence of the series,

especially if we have in mind a continuously updated portfolio to hedge a derivative.

Since the SPDR is intended to mimic the S&P 500, we can expect that both series

will have similar dynamic features, as long as linear dynamics are considered. However,

it is less likely that their nonlinear dynamics such as the volatility or the extreme return

dynamics are compatible.

Tables C.2 and C.3 report the Ljung-Box statistics of daily relative price changes and

holding period returns of the S&P 500 Index and the SPDR. Table C.2 suggests that the

white noise hypothesis at 5% significance level for the first 15 lags cannot be rejected

in both series. However, when more lags are included in the test statistics, the white

noise hypothesis at 5% significance level is rejected in both series. At higher lag, the

SPDR returns show stronger evidence against the white noise hypothesis. For example,

if 25 lags are included in the test, the white noise hypothesis is rejected by the geometric

return of the SPDR at 1% confidence level, but is not rejected by the geometric return

of the Index at the same significance level. Table C.3 shows the results for the two series

of holding period returns and suggests that the white noise hypothesis at 5% significance

level for the first 15 lags cannot be rejected. If more lags are included in the test, the

white noise hypothesis at 5% significance level is rejected for both series. However, up to

26

lag 10, the SPDR shows stronger evidence against the white noise hypothesis. For longer

lags, the S&P 500 shows stronger evidence against the white noise hypothesis18.

Let us now examine the risk dynamics. Figures C.14 and C.15 show the autocorrela-

tion functions of the squares of relative price changes and holding period returns of the

S&P 500 Index and the SPDR. As reported in the ARCH literature, there is more linear

serial dependence in the squared returns than in the returns. This is confirmed by the

Ljung-Box statistics for squared returns reported in Tables C.4 and C.5. The difference in

the autocorrelation functions between these series suggests that they have very different

risk dynamics. Figures C.16 and C.17, where different cross-correlation functions of the

index and the SPDR are compared with corresponding autocorrelation functions of the

index, confirm this result. The graphs show that these functions are clearly different. We

conclude that the risk dynamics of the SPDR and the S&P 500 are different.

As seen from the histograms of daily relative price changes and holding period returns,

the tail quantiles of all series are different and this can have important effects on European

call and put prices for which the price thresholds (i.e. strikes) play an important role. To

see the dynamics of the right tails, we plot the autocorrelation functions ofMax(exp(Rt)−

k, 0) and Max(exp(Rdt )− k, 0) of the S&P 500 Index and the SPDR in Figures C.18 and

C.19, respectively. We also plot the autocorrelation functions of Max(k − exp(Rt), 0)

and Max(k− exp(Rdt ), 0) for the S&P 500 Index and SPDR in Figures C.20 and C.21 to

18We also plot the autocorrelation (ACF) and partial autocorrelation function (PACF) for daily relativeprice changes and holding period returns of the S&P 500 Index and the SPDR up to 135 lags as wellas their cross correlation functions (XCF). For the sake of brevity, we do not report them here. It canbe easily seen from the graphs that, although the signs of the ACF and the PACF of the series arealmost always the same, the sizes are very different, which implies that they have different conditionalevolutions.

27

describe the dynamics of the left tails. For the right tails of index returns, since the 97.5%

quantile is about 0.025, the 95% quantile is about 0.02 and the 90% quantile is about

0.015, we pick k=1.015, 1.02 and 1.025 for the call, as shown in Figures C.18 and C.19.

For the same reason, we pick k=0.985, 0.98 and 0.975 for the put, as shown in Figures C.20

and C.21. The autocorrelation functions of Max(exp(Rt)−k, 0) and Max(k−exp(Rt), 0)

are very different for the S&P 500 Index and the SPDR. The SPDR has higher first-order

autocorrelation than the Index in all the plots, which implies that the SPDR has higher

volatility clustering than the index. The difference in the autocorrelation functions for

higher orders reflects that the two series have very different volatility clustering properties.

A similar analysis suggests that the holding period returns of the S&P 500 Index and the

SPDR also have different volatility properties.

The analysis shows that the S&P 500 index and the SPDR price have different histor-

ical distributions and different dynamics. Therefore, the SPDR price does not replicate

the S&P 500 perfectly. A similar analysis performed for the holding period returns of

the SPDR and the index suggests that the daily holding return of the SPDR does not

replicate the return of the Index effectively, either. This implies that the index cannot

be replicated by the SPDR using self-financing strategy.

The facts that S&P 500 is not traded in the market and cannot be replicated have a

lot of implications which are discussed in the next section.

28

1.3 The Effects of the Non-Tradability of the Index

1.3.1 Derivative Pricing

The non-tradability of the S&P 500 Index has significant implications on risk hedging

and pricing restrictions. For example, the well-known Black-Scholes model [See Black

and Scholes (1973) and Merton (1973b)] assumes that the underlying asset is tradable

and follows a geometric Brownian Motion process with constant volatility. Therefore,

the market is completed by the underlying asset itself. By the No-Arbitrage condition,

the market price of risk is determined uniquely by the price of the underlying asset. All

derivatives written on the underlying asset can be evaluated uniquely with this market

price of risk, combined with the terminal condition of the respective derivatives. But, if

the underlying asset is non-tradable, the underlying asset cannot be used as part of the

arbitrage strategy and hence the value of the underlying asset does not need to satisfy

the No-Arbitrage condition. The knowledge of the value of the underlying asset alone

does not reveal the price of risk. Therefore, the prices of options written on a non-

traded underlying asset, whose value follows a geometric Brownian Motion process, are

not constrained to satisfy the Black-Scholes formula. To evaluate the illiquid assets in

this framework, we need either to introduce an additional tradable asset to complete the

market, or to assume some form of risk-neutral distribution, i.e., of market price of risk.

Since the S&P 500 Index is not a tradable asset in the market, the index and its options

cannot be used in standard option pricing models that are based on the assumption that

the underlying asset is a security traded in the market.

29

The pricing formulas derived from general equilibrium models are also no longer valid.

If the investor cannot trade the underlying asset in the market, s/he has fewer choices for

hedging future risks, so s/he has to make different consumption and investment decisions,

which will affect the equilibrium prices.

In Chapter 3, we derive a coherent multi-factor model for pricing various derivatives

such as forwards, futures and European options written on the S&P 500. We consider

two cases when the underlying asset is tradable and when it is not. The model explains

why the prices of derivatives written on a tradable asset and a non-tradable asset can be

different.

In the next subsection, we study how the pricing parities are violated when the un-

derlying asset is not tradable.

1.3.2 Spot-Futures Parity and Put-Call Parity

In this subsection, we use the following notations:

- It is the value of a non-tradable underling asset at t;

- St is the value of a tradable underling asset at t;

- F it,T is the forward price at t of a contract written on the non-tradable asset expiring

at T , with t ≤ T ;

- F st,T is the forward price at t of a contract written on the tradable asset expiring at

T ;

30

- Git denotes the price at time t of a European call option written on the non-tradable

underlying asset with strike K and maturity T;

- Gst denotes the price at time t of a European call option written on the tradable


- H it denotes the price at time t of a European put option written on the non-tradable


- Hst denotes the price at time t of a European put option written on the tradable


- rt,T is the risk-free interest rate from t to T .

Let us first suppose that there is no dividend. Consider Portfolio A: At time t, let

us take a long position in the forward contract written on the tradable underlying asset

and invest an amount of cash equal to F st,T exp[−rt,T (T − t)] into the risk-free asset. The

value of this portfolio at T is ST . By the No-Arbitrage condition, the current value of

Portfolio A should be St. This implies the Spot-Futures Parity19 relationship for the

tradable underlying asset and the forward contract written on it,

F st,T = St exp[rt,T (T − t)]. (1.5)

Next consider Portfolio B, which is the same as Portfolio A except that the forward

contract is written on the non-tradable underlying asset and the invested cash amount is

19We follow the convention and call Equation (1.5) Spot-Futures Parity [see e.g. Ait-Sahalia and Lo(1993)], although it generally holds for the spot and forward contracts. Due to the mark-to-market rule,the futures contract is different from the forward contract in general. However, when the interest ratesare non-stochastic, the futures price and the forward price are equal[see Cox, Ingersoll and Ross (1981)and French (1983)], and Equation (1.5) holds also for the spot and futures contracts.

31

F it,T exp[−rt,T (T −t)]. The value of Portfolio B at T is IT . It is very tempting to conclude

that the value of Portfolio B at t is It. But this is not true because the underlying asset

is non-tradable. Let us consider the S&P 500 Index as an example. As discussed above,

when an investor buys a portfolio at t for indext, there is no guarantee that s/he can

sell it for indexT at T. In other words, the value at t of some asset which will be worth

indexT at T is not necessarily indext. Thus in general, the Spot-Futures Parity does not

hold for the non-tradable underlying asset and the forward contract written on it:

F it,T 6= It exp[rt,T (T − t)]. (1.6)

By the same token, let us consider Portfolio C: At time t, let us buy one European

call option with strike K written on the tradable underlying asset, sell one European

put option with strike K on the same asset, and invest an amount of cash equal to

K exp[−rt,T (T − t)] in the risk-free asset. The value of this portfolio at T is ST . By

the No-Arbitrage condition, the current value of Portfolio A should be St. This is the

Put-Call Parity:

Gst +K exp[−rt,T (T − t)] = Hs

t + St. (1.7)

Let us now consider Portfolio D, which is the same as Portfolio C except that the

options are written on a non-tradable asset. Although the value of this asset at T is IT ,

for the same reason as above, Equation (1.7) is not necessarily satisfied for options on

non-tradable assets:

Git +K exp[−rt,T (T − t)] 6= H i

t + It. (1.8)

32

Although, the Spot-Futures Parity and the Put-Call Parity do not generally hold for

the S&P 500 Index and its derivatives, the following equation still holds when the forward

contract and options have the same maturity date20.

Git +K exp[−rt,T (T − t)] = H i

t + F it,T exp[−rt,T (T − t)]. (1.9)

When dividends are taken into account, there are at least three reasons why the

Spot-Futures Parity and the Put-Call Parity may not hold for the S&P 500 Index and

its derivatives.

First, the component stocks of the index pay dividends while the S&P 500 Index

reflects the evolution of the market value of the 500 listed firms without reflecting cash

dividends paid out by those stocks. Therefore, the index is not self-financed and Equa-

tions (1.5) and (1.7) do not hold for the index and its derivatives.

Second, when the index is non-tradable, the Spot-Futures Parity and the Put-Call

Parity for dividend-paying tradable asset do not need to hold. For example, let us

consider a tradable underlying asset that pays a continuous dividend yield at a rate qt,T

per annum. If the dividend is reinvested continuously, then one share at t will become

exp[qt,T (T −t)] shares at T . Thus, if a portfolio pays ST at T and if qt,T is known ex-ante,

the value of the portfolio at t must be St exp[−qt,T (T − t)] to be consistent with the No-

Arbitrage condition. So the tradable dividend-paying underlying asset and its forward

20This can be seen by considering the following portfolio: At t, buy one European call option writtenon the index, sell one European put option, sell one forward contract written on the same asset andinvest an amount of cash equal to (K −F it,T ) exp[−rt,T (T − t)] into the risk-free asset. The value of thisportfolio at T is 0. By No-Arbitrage, the current value of Portfolio A should be 0.

33

price should satisfy the following relationship:

F st,T = St exp[(rt,T − qt,T )(T − t)], (1.10)

and similarly, the Put-Call Parity for options written on the underlying asset paying

dividend is:

Gst +K exp[−rt,T (T − t)] = Hs

t + St exp[−qt,T (T − t)]. (1.11)

However, for the same reason as above, the arbitrageurs cannot earn a risk-free arbitrage

return by trading a non-tradable asset. That is, even if Equations (1.10) and (1.11) do

not hold, there does not necessarily exist a risk-free arbitrage opportunity. So Equations

(1.10) and (1.11) do not have to hold for non-tradable assets under No-Arbitrage.

Third, the timing and size of dividends paid by the index component stocks are not

known ex-ante. Even if the index could be mimicked perfectly by its replicating portfolio,

Equations (1.10) and (1.11) do not need to hold for the index and its derivatives since the

arbitrage opportunity is not risk-free. Moreover, the uncertainty involved in the change

of index increases the uncertainty on the dividend. Therefore the Spot-Futures parity

and the Put-Call parity may not hold.

The Spot-Futures Parity and Put-Call Parity are widely used in derivative pricing.

The deviations from both parities by the S&P 500 Index and its derivatives implies that

the two parities cannot be used for various purposes. For example, to calculate the

dividends on indices, the Ivy DB for OptionMetrics Dataset assumes that the security

pays dividends continuously and the existence of a Put-Call relationship. The implied

index dividend is calculated from a linear regression model based on the Put-Call Parity.

34

If the Put-Call Parity does not hold for the S&P 500 Index and its options, the dividends

computed from this method may be incorrect and yield misleading results.

We have argued above that because the S&P 500 Index is not tradable and cannot

be mimicked perfectly by its replicating portfolios, the Spot-Futures Parity and Put-Call

Parity should not hold for the index and its derivatives. However, if investors traded as if

the index were tradeable, the Parities might hold in practice. In Chapter 2, we take the

Put-Call Parity as an example and test whether the Put-Call Parity holds for S&P 500

Index and its options in reality.

1.4 Conclusion

The S&P 500 Index is not a self-financed or a tradable portfolio. This is because the

index itself is not traded in the market and no portfolio can be constructed to perfectly

replicate the index due to the composition and maintenance of the index as well as

the frictions and the lack of perfect liquidity on the market. Although some mimicking

portfolios such as the SPDR can effectively mimic the linear dynamics of the index ,

the nonlinear and extreme risk dynamics of the SPDR are very different from those of

the index. If the index is non-tradable, investors cannot form a risk-free portfolio using

the index and futures contract or using the index and options. Hence, the No-Arbitrage

condition does not imply the Spot-Futures Parity or the Put-Call Parity for the index

and its derivatives. The non-tradability of the index also implies that the index and its

options cannot be used for testing the option pricing models based on the assumption

35

that the underlying asset is a security traded in the market. These consequences of the

non-tradability of S&P 500 Index are explained and tested in detail in Chapter 2 and

Chapter 3.

36

Chapter 2

The S&P 500 Index Options and the

Put-Call Parity

37

2.1 Introduction

The market for S&P 500 Index Options has grown very quickly since they appeared on

July 1, 1983. The options on S&P 500 are standardized financial contracts which have

deterministic issuing dates, maturity dates and strikes. Although this standardization

enhances the liquidity, it also creates seasonal trading patterns and may cause market

incompleteness on certain days. For example, the options that mature in a quarterly

cycle in March, June, July and December are more actively traded than other options

with the same time-to-maturity that mature in different months.

The S&P 500 Index Options are widely used by researchers for testing the option

pricing theories. However, their underlying asset, i.e., the S&P 500 Index, is an artificial

summary statistic constructed to reflect the evolution of the market [see Chapter 1]. It

is not a self-financed tradable portfolio and its modifications do not coincide with the

changes of a mimicking portfolio (tracker) such as the SPDR, due to the particular way

the S&P 500 Index is calculated and maintained. Therefore, pricing theories based on the

assumptions of tradable assets may not hold for the S&P 500 Index and its derivatives.

For example, the non-tradability of the index can help explain why the Put-Call Parity

does not hold for the S&P 500 Index and its options.

The Put-Call Parity implied by the No-Arbitrage condition was first formalized by

Stoll (1969) and Merton (1973a). It is based on the argument that in a frictionless market

the spot asset and the put and call options written on it can form a risk-free portfolio.

The presence of arbitrageurs ensures that there is no remaining arbitrage opportunity

38

and this portfolio will earn the risk-free rate. Using the daily and intradaily prices of the

S&P 500 and its options from 1986-1989, Kamara and Miller (1995) observed deviations

from the Put-Call Parity even after controlling for dividends and transaction costs. Ackert

and Tian (2001) found similar evidence using data from 1992 to 1994. Both articles

tested whether this was due to the ”liquidity risk”, that is, to the difficulty in buying

or selling a portfolio of 500 stocks and index options simultaneously before any adverse

price movement. The evidence found in these articles suggested that the ”liquidity risk”

can help explain the parity deviation. In both articles, the S&P 500 Index was a tradable

asset, which is not very liquid. However, the index is not only illiquid, it is not tradable

as we have seen in Chapter 1. As a consequence, investors cannot make risk-free arbitrage

transactions using the index and the options. Therefore, the No-Arbitrage condition does

not imply Put-Call Parity for the index and its options. Furthermore, the index is not

self-financed since it reflects the price changes of its component stocks without reflecting

the cash dividends. The uncertainty of the dividends can also help explain why the Put-

Call Parity does not hold for the S&P 500 Index. In this paper, we perform an empirical

analysis, based on the most liquid options to eliminate the liquidity risk. We use the

Put-Call Parity equation to derive and compute the Put-Call Parity implied dividends.

We find that the implied dividends depend on time, maturity and strike price. We also

analyze why deviations from the Put-Call Parity do not imply the existence of risk-free

arbitrage opportunities.

In Section 2.2, we consider the S&P 500 Index Options. We first describe the char-

acteristics of traded options and their issuing procedure. Then we provide stylized facts

39

concerning the S&P 500 options. We compute and analyze the implied dividends in

Section 2.3. Section 2.4 concludes.

2.2 S&P 500 Index Options

2.2.1 The Characteristics of Traded Options

The S&P 500 Index Options traded on the Chicago Board Options Exchange (CBOE) are

European options. They are among the most liquid exchange-traded options and are ex-

tensively used for testing the option pricing models. The exchange-traded S&P 500 Index

Options differ from over-the-counter options and have a standardized format described

below. This standardization is introduced to enhance the liquidity.

New S&P 500 Index Options are issued on the Monday following the third Friday of

each month. These new options have the same maturity date but different strikes. Table

C.6 shows the times-to-maturity of the options introduced each month. For example, the

options issued in March and September have a 12 month time-to-maturity, the options

issued in June and December have a 24 month time-to-maturity, and the options issued

in the other months have a time-to-maturity of 3 months. Based on this standardized

deterministic issuing procedure, there are generally eight maturity dates on any trading

day. The expiration months are the three near-term months followed by three additional

months from the March quarterly cycle, which are March, June, September and Decem-

ber, plus two additional months from June and December1. When options with a new

1In the product specification of the S&P 500 Index Options from CBOE, it is stated that the ex-piration months are the three near-term months followed by three additional months from the March

40

maturity are issued, the in-, at- and out-of-the-money strike prices are initially listed.

The underlying asset is the index level multiplied by 100. The index level is computed

using the last transaction prices of its component stocks. The options are quoted in

index points, where one point equals $100. The options have a minimum tick of 0.05

points for options trading below 3.00 and 0.10 points for the others. Strike price intervals

are 5 points and 25 points for long term contracts. The expiration date is the Saturday

following the third Friday of the expiration month. As the options are European, they

can only be exercised on the last business day before expiration. Trading in S&P 500

Index Options will generally terminate on the trading day preceding the day on which

the exercise-settlement value is computed, which is usually a Thursday. The exercise-

settlement value is computed using the opening (first) reported sales price in the primary

market of each component stock on the last business day before the expiration date, i.e.,

usually a Friday. The last reported sales price in the primary market is used in computing

the exercise-settlement value if a stock in the index does not open on the day on which

the exercise and settlement value is determined. Exercise will result in delivery of cash,

which amounts to the difference between the exercise-settlement value and the strike

price of the option multiplied by 100, on the business day following expiration. There are

no effective position and exercise limits. Purchases of puts or calls with 9 months or less

until expiration must be paid in full. Writers of uncovered puts or calls must maintain a

certain margin to cover their position if their investment value declines over time.

quarterly cycle (March, June, September and December). The data show that there are generally twoadditional longer term options traded in the market.

41

2.2.2 The Activity on the Option Market

The option data used in this paper are from the Wharton Research Data Services (WRDS)

and prepared by Ivy DB OptionMetrics. The data consists of the highest closing bid

prices, the lowest closing ask prices and the traded volumes of all the call and put options

with various strike prices and expiration dates for each trading day from January 2nd,

2003 to December 31st, 2003. There are 130,336 observations.

Figure C.22 shows the number of different times-to-maturity on each trading day

in 2003. As noted above, there are generally eight maturity dates for options on any

trading day. However, since expiring options are usually not traded at maturity, but

settled on these days, while new options are not issued until the next Monday, there are

seven maturity dates only on the last trading days before expiration dates2. This could

potentially cause market incompleteness, since there are fewer traded options on the last

trading days before expiration dates.

Because the exchanged traded options have fixed maturity dates, we observe a de-

clining pattern of times-to-maturity on each trading day, as shown in Figure C.23. This

figure and Table C.6 show that options traded in the first six months have the same

time-to-maturity pattern as those traded in the following six months. For example, the

pattern on the left side of Figure C.23 is the same as the pattern on the right side. We

2The exceptions are the following: on Jan 17, 2003, the data shows 8 maturity dates, one of whichis Jan 18, 2003. But the volume for options expiring on this date is 0. The same thing happens onSeptember 19, 2003. On Jan 2nd, the data does not show any options maturing on December 18, 2004,so there are only 7 maturity dates on that day. For June 19 and June 20, the data shows the optionmaturing on June 18, 2005, which usually will not be traded until the next Monday, although the volumeis 0 on both days. The data does not show options maturing on December 17, 2005 for December 22,2003.

42

conclude that there are two cycles per year. Figure C.23 also shows that new options

which are not issued from the March quarterly cycle will always expire in three months,

while the options issued in March and September will expire in a year and the options

issued in June and December will expire in 2 years. Therefore, all options with a time-to-

maturity longer than three months have expiration dates in the March quarterly cycle,

and all the options with a time-to-maturity longer than 1 year have expiration dates in

June and December.

Table C.7 presents the summary statistics of call and put options traded in 2003.

There are, on average, 517 options traded every day with different strike prices and

times-to-maturity. The daily number ranges from 422 to 572 with a standard deviation

of 28. For each call option with a given strike and time-to-maturity, we almost always

find a put option with the same strike and time-to-maturity. In the 252 trading days

in 2003, there are only two days when the number of call options is different from the

number of put options and that difference is just 1 option.

The market for S&P 500 Index Options is fast-growing and these options are among

the most actively traded derivatives. The average total daily traded volume in 1993 was

65,476 contracts [see Ait-Sahalia and Lo (1998)], and it increased to 137,143 contracts in

2003. The daily traded volume of S&P 500 Index Options ranges from 37,428 contracts

to 322,711 contracts, with a standard deviation of 43,213 contracts. The average daily

traded volume is higher for put options than for call options, while the opposite is true

for the average daily traded value. The average total daily traded value for all options

is $327,816,156 with a minimum of $51,359,310 and a maximum of $1,600,668,170. The

43

standard deviation is $197,363,809.

The second panel of Table C.7 describes the summary statistics of volumes and prices

in the sample of 130,336 observations. The options with the same maturity and strike

price, but traded on different days, are regarded as different options. The price is com-

puted as the average of the best closing bid and ask prices. More than 60% of the options

have zero traded volume in 2003 and only about 4% of the options have a traded volume

greater than 2,000 contracts. Prices move from $0.025 to $993.5 per 1/100 contract, with

a mean of $111.85 and a standard deviation of $146.53.

Figure C.24 displays the times-to-maturity of both call and put options with a traded

volume greater than 2,000 contracts on each day in 2003. By comparing this figure with

Figure C.23, we see again that a large number of options are not actively traded each

day, especially those with longer times-to-maturity. The options expiring in the March

quarterly cycle are more frequently traded after their times-to-maturity reduce to about

100 days. Figures C.25 and C.26 show strike prices and moneyness3 of options with high

traded volumes on each trading day. Most of the actively traded options are at- and

out-of-the-money. These are call options with strike prices equal to or higher than the

index level, and put options with strike prices equal to or lower than the index level.

This is consistent with the results reported by Kamara and Miller (1995) and Ackert and

Tian (2001). The observed asymmetry reflects the strong demand of portfolio managers

for protective calls and puts. Figure C.27 shows the prices of options with high traded

volumes on each day in 2003. The most actively traded options are those with low prices.

3The moneyness is defined as the strike price divided by the price of the underlying index.

44

This is not surprising since most options with high volumes are at- and out-of-money.

Figures C.28 to C.33 investigate the number and proportion of highly traded options

on each trading day, and their summary statistics are presented in Table C.8.

Figure C.28 shows the number of highly traded options on each trading day and their

daily relative frequencies of trades (in %). As shown in Table C.8, there are on average

20 options actively traded on each day, ranging from 3 options to 43 options with a

standard deviation of 6.57 options. On average, these actively traded options account for

only 3.89% of all traded options, with a minimum of 0.59%, a maximum of 7.62% and a

standard deviation of 1.21%. It can also be observed that the number of actively traded

options on the last trading day before the expiration date is always lower than that on the

previous day. There is no such regular decrease in the relative frequencies of trades (in

%) since the total number of options on these days are also lower. Although the actively

traded options account for only a small part of all traded options, their traded volumes

account on average for 61.65% of all the options, ranging from 34.24% to 85.65%, with a

standard deviation of 9.88%.

Figure C.29 shows the number of actively traded put options on each trading day and

its daily part of trades. Among those highly traded options, on average 58.92% are put

options , with a minimum of 28.57% and a maximum of 100%, as shown in the second

panel of Table C.8. The standard deviation is 11.11%.

Let us now examine the options in terms of their times-to-maturity on each day. All

the options traded in 2003 are classified into four categories for each trading day: T1

with the shortest time-to-maturity, T2 with the second shortest time-to-maturity, T3

45

with the third shortest time-to-maturity, and T4 with longer times-to-maturity. From

Figure C.22, there are generally eight traded maturity dates on each trading day. The

T1 options are mainly those options with the closest maturity date, except that on the

last trading day before the expiration date in each month, the options to expire the next

month are regarded as T1 since the options expiring this month are not traded on that

day. A similar rule applies to the T2 and T3 options, which usually include the options

with the second and third closest maturity date, except on the last trading day before

the expiration date in each month. The T4 options contain all the options with other

maturity dates. We focus our attention on the actively traded options.

As shown in the second panel of Table C.8, among actively traded options, about 46%

on average are T1, while T2 represents 25%, T3 13% and T4 16%, respectively. Thus

the most actively traded options are those with the shortest time-to-maturity.

We observe periodic patterns from Figures C.30 to C.33. For example, in Figure C.32,

the number of T3 options is mostly zero on the last trading day before the expiration date

in the months other than the March cycle, while the number of T3 options on the last

trading day before the expiration date in the March cycle is positive and higher than that

on the previous day. The T3 options on the last trading day before the expiration date in

the March cycle are the options with 3 months of time-to-maturity and to expire in the

March cycle. We observed in Figure C.24 that options expiring in the March quarterly

cycle seem to be more frequently traded in the last 100 days of their times-to-maturity.

On the other hand, on the last trading day before the expiration date in the months

other than the March cycle, the T3 options are the less actively traded options with

46

four or five months to maturity, since the 3-month options are yet to be introduced on

the following Monday. Therefore, it is not surprising to observe in Figure C.32 a higher

number of actively traded T3 options on the last trading day before the expiration date

in the March quarterly cycle than in the other months.

The first graph in Figure C.32 also shows that the T3 options expiring in the March

quarterly cycle (the first, fourth, seventh, tenth and thirteenth column of the graph)

have a higher number of actively traded options than the T3 options expiring in the

other months. One possible explanation is that the T3 options expiring in the March

quarterly cycle are issued much earlier before they become T3 options, and thus may have

a larger number of actively traded strike prices, while the T3 options expiring in the other

months are newly issued and may have a smaller number of actively traded strike prices.

However, as shown in the first graph of Figure C.36, although the T3 options expiring in

the March quarter cycle have a greater number of actively traded options with different

moneyness, the range of their moneyness is not very different from that of the other

T3 options. Moreover, the second and third graphs in Figure C.36 show that the total

volume and total value of the actively traded T3 options expiring in the March cycle

are higher as well. So there may be other reasons that explain the pattern observed in

the first graph of Figure C.32. For example, the S&P 500 Futures traded in the Chicago

Mercantile Exchange expire only in the March quarterly cycle. The more active trading

of the T3 options expiring in the March cycle could be caused by the interaction between

the S&P 500 option and the S&P 500 futures markets. A similar pattern can also be

observed in the second graphs of Figures C.30 and C.31 and in Figures C.34 and C.35,

47

that is, the options that mature in the March quarterly cycle are more actively traded

than other options with the same time-to-maturity that mature in different months.

We also observe a periodic pattern in the first graph of Figure C.33, where the number

of actively traded T4 options is generally higher during the days close to but before the

expiration day in the March cycle, and decreases sharply on the last trading day before

the expiration date in the March cycle. This observation is consistent with the pattern

observed in Figure C.32. During the days close to but before the expiration day in the

March cycle, the options maturing in the March cycle have less than 100 days of time-

to-maturity and are actively traded, while on the last trading day before the expiration

date in the March cycle, these options become T3.

In summary, the S&P 500 index options are standardized contracts, with determin-

istic issuing dates, and a limited number of maturity dates and strikes available at the

issuing. Although this standardization enhances liquidity, it also creates certain trading

seasonality. The S&P 500 index options are among the most actively traded options in

the world. However, most of these actively traded options are at- or out-of-the-money

options with times-to-maturity of less than three months, and hence with relatively low

prices. In the next section, we will use these option data to test the Put-Call Parity for

the S&P 500 Index.

48

2.3 The Put-Call Parity Implied Dividends of the

S&P 500 Index

2.3.1 Calculation of the Implied Dividends

The Put-Call Parity implied by No-Arbitrage was first formalized by Stoll (1969) and

Merton (1973a). When the underlying asset is tradable and the dividends paid by the

underlying asset are known ex-ante, the Put-Call Parity is expressed in the following

inequality form:

Ca − P b +K exp[−rt,T (T − t)]− St exp[−qt,T (T − t)] ≥ 0, (2.1)

P a − Cb −K exp[−rt,T (T − t)] + St exp[−qt,T (T − t)] ≥ 0, (2.2)

where Cb and Ca are the bid and ask prices of the call option at time t, P b and P a are

the bid and ask prices of the put option, K is the strike price, T is the maturity date,

rt,T is the risk-free interest rate from t to T , and qt,T is a continuous dividend yield paid

by the underlying asset. The two inequalities need to hold simultaneously. When the

first inequality is violated, arbitrageurs can generate earnings by buying the call, selling

the put and the underlying asset, and investing the net proceeds into a risk-free interest

account. When the second inequality is violated, arbitrage profits can be created by

selling the call, borrowing K exp[−rt,T (T − t)] at the risk-free interest rate, buying the

put, and the underlying asset.

The S&P 500 Index is an artificial number constructed to reflect the performance of

the entire market. As argued in Chapter 1, it is not a traded asset in the market. Due

49

to the frictions of the market, the illiquidity of the component stocks and the changes

of composition, the index cannot be replicated by any traded portfolio. Therefore, it is

impossible to form a riskless arbitrage position with the options and the index. That is,

the Put-Call Parity does not have to hold for the S&P 500 Index and its options. The

deviation from the Put-Call Parity by the S&P 500 Index and its options does not imply

that there exist arbitrage opportunities.

Furthermore, the dividends paid by the index component stocks are not known ex-

ante. Even if the index could represent the price of a tradable portfolio, Inequalities (2.1)

and (2.2) do not have to hold for the index and its options because the arbitrage is risky.

Testing whether Inequalities (2.1) and (2.2) hold empirically for the S&P 500 Index

and its options is difficult, if not impossible, in a model-free environment because the

dividend yield, qt,T , is not observable, and there is no reason to assume that the dividends

observed ex-post are the same as the expected future dividend [see Harvey and Whaley

(1991) for example]. However, the financial literature often assumes that the following

equation holds for the S&P 500 Index and its options [See e.g. Ait-Sahalia and Lo (1998)]:

Git +K exp[−rt,T (T − t)] = H i

t + It exp[−qt,T (T − t)], (2.3)

where It is the value of the S&P 500 index at time t, and Git and H i

t denote the average

of the bid and ask prices at time t of a European call option and a European put option

written on the index with strike K and maturity T, respectively. In this paper, we will

test whether Equation (2.3) holds for S&P 500 Index and its options.

50

Let us define the ”Implied Dividend”, Q, as

Q(t,K, T ) =−1

T − tlog

Git +K exp[−rt,T (T − t)]−H i

t

It, (2.4)

where T − t is measured in years.

The position, which consists of buying a European call option, selling a European

put option on the index with strike price K and maturity T, and investing an amount

of cash equal to K exp[−rt,T (T − t)] into the risk-free asset at t, will result in receiving

IT at T. Thus the market evaluates the current value of the future index IT as Git +

K exp[−rt,T (T − t)] −H it . Because the index is non-tradable and not self-financed, this

value is not necessarily the same as the current index level, It. The implied dividend,

Q, represents the annualized ratio between the current value of the future index IT and

the current index, It. If the Put-Call Parity (2.3) holds, the implied dividends should

not depend on the strike price of the options. In the next subsection, we analyze the

properties of the implied dividend.

2.3.2 Data and Empirical Results

In order to calculate the implied dividends, we have to consider put and call options

with the same strike price and expiration date that are both actively traded on the same

day. We select those options with a daily traded volume of more than 2,000 contracts.

There are 798 pairs of such put and call options. As shown in Table C.9, there are on

average only 3 pairs of actively traded put and call options with the same strike price and

maturity on each day. The daily number of actively traded pairs of options ranges from

0 to 10 pairs, with a standard deviation of 1.78 pairs. These options account on average

51

for only 1.22% of all the traded options, with a standard deviation of 0.67%. However,

in terms of traded volume, these option pairs account on average for about 22.25% of

all traded options, with a range from 0 to 72.99% and a standard deviation of 11.55%.

Figures C.37 and C.38 show the moneyness and times-to-maturity of these options on

each trading day in 2003. It is not surprising that the moneyness is around 1 and most

of the times-to-maturities are less than 100 days.

The data for the S&P 500 Index come from the Center for Research in Security Prices

(CRSP). We use the level of the Standard & Poor’s 500 Composite Index at the end of

the trading day. These data are collected from publicly available sources such as the

Dow Jones News Service, The Wall Street Journal or the Standard & Poor’s Statistical

Service. The index does not include dividends.

We use two proxies for the risk-free rate. The U.S. Treasury Bill rate is from the US

Treasury. For options with a time-to-maturity less (resp. more) than 28 days, we use

the closing over-the-counter market quotation on recently issued 4-week (resp. 13-week)

Treasury Bills. In 2003, the averages of the 4-week and 13-week Treasury Bill rates are

1.02% and 1.03%, with a standard deviation of 0.13% and 0.11%, respectively. Since the

investors generally use the LIBOR rate as a benchmark, we also consider the zero rates

provided by OptionMetrics as alternative proxies for the risk-free rates. They are based

on the BBA LIBOR rates and the Eurodollar strip implied future rates.

Let us first use the Treasury Bill rates as proxies for the risk-free rates. The implied

dividend, Q(t,K,T), is computed following Equation (2.4) and shown in Figure C.39. The

implied dividend is displayed on a daily basis, with the horizontal axis representing the

52

day when the options are traded. There are days without significant trades of put and

call options, while on other days there are up to 10 traded pairs. The summary statistics

are presented in the first column of Table C.10. The implied dividends range from −0.51

to 0.43 with a median of 1.40× 10−2 and a standard deviation of 6.26× 10−2. For a pair

of call and put options with time-to-maturity of 50 days, the ratio between the current

value of the future index and the current index value ranges from 94.27% to 107.24%

with a median of 99.81%. The t-statistic of 3.62 suggests that the null hypothesis of zero

dividend is rejected. That is, the current value of the future index is not the same as the

current index value.

In order to see whether the implied dividend depends on the strike price and time-

to-maturity, we consider i) implied dividends with the same trading day t and the same

time-to-maturity T-t, but at least two different strike prices K, and ii) implied dividends

with the same date t and the same strike K, but at least two different T-t.

In Figure C.40, the implied dividends computed from options traded on the same day

and with the same time-to-maturity but different strike prices share the same point on

the horizontal axis and are regarded as one observation. There are 218 such observations.

For each observation, there are at least two implied dividends computed with different

strike prices. As shown in Figure C.40, options with the same date t and time-to-

maturity T-t, but different strikes K, yield different implied dividends. Figure C.41

displays the difference between the maximum and the minimum of the implied dividends.

The summary statistics are presented in the second column of Table C.10. The difference

has a mean of 1.03 × 10−2 and a standard deviation of 3.79 × 10−3 with a range from

53

6.44 × 10−5 to 0.19. The t-statistic of 6.92 implies that the null hypothesis that the

options with the same date t and the same time-to-maturity T-t have the same implied

dividend is rejected.

In Figure C.42, the implied dividends are computed from options traded on the same

day and with the same strike price, but different expiration dates. There are 142 such

sets of observations. For each set, there are at least two implied dividends computed

with different expiration dates. As shown in Figure C.42, the options with the same

date t and the same strike K, but different times-to-maturity T-t, yield different implied

dividends. Figure C.43 displays the difference between the maximum and the minimum

of the implied dividends for each set. The summary statistics are presented in the third

column of Table C.10. The difference has a mean of 6.61×10−2 and a standard deviation

of 9.57× 10−2 with a range from 8.91× 10−6 to 0.51. The t-statistic of 13.06 means that

the null hypothesis that the options traded on the same day, with the same strike price,

will have the same implied dividend is rejected.

When the zero rates are used as proxies of the risk-free rates, the empirical results

are reported in Table C.11 and Figures C.44 to C.48. We restrict our attention to the

actively traded put and call options with time-to-maturity greater than 30 days. Indeed,

small errors in the prices of the options and the index translate into large variations in

implied dividends for small times-to-maturity. The results are similar to those where the

Treasury Bill rates are proxies of the risk-free rates. For example, the t-statistics in Table

C.11 show that the hypotheses that the implied dividend does not depend on either the

time-to-maturity, or the strike price, are rejected.

54

2.3.3 Discussions of the Empirical Results

The above empirical results show that the current value of the future index is not equal

to the current index value, and that the implied dividends depend not only on the time

and maturity, but also on the strike. Several explanations for these stylized facts can be

offered.

Firstly, since the index is non-tradable, and the changes of its composition as well

the dividends to be paid by its component stocks are uncertain ex-ante, the arbitrages

in this option market can be risky. The violation of Inequalities (2.1) and (2.2) does

not imply risk-free arbitrage opportunities. When the deviations become too large, a

traded portfolio of a small subset of stocks in the index is selected and traded against

the options. Because this arbitrage is not risk-free, the options prices can move freely in

a wide range without introducing arbitrage profits. This allows implied dividends to be

different for options with the same time t and the same maturity T.

Secondly, transaction costs may also play a role. Transaction costs for arbitrage trad-

ing include both the commissions and the bid-ask spreads. The commissions for arbitrage

traders may be small, while the bid-ask spreads can be large. The transaction costs for

trading options and Treasury Bills in the risky arbitrage mentioned above are easily de-

fined. They include the commissions and bid-ask spreads of trading the call options,

the put options and the Treasury Bills. However, it is hard to measure the transaction

costs of trading the stock portfolios, since the costs depend on the portfolios used by

the arbitrageurs. Without explicitly modeling the evolution process of the index and

55

its component stocks, specifying the dividend paying process, and making assumptions

about the arbitrageurs’ risk preferences, we do not know what portfolios are used in the

risky arbitrage and how to measure the relevant transaction costs. Moreover, in this

model-free framework, it is hard to distinguish the effects of the non-tradability of the

index, the uncertainty of the dividends and the transaction costs.

Thirdly, timing is of importance. In our paper, all prices and index levels are recorded

at the end of the trading day. However, the CBOE closes at 3:15pm Central Time, while

the underlying stock market closes at 3:00pm Central Time. Thus, the implied dividend

reflects the ratio between the current value of the future index at 3:15pm and the current

index level at 3:00pm. Any new information entering the market between 3:00pm and

3:15pm will be reflected in the option prices, but not in the index level. Whether this

timing effect is significant is not well understood. Harvey and Whaley (1991) find that the

non-synchronous price problem may induce negative first-order serial correlation in the

implied volatility changes from day to day, while Evnine and Rudd (1985) and Kamara

and Miller (2001) both find that intraday and daily closing data yield similar results for

testing the Put-Call Parity.

A similar remark can be made about the maturities. We use the closing over-the-

counter market quotation on recently issued 4-week (resp. 13-week) Treasury Bills for

options with time-to-maturity less (resp. more) than 28 days. Although these are among

the most actively traded Treasury Bills, there is a mismatch between the maturities of

the Treasury Bills and of the index options.

56

2.4 Conclusion

In this chapter, we analyzed some characteristics of the S&P 500 Index Options. The

S&P 500 options are standardized exchange traded options. This standardization en-

hances liquidity for short- and medium-term, at- and out-of-the-money options, but is

not sufficient to recover the Put-Call Parity. The S&P 500 Index is not a self-financed

tradable portfolio and cannot be replaced by a mimicking portfolio. This property of

the index may cause deviations from the Put-Call Parity, without producing arbitrage

opportunities. This paper empirically tests this argument by considering the series of the

Put-Call Parity implied dividends in 2003. The implied dividends depend significantly

on the strike, which indicates that the standard Put-Call Parity equalities taking into

account the dividends do not hold.

57

Chapter 3

Non-tradable S&P 500 Index and

the Pricing of Its Traded Derivatives

58

3.1 Introduction

As argued in Chapter 1, the S&P 500 Index is an artificial number constructed to reflect

the evolution of the market. It is not a self-financed or a tradable portfolio and it cannot

be replaced by a mimicking portfolio such as the SPDR due to the particular way the

S&P 500 Index is calculated and maintained. The non-tradability of the S&P 500 Index

has significant implications on risk hedging and pricing constraints. For example, the well-

known Black-Scholes model [See Black and Scholes (1973) and Merton (1973)] assumes

that the underlying asset is tradable and follows a geometric Brownian Motion process

with constant volatility. Therefore, the market is completed by the underlying asset itself

and the risk involved can be fully hedged by the underlying asset. By the No-Arbitrage

condition, the market price of risk is determined uniquely by the price of the underlying

asset. All derivatives written on the underlying asset can be evaluated uniquely with this

market price of risk, combined with the terminal condition of the respective derivatives.

If the underlying asset is non-tradable, the underlying asset cannot be used as part of

the arbitrage strategy and the value of the underlying asset does not need to satisfy the

No-Arbitrage condition. The risk associated with the underlying asset is not hedged by

itself and the expected return of the underlying asset under the risk-neutral probability

is not necessarily equal to the risk-free rate. Knowing only the value of the underlying

asset, we do not know the price of the risk. Therefore the prices of options written on

a non-traded underlying asset whose price follows a geometric Brownian Motion process

do not have to be evaluated by the Black-Scholes formula. Similar ideas apply to many

59

other models. For instance, the stochastic volatility models in Heston (1993) and Ball

and Roma (1994) assume that the expected return of the underlying asset is equal to the

risk-free rate under the risk-neutral probability, i.e., that the underlying asset is tradable

and the risk associated with the underlying asset is hedged by itself. In general, because

of the non-tradability of the S&P 500 Index, the prices of its options do not have to

satisfy the restrictions imposed by the pricing models that are based on the assumption

that the underlying asset is a security traded in the market.

In this paper we introduce a coherent multi-factor model for pricing various derivatives

such as forwards, futures and European options written on the non-tradable S&P 500

Index. The model illustrates the relationship between the index and its futures, and the

relationship between the index and its put and call options, when the underlying asset

is non-tradable. We also consider what the prices of the derivatives should be, if the

index were self-financed and tradable. The model explains why the prices of derivatives

written on a tradable asset and a non-tradable asset can be different. Moreover, the model

provides a framework to test whether the S&P 500 derivatives are priced by the investors

as if the index were self-financed and tradable. This model can be easily extended to

price derivatives written on other non-tradable indices such as a retail price index, a

meteorological index, or an index summarizing the results of a set of insurance companies.

The rest of the paper is organized as follows: In Section 3.2, we present a coherent

pricing model for pricing derivatives written on the S&P 500. The Spot-Futures Parity

and Put-Call Parity are also derived for the case of a non-tradable underlying index. In

Section 3.3, we derive the parameter restrictions which characterize the derivative pricing

60

if the index were tradable. This allows us to discuss how derivative prices can differ for

tradable and non-tradable underlying assets. We conclude in Section 3.4. Technical

results and details are gathered in the Appendices.

3.2 The Pricing Model

Under the absence of arbitrage opportunity (AAO), the market prices have to be com-

patible with a valuation system based on stochastic discounting [Harrison and Kreps

(1979)]. The pricing formulas can be written either in discrete time or in continuous

time, according to the assumptions of discrete or continuous trading (and information

sets). The modern pricing methodology requires a joint coherent specification of these

historical and risk neutral distributions. For this purpose, we follow the practice initially

introduced by Constantinides (1992), which specifies a parametric historical distribution

and a parametric stochastic discount factor.

3.2.1 Assumptions

3.2.1.1 Historical Dynamics of the Index

The value of the index at date t is denoted by It. We assume that the log-index satisfies a

diffusion equation with affine drift and volatility functions of K underlying factors {xk,t},

k = 1, · · · , K:

Assumption 3.1.

d log It = (µ0 +K∑k=1

µkxk,t)dt+ (γ0 +K∑k=1

γkxk,t)1/2dwt, (3.1)

where {µk} and {γk} , k = 0, · · · , K are constants, and {wt} is a Brownian motion.

61

The underlying factors summarize the dynamic features of the index. As seen in

Equation (3.2), they are assumed to be independent Cox, Ingersoll and Ross (CIR)

processes, independent of the standard Brownian motion {wt}. Since the CIR processes

are nonnegative, the volatility of the log-index is positive whenever parameters {γk}, k =

0, · · · , K, are positive. This positive parameter restriction is imposed in the rest of the

paper.

Assumption 3.2. The CIR processes {xk,t}, k = 1, · · · , K satisfy the stochastic differ-

ential equations:

dxk,t = ξk(ζk − xk,t)dt+ νk√xk,tdwk,t, k = 1, · · · , K, (3.2)

where ξk, ζk and νk are positive constants and {wk,t}, k = 1, · · · , K are standard inde-

pendent Brownian motions, independent of {wt}.

The condition ξkζk > 0 ensures the nonnegativity of the CIR process (for a positive

initial value x0 > 0), while the conditions ξk > 0 and ζk > 0 imply the stationarity of the

CIR process. The condition νk > 0 can always be assumed for identifiability reason.

This general specification of the index dynamics includes the Black-Scholes model

[Black and Scholes (1973)], when µk = γk = 0, k = 1, · · · , K, the stochastic volatility

model considered by Heston (1993) and Ball and Roma (1994), when K = 1 and x1

is interpreted as a stochastic volatility, or the model with stochastic dividend yield [see

Schwartz (1997) for example], when K = 1 and x1 appears in the drift only.

The transition distribution of the integrated CIR process is required for deriva-

tive pricing. This distribution is characterized by the conditional Laplace transform

Et[exp(−z∫ t+h

txk,τdτ)], where Et denotes the conditional expectation given the past

62

values of the process and z is a nonnegative constant (or more generally a complex num-

ber), which belongs to the domain of the existence of the conditional Laplace transform.

This domain does not depend on past factor realizations, that is, on the information

set. The conditional Laplace transform admits a closed form expression [see e.g. Cox,

Ingersoll and Ross (1985b)]. The conditional Laplace transform of the integrated CIR

process is an exponential affine function of the current factor value. It is given by

Et[exp(−z∫ t+h

t

xk,τdτ)] = exp[−Hk1 (h, z)xk,t −Hk

2 (h, z)], (3.3)

where

Hk1 (h, z) =

2z(exp[εk(z)h]− 1)

(εk(z) + ξk)(exp[εk(z)h]− 1) + 2εk(z),

Hk2 (h, z) =

−2ξkζkν2k

{log[2εk(z)] +h

2[εk(z) + ξk] (3.4)

− log[(εk(z) + ξk)(exp(εk(z)h)− 1) + 2εk(z)]},

εk(z) =√ξ2k + 2zν2

k .

This formula also holds for a complex number z = u+ iv, whenever u > −1 and v ∈ R.

63

The joint dynamics of factors and log-index can be represented by means of the

stochastic differential system:

d

x1,t

...

xK,t

log It

=

ξ1(ζ1 − x1,t)

...

ξK(ζK − xK,t)

µ0 +∑K

k=1 µkxk,t

dt

+

ν1√x1,t 0 · · · 0

0. . .

...

... νK√xK,t 0

0 · · · 0 (γ0 +∑K

k=1 γkxk,t)1/2

dw1,t

...

dwK,t

dwt

,

where both the drift vector and the volatility-covolatility matrix are affine functions of

the current values of the joint process (x1,t, · · · , xK,t, log It)′. Thus, the stacked process

(x1,t, · · · , xK,t, log It)′ is an affine process [see Duffie and Kan (1996)], and the conditional

Laplace transform of the integrated process Et[exp∫ t+h

t(z1x1,τ + · · ·+zKxK,τ +z log Iτ )dτ ]

will also admit an exponential affine closed form expression.

3.2.1.2 Specification of the Stochastic Discount Factor

The model is completed by a specification of a stochastic discount factor (SDF), which

is used later on to price all derivatives written on the index.

64

Assumption 3.3. The stochastic discount factor (SDF) for period (t, t+dt) is

Mt,t+dt = exp(dmt) = exp[(α0 +K∑k=1

αkxk,t)dt+ βd log It]

= exp{[α0 + βµ0 +K∑k=1

(αk + βµk)xk,t]dt+ β(γ0 +K∑k=1

γkxk,t)1/2dwt}. (3.5)

This SDF explains how to correct for risk when pricing derivatives. The “risk premia”

depend on the factors and index values, whereas the sensitivities of this correction with

respect to these risk variables are represented by the α and β parameters. The market

price of risk associated with wt is1 −β(γ0 +∑K

k=1 γkxk,t)1/2. This specification of the

SDF implicitly assumes that the market prices of the risk factors {wk,t}, k = 1, · · · , K

are 0. Equivalently, Equation (3.2) also describes the risk-neutral distribution of {xk,t},

k = 1, · · · , K. Under the risk-neutral probability, the joint dynamics of the underlying

factors and log-index can be represented by means of the stochastic differential system:

d

x1,t

...

xK,t

log It

=

ξ1(ζ1 − x1,t)

...

ξK(ζK − xK,t)

µ0+βγ0+∑

(µk+βγk)xk,t)

dt

+

ν1√x1,t 0 · · · 0

0. . .

...

... νK√xK,t 0

0 · · · 0√γ0+

∑γkxk,t

dw1,t

...

dwK,t

dw∗t

,

1This can be seen easily from the short rate computed in Equation (3.14).

65

where {wk,t}, k = 1, · · · , K, and {w∗t } are standard independent Brownian motions under

the risk-neutral probability. Thus only the last row is corrected for risk. This differential

stochastic system is still an affine process.

3.2.2 Pricing Formulas for European Derivatives Written on

the Index

As mentioned above, the arbitrage pricing proposes a valuation approach, which is

compatible with observed market prices and proposes coherent quotes for non-highly-

traded derivatives. More precisely, the value (price) at t of a European derivative paying

g(x1,t+h, · · · , xK,t+h, It+h) at time t+ h is

c(t, t+ h, g) = Et[exp(

∫ t+h

t

dmτ )g(x1,t+h, · · · , xK,t+h, It+h)]. (3.6)

The valuation formula is not assumed to be unique. Indeed, the SDF has been

parameterized by α0, α1, · · · , αK , β, but the parameter values have not been fixed ex-

ante. Thus, we propose implicitly different possible valuations and by comparing with

observed derivative prices, we estimate ex-post which one(s) is(are) compatible with

observed market prices.

The aim of this section is to derive explicit valuation formulas for European index

derivatives. All the formulas are derived from the valuation of European Index derivatives

with power payoff. Such derivatives are not traded or more generally quoted. But these

basic computations are used to derive:

• the risk-free term structure of interest rates

66

• the forward and futures prices of the index

• the prices of European options written on the index.

3.2.2.1 Power Derivatives Written on the Index

The following proposition is proved in Appendix B.1.

Proposition 3.1. The value at t of the European derivative paying exp[ulog(It+h)] =

(It+h)u at maturity t+h is

C(t, t+ h, u) = Et[(It+h)u exp(

∫ t+h

t

dmτ )]

= Et[exp(

∫ t+h

t

dmτ + u log It+h)]

= exp(u log It) exp[−hz0(u)−K∑k=1

Hk1 (h, zk(u))xk,t −

K∑k=1

Hk2 (h, zk(u))],

(3.7)

where

zk(u) = −αk − (β + u)µk −γk2

(β + u)2, ∀ k = 0, · · · , K, (3.8)

= zk(0) + ulk +γk2u(1− u), (3.9)

lk = −µk −1 + 2β

2γk, ∀ k = 0, · · · , K, (3.10)

and Hk1 (·, ·) and Hk

2 (·, ·) are given in equation(3.4).

Proposition 3.1 holds, if and only, if zk(u) > −1, ∀ k = 1, · · · , K. When we apply this

formula to different traded derivatives, i.e., different values of u, the inequalities above

imply restrictions on parameters α, β and γ.

3.2.2.2 The Risk-free Term Structure

The zero-coupon bonds correspond to a unitary payoff, and their prices B(t, t + h) cor-

respond to the special case of C(t, t+ h, u) where u = 0. The continuously compounded

67

risk-free interest rates are defined by r(t, t + h) = − 1h

logB(t, t + h). We make the

following proposition:

Proposition 3.2. The prices of the zero-coupon bonds are:

B(t, t+ h) = C(t, t+ h, 0)

= exp[−hz0(0)−K∑k=1

Hk1 (h, zk(0))xk,t −

K∑k=1

Hk2 (h, zk(0))], (3.11)

where zk(·) is defined in Equation (3.8), and Hk1 (·, ·) and Hk

2 (·, ·) are given in equa-

tion(3.4). We deduce the expressions of the interest rates:

r(t, t+ h) = −1

hlogB(t, t+ h)

= −1

h[−hz0(0)−

K∑k=1


K∑k=1

Hk2 (h, zk(0))]

= z0(0) +1

h

K∑k=1

Hk1 (h, zk(0))xk,t +

1

h

K∑k=1

Hk2 (h, zk(0)). (3.12)

The risk-free interest rates are affine functions of the CIR risk factors. This specifi-

cation is the standard affine term structure model introduced in Duffie and Kan (1996)

[see also Dai, Singleton (2000)]. It includes the one-factor CIR model [Cox, Ingersoll and

Ross (1985b)] as well as the multi-factor term structure model of Chen and Scott (1993).

As explained in subsection 3.2.2.1, the following restrictions are imposed on the pa-

rameters:

zk(0) = −αk − βµk −γk2β2 > −1, ∀ k = 1, · · · , K. (3.13)

The short rate is defined by r(t) = limh→0− 1h

logB(t, t+h). The following proposition

is proved in Appendix B.2.

68

Proposition 3.3. The short rate is given by

r(t) = limh→0−1

hlogB(t, t+ h) =

d[− logB(t, t+ h)]

dh|h=0

= z0(0) +K∑k=1

zk(0)xk,t. (3.14)

3.2.2.3 Forward Prices for the S&P 500 Index

A forward contract is an agreement to deliver or receive a specified amount of the under-

lying asset (or equivalent cash value) at a specified price and date. A forward contract

always has zero value when it is initiated. There is no money exchange initially or during

the life of the contract, except at the maturity date when the price paid is equal to the

specified forward price. The following proposition is proved in Appendix B.3.

Proposition 3.4. The forward prices are given by

f(t, t+ h) =C(t, t+ h, 1)

C(t, t+ h, 0)

= It exp{−hl0 −K∑k=1


K∑k=1

Hk1 (h, zk(0))xk,t

−K∑k=1

Hk2 (h, zk(1)) +

K∑k=1

Hk2 (h, zk(0))}, (3.15)

where zk(·) is defined in Equation (3.8), l0 is defined in Equation (3.10), and Hk1 (·, ·) and

Hk2 (·, ·) are given in equation(3.4).

In addition to the restrictions in (3.13), the following restrictions are imposed on the

parameters:

zk(1) = −αk − (β + 1)µk −γk2

(β + 1)2 = zk(0) + lk > −1, ∀ k = 1, · · · , K. (3.16)

69

3.2.2.4 Futures Prices

Let us now consider the price at t of a futures contract written on It+h. The major dif-

ference between a futures contract and a forward contract is the mark-to-market practice

for the futures. A futures contract has also zero value when it is issued and there is no

money exchange initially. However, at the end of each trading day during the life of the

contract, the party against whose favor the price changes must pay the amount of change

to the winning party. That is, a futures contract always has zero value at the end of each

trading day during the life of the contract. If the interest rate is stochastic, the forward

price and futures price are generally not the same [See Cox, Ingersoll and Ross (1981)

and French (1983)]. The following proposition is proved in Appendix B.4.

Proposition 3.5. The prices at t of futures written on It+h are given by

Ft,t+h = Et[exp(

∫ t+h

t

dmτ ) exp(

∫ t+h

t

rτdτ)It+h]

= It exp[−hl0 −K∑k=1

Hk1 (h, lk)xk,t −

K∑k=1

Hk2 (h, lk)], (3.17)

where lk is defined in Equation (3.10), and Hk1 (·, ·) and Hk


As explained earlier, in addition to the restrictions in (3.13), the following restrictions

are imposed on the parameters:

lk = −µk −1 + 2β

2γk > −1 ∀ k = 1 · · ·K. (3.18)

Propositions 3.4 and 3.5 show that

f(t, t+ h) = Et[exp(

∫ t+h

t

dmτ ) exp(r(t, t+ h)h)It+h]

70

and

Ft,t+h = Et[exp(

∫ t+h

t

dmτ ) exp(

∫ t+h

t

rτdτ)It+h].

Since the short rate is stochastic, the forward and futures prices are not equal in general.

A sufficient condition for the forward and futures prices to be identical is zk(0) = 0,

∀ k = 1 · · ·K, i.e., the interest rates are non-stochastic. This is Proposition 3 in Cox,

Ingersoll and Ross (1981).

3.2.2.5 European Call and Put Options Written on the Index

The prices of the European options are deduced by applying a transform analysis to

function C(t, t + h, u) computed for pure imaginary argument u [see Duffie, Pan and

Singleton (2000) and Appendix B.5].

Proposition 3.6.

i) The European call prices are given by

G(t, t+ h,X) = Et{exp(

∫ t+h

t

dmτ )[exp(log It+h)−X]+} (3.19)

=C(t, t+ h, 1)

2− 1

π

∫ ∞

0

Im[C(t, t+ h, 1− iv) exp(iv logX)]

vdv

−X{C(t, t+ h, 0)

2− 1

π

∫ ∞

0

Im[C(t, t+ h,−iv) exp(iv logX)]

vdv}

(3.20)

where X is the strike price, h is the time-to-maturity, i denotes the pure imaginary number

and Im(·) is the imaginary part of a complex number.

71

ii) The European put prices are given by

H(t, t+ h,X) = Et{exp(

∫ t+h

t

dmτ )[X − exp(log It+h)]+} (3.21)

= −C(t, t+ h, 1)

2+

1

π

∫ ∞

0

Im[C(t, t+ h, 1 + iv) exp(−iv logX)]

vdv

+X{C(t, t+ h, 0)

2− 1

π

∫ ∞

0

Im[C(t, t+ h, iv) exp(−iv logX)]

vdv}.

(3.22)

Again, the restrictions (3.13) and (3.16) are imposed2.

In particular, the relationship between the prices of European call and put options is

G(t, t+ h,X)− C(t, t+ h, 1) = H(t, t+ h,X)−XC(t, t+ h, 0) (3.23)

− 1

π

∫ ∞

0


vdv

+1

π

∫ ∞

0


vdv

+X

π

∫ ∞

0


vdv

− X

π

∫ ∞

0


vdv

Equation (3.23) provides the deviation to the Put-Call Parity due to the non-tradability

of the underlying index and shows that this deviation is stochastic.

3.2.3 Pricing Formulas for European Derivatives Written onFutures

3.2.3.1 Derivatives Written on Futures

As for derivatives written on the index, we first consider European derivatives written

on futures with exponential payoffs. More precisely, we introduce three different dates:

2Re(zk(1− iv)) > −1, Re(zk(−iv)) > −1, Re(zk(1 + iv)) > −1 and Re(zk(iv)) > −1,∀ k = 1, · · · ,K,where Re(·) denotes the real part of a complex number, should also hold in order for the pricing formulasto exist. Re(zk(1 − iv)) = Re(zk(1 + iv)) = −αk − (β + 1)µk − γk

2 (β + 1)2 + γk

2 v2 = zk(1) + γk

2 v2 and

Re(zk(−iv)) = Re(zk(iv)) = −αk − βµk − γk

2 β2 + γk

2 v2 = zk(0) + γk

2 v2. So the restrictions (3.13)and

(3.16)are sufficient.

72

• t is the current date

• t+h is the maturity date of the derivatives on futures

• t+h+m is the maturity date of the futures on which the derivatives are written.

CF (t, t + h, t + h + m,u) denotes the price at t of the European derivative paying

(Ft+h,t+h+m)u at t+ h. The following proposition is proved in Appendix B.6.

Proposition 3.7. The prices at t of the European derivatives paying (Ft+h,t+h+m)u at

t+ h are given by

CF (t, t+ h, t+ h+m,u)

=Et[exp

∫ t+h

t

dmτ (Ft+h,t+h+m)u]

= exp(u log It) exp{m(uµ0 −1 + 2β

2uγ0) + h[(β + u)µ0 +

(β + u)2

2γ0 + α0]

−K∑k=1

uHk1 (m, lk)hξkζk −

K∑k=1

uHk2 (m, lk)−

K∑k=1

Hk2 (h, pk(m,u))

−K∑k=1

[uHk1 (m, lk) +Hk

1 (h, pk(m,u))]xk,t} (3.24)

where

pk(m,u) = −αk−(β+u)µk−uHk1 (m, lk)ξk−

γk2

(β+u)2−u2

2[Hk

1 (m, lk)]2ν2k ∀ k = 1 · · ·K,

(3.25)

lk is given in equation(3.10), and Hk1 (·, ·) and Hk


Again, we impose (3.13) and (3.18) as well as the following restrictions on the param-

eters:

pk(m,u) = −αk − (β + u)µk − uHk1 (m, lk)ξk −

γk2

(β + u)2 − u2

2[Hk

1 (m, lk)]2ν2k > −1,

∀ k = 1, · · · , K. (3.26)

73

3.2.3.2 European Call Options Written on Futures

The following proposition is proved in Appendix B.7.

Proposition 3.8. The prices of European calls written on futures are given by

GF (t, t+ h, t+ h+m,X)

=Et{exp(

∫ t+h

t

dmτ )[exp(logFt+h,t+h+m)−K]+}

=CF (t, t+ h, t+ h+m, 1)

2

− 1

π

∫ ∞

0

Im[CF (t, t+ h, t+ h+m, 1− iv) exp(iv logX)]

vdv

−X{CF (t, t+ h, t+ h+m, 0)

2

− 1

π

∫ ∞

0

Im[CF (t, t+ h, t+ h+m,−iv) exp(iv logX)]

vdv} (3.27)

The parameters are subject to the restrictions in (3.13), (3.18) and3

pk(m, 1) = −αk − (β + 1)µk −Hk1 (m, lk)ξk −

γk2

(β + 1)2 − 1

2[Hk

1 (m, lk)]2ν2k > −1,

∀ k = 1, · · · , K. (3.28)

3.3 Parameter Restrictions for a Tradable Index

In Section 3.2, the pricing formulas are valid for tradable or non-tradable indices. In this

section, we derive the restrictions implied by the tradability of the underlying index.

When the benchmark index is a self-financed and tradable asset, the pricing formula

is also valid for the index itself. In this case, we have

It = Et[exp(

∫ t+h

t

dmτ )It+h] = C(t, t+ h, 1),

3Restrictions (3.13) imply pk(m, 0) = zk(0) > −1. The inequalities pk(m, 0) > −1 and pk(m, 1) > −1imply Re(pk(m,−iv)) > −1 and Re(pk(m, 1− iv)) > −1.

74

and

C(t, t+ h, 1) = It exp[−hz0(1)−K∑k=1


K∑k=1

Hk2 (h, zk(1))], (3.29)

where zk(·) is defined in Equation (3.8), and Hk1 (·, ·) and Hk

2 (·, ·) are given in equation

(3.4). zk(1) > −1 is imposed, ∀ k = 1, · · · , K.

By considering the expression of C(t, t + h, 1) and identifying the different terms in

the decomposition, we see that the dynamic parameters are constrained byHk

1 (h, zk(1)) = 0, ∀ k = 1, · · · , K, ∀h,

−hz0(1)−∑K

k=1Hk2 (h, zk(1)) = 0, ∀h,

(3.30)

or equivalently by the conditions shown in Proposition 3.9 (See the proof in Appendix

B.8).

Proposition 3.9. When the benchmark index is a self-financed and tradable asset, the

dynamic parameters are constrained by

zk(1) = αk + (β + 1)µk +γk2

(β + 1)2 = zk(0) + lk = 0, ∀ k = 0, · · · , K. (3.31)

These restrictions fix the parameters {αk}, k = 0, · · · , K of the SDF as functions of

the parameters of the index dynamics.

When the benchmark index is tradable, the risk-neutral dynamics of log It can also

be written as

d log It = [r(t)− γ0 +∑γkxk,t

2]dt+

√γ0+

∑γkxk,tdw

∗t , (3.32)

or

dItIt

= r(t)dt+√γ0+

∑γkxk,tdw

∗t . (3.33)

75

In other word, conditional on the underlying factors, the risk wt can be hedged by the

index and the short rate if the index is tradable.

If the benchmark index is tradable, the formulas of derivative prices can be simplified.

In particular, the forward price derived in Proposition 3.4 simplifies to the standard

formula:

f(t, t+ h) =It

B(t, t+ h), (3.34)

and the Spot-Futures Parity will hold for the index and its forward price.

3.4 Conclusion

In this paper we consider a coherent multi-factor affine model to price various derivatives

such as forwards, futures and European options written on the non-tradable S&P 500

Index, and derivatives written on the S&P 500 futures.

We consider both cases when the underlying index is self-financed and tradable and

when it is not, and show the difference between them. When the underlying asset is

self-financed and tradable, an additional arbitrage condition has to be introduced and

implies additional parameter restrictions. These restrictions can be tested in practice to

check whether the derivatives are priced as if the underlying index were self-financed and

tradable.

The S&P 500 Index is not the only non-tradable index on which various derivatives

have been written. Our model can be easily extended to price derivatives written on

76

other non-tradable indices such as a retail price index, a meteorological index, or an

index summarizing the results of a set of insurance companies.

77

Appendix A

Official Description of the Index

The S&P presents the computation of the S&P 500 Index in the following way1. The

S&P 500 Composite Index is calculated with divisors using the so-called “base-aggregative

approach” as follows:

Indext =

∑i pi(t),tqi(t),tDt

, (A.1)

where Dt is the base value or divisor at time t. That is, each component stock’s price is

multiplied by the number of outstanding common shares available to the public for that

company, and the resulting float-adjusted market values are summed for all 500 stocks

and divided by a predetermined base value. Equation (A.1) implies that

Indext+1

Indext=

∑i pi(t+1),t+1qi(t+1),t+1∑

i pi(t),tqi(t),t

Dt

Dt+1

(A.2)

The stocks in the index change from time to time because of mergers, acquisitions or

bankruptcies. For example, in 2005, there were 20 component changes to the S&P 500

Index. The changes are effective after the closure of the trading day. When the index

1Most of the material from this appendix is from www.standardandpoors.com.

78

composition changes, the index value should not change, as otherwise market moves would

be confused by index maintenance. The divisor is used to keep the index value unchanged.

The float adjusted market capitalization of the index is calculated before and after the

change. Before the change, we know the market capitalization2,∑

i pi(t),tqi(t),t, the index

level and the divisor of time t. After the change, we know the market capitalization with

the new stocks {i(t + 1)}, which is∑

i pi(t+1),tqi(t+1),t+1, and the index level (which will

not change); Thus, we can solve for the new divisor, that is,

Indext =

∑i pi(t),tqi(t),tDt

=

∑i pi(t+1),tqi(t+1),t+1

Dt+1

⇒Dt+1

Dt

=

∑i pi(t+1),tqi(t+1),t+1∑

i pi(t),tqi(t),t. (A.3)

The S&P performs the calculations after the market closes. The index opens the next

day with the new stock components and the new divisor. The index level changes only

if the prices of the stocks change.

We can derive from above that

Indext+1

Indext=

∑i pi(t+1),t+1qi(t+1),t+1∑

i pi(t),tqi(t),t

Dt

Dt+1

=

∑i pi(t+1),t+1qi(t+1),t+1∑i pi(t+1),tqi(t+1),t+1

.

The second equation is the same as Equation(1.1).

Equation (A.3) holds generally no matter whether t is a market closing time or not.

If t is not a closing time,

Dt+1

Dt

= 1 =

∑i pi(t+1),tqi(t+1),t+1∑i pi(t+1),tqi(t+1),t+1

=

∑i pi(t+1),tqi(t+1),t+1∑

i pi(t),tqi(t),t,

2Since changes only happen after the market closes, t denotes the closing time and t+1 the openingtime on the next day.

79

since {i(t+ 1)} is the same as {i(t)} and hence qi(t+1),t+1 is the same as qi(t),t for all i.

The divisor is also adjusted periodically (typically, several times per quarter), when

the total of outstanding shares have changed in one or more component index securities

due to secondary offering, repurchases, conversions or other corporate actions, that is,

{qi(t),t} is different from {qi(t+1),t+1}, although {i(t + 1)} is the same as {i(t)}. If the

number of shares of a stock changes, the divisor is adjusted in the same way as when

we have stock changes so the index level remains unchanged. The index can be adjusted

in this way when companies issue new shares or buy back their shares. The same basic

approach is used for stock price adjustments, such as when a company spins off a unit.

Adjustments to the divisor assure that changes in the index’s level reflect the changes in

the market and not the corporate actions. There are many other corporate actions that

require divisor adjustments, for instance, a change of the available float shares of index

securities. One common action that does not require a divisor adjustment is a stock split

when the number of shares increases and their price decreases in proportion.

The S&P 500 is maintained by the S&P Index Committee, that is, a team of Standard

& Poor’s economists and index analysts who meet on a regular basis.

i) Eligibility criteria for inclusion in the S&P 500 are:

• U.S. companies.

• Adequate liquidity and reasonable per-share price – the ratio of annual dollar

value traded to market capitalization should be 0.3 or greater. Very low stock

prices can affect a stock liquidity.

80

• Market capitalization of $4 Billion or more for the S&P 500. The market

capitalization of a potential addition to an index is looked at in the context of

its short- and medium-term historical trends, as well as those of its industry.

The range is reviewed from time to time to ensure consistency with market

conditions.

• Financial viability, usually measured as four consecutive quarters of positive

as-reported earnings. As-reported earnings are GAAP Net Income excluding

discontinued operations and extraordinary items.

• Public float of at least 50% of the stock.

• Maintaining sector balance for each index, as measured by a comparison of

the GICS sectors in each index and in the market, in the relevant market

capitalization ranges.

• Initial public offerings (IPOs) should be seasoned for 6 to 12 months before

being considered for addition to indices.

• Operating company and not a closed-end fund, holding company, partnership,

investment vehicle or royalty trust. Real Estate Investment Trusts are eligible

for inclusion in Standard & Poor’s U.S. indices.

ii) Eligibility criteria for deletions from the S&P 500 are:

• Companies involved in mergers, being acquired or significantly restructured

such that they no longer meet inclusion criteria.

81

• Companies which substantially violate one or more of the addition criteria.

The Standard & Poor’s believes turnover in index membership should be avoided

when possible. The addition criteria are for addition to an index, not for continued

membership. As a result, a company in an index that appears to violate the criteria for

addition to that index will not be deleted unless ongoing conditions warrant an index

change. When a company is removed from an index, the Standard & Poor’s will explain

the basis for the removal.

There are totally 176 index composition changes from Jan 5, 2000 to March 31, 2006.

In 162 of these changes, the deleted stock is immediately replaced with another stock

and the simultaneous deletion and addition are counted as a single change. In the other

14 changes, what typically happens is a stock is deleted from the index after the market

closes on the first day and another stock is added to the index after the market closes

on the next one or two days. The deletion and the addition are counted as two changes.

During the period when the old stock is deleted but the new stock is not added yet,

the index is usually calculated with the closing price of the deleted stock on the day the

deletion happens. For example, on January 3, 2006, the Standard & Poors announced

that Estee Lauder Companies, Inc. would be added to the S&P 500 after the close of

trading on January 4. Estee Lauder would take the place of S&P 500 constituent Mercury

Interactive Corp., which would be removed from the index after the close of trading on

January 3. On January 4, the S&P 500 is calculated with the closing price of Mercury

Interactive on January 3. Then, when Estee Lauder is added on January 4, the divisor

82

is adjusted to reflect the change of the index.

Changes to the S&P 500 Index are made on an as needed basis. There is no annual or

semi-annual reconstitution. Rather, changes in response to corporate actions and market

developments can be made at any time. Constituent changes are typically announced two

to five days before they are scheduled to be implemented. Announcements are available

to the public via the web site, www.indices.standardandpoors.com, before or at the same

time they are available to clients or the affected companies.

In cases where there is no achievable market price for a stock being deleted, it can be

removed at a zero or minimal price at the index committee’s discretion, in recognition of

the constraints faced by investors in trading bankrupt or suspended stocks.

The index committee also lays down policies about share changes. Changes in a com-

pany’s outstanding shares of less than 5% due to its acquisition of another company in

the same headline index (e.g., both are in the S&P 500) are made as soon as reasonably

possible. All other changes of less than 5% are accumulated and made quarterly on the

third Friday of March, June, September, and December; they are usually announced two

days prior. Changes in a company’s outstanding shares of 5% or more due to mergers, ac-

quisitions, public offerings, private placements, tender offers, Dutch auctions or exchange

offers are made as soon as reasonably possible. Other changes of 5% or more (due to,

for example, company stock repurchases, redemptions, exercise of options, warrants, con-

version of preferred stock, notes, debt, equity participations or other recapitalizations)

are made weekly, and are announced on Tuesday for implementation after the close of

trading on Wednesday. In the case of certain rights issuances, in which the number of

83

rights issued and/or terms of their exercise are deemed substantial, a price adjustment

and share increase may be implemented immediately. Corporate actions such as stock

splits, stock dividends, spinoffs and rights offerings are applied after the close of trading

on the day prior to the ex-date.

Changes in IWF’s of more than ten percentage points caused by corporate actions

(such as M&A activity, restructurings or spinoffs) are made as soon as reasonably possible.

Other changes in IWF’s are made annually, in September, when IWF’s are reviewed.

84

Appendix B

Proofs of Propositions

B.1 Proof of Proposition 3.1

The price of the call option is:

C(t, t+ h, u)

=Et[exp(

∫ t+h

t

dmτ + u log It+h)]

=Et{exp[

∫ t+h

t

dmτ + u(log It +

∫ t+h

t

d log Iτ )]}

= exp(u log It)Et[exp(

∫ t+h

t

(dmτ + ud log Iτ )]

= exp(u log It)Et{exp

∫ t+h

t

([α0 + βµ0 +K∑k=1

(αk + βµk)xk,τ ]dτ

+ β(γ0 +K∑k=1

γkxk,τ )1/2dwτ + u[(µ0 +

K∑k=1

µkxk,τ )dτ + (γ0 +K∑k=1

γkxk,τ )1/2dwτ ])}


∫ t+h

t

([α0 + (β + u)µ0 +K∑k=1

(αk + (β + u)µk)xk,τ ]dτ

+ (β + u)(γ0 +K∑k=1

γkxk,τ )1/2dwτ )}

85


∫ t+h

t

[α0 + (β + u)µ0 +K∑k=1


× Et(exp[(β + u)

∫ t+h

t

(γ0 +K∑k=1

γkxk,τ )1/2dwτ ] | Xk,τ )},

where Xk,τ denotes the set {xk,τ}k=1···Kτ=t···t+h.

Since exp[(β + u)

∫ t+h

t

(γ0 +K∑k=1

γkxk,τ )1/2dwτ ] | Xk,τ

∼ LN(0, (β + u)2

∫ t+h

t

(γ0 +K∑k=1

γkxk,τ )dτ),

we deduce that:

C(t, t+ h, u)


∫ t+h

t

[α0 + (β + u)µ0 +K∑k=1


× exp[(β + u)2

2

∫ t+h

t

(γ0 +K∑k=1

γkxk,τ )dτ ]}


∫ t+h

t

([α0 + (β + u)µ0 +K∑k=1


+(β + u)2

2(γ0 +

K∑k=1

γkxk,τ )dτ)}

= exp(u log It) exp

∫ t+h

t

[α0 + (β + u)µ0 +(β + u)2

2γ0]dτ

× Et{exp

∫ t+h

t

K∑k=1

[(αk + (β + u)µk +(β + u)2

2γk)xk,τ ]dτ}

= exp(u log It) exp{h[α0 + (β + u)µ0 +(β + u)2

2γ0]}

× Et{expK∑k=1

∫ t+h

t

[αk + (β + u)µk +(β + u)2

2γk]xk,τdτ}


2γ0]}

86

×K∏k=1

Et{exp−∫ t+h

t

−[αk + (β + u)µk +(β + u)2

2γk]xk,τdτ},

since factors {xk,t}, k = 1, · · · , Kare independent,


2γ0]}

×K∏k=1

exp[−Hk1 (h, zk(u))xk,t −Hk

2 (h, zk(u))]

= exp(u log It) exp{h[α0 + (β + u)µ0 +γ0

2(β + u)2]

−K∑k=1

Hk1 (h, zk(u))xk,t −

K∑k=1

Hk2 (h, zk(u))},

where

zk(u) = −αk − (β + u)µk −γk2

(β + u)2,

and Hk1 (·, ·) and Hk

2 (·, ·) are given in equation(3.4)


The instantaneous interest rate is defined by:

r(t) = limh→0−1

hlogB(t, t+ h) =


dh|h=0 .

We have:

− logB(t, t+ h) = −h(α0 + βµ0 +γ0

2β2) +

K∑k=1


K∑k=1

Hk2 (h, zk(0)).

We deduce that:


dh= −α0 − βµ0 −

γ0

2β2 +

K∑k=1

dHk1 (h, zk(0))

dhxk,t +

K∑k=1

dHk2 (h, zk(0))

dh,

where

87

Hk1 (h, zk(0)) =

2zk(0)(exp[εk(zk(0))h]− 1)

(εk(zk(0)) + ξk)(exp[εk(zk(0))h]− 1) + 2εk(zk(0)),

Hk2 (h, zk(0)) =

−2ξkζkν2k

{log[2εk(zk(0))] +h

2[εk(zk(0)) + ξk]

− log[(εk(zk(0)) + ξk)(exp[εk(zk(0))h]− 1) + 2εk(zk(0))]}.

Let us denote (εk(zk(0)) + ξk)(exp[εk(zk(0))h]− 1) + 2εk(zk(0)) ≡ A. We get:

dHk1 (h, zk(0))

dh|h=0

=2zk(0) exp[εk(zk(0))h]εk(zk(0))A

A2

− 2zk(0)(exp[εk(zk(0))h]− 1)(εk(zk(0)) + ξk) exp[εk(zk(0))h]εk(zk(0))

A2|h=0

=2zk(0)εk(zk(0))2εk(zk(0))− 0

[2εk(zk(0))]2= zk(0),

and

dHk2 (h, zk(0))

dh|h=0

=−2ξkζkν2k

{1

2[εk(zk(0)) + ξk]−

1

A(εk(zk(0)) + ξk) exp[εk(zk(0))h]εk(zk(0))} |h=0

=−2ξkζkν2k

{1

2[εk(zk(0)) + ξk]−

(εk(zk(0)) + ξk)εk(zk(0))

2εk(zk(0))} = 0.

We deduce:

r(t) = −α0 − βµ0 −γ0

2β2 +

K∑k=1

zk(0)xk,t.


Since

E[exp(

∫ t+h

t

dmτ )(f(t, t+ h)− It+h)] = 0,

88

we get:

B(t, t+ h)f(t, t+ h) = E[exp(

∫ t+h

t

dmτ )It+h],

and

f(t, t+ h) =C(t, t+ h, 1)

C(t, t+ h, 0).


Since Et[

∫ t+h

t

(exp

∫ t+τ

t

dms)dFτ ] = 0, we get:

Ft,t+h

=Et[exp(

∫ t+h

t

dmτ ) exp(

∫ t+h

t

rτdτ)It+h]

=Et{exp[

∫ t+h

t

(dmτ + rτdτ) + log It +

∫ t+h

t

d log Iτ ]}

=ItEt[exp(

∫ t+h

t

(dmτ + rτdτ + d log Iτ )]

=ItEt{exp

∫ t+h

t

[(α0 + βµ0 +K∑k=1

(αk + βµk)xk,τ )dτ

+ β(γ0 +K∑k=1

γkxk,τ )1/2dwτ + (µ0 +

K∑k=1

µkxk,τ )dτ

+ (γ0 +K∑k=1

γkxk,τ )1/2dwτ + (−α0 − βµ0 −

γ0

2β2 +

K∑k=1

zk(0)xk,τ )dτ ]}

=ItEt{exp

∫ t+h

t

[(α0 + (β + 1)µ0 − α0 − βµ0 −γ0

2β2)dτ

+K∑k=1

(αk + βµk + µk + zk(0))xk,τdτ + (β + 1)(γ0 +K∑k=1

γkxk,τ )1/2dwτ ]}

=It exp[h(µ0 −γ0

2β2)]Et{exp

∫ t+h

t

K∑k=1

(αk + (β + 1)µk + zk(0))xk,τdτ

× Et[exp

∫ t+h

t

(β + 1)(γ0 +K∑k=1

γkxk,τ )1/2dwτ | Xk,τ ]}.

89

Since exp

∫ t+h

t

(β + 1)(γ0 +K∑k=1

γkxk,τ )1/2dwτ | Xk,τ

∼ LN(0, (β + 1)2

∫ t+h

t

(γ0 +K∑k=1

γkxk,τ )dτ),

we get:

Ft,t+h


2β2)]

× Et{exp

∫ t+h

t

[K∑k=1

(αk + (β + 1)µk + zk(0))xk,τdτ +(β + 1)2

2(γ0 +

K∑k=1

γkxk,τ )dτ ]}


2β2 +

(β + 1)2

2γ0)]

× Et{exp

∫ t+h

t

[K∑k=1

(αk + (β + 1)µk + zk(0) +(β + 1)2

2γk)xk,τ ]dτ}


2β2 +

(β + 1)2

2γ0)]

×K∏k=1

Et{exp−∫ t+h

t

−[αk + (β + 1)µk + zk(0) +(β + 1)2

2γk]xk,τdτ}


2β2 +

(β + 1)2

2γ0)]

K∏k=1

exp[−Hk1 (h, lk)xk,t −Hk

2 (h, lk)]

=It exp[h(µ0 +1 + 2β

2γ0)−

K∑k=1

Hk1 (h, lk)xk,t −

K∑k=1

Hk2 (h, lk)],

where

lk = −αk − (β + 1)µk − zk(0)− γk2

(β + 1)2

= −µk −1 + 2β

2γk.

90


Let us first consider the call option with price G(t, t+ h,X). Its price is given by:

G(t, t+ h,X)

= Et{exp(

∫ t+h

t

dmτ )[exp(log It+h)−X]+}

= Et{exp(

∫ t+h

t

dmτ )[exp(log It+h)−X]1− log It+h≤− logX}

= A1,−1(− logX;x1,t, · · · , xK,t, log It, h)−XA0,−1(− logX;x1,t, · · · , xK,t, log It, h),

whereAa,b(y;x1,t, · · · , xK,t, log It, h) = Et[exp(

∫ t+h

t

dmτ ) exp(a log It+h)1b log It+h≤y].

The Fourier-Stieltjes transform of Aa,b(y;x1,t, · · · , xK,t, log It, h) is

∫<

exp(ivy)dAa,b(y;x1,t, · · · , xK,t, log It, h)

= Et{exp(

∫ t+h

t

dmτ ) exp[(a+ ivb) log It+h]} = C(t, t+ h, a+ ivb).

We deduce that:

Aa,b(y;x1,t, · · · , xK,t, log It, h)

=C(t, t+ h, a)

2− 1

π

∫ ∞

0

Im[C(t, t+ h, a+ ivb) exp(−ivy)]

vdv

[see Duffie, Pan and Singleton (2000), p1352].

By substitution we get the call price

G(t, t+ h,X) =C(t, t+ h, 1)

2− 1

π

∫ ∞

0


vdv

−X{C(t, t+ h, 0)

2− 1

π

∫ ∞

0


vdv}.

91

Similarly, for the put option with price H(t, t+ h,X), we have

H(t, t+ h,X)

= Et{exp(

∫ t+h

t

dmτ )[X − exp(log It+h)]+}

= Et{exp(

∫ t+h

t

dmτ )[X − exp(log It+h)]1 log It+h≤logX}

= −A1,1(logX;x1,t, · · · , xK,t, log It, h) +XA0,1(logX;x1,t, · · · , xK,t, log It, h)

= −C(t, t+ h, 1)

2+

1

π

∫ ∞

0


vdv

+X{C(t, t+ h, 0)

2− 1

π

∫ ∞

0


vdv}.


We have:

Ft+h,t+h+m = It+h exp[m(µ0 −1 + 2β

2γ0)−

K∑k=1

Hk1 (m, lk)xk,t+h −

K∑k=1

Hk2 (m, lk)],

logFt+h,t+h+m = log It+h +m(µ0 −1 + 2β

2γ0)−

K∑k=1

Hk1 (m, lk)xk,t+h −

K∑k=1

Hk2 (m, lk).

Therefore,

CF (t, t+ h, t+ h+m,u)

= Et(exp

∫ t+h

t

dmτ (Ft+h,t+h+m)u)

= Et[exp

∫ t+h

t

dmτ exp(u logFt+h,t+h+m)]

= Et[exp(

∫ t+h

t

dmτ + u logFt+h,t+h+m)]

92

= Et{exp(

∫ t+h

t

[(α0 + βµ0 +K∑k=1

(αk + βµk)xk,τ )dτ + β(γ0 +K∑k=1

γkxk,τ )1/2dwτ ]

+ u[log It +

∫ t+h

t

d log It +m(µ0 −1 + 2β

2γ0)

−K∑k=1

Hk1 (m, lk)(xk,t +

∫ t+h

t

dxk,τ )−K∑k=1

Hk2 (m, lk)])}

= exp[u log It + um(µ0 −1 + 2β

2γ0)− u

K∑k=1

Hk1 (m, lk)xk,t − u

K∑k=1

Hk2 (m, lk)]

× Et{exp

∫ t+h

t

[(α0 + βµ0 +K∑k=1

(αk + βµk)xk,τ )dτ + β(γ0 +K∑k=1

γkxk,τ )1/2dwτ

+ u(µ0 +K∑k=1

µkxk,τ )dτ + u(γ0 +K∑k=1

γkxk,τ )1/2dwτ

− uK∑k=1

Hk1 (m, lk)(ξk(ζk − xk,τ )dτ + νk

√xk,τdwk,τ )]}

= exp[u log It + um(µ0 −1 + 2β

2γ0)− u

K∑k=1

Hk1 (m, lk)xk,t − u

K∑k=1

Hk2 (m, lk)]

× Et{exp

∫ t+h

t

[(α0 + βµ0 + uµ0 − uK∑k=1

Hk1 (m, lk)ξkζk)dτ

+K∑k=1

(αk + βµk + uµk + uHk1 (m, lk)ξk)xk,τ )dτ

+ (β + u)(γ0 +K∑k=1

γkxk,τ )1/2dwτ − u

K∑k=1

Hk1 (m, lk)νk

√xk,τdwk,τ ]}

= exp{u[log It +m(µ0 −1 + 2β

2γ0)−

K∑k=1

Hk1 (m, lk)xk,t −

K∑k=1

Hk2 (m, lk)]

+ h[α0 + (β + u)µ0 − uK∑k=1

Hk1 (m, lk)ξkζk]}

× Et{exp(

∫ t+h

t

K∑k=1

[αk + (β + u)µk + uHk1 (m, lk)ξk]xk,τdτ)

× Et(exp

∫ t+h

t

[(β + u)(γ0 +K∑k=1

γkxk,τ )1/2dwτ − u

K∑k=1

Hk1 (m, lk)νk

√xk,τdwk,τ ] | Xk,τ )}

93

= exp{u log It + (um+ hβ + hu)µ0 −1 + 2β

2umγ0 + hα0

−K∑k=1

uHk1 (m, lk)(xk,t + hξkζk)−

K∑k=1

uHk2 (m, lk)}

× Et{exp(

∫ t+h

t

K∑k=1

[αk + (β + u)µk + uHk1 (m, lk)ξk]xk,τdτ)

× (exp

∫ t+h

t

((β + u)2

2(γ0 +

K∑k=1

γkxk,τ ) +K∑k=1

u2

2[Hk

1 (m, lk)]2ν2kxk,τ )dτ)}

= exp{u log It + (um+ hβ + hu)µ0 −1 + 2β

2umγ0 + hα0

−K∑k=1


K∑k=1

uHk2 (m, lk)}

× Et{exp

∫ t+h

t

((β + u)2

2γ0

+K∑k=1

[αk + (β + u)µk + uHk1 (m, lk)ξk +

(β + u)2

2γk +

u2

2[Hk

1 (m, lk)]2ν2k ]xk,τ )dτ}

= exp{u log It + (um+ hβ + hu)µ0 +(β + u)2h− (1 + 2β)um

2γ0 + hα0

−K∑k=1


K∑k=1

uHk2 (m, lk)}

×K∏k=1

Et[exp−∫ t+h

t

−(αk + (β + u)µk + uHk1 (m, lk)ξk

+(β + u)2

2γk +

u2

2[Hk

1 (m, lk)]2ν2k)xk,τdτ ]


2γ0 + hα0

−K∑k=1


K∑k=1

uHk2 (m, lk)}

×K∏k=1

exp[−Hk1 (h, pk(m,u))xk,t −Hk

2 (h, pk(m,u))]


2γ0 + hα0

94

−K∑k=1


K∑k=1

uHk2 (m, lk)−

K∑k=1

Hk2 (h, pk(m,u))

−K∑k=1

[uHk1 (m, lk) +Hk

1 (h, pk(m,u))]xk,t}

= exp(u log It) exp{m(uµ0 −1 + 2β

2uγ0) + h[(β + u)µ0 +

(β + u)2

2γ0 + α0]

−K∑k=1


K∑k=1

uHk2 (m, lk)−

K∑k=1

Hk2 (h, pk(m,u))

−K∑k=1

[uHk1 (m, lk) +Hk

1 (h, pk(m,u))]xk,t},

where

pk(m,u) = −αk − (β + u)µk − uHk1 (m, lk)ξk −

γk2

(β + u)2 − u2

2[Hk

1 (m, lk)]2ν2k ,

lk is given in equation(3.10), and Hk1 (·, ·) and Hk



The price of the call optioin written on the futures is:


= Et{exp(

∫ t+h

t

dmτ )[exp(logFt+h,t+h+m)−X]+}

= Et{exp(

∫ t+h

t

dmτ )[exp(logFt+h,t+h+m)−X]1− logFt+h,t+h+m≤− logX}

= A1,−1(− logX;x1,t, · · · , xK,t, logFt,t+h+m, h)

−XA0,−1(− logX;x1,t, · · · , xK,t, logFt,t+h+m, h),

where

Aa,b(y;x1,t, · · · , xK,t, logFt,t+h+m, h)

95

=Et[exp(

∫ t+h

t

dmτ ) exp(a logFt+h,t+h+m)1b logFt+h,t+h+m≤y].

The Fourier-Stieltjes transform of Aa,b(y;x1,t, · · · , xK,t, logFt,t+h+m, h) is:∫<

exp(ivy)dAa,b(y;x1,t, · · · , xK,t, logFt,t+h+m, h)

= Et{exp(

∫ t+h

t

dmτ ) exp[(a+ ivb) logFt+h,t+h+m]} = CF (t, t+ h, t+ h+m, a+ ivb).

Therefore, we have:

Aa,b(y;x1,t, · · · , xK,t, logFt,t+h+m, h)

=CF (t, t+ h, t+ h+m, a)

2− 1

π

∫ ∞

0

Im[CF (t, t+ h, t+ h+m, a+ ivb) exp(−ivy)]

vdv,

and


=CF (t, t+ h, t+ h+m, 1)

2− 1

π

∫ ∞

0

Im[CF (t, t+ h, t+ h+m, 1− iv) exp(iv logX)]

vdv

−X{CF (t, t+ h, t+ h+m, 0)

2

− 1

π

∫ ∞

0

Im[CF (t, t+ h, t+ h+m,−iv) exp(iv logX)]

vdv}.


The first restriction in Equation (3.30) holds, if and only, if

zk(1) = −αk − (β + 1)µk −γk2

(β + 1)2 = 0, ∀k = 1, · · · , K.

This implies εk(zk(1)) =|ξk |, ∀k = 1, · · · , K, and

Hk2 (h, zk(1)) =

−2ξkζkν2k

{log|2ξk |+h

2(|ξk |+ξk)− log[(|ξk |+ξk)(exp(|ξk |h)− 1) + 2 |ξk |]}

96

= 0, no matter if ξk > 0 or ξk < 0, ∀k = 1, · · · , K.

This, with the second restriction in Equation (3.30), implies that

α0 + (β + 1)µ0 +γ0

2(β1)

2 = 0.

Therefore, Equation (3.30) is equivalent to

αk + (β + 1)µk +γk2

(β1)2 = 0, ∀k = 0, · · · , K.

97

Appendix C

Tables and Figures

98

Table C.1: Summary Statistics of the SPDR Traded from Jan 2, 2001 to Dec 30, 2005.

Daily Traded Volume (1) No. of Publicly Held Shares (2) (1) as a % of (2)a

Mean 38,740,562 Shares 336,267,000 Shares 11.20%

Median 37,608,250 Shares 374,888,000 Shares 10.55%

Stddev 21,223,754 Shares 111,369,000 Shares 4.73%

Min 3,303,100 Shares 149,422,000 Shares 2.21%

Max 141,120,800 Shares 471,080,000 Shares 39.16%

Daily Traded Valueb (3) Value of Publicly Held Sharesc (4) (3) as a % of (4)d

Mean $4,216,321,787 $36,606,862,398 11.20%

Median $3,852,668,649 $39,438,671,390 10.55%

Stddev $416,685,383 $12,345,657,946 4.73%

Min $409,914,710 $14,535,772,160 2.21%

Max $16,821,599,360 $58,654,170,800 39.16%

aFor each day, the daily traded volume as a percentage of the number of publicly held shares iscalculated first, then the statistics are calculated.

bThe daily traded value is computed as the daily traded volume multiplied by the closing price.cThe value of publicly held shares is computed as the number of publicly held shares multiplied by

the closing price.dFor each day, the daily traded value as a percentage of the value of publicly held shares is calculated

first, then the statistics are calculated. So it is not surprising that the numbers in these columns areexactly the same as the ones above.

99

Table C.2: Ljung-Box Statistics for Daily Relative Price Changes of the S&P 500 Index

and the SPDR ( XR and DRR) from Jan 2, 2001 to Dec 30, 2005.

XR DRR

Lags Qstat p-V alue Qstat p-V alue

5 2.78 .734 2.80 .730

10 11.03 .355 14.32 .159

15 20.20 .165 22.72 .090

20 34.46 .023 36.78 .012

25 40.83 .024 45.64 .007

30 49.41 .014 57.16 .002

35 61.91 .003 68.60 .001

40 64.74 .008 71.95 .001

45 75.51 .003 83.13 .000

50 78.68 .006 85.95 .001

Table C.3: Ljung-Box Statistics for Daily Holding Period Returns of the S&P 500 Index

and the SPDR ( XR and DRRd) from Jan 2, 2001 to Dec 30, 2005.

XR DRRd


5 2.78 .734 2.83 .726

10 11.03 .355 11.18 .343

15 20.20 .165 19.15 .207

20 34.46 .023 32.22 .041

25 40.83 .024 38.10 .045

30 49.41 .014 48.03 .020

35 61.91 .003 61.20 .004

40 64.74 .008 65.01 .007

45 75.51 .003 74.68 .004

50 78.68 .006 76.87 .009

100

Table C.4: Ljung-Box Statistics for Squared Daily Relative Price Changes of the S&P 500

Index and the SPDR ( XR2 and DRR

2) from Jan 2, 2001 to Dec 30, 2005.

XR2

DRR2


5 379.98 0.000 393.33 0.000

10 642.35 0.000 683.72 0.000

15 841.33 0.000 909.53 0.000

20 988.01 0.000 1062.17 0.000

25 1058.96 0.000 1143.93 0.000

30 1159.51 0.000 1243.25 0.000

35 1235.91 0.000 1321.98 0.000

40 1308.42 0.000 1407.93 0.000

45 1394.45 0.000 1505.98 0.000

50 1540.66 0.000 1661.52 0.000

Table C.5: Ljung-Box Statistics for Squared Daily Holding Period Returns of the S&P 500

Index and the SPDR ( XR2 and DRR

2) from Jan 2, 2001 to Dec 30, 2005.

XR2

DRR2


5 379.98 0.000 367.30 0.000

10 642.35 0.000 679.47 0.000

15 841.33 0.000 869.90 0.000

20 988.01 0.000 1009.88 0.000

25 1058.96 0.000 1075.38 0.000

30 1159.51 0.000 1162.60 0.000

35 1235.91 0.000 1230.64 0.000

40 1308.42 0.000 1307.56 0.000

45 1394.45 0.000 1394.66 0.000

50 1540.66 0.000 1556.68 0.000

101

Tab

leC

.6:

Tim

es-t

o-M

aturi

tyof

the

S&

P50

0In

dex

Opti

ons

Mon

th01

0203

0405

0607

0809

1011

1201

0203

0405

0607

0809

1011

1201

0203

0405

06D

ec-J

an1m

2m3m

6m9m

12m

18m

24m

Jan-

Feb

1m2m

3m5m

8m11

m17

m23

mFe

b-M

ar1m

2m3m

4m7m

10m

16m

22m

Mar

-Apr

1m2m

3m6m

9m12

m15

m21

mA

pr-M

ay1m

2m3m

5m8m

11m

14m

20m

May

-Jun

1m2m

3m4m

7m10

m13

m19

mJu

n-Ju

l1m

2m3m

6m9m

12m

18m

24m

Jul-

Aug

1m2m

3m5m

8m11

m17

m23

mA

ug-S

ep1m

2m3m

4m7m

10m

16m

22m

Sep-

Oct

1m2m

3m6m

9m12

m15

m21

mO

ct-N

ov1m

2m3m

5m8m

11m

14m

20m

Nov

-Dec

1m2m

3m4m

7m10

m13

m19

m

Not

e: 1.T

he

left

colu

mn

show

sth

etr

adin

gm

onth

.E

ach

item

repre

sents

the

per

iod

from

the

firs

ttr

adin

gday

afte

rth

eop

tion

expir

atio

ndat

eof

one

mon

thto

the

opti

onex

pir

atio

nday

ofth

enex

tm

onth

.F

orex

ample

,“D

ec-J

an”

mea

ns

from

the

Mon

day

follow

ing

the

thir

dF

riday

ofD

ecem

ber

toth

eSat

urd

ayfo

llow

ing

the

thir

dF

riday

ofth

eJan

uar

y.

2.T

he

top

row

show

sth

eex

pir

atio

nm

onth

.F

orex

ample

,th

efirs

tro

wca

nb

ere

adas

:O

pti

ons

trad

edduri

ng

the

per

iod

from

the

Mon

day

follow

ing

the

thir

dF

riday

ofD

ecem

ber

(the

day

onw

hic

ha

set

ofop

tion

sw

ith

new

expir

atio

ndat

ear

ein

troduce

d)

toth

eth

ird

Fri

day

ofJan

uar

yw

ill

expir

ein

Jan

uar

y(0

1),

Feb

ruar

y(0

2),

Mar

ch(0

3),

June

(06)

,Sep

tem

ber

(09)

and

Dec

emb

er(1

2)of

the

sam

eye

aran

dJune

(06)

and

Dec

emb

er(1

2)in

the

follow

ing

year

.T

he

tim

e-to

-mat

uri

tyfo

rea

chof

them

onth

eM

onday

follow

ing

the

thir

dF

riday

ofD

ecem

ber

is1

mon

th,

2m

onth

s,3

mon

ths,

6m

onth

s,9

mon

ths,

12m

onth

s,18

mon

ths

and

24m

onth

s,re

spec

tive

ly.

3.It

can

easi

lyb

ese

enth

atop

tion

str

aded

inth

efirs

tsi

xm

onth

shav

eth

esa

me

tim

es-t

o-m

aturi

tyas

thos

etr

aded

inth

ese

cond

six

mon

ths,

soth

ere

are

two

cycl

esp

erye

ar.

102

Tab

leC

.7:

Sum

mar

ySta

tist

ics

ofth

eO

pti

ons

Tra

ded

from

Jan

2,20

03to

Dec

31,

2003

(252

day

s).

Dai

ly#

of

Opt

ions

Dai

lyT

otal

Tra

ding

Vol

ume

(Con

trac

ts)

Dai

lyT

otal

Tra

ding

Val

uea

($)

Cal

lP

utB

oth

Cal

lP

utB

oth

Mea

n51

7.21

57,4

64.9

279

,678

.26

137,

143.

1917

8,76

4,12

4.37

149,

052,

032.

0032

7,81

6,15

6.37

Med

ian

518.

0053

,988

.00

76,3

18.5

013

1,86

9.00

133,

965,

026.

2513

0,61

7,44

3.75

286,

344,

682.

50

Stdd

ev28

.05

23,2

66.5

025

,232

.18

43,2

12.9

116

6,54

1,44

1.06

76,6

46,5

31.5

619

7,36

3,80

9.44

Min

422.

007,

819.

0029

,609

.00

37,4

28.0

017

,861

,787

.50

21,5

85,5

67.5

051

,359

,310

.00

Max

572.

0016

3,80

7.00

164,

600.

0032

2,71

1.00

1,43

0,18

7,71

7.50

523,

773,

205.

001,

600,

668,

170.

00

(Con

tinued

onnex

tpag

e)

aD

aily

trad

edva

lue

isco

mpu

ted

asda

ilytr

aded

volu

me

tim

esth

eav

erag

eof

best

clos

ing

bid

and

ask

pric

esm

ulti

plie

dby

100.

103

(Table C.7 continued)

Traded Volume of Each Optiona Price of Each Optionb

Mean 265 Contracts $111.85

Mode 0 Contracts $0.25

Stddev 1,053 Contracts $146.53

Min 0 Contracts $0.025

10% quantile 0 Contracts $0.25




Median 0 Contracts $47.10





95% quantile 1,545 Contracts $427.80





Max 64,611 Contracts $993.50

aThere are 130,336 observations. Options with the same maturity date and strike price, but tradedon different days are regarded as different options.

bThe price is computed as the average of bid and ask prices. The value of each contract is equal tothe price multiplied by 100.

104

Tab

leC

.8:

Sum

mar

ySta

tist

ics

ofth

eA

ctiv

ely

Tra

ded

Opti

ons

(wit

htr

aded

volu

me≥

2,00

0co

ntr

acts

)fr

omJan

2,20

03

toD

ec31

,20

03(2

52day

s).

Dai

lyN

umbe

rof

Act

ivel

yT

rade

d

Opt

ions

Dai

lyT

rade

dV

olum

eof

Act

ivel

y

Tra

ded

Opt

ions

Dai

lyT

rade

dV

alue

ofA

ctiv

ely

Tra

ded

Opt

ions

Dai

lyN

umbe

rA

sa

%of

All

the

Opt

ions

Num

ber

of

Con

trac

ts

As

a%

ofA

llth

e

Opt

ions

Dai

lyT

radi

ngV

alue

($)

As

a%

ofA

llth

e

Opt

ions

Mea

n20

.15

3.89

%87

,842

.46

61.6

5%19

6,60

4,06

2.57

53.5

3%

Med

ian

20.0

03.

87%

83,0

57.0

061

.62%

150,

121,

532.

5053

.33%

Stdd

ev6.

571.

21%

39,7

26.0

09.

88%

183,

475,

723.

4216

.82%

Min

3.00

0.59

%16

,301

.00

34.2

4%8,

874,

810.

0014

.80%

Max

43.0

07.

62%

246,

621.

0085

.65%

1,40

0,20

9,01

7.50

93.2

4%

Dai

lyA

ctiv

ely

Tra

ded

Put

Opt

ions

Dai

ly#

ofA

ctiv

ely

Tra

ded

Opt

ions

Dai

ly%

ofA

ctiv

ely

Tra

ded

Opt

ions

#of

Con

trac

ts%

T1

T2

T3

T4

T1

T2

T3

T4

Mea

n11

.73

58.9

2%9.

145.

162.

623.

2446

.35%

25.0

2%12

.67%

15.9

7%

Med

ian

11.0

058

.82%

9.00

4.00

1.00

2.00

45.0

8%22

.40%

7.28

%13

.33%

Stdd

ev4.

0611

.11%

3.69

3.70

2.97

2.79

15.2

8%15

.74%

13.6

5%12

.41%

Min

2.00

28.5

7%1.

000.

000.

000.

0015

.00%

0.00

%0.

00%

0.00

%

Max

28.0

010

0.00

%20

.00

19.0

013

.00

15.0

094

.12%

66.6

7%56

.25%

66.6

7%

105

Table C.9: Summary Statistics of the Actively Traded Call and Put Options with the

Same Strike Prices and Times-to-Maturity from Jan 2, 2003 to Dec 31, 2003 (252 days).

Daily Number of Op-

tions

Daily Traded Vol-

ume

Daily Traded Value

Number

of Pairs

As a %

of All the

Options

Number of

Contracts

As a %

of All the

Options

Traded Value

($)

As a %

of All the

Options

Mean 3.17 1.22% 33,320.52 22.25% 101,492,573.96 28.22%

Median 3.00 1.16% 26,833.50 20.93% 67,370,921.25 24.62%

Stddev 1.78 0.67% 26,261.42 11.55% 109,100,474.65 16.77%

Min 0.00 0.00% 0.00 0.00% 0.00 0.00%

Max 10.00 3.95% 205,883.00 72.99% 858,955,617.50 89.98%

Table C.10: Summary Statistics of Implied Dividends, Q(t,K,T-t), with Treasury Bill

Rates as Proxies for the Risk-free Rates .

Implied

Q(t,K,T-t)

maxQ(t,K, T − t) −

minQ(t,K, T − t) with

same t and T-t but

different K



same t and K but different

T-t

# of Observations 798 218 142

Mean 8.02× 10−3 1.13× 10−2 6.61× 10−2

Median 1.40× 10−2 3.79× 10−3 2.71× 10−2

Stddev 6.26× 10−2 2.41× 10−2 9.57× 10−2

Min −0.51 6.44× 10−5 8.91× 10−6

Max 0.43 0.19 0.51

t-statistic 3.62 6.93 8.23

106

Table C.11: Summary Statistics of Implied Dividends, Q(t,K,T-t), with Zero Rates as

Proxies for the Risk-free Rates .

Implied

Q(t,K,T-t)



same t and T-t but

different K



same t and K but different

T-t

# of Observations 423 109 33

Mean 1.68× 10−2 2.43× 10−3 5.17× 10−3

Median 1.68× 10−2 1.62× 10−3 4.13× 10−3

Stddev 1.12× 10−2 2.42× 10−3 4.11× 10−3

Min −2.93× 10−2 4.03× 10−6 3.76× 10−4

Max 6.62× 10−2 1.10× 10−2 1.57× 10−2

t-statistic 30.80 10.49 7.23

107

Figure C.1: Bid-Ask Spread and Trading Volume (shares) of the SPDR from Jan 2, 2001

to Dec 30, 2005 (1,256 observations)

0 200 400 600 800 1000 1200 1400−1

0

1

2

3

4

5

6

7

Date

Bid

−A

sk S

prea

d of

SP

DR

Jan 2, 2001 to Dec 30, 2005

0 200 400 600 800 1000 1200 1400−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Date

Bid

−A

sk S

prea

d of

SP

DR

Jun 1, 2001 to Dec 30, 2005

0 200 400 600 800 1000 1200 14000

5

10

15x 10

7

Date

Vol

ume

Jan 2, 2001 to Dec 30, 2005

Ask-Bid of SPDR Trading Volume

Mean 0.13 38,740,562 shares

cov 0.27 −3.4513× 106

−3.4513× 106 4.5045× 1014

108

Figure C.2: Histograms of Bid-Ask Spread and Trading Volume (shares) of the SPDR

from Jan 2, 2001 to Dec 30, 2005

−1 0 1 2 3 4 5 6 70

100

200

300

400

500

600

700

800

900

1000

Spread

Fre

quen

cy

SPDR

0 5 10 15

x 107

0

5

10

15

20

25

30

35

40

Volume

Fre

quen

cy

SPDR

−1

0

1

2

3

4

5

6

7

0

5

10

15

x 107

0

20

40

60

80

SpreadVolume

10

20

30

40

50

60

70

Frequency

Quantiles of the Bid-Ask Spread and Trading Volume:

Prob. 0 1% 2.5% 5% 10% 15% 20%Spread −0.52 −0.08 −0.05 −0.03 −0.02 −0.01 0Volume 3.30× 106 6.06× 106 7.18× 106 8.42× 106 1.14× 107 1.48× 107 1.78× 107

Prob. 25% 30% 35% 40% 45% 50% 55%Spread 0 0 0.01 0.01 0.01 0.01 0.01Volume 2.14× 107 2.61× 107 2.98× 107 3.26× 107 3.51× 107 3.76× 107 4.00× 107

Prob. 60% 65% 70% 75% 80% 85% 90%Spread 0.01 0.02 0.02 0.03 0.04 0.05 0.08Volume 4.32× 107 4.60× 107 4.92× 107 5.18× 107 5.51× 107 5.94× 107 6.53× 107

Prob. 95% 97.5% 99% 100%Spread 1 1.76 2.98 6.5Volume 7.49× 107 8.66× 107 1.00× 108 1.41× 108

109

Figure C.3: The S&P 500 Index and the SPDR × 10 from Jan 2, 2001 to Dec 30, 2005

(1,256 observations).

0 200 400 600 800 1000 1200 1400700

800

900

1000

1100

1200

1300

1400

Date

Leve

l

SPX

0 200 400 600 800 1000 1200 1400700

800

900

1000

1100

1200

1300

1400

Date

Leve

l

SPDR×10

S&P 500 SPDR× 10

Mean 1097.94 1101.20

Var/Cov 15849.59 15779.02

15779.02 15714.26

Skewness -0.53 -0.54

Kurtosis 2.52 2.54

110

Figure C.4: Level Difference between the S&P 500 Index and the SPDR× 10 from Jan

2, 2001 to Dec 30, 2005.

0 200 400 600 800 1000 1200 1400−20

−15

−10

−5

0

5

10

15

20

Date

Leve

l

SPDR×10−SPX

SPDR× 10− S&P 500

Mean 3.26

Var 5.81

skewness -1.26

kurtosis 14.34

Figure C.5: Level Difference between the S&P 500 Index and the SPDR× 10 from Feb

1, 1993 to Dec 30, 2005.

0 500 1000 1500 2000 2500 3000 3500−30

−25

−20

−15

−10

−5

0

5

10

15

20

Number of Observations

Leve

l

SPDR×10−SPX

SPDR× 10− S&P 500

Mean 2.28

Var 6.35

skewness -6.26

kurtosis 1.32

111

Figure C.6: Daily Relative Price Changes (log xt− log xt−1) of the S&P 500 Index (XRt)

and the SPDR (DRRt) from Jan 2, 2001 to Dec 30, 2005 (1,255 observations).

0 200 400 600 800 1000 1200 1400−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Date

Geo

met

ric R

etur

n

XR

0 200 400 600 800 1000 1200 1400−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Date

Geo

met

ric R

etur

n

DRR

S&P 500 SPDR

Mean −2.20× 10−5 −2.74× 10−5

Var/Cov 1.31× 10−4 1.30× 10−4

1.30× 10−4 1.34× 10−4

Skewness 1.73× 10−1 1.29× 10−1

Kurtosis 5.41 5.34

112

Figure C.7: Daily Holding Period Returns (log(xt + dt)− log xt−1) of the S&P 500 Index

(XRdt ) and the SPDR (DRR

dt ) from Jan 2, 2001 to Dec 30, 2005 (1,255 observations).

0 200 400 600 800 1000 1200 1400−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Date

Geo

met

ric R

etur

n

XR

0 200 400 600 800 1000 1200 1400−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Date

Geo

met

ric R

etur

n

DRR

S&P 500 SPDR

Mean −2.20× 10−5 3.75× 10−5

Var/Cov 1.31× 10−4 1.29× 10−4

1.30× 10−4 1.34× 10−4

Skewness 1.73× 10−1 6.66× 10−2

Kurtosis 5.41 5.34

113

Figure C.8: Daily Relative Price Change Difference between the S&P 500 Index and the

SPDR from Jan 2, 2001 to Dec 30, 2005.

0 200 400 600 800 1000 1200 1400−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

Date

Geo

met

ric R

etur

nDR

R−XR

DRRt −X Rt

Mean −5.34× 10−6

Var 4.52× 10−6

skewness 5.02× 10−1

kurtosis 3.42× 10

Figure C.9: Daily Holding Period Return Difference between the S&P 500 Index and the

SPDR from Jan 2, 2001 to Dec 30, 2005.

0 200 400 600 800 1000 1200 1400−0.01

−0.005

0

0.005

0.01

0.015

Date

Geo

met

ric R

etur

n

DRR−

XR

DRRdt −X Rdt

Mean 5.96× 10−5

Var 5.86× 10−6

skewness 2.07× 10−2

kurtosis 4.53

114

Figure C.10: Histogram of Daily Relative Price Changes of the S&P 500 Index (XR)and

the SPDR (DRR) and Daily Holding Period Returns of the SPDR (DRRd)from Jan 2,

2001 to Dec 30, 2005.

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

10

20

30

40

50

60

70

80

Geometric Return

Fre

quen

cy

XR

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

10

20

30

40

50

60

70

80

Geometric Return

Fre

quen

cy

DRR

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

10

20

30

40

50

60

70

80

90

Geometric Return

Fre

quen

cy

DRRd

Quantiles of the Geometric Returns:

Prob. 0 1% 2.5% 5% 10%XR −5.05× 10−2 −3.00× 10−2 −2.40× 10−2 −1.82× 10−2 −1.40× 10−2

DRR −4.82× 10−2 −3.02× 10−2 −2.39× 10−2 −1.93× 10−2 −1.41× 10−2

DRRd −5.37× 10−2 −3.19× 10−2 −2.46× 10−2 −1.98× 10−2 −1.32× 10−2

Prob. 15% 20% 25% 30% 35%XR −1.03× 10−2 −8.18× 10−3 −6.43× 10−3 −4.80× 10−3 −3.16× 10−3

DRR −1.04× 10−2 −8.08× 10−3 −6.49× 10−3 −4.66× 10−3 −3.33× 10−3

DRRd −1.03× 10−2 −8.41× 10−3 −6.42× 10−3 −4.49× 10−3 −2.85× 10−3

Prob. 40% 45% 50% 55% 60%XR −1.88× 10−3 −7.71× 10−4 3.60× 10−4 1.30× 10−3 2.29× 10−3

DRR −2.02× 10−3 −7.61× 10−4 3.26× 10−4 1.41× 10−3 2.37× 10−3

DRRd −1.80× 10−3 −3.32× 10−4 7.06× 10−4 1.59× 10−3 2.72× 10−3

Prob. 65% 70% 75% 80% 85%XR 3.42× 10−3 4.61× 10−3 5.95× 10−3 7.64× 10−3 9.89× 10−3

DRR 3.50× 10−3 4.60× 10−3 6.07× 10−3 7.61× 10−3 9.74× 10−3

DRRd 3.65× 10−3 4.80× 10−3 6.17× 10−3 7.70× 10−3 1.01× 10−2

Prob. 90% 95% 97.5% 99% 100%XR 1.28× 10−2 1.74× 10−2 2.27× 10−2 3.48× 10−2 5.57× 10−2

DRR 1.31× 10−2 1.85× 10−2 2.27× 10−2 3.45× 10−2 6.00× 10−2

DRRd 1.29× 10−2 1.75× 10−2 2.21× 10−2 3.39× 10−2 5.79× 10−2

115

Figure C.11: Histogram of Relative Price Change Difference between the S&P 500 Index

and the SPDR and Histogram of Daily Holding Period Return Difference between the

S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005.

−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.0250

50

100

150

200

250

300

Geometric Return Difference

Fre

quen

cy

DRR−

XR

−0.01 −0.005 0 0.005 0.01 0.0150

10

20

30

40

50

60

Geometric Return Difference

Fre

quen

cy

DRRd−

XR

Quantiles of Geometric Return Difference:

Prob. 0 1% 2.5% 5% 10%DRR−X R −1.78× 10−2 −7.47× 10−3 −4.19× 10−3 −2.66× 10−3 −1.34× 10−3

DRRd −X R −9.32× 10−3 −6.44× 10−3 −5.21× 10−3 −3.93× 10−3 −2.88× 10−3

Prob. 15% 20% 25% 30% 35%DRR−X R −9.65× 10−4 −6.45× 10−4 −5.00× 10−4 −3.54× 10−4 −2.34× 10−4

DRRd −X R −2.09× 10−3 −1.63× 10−3 −1.26× 10−3 −9.21× 10−4 −6.12× 10−4

Prob. 40% 45% 50% 55% 60%DRR−X R −1.33× 10−4 −4.52× 10−5 4.17× 10−5 1.39× 10−4 2.48× 10−4

DRRd −X R −4.01× 10−4 −1.71× 10−4 4.48× 10−5 2.77× 10−4 5.82× 10−4

Prob. 65% 70% 75% 80% 85%DRR−X R 3.39× 10−4 4.18× 10−4 5.41× 10−4 6.84× 10−4 8.95× 10−4

DRRd −X R 7.93× 10−4 1.09× 10−3 1.40× 10−3 1.78× 10−3 2.31× 10−3

Prob. 90% 95% 97.5% 99% 100%DRR−X R 1.35× 10−3 2.31× 10−3 3.44× 10−3 5.91× 10−3 2.23× 10−2

DRRd −X R 2.96× 10−3 3.76× 10−3 4.87× 10−3 6.82× 10−3 1.01× 10−2

116

Figure C.12: Squared Relative Price Changes of the S&P 500 Index (XR2t ) and the SPDR

(DRR2t ) from Jan 2, 2001 to Dec 30, 2005.

0 200 400 600 800 1000 1200 14000

0.5

1

1.5

2

2.5

3

3.5x 10

−3

Date

Squ

are

of G

eom

etric

Ret

urn

XR2

0 200 400 600 800 1000 1200 14000

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3

Date

Squ

are

of G

eom

etric

Ret

urn

DRR2

XR2t DRR

2t

Mean 1.31× 10−4 1.34× 10−4

Var/Cov 7.56× 10−8 7.47× 10−8

7.47× 10−8 7.81× 10−8

Skewness 5.11 5.17

Kurtosis 3.83× 10 4.12× 10

Figure C.13: Squared Daily Holding Period Returns of the S&P 500 Index and the SPDR

from Jan 2, 2001 to Dec 30, 2005.

0 200 400 600 800 1000 1200 14000

0.5

1

1.5

2

2.5

3

3.5x 10

−3

Date

Squ

are

of G

eom

etric

Ret

urn

XR2

0 200 400 600 800 1000 1200 14000

0.5

1

1.5

2

2.5

3

3.5x 10

−3

Date

Squ

are

of G

eom

etric

Ret

urn

DRR2

XR2t DRR

2t

Mean 1.31× 10−4 1.34× 10−4

Var/Cov 7.56× 10−8 7.42× 10−8

7.47× 10−8 7.78× 10−8

Skewness 5.11 5.10

Kurtosis 3.83× 10 3.94× 10

117

Figure C.14: Autocorrelation Function for Squared Daily Relative Price Changes of the


0 20 40 60 80 100 120 140−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Lag

AC

FXR2

DRR2

Figure C.15: Autocorrelation Function for Squared Daily Holding Period Returns of the


0 20 40 60 80 100 120 140−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Lag

AC

F

XR2

DRR2

118

Figure C.16: Cross-Correlation Function for Squared Relative Price Changes of the

S&P 500 Index and the SPDR and Autocorrelation Function for Squared Relative Price

Changes of the S&P 500 Index from Jan 2, 2001 to Dec 30, 2005.

−140 −120 −100 −80 −60 −40 −20 0−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Lag

XC

F/A

CF

ACF of XR2

XCF of XR2

t &

DRR2

t+lag

0 20 40 60 80 100 120 140−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Lag

XC

F/A

CF

ACF of XR2

XCF of XR2

t &

DRR2

t+lag

119

Figure C.17: Cross-Correlation Function for Squared Daily Holding Period Returns of the

S&P 500 Index and the SPDR and Autocorrelation Function for Squared Daily Holding

Period Returns of the S&P 500 Index from Jan 2, 2001 to Dec 30, 2005.

−140 −120 −100 −80 −60 −40 −20 0−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Lag

XC

F/A

CF

ACF of XR2

XCF of XR2

t &

DRR2

t+lag

0 20 40 60 80 100 120 140−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Lag

XC

F/A

CF

ACF of XR2

XCF of XR2

t &

DRR2

t+lag

120

Figure C.18: Autocorrelation Function of Max(exp(Rt) − k, 0) for Daily Relative Price

Changes of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005.

0 20 40 60 80 100 120 140

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Lag

AC

F o

f [ex

p(R

t)−k]

+

k=1.015

XR

t

DRR

t

0 20 40 60 80 100 120 140−0.02

0

0.05

0.1

0.15

0.2

Lag

AC

F o

f [ex

p(R

t)−k]

+

k=1.02

XR

t

DRR

t

0 20 40 60 80 100 120 140

0

0.05

0.1

0.15

0.2

0.25

Lag

AC

F o

f [ex

p(R

t)−k]

+

k=1.025

XR

t

DRR

t

121

Figure C.19: Autocorrelation Function ofMax(exp(Rdt )−k, 0) for Holding Period Returns

of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005.

0 20 40 60 80 100 120 140

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Lag

AC

F o

f [ex

p(R

t)−k]

+

k=1.015

XR

t

DRR

td

0 20 40 60 80 100 120 140

0

0.05

0.1

0.15

0.2

Lag

AC

F o

f [ex

p(R

t)−k]

+

k=1.02

XR

t

DRR

td

0 20 40 60 80 100 120 140

0

0.05

0.1

0.15

0.2

Lag

AC

F o

f [ex

p(R

t)−k]

+

k=1.025

XR

t

DRR

td

122

Figure C.20: Autocorrelation Function of Max(k − exp(Rt), 0) for Daily Relative Price

Changes of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005.

0 20 40 60 80 100 120 140−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Lag

AC

F o

f [k−

exp(

Rt)]

+

k=0.985

XR

t

DRR

t

0 20 40 60 80 100 120 140−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Lag

AC

F o

f [k−

exp(

Rt)]

+

k=0.98

XR

t

DRR

t

0 20 40 60 80 100 120 140

0

0.05

0.1

0.15

0.2

0.25

Lag

AC

F o

f [k−

exp(

Rt)]

+

k=0.975

XR

t

DRR

t

123

Figure C.21: Autocorrelation Function ofMax(k−exp(Rdt ), 0) for Holding Period Returns

of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005

0 20 40 60 80 100 120 140

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Lag

AC

F o

f [k−

exp(

Rt)]

+

k=0.985

XR

t

DRR

td

0 20 40 60 80 100 120 140

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Lag

AC

F o

f [k−

exp(

Rt)]

+

k=0.98

XR

t

DRR

td

0 20 40 60 80 100 120 140

0

0.05

0.1

0.15

0.2

0.25

Lag

AC

F o

f [k−

exp(

Rt)]

+

k=0.975

XR

t

DRR

td

124

Figure C.22: Number of Times-to-Maturity on Each Trading Day in 2003

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/197

7.2

7.4

7.6

7.8

8

8.2

8.4

8.6

8.8

9

date

Num

ber

of M

atur

ities

Number of Maturities on Each Trading Day

Note: The label on the x-axis shows the Friday before expiration date in each month except for April17, which is the Thursday, since there was no trading on April 18.

Figure C.23: Times-to-Maturity on Each Trading Day in 2003

01/17 02/21 03/21 04/17 05/16 06/2007/02 07/18 08/15 09/19 10/17 11/21 12/190

100

200

300

400

500

600

700

800

date

Tim

e to

Mat

urity

Time to Maturities on Each Trading Day

125

Figure C.24: Times-to-Maturity of Options with High Volume on Each Trading Day in

2003

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

100

200

300

400

500

600

700

800

date

Tim

e to

Mat

urity

Call and Put Options with Daily Volume >= 2000

callput

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

100

200

300

400

500

600

700

800

date

Tim

e to

Mat

urity

Call Options with Daily Volume >= 2000

call

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

100

200

300

400

500

600

700

date

Tim

e to

Mat

urity

Put Options with Daily Volume >= 2000

put

126

Figure C.25: Strike Prices of Options with High Volume on Each Trading Day in 2003

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/19400

600

800

1000

1200

1400

1600

1800

date

SP

X &

Str

ike


SPXcall

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/19400

600

800

1000

1200

1400

1600

1800

date

SP

X &

Str

ike


SPXput

Figure C.26: Moneyness of Options with High Volume on Each Trading Day in 2003

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

date

Mon

eyne

ss


call

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

date

Mon

eyne

ss


put

Figure C.27: Prices of Traded Options with Traded Volume ≥ 2,000 Contracts in 2003

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

100

200

300

400

500

600

700

800

900

date

Opt

ion

Pric

es

Prices of Call and Put Options with Daily Volume >= 2000

127

Figure C.28: Number and Proportion of Highly Traded Options on Each Trading Day in

2003

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

5

10

15

20

25

30

35

40

45

date

Num

ber

of O

ptio

nsNumber of Highly Traded Options

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

date

Pro

port

ion

Proportion of Highly Traded Options

128

Figure C.29: Number of Highly Traded Put Options and Its Proportion in Highly Traded

Options on Each Trading Day in 2003

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

5

10

15

20

25

30

date

Num

ber

of O

ptio

nsNumber of Highly Traded Put Options

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

date

Pro

port

ion

Highly Traded Put as a Proportion of Highly Traded Options

129

Figure C.30: Number of Actively Traded Options with Time-to-Maturity T1 and Its

Proportion in the Actively Traded Options on Each Trading Day in 2003

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

2

4

6

8

10

12

14

16

18

20

date

Num

ber

of O

ptio

nsNumber of Options with Daily Volume >= 2000 and Time to Maturity T1

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

date

Pro

port

ion

Proportion of Options with Daily Volume >= 2000 and Time to Maturity T1

130



01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

2

4

6

8

10

12

14

16

18

20

date

Num

ber

of O

ptio


01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

date

Pro

port

ion


131



01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

2

4

6

8

10

12

14

16

18

20

date

Num

ber

of O

ptio


01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

date

Pro

port

ion


132



01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

2

4

6

8

10

12

14

16

18

20

date

Num

ber

of O

ptio


01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

date

Pro

port

ion


133

Figure C.34: Moneyness, Total Volume and Total Value of Actively Traded Options with

Time-to-Maturity T1 on Each Trading Day in 2003

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

date

Mon

eyne

ss

Moneyness of Options with Daily Volume >= 2000 and Time to Maturity T1

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

2

4

6

8

10

12

14x 10

4

date

Vol

ume

of O

ptio

ns

Daily Total Volume of Options with Daily Volume >= 2000 and Time to Maturity T1

01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

1

2

3

4

5

6

7x 10

8

date

Val

ue o

f Opt

ions

Daily Total Value of Options with Daily Volume >= 2000 and Time to Maturity T1

134



01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

date

Mon

eyne

ss


01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

2

4

6

8

10

12x 10

4

date

Vol

ume

of O

ptio

ns


01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

1

2

3

4

5

6

7x 10

8

date

Val

ue o

f Opt

ions


135



01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

date

Mon

eyne

ss


01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

5

10

15x 10

4

date

Vol

ume

of O

ptio

ns


01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

1

2

3

4

5

6

7

8x 10

8

date

Val

ue o

f Opt

ions


136

Figure C.37: Moneyness of Actively Traded Call and Put Options with the Same Strike

Prices and Times-to-Maturity on Each Trading Day in 2003

01/17 02/21 03/21 04/18 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

date

Mon

eyne

ss

Call and Put Options with the Same Strike Prices and Time to Maturities That Have Daily Volume >= 2000

Figure C.38: Time-to-Maturity of Actively Traded Call and Put Options with the Same

Strike Prices and Times-to-Maturity on Each Trading Day in 2003

01/17 02/21 03/21 04/18 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190

100

200

300

400

500

600

700

date

Tim

e to

Mat

urity

Call and Put Options with the Same Strike Prices and Time to Maturities That Have Daily Volume >= 2000

137

Figure C.39: Implied Dividends, Q(t,K,T-t), with Treasury Bill Rates as Proxies for the

Risk-free Rates

252−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

t

Impl

ied

Div

iden

d, Q

Implied Dividend, Q(t,K,T−t)

Figure C.40: Implied Dividends, Q(t,K,T-t), with the same t and T-t, but different K,

and with Treasury Bill Rates as Proxies for the Risk-free Rates

218−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

(t, T−t)

Impl

ied

Div

iden

d, Q

Implied Dividend, Q(t,K,T−t), with same t and T−t but different K

138

Figure C.41: Difference between the maximum and minimum Q(t,K,T-t), with the same

t and T-t, but different K, and with Treasury Bill Rates as Proxies for the Risk-free Rates

2180

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

(t, T−t)

Diff

eren

ce o

f Im

plie

d D

ivid

end

Difference between the maximum and minimum Q(t,K,T−t), with same t and T−t but different K

Figure C.42: Implied Dividends, Q(t,K,T-t), with the same t and K, but different T-t,

and with Treasury Bill Rates as Proxies for the Risk-free Rates

142−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

(t, K)

Impl

ied

Div

iden

d, Q

Implied Dividend, Q(t,K,T−t), with same t and K but different T−t

139


t and K, but different T-t, and with Treasury Bill Rates as Proxies for the Risk-free Rates

1420

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(t, K)

Diff

eren

ce o

f Im

plie

d D

ivid

end

Difference between the maximum and minimum Q(t,K,T−t), with same t and K but different T−t

Figure C.44: Implied Dividend, Q(t,K,T-t), with Zero Rates as Proxies for the Risk-free

Rates

252−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

t

Impl

ied

Div

iden

d, Q

Implied Dividend, Q(t,K,T−t)

140

Figure C.45: Implied Dividends, Q(t,K,T-t), with the same t and T-t, but different K,

and with Zero Rates as Proxies for the Risk-free Rates

109−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

(t, T−t)

Impl

ied

Div

iden

d, Q

Implied Dividend, Q(t,K,T−t), with same t and T−t but different K


t and T-t, but different K, and with Zero Rates as Proxies for the Risk-free Rates

1090

0.002

0.004

0.006

0.008

0.01

0.012

(t, T−t)

Diff

eren

ce o

f Im

plie

d D

ivid

end

Difference between the maximum and minimum Q(t,K,T−t), with same t and T−t but different K

141

Figure C.47: Implied Dividends, Q(t,K,T-t), with the same t and K, but different T-t,

and with Zero Rates as Proxies for the Riskfree Rates

33−0.02

−0.01

0

0.01

0.02

0.03

0.04

(t, K)

Impl

ied

Div

iden

d, Q

Implied Dividend, Q(t,K,T−t), with same t and K but different T−t


t and K, but different T-t, and with Zero Rates as Proxies for the Risk-free Rates

330

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

(t, K)

Diff

eren

ce o

f Im

plie

d D

ivid

end

Difference between the maximum and minimum Q(t,K,T−t), with same t and K but different T−t

142

Bibliography

[1] Ackert, L., and Y. S., Tian (2001), “Efficiency in Index Options Markets and Trading

in Stock Baskets” Journal of Banking and Finance, 25, 1607-1634.

[2] Ait-Sahalia, Y., and A., Lo (1998), “Nonparametric Estimation of State-Price Den-

sities Implicit in Financial Asset Prices”, The Journal of Finance, vol 53, 499-547.

[3] American Stock Exchange Website (2006), “Standard & Poor’s Depositary Receipts

Propectus”, www,amex.com.

[4] Ball, C., and A., Roma (1994), “Stochastic Volatility Option Pricing”, Journal of

Financial and Quantitative Analysis, 29, 589-607.

[5] Beaulieu, M. C., and I. G., Morgan (2000), “High-Frequency Relationships Between

the S&P 500 Index, S&P 500 Futures, and S&P Depositary Receipts”, Manuscript.

[6] Beneish, M. D., and R. E., Whaley (1996), “An Anatomy of the “S&P Game”: The

Effect of Changing the Rules”, The Journal of Finance, Vol. 51, 1909-1929.

[7] Black, F., and M., Scholes (1973), “The Pricing of Options and Corporate Liabili-

ties”, Journal of Political Economy, 81, 637-659.

143

[8] Blitzer, D., (2004), “S&P U.S. Indices Methodology”, September ,

www.standardandpoor.com.

[9] Blitzer, D., (2005), “Index Mathematics, a Very Short Course”,

www.standardandpoor.com.

[10] Brennan, M., and E., Schwartz (1990), “Arbitrage in Stock Index Futures”, Journal

of Business, Vol. 63, 7-31.

[11] Chen, R., and L., Scott (1993), “Maximum Likelihood Estimation of a Multi-Factor

Equilibrium Model of the Term Structure of Interest Rates”, Journal of Fixed In-

come, December, 1993, 14-31.

[12] Chicago Board of Exchange (CBOE) (2004), “Product Specification: S&P 500 Index

Options”; The early www.cboe.com.

[13] Constantinides, G. (1992), “A Theory of the Nominal Term Structure of Interest

Rates”, Review of Financial Studies, 5, 531-552.

[14] Cornell, B. (1985), “Taxes and the Pricing of Stock Index Futures: Empirical Re-

sults”, Journal of Futures Markets, 5, 89-101.

[15] Cornell, B., and K., French (1983), “Taxes and the Prices of Stock Index Futures”,

Journal of Finance, 38, 675-694.

[16] Cox, J., Ingersoll, J., and S., Ross (1981), “The Relation between Forward Prices

and Future Prices”, Journal of Financial Economics, 9, 321-346.

144

[17] Cox, J., Ingersoll, J., and S., Ross (1985a), “An Intertemporal General Equilibrium

Model of Asset Prices”, Econometrica, 53, 363-384.

[18] Cox, J., Ingersoll, J., and S., Ross (1985b), “A Theory of the Term Structure of

Interest Rates”, Econometrica, 53, 385-407.

[19] Cusick, P. A. (2002), “Price Effect of Addition or Deletion from the Standard &

Poor’s 500 Index”, Financial Markets, Institutions & Instruments, V. 11, No. 4,

November.

[20] Dai, Q., and K., Singleton (2000), “Specification Analysis of Affine Term Structure

Models”, Journal of Finance, Vol 55, 1943-1978.

[21] Dai, Q., and K., Singleton (2003), “Term Structure Dynamics in Theory and Real-

ity”, The Review of Financial Studies, 16, 631-678.

[22] De Jong, F. and T., Nijman (1997), “High-frequency Analysis of Lead-lag Relation-

ships between Financial Markets”, Journal of Empirical Finance, 4, 259-277.

[23] Duffie, D., and R., Kan (1996), “A Yield-Factor Model of Interest Rates”, Mathe-

matical Finance, 6, 379-406.

[24] Duffie, D., Pan, J., and K., Singleton (2000), “Transform Analysis and Asset Pricing

for Affine Jump-Diffusions”, Econometrica, 68, 1343-1376.

[25] Elton, E.J., Gruber, M.J., Comer, G., and K., Li (2002), “Spiders: Where Are the

Bugs?”, Journal of Business, vol. 75, no. 3, 453-472.

145

[26] Evine, J., and A., Rudd (1985), “Index Options The Early Evidence”, Jounal of

Finance, 40, 743-755.

[27] Figlewski, S. (1984a), “Hedging Performance and Basis Risk in Stock Index Futures”,

Journal of Finance, Vol. 39, 657-669.

[28] Figlewski, S. (1984b), “Explaining the Early Discounts on Stock Index Futures: The

Case for Disequilibrium”, Financial Analysts Journal, Vol. 40, Issue 4, 43-67.

[29] French, K. R. (1983), “A Comparison of Futures and Forward Prices”, Journal of

Financial Economics, Vol. 12, 311-342.

[30] Gastineau, G. (2001), “Exchange Traded Funds: An Introduction”, Journal of Port-

folio Management, 27, pp. 88-96.

[31] Gourieroux, C., and J., Jasiak (2001), “Financial Econometrics” Princeton Univer-

sity Press.

[32] Gourieroux, C., Jasiak, J., and P., Xu (2008), “Non-tradable S&P 500 Index and

the Pricing of its Traded Derivatives”, working paper.

[33] Harrison, J., and D., Kreps (1979), “Martingales and Arbitrage in Multiperiod Se-

curities Markets”, Journal of Economic Theory, 20, 381-408.

[34] Harris, L., Sofianos, G., and J., E., Shapiro (1994), “Program Trading and Intraday

Volatility”, The Review of Financial Studies, vol 7, No. 4, 653-685.

146

[35] Harvey, C., and R., Whaley (1991), “S&P 100 Index Option Volatility”, The Journal

of Finance, Vol. 46, No. 4, 1551-1561.

[36] Harvey, C., and R., Whaley (1992a), “Dividends and S&P 100 Index Option Valu-

ation”, The Journal of Futures Markets, Vol. 12, No. 2, 123-137.

[37] Harvey, C., and R., Whaley (1992b), “Market volatiltity Prediction and the Effi-

ciency of the S&P 100 Index Option Market”, Journal of Financial Economics, 31,

43-73.

[38] Heston, S.(1993) “A Closed-Form Solution for Options with Stochastic Volatility

with Applications to Bond and Currency Options”, The Review of Financial Studies,

6, 327-343.

[39] Kamara, A., and T. W., Miller, Jr. (1995), “Daily and Intradaily Tests of European

Put-Call Parity” Journal of Financial and Quantitative Analysis, Vol 30, No 4, 519-

539.

[40] Lynch, A. W., and R. R., Mendenhall (1997), “New Evidence on Stock Price Effects

Associated with Changes in the S&P 500 Index”, Journal of Business, Vol 70, no3,

351-384.

[41] Mackinlay, A. C., and K., Ramaswamy (1988), “Index-Futures Arbitrage and the

Behavior of Stock Index Futures Prices”, The Review of Financial Studies, Vol 1, 2,

137-158.

147

[42] Merton, R. C., (1973a), “The Relationship Between Put and Call Option Prices:

Comment”, Journal of Finance, 28, 183-184.

[43] Merton, R. (1973b), “The Theory of Rational Option Pricing”, Bell Journal of Eco-

nomics and Management Science, 4, 141-183.

[44] Poterba, J. M. and J. B., Shoven (2002), “Exchange Traded Funds: A New Invest-

ment Option for Taxable Investors”, NBER Working Paper Series, Working Paper

8781.

[45] Schwartz, E. (1997), “The Stochastic Behavior of Commodity Prices: Implications

for Valuation and Hedging”, Journal of Finance, vol 52, 923-973.

[46] Standard and Poor’s Website (2004), “S&P Global Indices Methodology”, May,

www.standardandpoors.com.

[47] Standard and Poor’s Website (2006), “S&P U.S. Indices Methodology”, March,

www.standardandpoors.com.

[48] Stoll, H. (1969), “The Relationship between Put and Call Option Prices”, The Jour-

nal of Finance, Vol. 24, No. 5, 802-824.

[49] Vanguard Website (2006), “Vanguard 500 Index Fund Propectus”,

www.vanguard.com.

[50] Xu, P. (2006), “Is S&P 500 Index tradable?”, Manuscript.

[51] Xu, P. (2007), “S&P 500 Index Options and Put-Call Parity”, Manuscript.

148

nontradable market index and its derivatives · therefore, the spot-futures parity and the put-call...

Documents