An Empirical Study on theImplied Volatility Function of S&P 500 Options

by

Shiheng Wang

An essay submitted to the Department of Economicsin partial fulfillment of the requirements for

the degree of Master of Arts

Queen’s UniversityKingston, Ontario, Canada

August 2002

copyright c©Shiheng Wang 2002

Abstract

A better understanding of the empirical dynamics of Black-Scholes implied volatil-

ity surface has long been of considerable interest to both practitioners and aca-

demics. Basing on some findings about the ad hoc Black-Scholes valuation ap-

proach suggested in Dumas, Flemming and Whaley (1998), this essay studies the

empirical performance of various volatility function forms that characterize the

regularities underlying the Black-Scholes implied volatility skew or smile.

Because the volatility function suggested in Dumas, Flemming and Whaley (1998)

is unstable over time, we propose a new class of dynamics implied volatility function

that separates a time-invariant implied volatility function from the random factors

that drive changes in the individual implied volatilities. The random factors are

incorporated through the at-the-money implied volatilities which are modelled as

a function of lagged volatility and a non-linear function of the underlying asset

return. This dynamic model is found to greatly improve the pricing performance.

Acknowledgements

I would like to thank Professor Wulin Suo for his invaluable guidance and patience

in advising me with this essay. I am also grateful to Yu Du and Jun Yuan for

useful comments and assistance in mathematics. Finally, I dedicate this essay to

my families who always support and encourage me in my endeavors.

1

Contents

1 Introduction 3

2 Black-Scholes Implied Volatility Function 8

2.1 Modelling Implied Volatility Smile . . . . . . . . . . . . . . . . . . 8

2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Estimated Parameters and In-Sample Pricing Fit . . . . . . . . . . 14

2.5 Out-of-Sample Forecasting Performance . . . . . . . . . . . . . . . . 16

3 Dynamic Black-Scholes Implied Volatility Function 19

3.1 A Brief Review of Derman’s “Sticky” Models of Implied Volatility . 20

3.2 Dynamic Black-Scholes Implied Volatility Function . . . . . . . . . 24

3.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4 Estimation and Forecasting Results . . . . . . . . . . . . . . . . . . 28

4 Conclusion 32

2

1 Introduction

During the last two decades the market for financial derivatives has experienced

rapid growth and many innovative product creations. For valuing these instru-

ments, the Black-Scholes (1973) model is often applied as a starting point. Black-

Scholes model assumes that the underlying asset is traded in a frictionless market

and the asset’s price follows a geometric Brownian motion with constant volatility.

To recover the volatility from option price, we invert the Black-Scholes call option

formula by replacing St with St − PV D,

C = (St − PV D)N(d1)−Ke−rτN(d2), (1)

d1 =ln[(St − PDV )/K] + (r + 0.5σ2)τ

σ√

τ, (2)

d2 = d1 − σ√

τ , (3)

where St is the spot price of underlying asset, K is the strike price, r is the risk free

interest rate, τ is time to maturity of the option, σ is the volatility of underlying as-

set price and PV D is the present value of cash dividends summed over option’s life.

For all options on the same underlying asset, the volatilities σ implied by inverting

the Black-Scholes valuation formula (hereon referred to as “Black-Scholes implied

volatility” or “BSIV” for short) would be identical regardless of strike price and

time to maturity of individual options. In practice, however, Black-Scholes im-

plied volatilities tend to differ across strike prices and time to maturity1. S&P

500 option Black-Scholes implied volatilities, for example, form a “smile” pattern

1Rubinstein(1994), Dumas, Flemming and Whaley (1998), and Bakshi, Cao and Chen(1997)examined the S&P 500 index option market. Taylor and Xu(1993) performed similar investi-gations for Philadelphia Exchange foreign currency option market. Duque and Paxson (1993)examined the stock options traded at the London International Financial Futures Exchange,Heynen(1993) investigated the European Options Exchange.

3

prior to the October 1987 market crash. Options that are deep in- and out-of-

the-money have higher implied volatilities than at-the-money options. After the

crash, a “skew” appears—the implied volatilities decrease monotonically as the

strike price rises relative to the index level, with the rate of decrease increasing for

options with shorter time to maturity. Therefore, practitioners commonly use dif-

ferent volatilities for different strike prices and maturities in order to take account

of the deviation from the Black-Scholes constant volatility assumption.

A number of new approaches have been developed to account for the nonconstant

volatility, among which a very representative one is the “deterministic volatility

function” method proposed by Derman and Kani (1994), Dupire (1994), Rubin-

stein (1994) and later empirically tested by Dumas, Flemming and Whaley (1998)

(hereon referred to as “DFW”). In this method, the volatility is assumed to be

a deterministic function of asset price and/or time. This is the “simplest method

that preserves the arbitrage argument underlying the Black-Scholes model” (DFW,

1998, p.2065). Stochastic volatility or jump diffusion models require additional as-

sumptions about investor preferences for risk or additional securities which can

be used to hedge volatility or jump risk. Thus they are difficult for practitioners

to implement in practice. In contrast, deterministic volatility function approach

assumes the volatility rate is a flexible but deterministic function of asset price

and time, and only the parameters governing the volatility processes need to be

estimated.

However, DFW (1998) found that an ad hoc Black-Scholes valuation method

(hereon referred to as “AHBS” method) even outperformed the deterministic

4

volatility function model in hedging performance. In the AHBS method, the Black-

Scholes implied volatility is constructed as a quadratic function of asset price S

and time t, in part because the Black-Scholes implied volatilities for S&P 500 op-

tions tend to have a parabolic shape. At date t, the suggested function is plugged

into Black-Scholes valuation formula to back out the parameters of the function

by minimizing differences between reported option prices and model prices. The

estimated parameters are used to calculate the volatility level at date t + 7, which

is then applied to Black-Scholes formula to predict the option price at t + 7.

Since the deterministic volatility function approach has not been found to be more

successful than the AHBS method, it may be useful to re-examine the AHBS

method in details. The main reason for this investigation is that AHBS method is

a “variation of what is applied in practice as a means of predicting option prices”

(DFW, 1998, p.2086). To account for the skew pattern in Black-Scholes implied

volatilities, many market practitioners simply smooth the implied volatility rela-

tion across strike prices and time to maturity, and then value option using the

smoothed relation. The AHBS method operationalizes such practice. Same as

the deterministic volatility function, it’s simple in parameter estimation compared

to the stochastic volatility models. Moreover, it is meaningful to investigate the

AHBS methods in more detail if the following points are considered:

First, in their empirical test, DFW (1998) found that the volatility functions ap-

plied to AHBS methods are unstable over time. That is, the volatility function

implied by today’s option price is not the same one embedded in option price to-

morrow. To apply the AHBS method, we need to re-estimate the volatility function

5

every day or every week so as to remedy the observed functional instability. Before

we follow such practice, we need to first find out what the best volatility function

form to perfectly characterize the regularities underlying the Black-Scholes implied

volatility skew or smile. Although DFW (1998) proposed some volatility functions

in their research, those function forms are restrictive and arbitrary to a certain

extent, since DFW’s objective was to test the stability of the suggested functions

rather than to find a function form which best captures the daily volatility pattern.

Second, DFW (1998) only utilized the current underlying asset price S and time t

as variables in the volatility functions when implementing AHBS method. Actu-

ally, in existing literature, Black-Scholes implied volatilities are found to depend-

ing on a wide variety of contract specific variables, such as strike price K(Bates,

1995), proportional moneyness S/K − 1 (Clewlow, 19992) and maturity adjusted

proportional moneyness 1√τ

ln(K/F ) (where F is the forward price of S) (Naten-

berg, 1994).

Lastly, DFW’s (1998) conclusion of functional instability was based on a method-

ology which examined whether the levels of Black-Scholes implied volatilities for

options with a certain time to maturity (i.e. 90 days) will predict the levels of

volatilities of options with a different time to maturity (i.e. 83 days). If the rel-

ative shape of Black-Scholes implied volatility surfaces depends on the time to

maturity, the volatility function may remain stable over time when maturity of

all options are controlled to be the same. Then we don’t need to continuously

update the function every day or every week . The stable function could thus be

2Their study applied principal component analysis to first differences of Black-Scholes impliedvolatilities for fixed maturity buckets, across both strike and moneyness metrics.

6

implemented to AHBS valuation method for long-term forecasting or hedging.

The objectives of this essay are twofold. First, by incorporating different contract

specific variables in the volatility functions that are applied to AHBS valuation

method, we aim at finding out which competitive function could best characterize

the regularities in the skew pattern of the Black-Scholes implied volatility. Second,

by controlling all options to the same maturity, we investigate whether a stable

volatility function exists. This hypothesis is tested by comparing the out-of-sample

predicting performance of the assumed stable volatility function with the known

unstable volatility function, both of which are applied to the AHBS valuation

method. If the former model could outperform the latter one, this may prove from

one angle that stable volatility function exists under certain restrictions, or at least

the stability of volatility functions could be improved under certain assumptions.

This essay is organized as follows. Section 2 proposes competitive volatility func-

tions for the Black-Scholes implied volatility skew. Tests on both in-sample fit-

ting and out-of-sample forecasting accuracy are carried out by employing AHBS

valuation methods. Section 3 investigates a richer structure of Black-Scholes im-

plied volatility time series for options with given time to maturity, and presents a

volatility functional form which is assumed to be stable over time. Comparisons

of out-of-sample pricing accuracy with unstable volatility function are performed.

Section 4 concludes the paper.

7

2 Black-Scholes Implied Volatility Function

2.1 Modelling Implied Volatility Smile

As introduced in Section 1, in our study we call the volatility recovered from option

price by inverting the Black-Scholes (1973) formula “Black-Scholes implied volatil-

ity”. The Black-Scholes implied volatility smile is often modelled by a quadratic

regression of the form

σ = α0 + α1X + α2X2 (4)

where X is a contract specific state variable.

In principal, with a suitable choice of X , this function is able to capture a smile

as well as a skew pattern. The first objective of our study is to find out what

function form that applied to AHBS valuation approach could best fit reported

option prices. In our study, we call function (4) “Black-Scholes implied volatility

function” (hereon referred to as “BSIV function”). In existing literature, the fol-

lowing functions have been suggested to describe the volatility skew:

Model 1: σ(K, τ) = α0 + α1K100

+ α2(K100

)2

Model 2 : σ(K, τ) = α0 + α1Kt

St+ α2(

Kt

St)2

Model 3: σ(K, τ) = α0 + α1K100

+ α2(K100

)2 + α3τ + α4(K100

)τ

Model 4: σ(K, τ) = α0 + α1ln(K/F )√

τ+ α2[

ln(K/F )√τ

]2

where σ(K, τ) is Black-Scholes implied volatility for the option with strike price

K and time to maturity τ ; Kt is the strike price K discounted to the value at time

t; and F is the forward price of St.

8

Model 1 was utilized by Shimko (1993) and Bates (1995) to describe the relation

between Black-Scholes implied volatilities and strike prices. It was also proved to

be the best model in DFW’s study with respect to the out-of-sample forecasting

performance (We make a revision by replacing K with K/100 to avoid the esti-

mated coefficient being too small). Similar to mode 1, model 2 tries to capture

the connection between implied volatilities and moneyness. Rosenberg (2000) used

proportional moneyness KSt−1 as the variable in the volatility function. In contrast,

we replace K by Kt to ensure a consistent definition of moneyness Kt/St at differ-

ent time points. Model 3, proposed by DFW (1998), was the best model in their

study in respect of in-sample fitting, but performed poor in out-of-sample fore-

casting. DFW (1998) argued that the time variable may be an important cause of

overfitting at the estimation stage. Model 4, put forward first by Natenberg (1994)

and later adopted by Tompkin (1999), is expected to reduce the smile’s dependence

on time to maturity and thus overcome the deficiency inherited in model 3. As

the shape of the volatility smile depends on the option maturity and in most cases

the smile becomes less pronounced as the option maturity increases. Define the

volatility smile as the relationship between implied volatility and ln(K/F )√τ

usually

makes the smiles much less dependent on time to maturity.3 This is consistent

with Taylor and Xu (1993), who demonstrated that a more complex valuation

model (such as jump diffusion) can generate time-dependence in the skew even if

volatility is constant over time.

Although every function has been proved to capture certain characteristics of

3For more detailed discussion of this function, see Natenberg, Option Pricing and Volatility:Advanced Trading Strategies and Techniques, second edition, Chicago (1994). This approachmakes the assumption that the volatility of an option depends on the number of standard devi-ation lnK is away from the mean of lnF .

9

volatility skew in different sample periods or for different derivative products, few

studies made an investigation into which model best characterizes Black-Scholes

volatility skew, based on the same derivative product and under the same sample

period. It’s therefore meaningful for us to carry out such a test on the S& P 500

index options.

2.2 Methodology

Since the BSIV functions implemented in AHBS valuation method were found to

be unstable in DFW’s (1998) empirical study, we re-estimate it on a daily ba-

sis. The logic of our test is straightforward. First, we use today’s option prices

to estimate the parameters of each competitive function. Then, we step one day

forward in time—using the estimated parameters to forecast the Black-Scholes im-

plied volatilities and option prices on the second day. We repeat such procedures

every day for three months in 1996. This approach is similar to those employed

by DFW (1998) who re-estimated the volatility functions week to week.

We estimate each competitive model by minimizing the sum of squared dollar errors

between the reported option price and model price. Let Cmak(K, τ) be reported

price for the option with strike price K and time to maturity τ and Cest(α, K, τ)

the option’s model price determined by plugging the BSIV function into Black-

Scholes valuation formula. The objective loss function with respect to parameters

α of each competitve function is in the form of

Loss(α) = minn∑

i=1

(Cmak − Cest)2 (5)

Backing out the structural parameters by using the loss function is a common

10

approach in the existing literature. For example, Bates (1996), Bakshi, Cao and

Chen (1997), and Longstaff (1995) all used this approach. We employ the mini-

mized loss as the measure of in-sample fitting.

To assess the out-of-sample predicting accuracy, we use two measures on each day:

(1) the root mean squared error (RMSE) is the square root of the average squared

deviation of the reported option price from the model price, i.e.

RMSE =

√√√√ 1

n

n∑i=1

(Cmak − Cest)2 (6)

(2) the mean relative error (MRE) is the average ratio of the absolute pricing error

to market price, i.e.

MRE =1

n

n∑i=1

(|Cmak − Cest|/Cmak) (7)

2.3 Data

The daily record on the last bid-ask quote of S&P 500 call options from March

1, 1996 to May 31, 1996 is obtained from the Chicago Board Option Exchange

(CBOE). The daily record of closing quote of S&P 500 index level is obtained

from the Center for Research in Security Prices (CRSP) database.4

To proxy for the risk-free rate, the rate on T-bills of comparable maturity is used.

The data on the daily T-bill bid and ask discount with maturities up to three

months are hand-collected from the Wall Street Journal. The average of the bid

4Option prices and closing index levels are nonsynchronous, either because their closing quoteare simply recorded at different times or because closing prices of at least some of five hundredstocks that constitute the index are stale. Assuming such random errors in index prices to be0.25%, Jorion (1995) estimates that the error in measured implied volatility is about 1.2%.

11

and ask T-bill discounts is used and converted to an annualized interest rate. Since

T-bills mature every Thursdays while index options expire on the third Friday of

each month, we utilize two T-bill rates straddling an option’s maturity date to

obtain the interest rate corresponding to the option’s maturity. This is done for

each contract on each day in our sample.

The daily cash dividends for the S&P 500 index are collected from the S&P 500

Information Bulletin. The actual cash dividend paid during the option’s life are

used to proxy for expected dividends and are summed over the option’s life.

PV D =n∑

i=1

e−ritiDi, (8)

Where Di is the ith cash dividend payment, ti is the time to ex-dividend from

current date t, and ri is the ti-period risk free interest rate.

The forward price of S&P 500 index is therefore

F = (St − PV D)erτ (9)

Three exclusive filters are applied to the option data. First, we eliminate options

with fewer than 7 or more than 90 days to expiration. The options with maturity

fewer than 7 days usually have relatively small time premium, hence the estima-

tion of volatility is extremely sensitive to nonsynchronous option prices and other

possible measurement errors. Options with more than 90 days to maturity are

not actively traded. Second, we eliminate options whose absolute “moneyness”,

|K/S| − 1, is greater than 10 percent. Like extremely short-term options, deep in-

and out-of-the money options have small time premiums and hence contain little

information about the volatility function. Nor are they actively traded. Finally,

quotes not satisfying the arbitrage restriction

12

Ct ≥ max(0, St −Ke−rτ − PDV ) (10)

are taken out of the sample.

We divide the option data into several categories based on moneyness and time to

maturity. Define S−K as the time-t intrinsic value of a call, where S is dividend-

excluded. A call option is then said to be at-the-money (ATM) if 0.97≤K/S≤1.03,

out-of-the-money (OTM) if K/S>0.97, and in-the-money (ITM) if K/S<1.03. A

finer partition results in six moneyness categories. By the time to maturity, an

option contract can be classified as (i) short-term (7-45 days) and (ii) long-term

(45-90 days). The proposed moneyness and maturity classification produce 12 cat-

egories for which the empirical results are reported.

Table 1 describes certain properties of the call options used in our study. The

average closing bid-ask midpoint option price, the average Black-Scholes implied

volatility and number of observations are reported for each moneyness-maturity

category.

There are 1924 call options included in our study, with OTM and ATM options

respectively taking up 47 percent and 45 percent of the whole sample. The average

option price ranges from $0.68 for the short-term deep OTM option, to $67.13 for

the long-term deep ITM option.

Clearly, regardless of time to maturity, the Black-Schole implied volatility exhibits

a strong skew pattern (as the call option goes from deep ITM to ATM and then to

13

deep OTM, with the deepest ITM implied volatilities taking the highest values).

Furthermore, the volatility skew is stronger for short-term options, indicating that

short-term options are more severely mispriced by the Black-Scholes model. These

findings of clear moneyness-related and maturity-related bias associated with the

Black-Scholes, are consistent with those in the existing literature (Bates, 1996;

Bakshi, Cao and Chen, 1997).

Table 1: Sample Properties of Closing Call Option QuoteMarch 1, 1996 to May 31, 1996

Moneyness Days to Maturity Number ofK/S 7-45 days 46-90 days samples

ITM [0.90, 0.94) $49.07 $67.13(0.296) (0.205) [50]

[0.94, 0.97) $32.82 $42.55(0.211) (0.187) [115]

ATM [0.97, 1.00) $17.53 $24.64(0.166) (0.158) [312]

[1.00, 1.03) $7.70 $16.89(0.152) (0.155) [546]

OTM [1.03, 1.06) $2.14 $8.10(0.146) (0.145) [465]

[1.06, 1.10] $0.68 $3.15(0.145) (0.137) [436]

2.4 Estimated Parameters and In-Sample Pricing Fit

The estimation procedures described in the Section 2.2 are separately done for

each competitive function on each day. Table 2 reports the average and standard

error of each estimated parameter as well as the average loss for each competitive

function. These reported statistics are informative and several observations are in

14

order:

First, all models basically grasp the skew pattern of Black-Scholes implied volatil-

ity, with all α1<0 (which represents tilt of the skew) and α2> 0 (which represents

curvature of the skew).

Table 2: Mean and Standard Deviation of Estimated Parameters

Coefficient Competitive ModelsEstimate model 1 model 2 model 3 model 4

α0 7.62 7.63 7.82 0.16(std err) (6.89) (8.74) (8.23) (0.01)

α1 -2.19 -13.30 -2.26 -0.78(std err) (2.58) (16.94) (2.47) (2.33)

α2 0.16 6.32 0.17 1.09(std err) (0.19) (8.28) (0.19) (1.92)

α3 -1.51(std err) (1.67)

α4 0.06(std err) (0.32)Loss(α) 35.27 28.92 29.03 27.15

Second, it’s obvious there is considerable variation in coefficient estimates, which is

suggested by the standard error of coefficient estimates. It demonstrates that even

within a very short time interval (overnight), infusion of new information could

produce great change in Black-Scholes implied volatility surface which leads to the

instability of volatility function. This result is consistent with that of DFW (1998).

Lastly, based on the minimized loss, model 2, model 3 and model 4 greatly im-

proves the in-sample fitting compared to model 1. The average minimized loss of

15

model 1 reaches 35.27, while the average minimized loss of the other three models

are all around 28. Model 4 slightly dominates both model 2 and model 3. Model

1’s poor performance is in expectation, since it contains the least information, with

only the strike price taken into consideration. By expressing the underlying asset’s

effect in the form of moneyness Kt/St, model 2 obviously infuses more information

than model 1. More importantly, model 2 actually describes the inverse relation

between index return and volatility changes. Although most empirical studies use

index return to measure volatility, DFW (1998) and Derman (1999) found that the

inverse relation is also apparent between the index return and Black-Scholes im-

plied volatility. The addition of time to maturity variable τ to model 3 appears to

be important, as its in-sample fitting is also greatly improved compared to model

1. But most of the incremental power comes from the cross-product term K100

τ ,

as we find that little difference in explanatory power occurs if we estimate model

3 without the variable τ . And it does no better than model 2, which does not

consider time to maturity. Model 4 incorporates all the advantage of model 2 and

model 3. It includes all the variables to which the implied volatilities are sensitive.

Furthermore, its superiority to other models may proves from a different angle

that the volatility of an option depends on the number of standard deviations lnK

is away from the mean of lnF . Compared to model 3, it also enjoys the advantage

of incorporating all related factors in a parsimonious function form.

2.5 Out-of-Sample Forecasting Performance

We have shown that the in-sample fit of daily option prices is increasingly better

as we move from model 1 to 3, and then to 2 and finally to 4. But in-sample fitting

actually reflects a model’s static performance and a good in-sample fit does not

16

necessarily guarantee an good out-of-sample fit. For example, one may argue that

model 3 outperforms model 1 owing to extra parameters. The presence of more

parameters may actually cause overfitting and therefore penalize the model if the

extra parameters do not improve its structural fitting. For this purpose, we rely on

the current day’s estimated parameters to compute next day’s model based option

prices. Next, compute RMSE and MRE every day to obtain the average RMSE

and MRE. Table 3 reports each model’s average RMSE and MRE with respect to

the same classification of moneyness and maturity specified in Table 1.

Two features are noticeable across all models in the forecasting results: First, for a

given maturity, the absolute pricing error measurement RMSE typically increases

from OTM to ATM and then to ITM options. But by percentage pricing error

measurement MRE, OTM options are mispriced to the greatest extent. One pos-

sible reason for this feature is that the loss function (5) is constructed in favor of

more expensive options. The value of call options is a decreasing function of strike

prices, and ITM options generally have the highest prices. To fit the options in

an overall level, loss function (5) obviously puts first priority on fitting the most

expensive option. So, even though OTM options are priced with the least absolute

error, they are actually mispriced with the largest relative error.

Second, for a given moneyness category, RMSE typically increases from short to

long term options. But MRE shows that the short-term options are mispriced to

a larger degree. The reason for this result is also inherited in the characteristics of

the loss function, which is a more favorable treatment of long-term options, whose

values are generally higher than short-term ones.

17

Table 3: Average Forecasting Error of CompetitiveBlack-Scholes Implied Volatility Functions

Days to Moneyness Competitive ModelsMaturity K/S model 1 model 2 model 3 model 4

RMSE 7-45 [0.90, 0.94) 2.87 2.57 2.93 2.43[0.94, 0.97) 2.37 1.95 2.41 1.87[0.97, 1.00) 1.79 1.55 1.84 1.62[1.00, 1.03) 1.34 1.09 1.39 1.11[1.03, 1.06) 0.66 0.47 0.71 0.40[1.06, 1.10] 0.22 0.16 0.21 0.13

46-90 [0.90,0.94) 3.38 2.97 3.42 2.87[0.94, 0.97) 2.79 2.42 2.84 2.34[0.97, 1.00) 2.15 1.82 2.17 1.87[1.00, 1.03) 1.59 1.24 1.63 1.29[1.03, 1.06) 1.17 0.89 1.12 0.74[1.06, 1.10] 0.79 0.64 0.75 0.55

MRE 7-45 [0.90, 0.94) 0.057 0.053 0.061 0.047[0.94, 0.97) 0.074 0.059 0.070 0.055[0.97, 1.00) 0.102 0.086 0.105 0.092[1.00, 1.03) 0.173 0.140 0.174 0.141[1.03, 1.06) 0.307 0.220 0.331 0.187[1.06, 1.10] 0.335 0.235 0.335 0.201

46-90 [0.90, 0.94) 0.049 0.042 0.048 0.040[0.94, 0.97) 0.064 0.054 0.063 0.052[0.97, 1.00) 0.081 0.056 0.066 0.055[1.00, 1.03) 0.094 0.075 0.094 0.076[1.03, 1.06) 0.142 0.105 0.138 0.091[1.06, 1.10] 0.251 0.203 0.239 0.174

Next we move to detailed analysis on the out-of-sample pricing accuracy of com-

petitive functions. First, model 1, which performs poorest in in-sample fitting,

outperforms model 3 for almost all option categories except for the the short-term

deep OTM options with monyness 1.06≤K/S≤ 1.10 and long-term OTM options

with moneyness 1.03≤K/S≤ 1.10. The deterioration of model 3’s performance

18

proves what DFW (1998) found: time to maturity may only serves to overfit the

data at the estimation stage.

Second, the inclusion of the index level in the volatility function (the index level

is expressed in forward price in model 4) make both model 2 and model 4 greatly

outperform model 1 and 3. This may suggest that the the stability of volatility

functions may be greatly improved by utilizing moneyness Kt/St or maturity ad-

justed moneyness ln(K/F )√τ

as state variables. Further comparison between model 2

with model 4 reveals that regardless of time to maturity, model 4 dominates model

3 in both ITM and OTM categories, but does worse for pricing ATM options.

Results in Section 2.4 and 2.5 suggest two possible avenues for us: employing time

to maturity as one of variables to construct more elaborate volatility functions

may be unnecessary; and, the stability of BSIV functions is improved most by a

quadratic regression of maturity adjusted monyness.

3 Dynamic Black-Scholes Implied Volatility Func-

tion

In last section we identify a Black-Scholes implied volatility function which greatly

improves both the in-sample fitting and out-of-sample pricing accuracy when ap-

plied to the AHBS valuation approach. However, to implement such a function,

we have to re-estimate it on a daily basis to remedy the observed functional insta-

bility. One may ask whether we could filter the random factors that produce the

instability and whether a stable BSIV function exists after we filter the random

19

factors. If the answer is yes, we could avoid continuously updating the function

everyday. What we need to do is to model the random factors and the time-

invariant BSIV function, respectively. We could call this kind of model “dynamic

BSIV function”. This is a potentially promising approach, because, as noted in

DFW (1998), an important reason for the failure of the deterministic volatility

function is that they were unable to replicate the stylized fact that “new market

information induces a shift in the level of overall market volatility from week to

week” (p.2081). This is also the main deficiency inherited in the BSIV function we

discussed in Section 2. A dynamic BSIV function that overcomes this deficiency

may be able to extract some useful information embedded in the cross-section of

option prices, while preserving the key dynamics that drive changes in individual

option implied volatilities.

Next we give a brief review of Derman’s “sticky” volatility models which sheds

light on the structural form of dynamic BSIV functions we will discuss later.

3.1 A Brief Review of Derman’s “Sticky” Models of Im-plied Volatility

Figure 1 shows the 45 days to maturity Black-Scholes implied volatilities of call

options with strikes 660, 670, 680, 690, 700 on S&P 500 for the period from March

1, 1996 to 31 December, 1996.5 Two features are noticeable: first, the volatility

skew is always negative. At any time, implied volatilities increase monotonically

as the strike levels decrease. Second, all time series of fixed strike option implied

5The 45 days to maturity implied volatility for a particular strike is obtained by interpolationfrom the closing bid-ask midpoint prices of options with maturity straddling the 45 days tomaturity.

20

volatilities have a similar trending, they move approximately linearly away from

the at-the-money implied volatility series. The greater the strike price, the smaller

the deviation.

Observation of data similar to these, but on S&P 500 index options 3-month volatil-

ities of different sample periods, has motivated Derman (1999) to investigate the

systematic connections between changes in index level and the volatility recovered

from option prices. He argued that the market view of the future volatility struc-

ture can be determined from current option price via the implied volatility tree

model, as illustrated in Derman (1996), much as forward rates can be extracted

from the market’s current bond yield. Your personal view of future volatilities can

be used to produce your own future volatility tree. The chosen tree determines all

future options values. Derman formulated three different types of market regimes

in which different volatility trees exist:

(a) Range-bounded regime, where future index moves are likely to be constrained

within a certain range and there is no significant change in the future volatility;

(b) Trending regime, where the level of the market is undergoing some significant

change, without a significant change in future volatility;

(c) Jumpy regime, where the probability of jumps in the index level is particularly

high, so future volatility increases.

Different linear parameterization of the volatility skew for pricing and hedging

options applies in each regime. These are known as “sticky” volatility models, be-

cause each parameterization implied a different type of “stickiness” for the volatil-

21

ity in a binomial tree. Denote by σ(atm, τ) the volatility of the τ -maturity ATM

option, σ0 and S0 the initial implied volatility and price used to calibrate the tree:

(a) In a range-bounded market, Derman proposed that skews are parameterized

by “sticky strike” model:

σ(K, τ) = σ0 − b(τ)(K − S0) (11)

So, for the given current skew which is determined by current option price and

index level S0, the future volatility of an option with a particular strike σ(K, τ)

remain unchanged as index moves. This model attribute to each option of a def-

inite strike a different volatility tree. As the index moves, all that happens is the

root of each tree is moved to the new index level. The same tree is still used to

price the option with fixed strike. Since σ(atm, τ) = σ0− b(τ)(S−S0), this model

implies that σ(atm, τ) decreases as the index increases. The rules are illustrated

graphically in Figure 2.

(b) For a stable trending market, skews are parameterized by the “stick moneyness”

model:

σ(K, τ) = σ0 − b(τ)(K − S) (12)

So, fixed strike volatility σ(K, τ) increases with the index level S, but the volatility

stays constant with respect to the moneyness of the option. The option that is

10% out of the money after the index moves should have the same implied volatil-

ity as the 10% OTM option before the index moves. That is, it is the moneyness

of the option that determines the future volatility in the tree. You can visualize

this in Figure 3: as the index moves, the moneyness of the option changes and we

consequently move to a tree of different volatility structure, the one corresponding

22

to the current option moneyness. Since σ(atm, τ) = σ0, this model implies that

σ(atm, τ) is independent of the index.

(c) In jumpy markets, skews are parameterized by the “sticky tree” model:

σ(K, τ) = σ0 − b(τ)(K + S) + 2b(τ)S0 (13)

So fixed strike future volatility σ(K, τ) decreases as the index increases. In this

model, the future volatilities are no longer constant regardless of strike or money-

ness. However, current volatility skew determines one unique tree for the evolution

of future volatilities (compared to multiple trees in (a) and (b)) that can be used

to price all options. Since σ(atm, τ) = σ0 − 2b(τ)(S − S0), the σ(atm, τ) also

decreases as index increases, twice as fast as the fixed strike volatilities. Figure 4

illustrate how the unique tree is built.

In fact, Derman’s models yield the same relationship between fixed strike volatility

deviation from at-the-money volatility and the current index price, namely,

σ(K, τ)− σ(atm, τ) = −b(τ)(K − S) (14)

with different specification for the behavior of at-the-money volatility in relation

to the index in each regime, namely,

(a)Range-bounded: σ(atm, τ) = σ0 − b(τ)(S − S0)

(b) Stable trending: σ(atm, τ) = σ0

(c) Jumpy market: σ(atm, τ) = σ0 − 2b(τ)(S − S0)

So all three models imply the same, positive correlation between the index level S

and the deviation σ(K, τ)− σ(atm, τ).

23

Derman’s models provide insight on how to remedy the instability of BSIV func-

tion in two respects:

First, both DFW (1998) and our study in section 2 examine whether the levels of

Black-Scholes implied volatilities of today’s options with a certain time to maturity

(τ) could predict the levels of volatilities of tomorrow’s options with different time

to maturity (τ − 1). The conclusion of instability is made on the empirical results

for answering this question. However, the stability of BSIV function or the rela-

tive shape of volatility surface may depend on the maturity. In other words, stable

function may exist if the maturity of all options are controlled to be the same. All

sticky volatility models are constructed on the restriction of same time to maturity.

More importantly, this model set up a time-invariant volatility function—the pos-

itive correlation between the index level S and the deviation σ(K, τ)− σ(atm, τ),

at the same time identifying and explicitly modelling the random factor σ(atm, τ)

which drive changes in the individual implied volatility.

3.2 Dynamic Black-Scholes Implied Volatility Function

Further investigation on our data raises some doubt on the validity of a linear

parameterization given by equation (14). For most of the trading days from March

1, 1996 to 31 December 31, 1996, the data reveals that the volatility skew, with

respect to the strike price, actually implies a similar skew of the σ(K, τ)−σ(atm, τ).

The skew becomes smoother with the increase in time to maturity. Figure 5-1

and 5-2 illustrate typical patterns in S&P 500 implied volatilities skew and the

skew of σ(K, τ)−σ(atm, τ). If Derman’s linear parameterization is valid, then the

24

σ(K, τ)−σ(atm, τ) should be a negative straight line with slope −b(τ) rather than

a curve in figure 5-2. In this connection, we should take the tilt and curvature

into consideration when modelling this skew pattern. Considering we proved in

Section 2 that maturity adjusted moneyness ln(K/F )√τ

could best characterize the

Black-Scholes volatility skew, we refine equation (14) as:

σ(K, τ)− σ(atm, τ) = α0 + α1ln(K/F )√

τ+ α2[

ln(K/F )√τ

]2 (15)

As ATM option implied volatility has different sensitivity to changes in index level

under different market regime, to implement Derman’s sticky models, we have

to first identify which volatility regime is likely to prevail in the near future. In

fact, equation(14) is a generalization of all market regimes, regardless of how ATM

option implied volatility trends in individual market regimes. Then it’s valid to

generalize the process of ATM option implied volatility in one model.

Two properties of Black-Scholes implied volatility provide insight on how the pro-

cess of at-the-money option implied volatility could be described. First, an inverse

time series relation exists between the stock returns and volatility changes (This

feature is also obvious in Figure 1). A more interesting feature of this negative

relation is the leverage or asymmetric effect of bad news and good news on the

volatility. An unexpected drop in the underlying asset price (bad news) increases

volatility by a larger amount than an unexpected rise in price (good news) de-

creases the volatility. This asymmetric effect was first discovered by Black (1976)

and confirmed by findings of French, Schwert and Stambaugh (1987) and Schw-

ert (1990). Second, volatility tends to be mean-reverting. If current volatility is

25

historically low, then there is an expectation that it will increase. If the current

volatility is historically high, there is an expectation that it will decrease. Consid-

ering these properties, we could specify the ATM option implied volatility in the

process of

σ(atm, t) = ω + β1σ(atm, t− 1) + β2 max(0,−Rt) (16)

where Rt = St/St−1 − 1.

The first lag of at-the-money volatility is to incorporate the volatility mean re-

version, thus β1 is expected to be less than 1. The most recent asset return is

to incorporate the asymmetric effect of index return on implied volatility and β2

is expected to be bigger than zero. When index level rises, implied volatility is

expected to drop, then β1< 1 and max(0,−Rt) = 0 is consistent with such inverse

relation. When index level drops, implied volatility is expected to rise, this is

consistent with max(0,−Rt)>0.

To this end, the dynamic BSIV function is defined so that each option’s Black-

Scholes implied volatility depends on the level of the ATM volatility. This is

accomplished by identifying a time-invariant function (15). The ATM implied

volatility is treated as the random factor for the implied volatility function dy-

namics and modelled by equation (16).

3.3 Methodology

Following our basic assumption of given time to maturity, we restrict our analysis

to BSIV function with 30, 45 and 60 calendar days to maturity, which is the most

26

typical and relevant period length in our study samples.

Obviously, options with the desired time to maturity are not always available, so

we use the linear interpolation of the two latest options whose time to maturity

straddle the given days to maturity:

σ(K, τ0 = 30, 45, 60) =τ0 − τ1

τ2 − τ1

σ(K, τ2) +τ0 − τ2

τ1 − τ2

σ(K, τ1) (17)

Where τ1 is the latest available expiration date before or at τ0 and τ2 is the earliest

available expiration date after τ0.

The ATM Black-Scholes implied volatility, with given time to maturity on every

day, is also obtained by linear interpolation of two options with strikes straddle

the index level.

S&P 500 closing call option data from March 1, 1996 to December 31, 1996 is also

obtained from CBOE and the same data processing procedures as Section 2.3 are

performed. We employ the data on March 1, 1996 through December 23, 1996

to estimate the dynamic BSIV function, and then use the estimated parameter to

forecast the volatilities of options with the same maturity on the last five trad-

ing days during the year. Finally, by interpolating the forecasted volatilities with

given maturity, we get the volatilities with specific maturity, which are applied to

Black-Scholes formula to predict option prices.

Using a standard Ordinary Least Square(OLS) estimation approach to estimate

equation (15), we face the problems of heteroskedasticity and serial correlation. To

address the problem of heteroskedasticity, we use the Newey-West (1987) weight-

27

ing covariance matrix. However, the Newey-West estimator fails to correct for

the serial correlation and the model was re-specified along the lines suggested

by Hendry and Mizon (1978) and Mizon (1995) by including additional variables

[(σ(K, τ)− σ(atm, τ)]t−1 in the model.6 In summary, we employ the Newey-West

approach to estimate the re-specified equation (15):

[σ(K, τ)−σ(atm, τ)]t = γ0+γ1ln(K/F )√

τ+γ2[

ln(K/F )√τ

]2+γ3[σ(K, τ)−σ(atm, τ)]t−1

(18)

Compared to the loss functions adopted in estimating the unstable BSIV functions

in Section 2, we use a different approach in estimating the dynamic BSIV function.

Therefore we could not make a comparison on the in-sample goodness of fit. For

out-of-sample pricing accuracy, we compare it with the best BSIV function (model

4) we identified in Section 2. The same re-estimation and forecasting procedures

as Section 2 are carried out for model 4.

3.4 Estimation and Forecasting Results

Table 4 reports estimates for the ATM implied volatility process used in the dynam-

ics BSIV function. This model effectively characterizes S&P 500 ATM volatility

dynamics with the adjusted R2 around 80% for all maturities. As expected, lagged

implied volatilities are helpful in predicting future implied volatility , as all β1 are

less than 1. In addition, the leverage effect is apparent. A negative return results

in an increase in ATM implied volatility, which is reflected in all β2 significantly

greater than zero.

6They argue that the existence of serial correlation may indicate model misspecification, andthus, the model should be re-specified instead of adopting the faulty alternative of correctingfor serial correlation. They demonstrate that this simple re-specification of equation (15) fromstatic one to a dynamic equation (18) will yield more consistent estimates.

28

Table 4: Estimation of At-the-money Implied Volatility Functions

Coefficient Days to Maturity30 days 45 days 60 days

ω 0.0364 0.0372 0.0416(t-value) (5.33) (5.32) (6.26)

β1 0.7481 0.7454 0.7552(t-value) 16.92 16.85 (17.79)

β2 1.0096 1.3294 1.4296(t-value) (5.92) (7.54) (8.05)

Adjusted R2 0.7781 0.8133 0.8029

Table 5: Estimation of Dynamics Black-ScholesImplied Volatility Function

Coefficient Equation (15) Equation (18)30 days 45 days 60 days 30 days 45 days 60 days

γ0 -0.0002 -0.0023 -0.0019 -0.0002 -0.0022 -0.0017(t-value) (-4.93) (-7.29) (-5.65) (-4.67) (-6.94) (-5.09)

γ1 -0.2044 -0.2152 -0.2248 -0.1892 -0.2068 -0.2066(t-value) ( 69.37) (88.31) (70.75) (45.89) (54.29) (45.10)

γ2 0.5339 0.4280 0.3669 0.4968 0.4122 0.3410(t-value) (31.26) (25.59) (14.91) (27.01) (23.42) (13.68)

γ3 0.0878 0.0432 0.0907(t-value) (5.25) (2.86) (5.45)

Durbin-Waston 1.62 1.82 1.62 2.01 2.07 2.03Adjusted R2 0.7290 0.7807 0.7321 0.7328 0.7815 0.7359

To test the validity of including a lagged dependent variable in equation (15), we

apply Newey-West method to both equation (15) and equation (18), with the re-

sults reported in Table 5. The inclusion of [σ(K, τ)− σ(atm, τ)]t−1 does not lend

itself to significant economic interpretation, demonstrated by minor changes in the

adjusted R2. Moreover, there are no material changes in all parameter estimates

and the serial correlation has been well accounted, as the Durbin-Waston statistics

29

are all around 2 in equation (18). So the re-specified function proved to be effective.

For model 4, we re-estimate the function everyday from December 23, 1996 to De-

cember 30, 1996 and then use today’s parameter estimates to forecast tomorrow’s

volatilities, which are applied to forecast option prices. RMSE and MRE are then

calculated everyday and averaged over the last five days. For dynamic BSIV func-

tion, we use the estimation result in Table 5 to forecast all the call option closing

prices on the last five days at one time. Table 6 compares the average pricing error

of the dynamic BSIV function and the model 4.

Although the results of forecasting accuracy for ITM options are mixed between the

two models, with each model outperforming the other in some maturity categories,

the dynamic BSIV function is clearly superior to model 4 in pricing ATM and OTM

options, with the pricing error averagely decreasing 4 percent for OTM options.

Actually, our estimation method is a more favorable treatment for model 4, which

incorporates the new information everyday by the re-estimation. Therefore we

would expect it to price the options more accurately. One possible reason for the

better performance of the dynamic BSIV function is that the effect of random

factors on both the σ(K, τ) and σ(atm, τ) are partially cancelled out by σ(K, τ)−

σ(atm, τ), the skew of σ(K, τ)− σ(atm, τ) thus may remain relatively stable over

time, and isn’t affected by the arrival of new information.

30

Table 6: Average Forecasting Error of Dynamic Black-ScholesImplied Volatility Function

Days to Moneyness Competitive ModelsMaturity K/S Dynamic Function Model 4

RMSE 7-45 [0.90, 0.94) 2.41 2.27[0.94, 0.97) 1.74 1.69[0.97, 1.00) 1.40 1.32[1.00, 1.03) 0.76 0.80[1.03, 1.06) 0.32 0.41[1.06, 1.10] 0.17 0.22

46-90 [0.90, 0.94) 2.73 2.64[0.94, 0.97) 2.13 2.20[0.07, 1.00) 1.86 1.80[1.00, 1.03) 1.33 1.36[1.03, 1.06) 0.72 0.81[1.06, 1.10] 0.40 0.52

MRE 7-45 [0.90, 0.94) 0.045 0.042[0.94, 0.97) 0.053 0.051[0.97, 1.00) 0.084 0.079[1.00, 1.03) 0.096 0.103[1.03, 1.06) 0.135 0.172[1.06, 1.10] 0.173 0.223

46-90 [0.90, 0.94) 0.039 0.038[0.94, 0.97) 0.051 0.052[0.97, 1.00) 0.063 0.061[1.00, 1.03) 0.067 0.068[1.03, 1.06) 0.105 0.119[1.06, 1.10] 0.112 0.147

31

4 Conclusion

Of considerable interest to both practitioners and academics is a better under-

standing of the empirical dynamics of implied volatility surface. Basing on some

findings about the ad hoc Black-Scholes valuation approach suggested in Dumas,

Flemming and Whaley (1998), our study investigates and proposes some volatility

function forms which are applied to this valuation approach.

Basing on the instability inherited in the volatility function, we first investigate

what function form best characterizes the Black-Scholes implied volatility skew

and thus reduce the fitting or forecasting error in option prices. We suggest four

competitive implied volatility functions and re-estimate each of them day to day.

The results of both in-sample fitting and out-of-sample pricing tests demonstrate

that a quadratic function of maturity adjusted moneyness ln(K/F )√τ

could reduce the

maturity-related and moneyness-related pricing error to the largest degree.

Restricted by the assumption of given time to maturity, we propose and implement

a dynamic Black-Scholes implied volatility function which is expected to remedy

the instability of previous suggested volatility functions. This dynamic function

separates a time-invariant implied volatility relation from the random factors that

drive changes in the individual implied volatilities. The random factors are incor-

porated through the ATM implied volatilities which are modelled as a function

of lagged volatility and a non-linear function of the underlying asset return. The

out-of-sample predicting results proves that the dynamic function outperforms the

known unstable BSIV function, especially in reducing the pricing error of OTM

options. This result provides strong evidence that stable BSIV function in AHBS

32

valuation method may exist or at least the stability of the volatility function could

be greatly improved under certain restrictions.

It is clear that the dynamics of option prices have not been fully explained by the

suggested Black-Scholes implied volatility function, which offer promising areas for

future research.

33

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36

Figure 1: Fixed Strike Black-Scholes Implied Volatility Time Series from March 1, 1996 to December 31, 1996 The figure graphs the time series of S&P 500 index level, the at-the-money Black-Scholes implied volatility and the Black-Scholes implied volatility for options with strike 660, 670, 680, 690 and 700. The period extends from March 1, 1996 to December 31, 1996. All the options are of 45 days to maturity. The 45-day Black-Scholes implied volatility for a particular strike is obtained by interpolation from the two latest options whose time to maturity straddles the 45 days to maturity.

0.05

0.1

0.15

0.2

0.25

0.3

0.35

960301

960401

960501

960531

960701

960731

960829

960930

961029

961127

961230

Implied Volatility

600

650

700

750

800

Index Level

ATM K=660K=670K=680K=690K=700Index Level

Figure 2: The Evolution of Fixed Strike Volatility in the Stick-Strike Model

Each binomial tree represents the evolution of future volatility which is determined by current index level and particular strike. The size of each binomial fork in the tree is intended to represent the magnitude of the future instantaneous volatility. Thus, the binomial trees of a same row illustrate the changes in volatility structure of the fixed strike option when index level moves. We assume the center column match the current negative volatility skew for strikes of 90, 100 and 110, when index level is 100. In this model, when index level move from 100 to 90 or from 100 to 110, all trees in the same row (or for fixed strike) have the same volatility structure, except that the root of the tree is relocated to the new index level. That is, the volatilities keep constant for a given strike. When we move from the top left to the bottom right in this figure, we could see that at-the-money implied volatility decreases as index level increases.

Index 90

90

100

110

Strike100 110

Current Trees

Figure 3: The Evolution of Fixed Strike Volatility in the Stick-Moneyness Model

Each binomial tree represents the evolution of future volatility which is determined by current index level and particular strike. The size of each binomial fork in the tree is intended to represent the magnitude of the future instantaneous volatility. Thus, the binomial trees of a same row illustrate the changes in volatility structure of the fixed strike option when index level moves. We assume the center column match the current negative volatility skew for strikes of 90, 100 and 110, when index level is 100. In this model, the tree has a same structure for fixed moneyness as the index level moves. So when we move from the top left to the bottom right in this figure, we could see that at-the-money implied volatility keep constant as index level increases.

IndexStrike

100

110

100 110Current Trees

90

90

Figure 4: The Evolution of Fixed Strike Volatility in the Stick-Implied Tree Model

Each binomial tree represents the evolution of future volatility which is determined by current index level and particular strike. The size of each binomial fork in the tree is intended to represent the magnitude of the future instantaneous volatility. We assume the center column match the current negative volatility skew for strikes of 90, 100 and 110, when index level is 100. In this model, current skew determines there is only one volatility tree for all the options regardless of moneyness or strike. As the index moves, we simply slide along the tree to node at corresponding index level.

Index 90 100 110Strike Current Tree

90

100

110

Figure 5-1: Black-Scholes Implied Volatility on May 16, 1996 The figure graphs the Black-Scholes Implied Volatilities for options with 30, 45 and 60 days to maturity on May 16, 1996. The implied volatility of option with given time to maturity is obtained by interpolation from the two latest options whose time to maturity straddle the given time to maturity.

0.1000

0.1400

0.1800

0.2200

0.2600

60061

062

062

563

063

564

064

565

065

566

066

567

067

568

068

569

069

570

0

Strike

Impl

ied

Vola

tility

30 days45 days60 days

Figure 5-2: Black-Scholes Implied Volatility Deviation on May 16, 1996 The figure graphs the deviation σ(K ,τ) - σ(atm ,τ) for options with 30, 45 and 60 days to maturity on May 16, 1996.

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