1

Option Pricing under ARMA Processes

Theoretical and Empirical prospective

Chou-Wen Wang

2

Astract Motivated by the empirical findings that asset r

eturns or volatilities are predictable, this paper, extending Huang and Wu (2007), studies the pricing of European Futures options, the instantaneous changes of which depend upon an ARMA process.

An ARMA process transforms to an MA process with new MA orders depending on the observed time span under a risk-neutral probability measure.

3

Astract The ARMA pricing formula is similar to that of Black and

Scholes, except that the total volatility input depends upon the AR and MA parameters.

The ARMA(1,1) model has a competitive fit in-sample and provides the similar out-of-sample performance to ad hoc models.

Therefore, for parameter parsimony purpose, the ARMA(1,1) model is a good candidate for pricing TAIEX options both in-sample and out-of-sample.

4

Introduction The Taiwan Stock Exchange Capitalization Weighted

Stock Index (TAIEX) is the most widely quoted of all indices in Taiwan Stock Exchange Corporation (TSEC).

Both futures and option contracts on the TAIEX are traded on the Taiwan Futures Exchange (TAIFEX).

The average daily trading volume in 2007 TAIEX futures: 47,827 contracts TAIEX options: 374,841 contracts

These derivatives contracts play an important role in Taiwan financial market.

5

Introduction The objective of this paper is to study how the TAIEX options

are priced. The TAIEX options have several features. (1) the TAIEX options are European-style(2) both the TAIEX options and futures are cash settled(3) the expiration days of the TAIEX futures coincide with those

of the TAIEX options(4) the TAIEX options and futures are traded side by side on the

same exchange involving the same clearing house Therefore, the TAIEX options can be priced as if they are

European-style futures options with both the option and futures share the same maturity.

The last feature allows us to bypass the difficult task of determining the appropriate dividend yield for the TAIEX.

6

Literature Review From empirical data, stock dynamics under the physical measure

follow a more complicated process than geometric Brownian motion. Hence, various extensions of the standard BSM model have been proposed.

Fama (1965) finds that the first-order autocorrelations of daily returns are positive for 23 of 30 Dow Jones Industrials.

Fisher (1966) suggests that the autocorrelations of monthly returns on diversified portfolios are positive and larger than those for individual stocks.

Gençay (1996) uses the daily Dow Jones Industrial Average Index from 1963 to 1988 to examine predictability of stock returns with buy-sell signals generated from the moving average rules.

7

Literature Review Lo and MacKinlay (1988) find that weekly returns on portfolios of NYSE

stocks grouped according to size show positive autocorrelation.

Conrad and Kaul (1988) also present positive autocorrelations of Wednesday-to-Wednesday returns for size-grouped portfolios of stocks.

Lo and MacKinlay (1990) report positive serial correlations in weekly returns for indices and portfolios and negative serial correlations for individual stocks.

Chopra, Lakonishok and Ritter (1992), De Bondt and Thaler (1985), Fama and French (1988), French and Roll (1986), Jegadeesh (1990), Lehmann (1990) and Poterba and Summers (1988) all find negatively serial correlations in returns of individual stocks or various portfolios.

=>Those evidence documents the predictability of financial asset returns

8

Literature Review The value of an option depend on the log-price dynamics of underly

ing.

The stock price process under BSM assumptions is a geometric Brownian motion.

Distinguishing between the risk-neutral and true distributions of underlying asset return process, Grundy (1991) shows that the Black-Scholes formula still holds, even though the underlying asset returns follow an Ornstein-Uhlenbeck (O-U) process.

Along this line of research, Lo and Wang (1995) followed price options on an asset with a trending O-U process. They showed that as long as an Ito process with a constant diffusion coefficient describes the underlying asset’s log-price dynamics, the Black-Scholes formula yields the correct option price regardless of the specification and arguments of the drift.

9

Goals Liao and Chen (2006) derive the closed-form formula for a MA(1) option

on an asset. The first-order MA parameter is significant to option values even if the autocorrelation between asset returns is weak.

Huang and Wu (2007) examine that the BSM formula still holds when asset returns follow a ARMA process.

The first contribution of the paper is to extend Huang and Wu (2007) model to derive the closed-form formula for futures options where the index returns follow ARMA(p,q) process.

We focus on the ARMA(1,1) and ARMA(2,2) models and apply the model to the TAIEX options. As a benchmark model, following Heston and Nandi (2000) and Duan, Popova and Ritchken (2002), the ad hoc BS model of Dumas et al (1998) is chosen.

10

Goals Incorporating the concept of ad hoc BS model, we also

construct an ad hoc ARMA(1,1) model in which each option has its own implied volatility depending on the AR parameter, MA parameter, the strike price and time to maturity.

The empirical study uses 195 sets of option data sampled weekly from 2003 to 2006.

The empirical results shows that all of the ARMA-type models has a better performance for in-sample and the ad hoc ARMA(1,1) has a overall better out-of-sample fit. However, the ARMA(1,1) model has a competitive fit in-sample and provides the similar out-of-sample performance to ad hoc models.

11

Model Setup The dynamics of the instantaneous asset return is defined

as follows

(1)

It is worth noting that Equation (1) reduces to the continuous-time MA(1) process in Liao and Chen (2006) when the AR and MA coefficients are all zero except for

and when the time interval approaches zero.

p , q : the AR and MA orders

: AR coefficients10

ij : MA coefficients with

10 1

q

j

Pjtj

p

j jt

jtj

t

t ZS

S

S

S

01 11

lnln

12

Lemma1: the dynamics of the stock price lag for m periods

Lemma. Assume that the underlying stock price process S satisfies

Equation (1). Given that , and where and , repeated substitution in Equation (1) for m times yields

(3)

where (4)

(5)

qm

j

Pjntj

p

j jmnt

jmntj

m

jnt

nt ZmS

Smyjy

S

S

01

1

1

01

1 ln1ln

kjif

kjif

j

-k1-kyk

j

jq)kj(k-jj

111

p1,...,jfor,ky1kyky 1jp)j(jj 111

13

Autocorrelation

The variance of instantaneous stock returns Rt+n+1 at time t conditional on the time-t information set

the conditional autocorrelation coefficient is given by

n

i

2i

2ntt iRVar

01

gn

i

2i

n

i

2i

n

igii

gntntt

gii

giiR,RCorr

00

011

14

Martingale Property of an ARMA Process

By summing up the Equation (3) for m=n and n=0,…,N-1, the stock price dynamic takes the following equivalent form:

Conditioning on , for i=1,…,p is a realized stock return. In addition, the last term in the right hand side of Equation (8) can be rewritten as following form:

)8(,ln1ln1

0 01

1

11

0

1

0 01

N

j

jq

i

Pijti

P

i it

itN

ji

N

i

i

jt

Nt ZjS

Sjyjy

S

S

t

N

i

Pit

Ni

q

i

Pit

N

jji

N

j

jq

i

Pijti ZBZjZj

111

1

0

1

0 01

iN

xx

Ni xB

0

,

15

Martingale Property of an ARMA Process The dynamics of the stock prices are equivalent to t

he following Itô integral equation

where

)[ 1 nn t,tt

(Conditioning on , is measurable )

,

),( NtY t

q

i

Pit

N

jji

P

i it

itN

ji

N

i

i

j

ZjS

SjyrjyNtY

11

1

01

11

0

1

0 01 ln1),(

t

N

i

Pit

Ni

t

Nt ZBNtYrNS

S

1

),(ln

16

Martingale Property of an ARMA Process

As the paths of stock price and the standard normal random variables prior to the time t are known, is

-measurable. The mean of stock return during N time intervals conditional on is

the conditional variance which depends on the time-to-

maturity N satisfies

),( NtY

t

),()(ln NtYrNSSE ttNtP

N

i

NittNtPN BSSVarV

1

2)(ln

17

Local Risk-Neutralization Principle of Duan (1995)

Assumption 2. The local risk-neutralized probability measure Q, which is

defined over the period from 0 to a finite integer T, satisfies the local risk-neutral valuation relationship (LRNVR), that is,

(1) Q and P are mutually absolutely continuous;

(2) for i=1,…,N; and

(3) for all i, almost surely with respect to P.

tr

ttQ SeSE 1

tttPtttQ SSVarSSVar )(ln)(ln 11

18

Martingale Property of an ARMA Process

Proposition 1. With respect to local risk-neutralized probability

measure Q, the asset price process, conditional on , with ARMA relation obeys

Where , i=1,…,N; satisfies , , n=1,…,N.

N

i

Qit

NiN

t

Nt ZBVrNS

S

12

1ln

ti

Pit

Qit ZZ

n

ti

n

i

ni

tn VBntY

2

1),(

1 1

1

19

TAIEX futures Price Under the cost-of-carry model, the time-t price of the TAIEX

futures maturing at time t+U, denoted by , satisfies

under the local risk-neutralized probability measure Q, the price of TAIEX futures with delivery date t+U satisfies

UttF

Urt

Utt eSF

N

i

Qit

NiN

Utt

N

i

Qit

NiNtNt

UtNt KVFKVrUSNUrSF

11 2

1ln

2

1ln)(lnln

20

ARMA-type Futures Options

Proposition 2. Assuming that the dynamics of the underlying stock prices

are given by Equation (1), the closed-form solutions for the ARMA( p, q) -type Futures options are as follows:

(17)

(18)Where

,

N,tdX-N,tdFeC Utt

Nr-t 21

N,tdFN,tdXeP Utt

Nr-t 12

n

nX

F

n,tdn

n

Utt

2

1

2

1ln

n-n,tdn,td n2 1

n

iit

ni

nn K

nn

V

1

22 1

21

Properties of ARMA option formula

Inspection of Equations (17) and (18) shows that the closed-form solution for an ARMA( p, q)-type European option is the same as the Black-Scholes-Merton (BSM) formula, except that the volatility function depends upon the AR and MA parameters.

The implied volatility estimated from the BS formula can be successfully interpreted as one calculated from an ARMA( p, q)-type option formula.

Specifically, this finding demonstrates that the BSM implied volatility is also valid -- even if the stock returns follow an ARMA process.

22

Empirical Test : Data The empirical test is performed with daily closing

prices of TAIEX futures and option data from 2 January 2003 to 10 January 2007 obtained directly from the TAIFEX.

Both the futures and options prices are quoted in index points. We construct data sets by using the option prices adjacent Wednesdays. Altogether, there are 195 data sets.

For each data set, it includes spot month, the next two calendar months, and the next two quarterly months.

23

Empirical Test : Data To avoid liquidity-related biases, some filtering rules are applied t

o the raw sample.

1. First, option prices that are less than 1 are not used to mitigate the impact of price discreteness (the minimum tick for TAIEX option is 0.1). see Lim and Guo (2000) use similar rule for S&P 500 futures option data.

2. Second, in term of maturity, as in Dumas et al (1998) this empirical test includes only options with time to maturity greater than six days and less than one hundred days.

3. Third, a transaction must satisfy the no-arbitrage put-call parity relationship (eight points error)

4. Fourth, the deep-in-the-money and deep-out-of-the-money options are also excluded.

24

Empirical Test : Data The data set consists of 11,664 observations,

5,938 call prices and 5,726 put prices.

The strike prices range from 4000 to 8200 with average value 6123.77.

The futures prices in the sample period range from 4125 to 7989 with average value 6131.20.

The average option price is 132.21.

25

Empirical Test : Parameter Estimation For parameter parsimony purpose, the empirical

analysis focuses mainly on the two lags version of the ARMA model.

For each ARMA price, using the 194 sets of weekly option prices, the 194 sets of in-sample AR and MA parameter values are obtained by minimizing the sum of squared errors between ARMA(1,1) or ARMA(2,2) model option values and market option prices, allowing the parameter to change every week.

For each set of in-sample model parameter estimates, following one week of data are used to examine one-week out-of-sample performance.

26

Ad hoc BS and ARMA model As a benchmark model, following Heston and Nandi (2000)

and Duan, Popova and Ritchken (2002), the ad hoc BS model of Dumas et al (1998) is chosen.

is the BS implied volatility for an option of strike X and time to maturity .

The total volatility function of ARMA(1,1) model incorporati

ng the concept of ad hoc BS model

XaaaXaXaaBS 52

432

210 BS

N

i

iNN

iit

iN

N XNN 1

2

1

111

1

2

1

111

2 ),(1

1)(1

1

1

1)(1

1

XbbbXbXbbX 52

432

210),(

27

Table 1 In-sample model comparison

Panel A reports the aggregate (across 2003,2004,2005 and 2006) in-sample percentage valuation errors for all options by various models from weekly (every week) estimation by minimizing the sum of squared errors between model option values and market option prices for Ad hoc BS, ARMA(1,1), ARMA(2,2) and Ad hoc ARMA(1,1) models. RMSE is the ratio of root mean squared out-of-sample valuation errors to the average option price. MAE is the ratio of the mean absolute error to the average option price. Average premium is

the average option price in the sample.

RMSE MAEAverage

premiumNumber of observations

Panel A: Aggregate valuation errors across all years

Ad hoc BS 5.16% 3.58% 132.2112 11578

ARMA(1,1) 5.43% 3.50% 132.2112 11578

ARMA(2,2) 4.92% 3.38% 132.2112 11578

Ad hoc ARMA(1,1)

4.83% 3.23% 132.2112 11578

28

Table 2 Out-of-sample percentage valuation errors

RMSE MAEAverage

Premium

Number of

observations 100% 99% 100% 99%

Panel A: Aggregate percentage valuation errors across all years

Ad hoc BS 9.20% 7.87% 6.22% 5.79% 132.3840 11523

ARMA(1,1) 9.33% 8.08% 6.48% 6.05% 132.3840 11523

ARMA(2,2) 9.41% 8.06% 6.46% 6.02% 132.3840 11523

Ad hoc ARMA(1,1)

9.26% 7.76% 6.22% 5.76% 132.3840 11523

29

Table 3 Out-of-sample percentage valuation errors

RMSE MAEAverage

premium

Number Of

observations 95% 90% 95% 90%

Panel A: Aggregate valuation errors across all years

Ad hoc BS 6.41% 5.41% 4.95% 4.31% 132.3840 11523

ARMA(1,1) 6.67% 5.69% 5.22% 4.57% 132.3840 11523

ARMA(2,2) 6.63% 5.67% 5.19% 4.55% 132.3840 11523

Ad hoc ARMA(1,1)

6.37% 5.46% 4.97% 4.37% 132.3840 11523

30

Panel B: Percentage valuation errors by years

2003RMSE100%

RMSE99%

MAE100%

MAE99%

Average premium

Number ofobservations

Ad hoc BS 8.78% 8.00% 6.42% 6.07% 128.5802 2264

ARMA(1,1) 9.17% 8.40% 6.95% 6.60% 128.5802 2264

ARMA(2,2) 9.20% 8.44% 6.92% 6.57% 128.5802 2264

Ad hoc ARMA(1,1)

8.76% 7.88% 6.47% 6.10% 128.5802 2264

2004

Ad hoc BS 11.76% 9.76% 7.70% 7.13% 152.9901 2404

ARMA(1,1) 11.40% 9.72% 7.67% 7.14% 152.9901 2404

ARMA(2,2) 11.89% 9.93% 7.82% 7.24% 152.9901 2404

Ad hoc ARMA(1,1)

11.86% 9.59% 7.60% 6.99% 152.9901 2404

Table 2B Out-of-sample percentage valuation errors

31

Table 4 Out-of-sample valuation errors for call options

Days to Expiration

<=30 >30

Moneyness ModelAverage premium

RMSE % Error

Average premium

RMSE % Error

[0.95,0.99) Ad hoc BS 225.41 4.65% 246.84 5.32%

ARMA(1,1) 225.41 4.65% 246.84 5.36%

ARMA(2,2) 225.41 4.61% 246.84 5.52%

Ad hoc ARMA(1,1)

225.41 4.55% 246.84 5.54%

[0.99,1.01) Ad hoc BS 105.44 10.72% 156.81 8.77%

ARMA(1,1) 105.44 11.54% 156.81 8.69%

ARMA(2,2) 105.44 11.43% 156.81 8.68%

Ad hoc ARMA(1,1)

105.44 10.74% 156.81 8.72%

[1.01,1.05] Ad hoc BS 43.66 24.30% 86.49 15.77%

ARMA(1,1) 43.66 25.47% 86.49 15.62%

ARMA(2,2) 43.66 25.22% 86.49 15.59%

Ad hoc ARMA(1,1)

43.66 23.50% 86.49 15.78%

32

Table 5 Out-of-sample valuation errors for Put options

Days to Expiration

<=30 >30

Moneyness ModelAverage premium

RMSE % Error

Average premium

RMSE % Error

[0.95,0.99) Ad hoc BS 44.15 22.31% 86.80 18.06%

ARMA(1,1) 44.15 22.79% 86.80 17.67%

ARMA(2,2) 44.15 22.77% 86.80 18.47%

Ad hoc ARMA(1,1) 44.15 22.32% 86.80 18.87%

[0.99,1.01) Ad hoc BS 104.96 10.70% 153.92 9.55%

ARMA(1,1) 104.96 11.47% 153.92 9.54%

ARMA(2,2) 104.96 11.34% 153.92 9.76%

Ad hoc ARMA(1,1) 104.96 10.66% 153.92 9.58%

[1.01,1.05] Ad hoc BS 217.43 5.02% 236.67 6.46%

ARMA(1,1) 217.43 5.30% 236.67 6.19%

ARMA(2,2) 217.43 5.28% 236.67 6.49%

Ad hoc ARMA(1,1) 217.43 5.03% 236.67 6.57%

33

CONCLUSIONS Motivated by the empirical findings that asset returns or volatilities

are predictable, this paper studies the pricing of European Futures option under an autoregressive moving average (ARMA) process.

Then, we focus on the ARMA(1,1), ad hoc ARMA(1,1) and ARMA(2,2) models and apply the model to the TAIEX options. As a benchmark model, the ad hoc BS model of Dumas et al (1998) is chosen.

From the empirical study, the ARMA-type models is found to has a better performance for in-sample and the ad hoc ARMA(1,1) has a overall better out-of-sample fit.

However, the ARMA(1,1) model has a competitive fit in-sample and provides the similar out-of-sample performance to ad hoc ARMA(1,1) model.

Therefore, for parameter parsimony purpose, the ARMA(1,1) model is a good candidate for pricing TAIEX options both in-sample and out-of-sample.