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Journal of Mathematical Finance, 2012, 2, 159-174http://dx.doi.org/10.4236/jmf.2012.22017 Published Online May 2012 (http://www.SciRP.org/journal/jmf)Option Pricing Applications of Quadratic Volatility ModelsSrimantoorao S. Appadoo1*, Aerambamoorthy Thavaneswaran2, Saman Muthukumarana21Department of Supply Chain Management, University of Manitoba, Winnipeg, Canada2Department of Statistics, University of Manitoba, Winnipeg, CanadaEmail: *appadoo@cc.umanitoba.caReceived January 20, 2012; revised March 1, 2012; accepted March 16, 2012ABSTRACTRecently there has been a surge of interest in higher order moment properties of time varying volatility models. VariousGARCH-type models have been developed and successfully applied in empirical finance. Moment properties are important because the existence of moments permit verification of how well theoretical models match stylized facts suchas fat tails in most financial data. In this paper, we consider various types of random coefficient autoregressive (RCA)models with quadratic generalized autoregressive conditional heteroscedasticity (GARCH) errors and study the moments, mean, variance and kurtosis. We also consider the Black-Scholes model with RCA GARCH volatility and showthat these moments can be used to evaluate the call price for European options.Keywords: Garch Processes; RCA Models; Garch Models; Time Varying Volatility; Kurtosis1. IntroductionIt is well-known that many financial time series such asstock returns exhibit leptokurtosis and time-varying volatility [1]. The generalized autoregressive conditional heteroscedasticity (GARCH) and the random coefficientautoregressive (RCA) models have been extensively usedto capture the time-varying behaviour of the volatility.Studies using GARCH models commonly assume thatthe time series is conditionally normally distributed; however, the kurtosis implied by the normal GARCH tendsto be lower than the sample kurtosis observed in manytime series Bollerslev [1]. Thavaneswaran et al. [2] usean ARMA representation to derive the kurtosis of variousclasses of GARCH models such as power GARCH, nonGaussian GARCH, non-stationary and random coefficient GARCH. Recently, Thavaneswaran et al. [3], Appadoo et al. [4] have extended the results to stationaryRCA processes with GARCH errors and Paseka et al. [5]further extended the results to RCA processes with stochastic volatility (SV) errors.Leptokurtosis is commonly observed in financial timeseries, as well as in currency and commodity markets.The opening and closure of the markets, time-of-the-dayand day-of-the-week effects, weekends and vacation periods cause changes in the trading volume that translatesinto regular changes in price variability. Financial, currency, and commodity data also respond to new information entering into the market, which usually have largekurtosis. Recently, there has been growing interest in*Corresponding author.Copyright 2012 SciRes.using volatility models [3,4]. Most of the studies useGARCH models with dummy variables in the volatilityequation, and a few of them have been extended to amore flexible form such as the RCA GARCH. However,even though much research has been performed on volatility models applied to market data such as stock returns,more general specifications accounting for RCA withGARCH errors have been little explored. First we derivethe kurtosis of a simple time series model with behaviourin the mean. Then we introduce various classes of RCAGARCH models and study the moments and discuss applications in option pricing. We extend the results forRCA GARCH volatility models to RCA quadratic GARCHmodels. The RCA GARCH model is appropriate for timeseries where significant autocorrelation exists. Optionpricing with RCA model with quadratic GARCH errorsis also discussed in some detail. The moments derivedfor the RCA GARCH volatility models provide more accurate estimates of market data behaviour and help investors, decision makers, and other market participantsdevelop improved trading strategies. The rest of the paper is organized as follows. In rest of Section 1, we present results on standard GARCH models. These resultsare interesting for their own sake. In Section 2, we derivethe higher order moments of some RCA models withGARCH errors, and in Section 3 we discuss some optionpricing applications with RCA models with GARCHerrors.GARCH ModelsConsider the general class of GARCH (P,Q) model forJMF

S. S. APPADOO ET AL.160the time series yt, where t ht Z t ,PQi 1j 1(1.1)ht i t2 i j ht j(1.2)where Zt is a sequence of independent, normally distributed random variables with zero mean, unit variance. Letut t2 ht be the martingale difference and let u2 bethe variance of ut , (2.12) and (1.2) could be written as:PQi 1j 1 t2 ut i t2 i j ht j(1.3)QQP ij 2j 1 i B j B t j B ut ,j 1j 1 i 1 (1.4) B t2 B ut .(1.5)whereRQi 1j 1 B 1 i B i , i i i , B 1 j B joutside of the unit circle. i 0 i2 where the i sare obtained from the relation B B with B 1 i 1 i B i . The assumption sensure that theut s are uncorrelated with zero mean and finite varianceand that the t2 process is weakly stationary. In thiscase, the autocorrelation function of t2 will be exactlythe same as that for astationary ARMA(R,Q) model. Forany random variable ε with finite fourth moments, theE 4and if the process {Zt}2 Var is normal then the process {εt} defined by Equations (1.3)and (1.4) is called a normal GARCH (p, q) process. The kurtosis ofthe GARCHprocess is denoted by K whenit exists.kurtosis defined bynmating functions of the form g n bt i ht [6], wherei 2pht yt E yt Ft y1 yt yt i and bt i is a funci 1tion of y1 , y2 , , yt i and possibly the known parameters 1 , , p (i.e. We assume that the fitted model isavailable). If we restrict ourselves to a class of estimatingfunctions of the above form then we can forecast the future value of yn 1 based on the observed valuesy1 , y2 , , yn as yˆ n 1 E yn 1 yn , yn 1 , . That is,whether we have an AR(p) model or RCA(p) model wewill get the same linear predictor of yn 1 . However, forthe RCA model under consideration, we havepand R max(P,Q).We shall make the following stationarity assumptions for t2 which has an ARMA(R,Q)representation.All the zeroes of the polynomial Φ(B) lie where the parameters θi, i 2, ···, p, are assumed to beknown, et and bi t are zero mean square integrableindependent processes and the variances are denoted by e2 and b2 . bi t ı 1, 2, s are independent ofet and yt i and may be thought of as incorporatingstructural changes. In order to motivate nonlinear forecasts for nonlinear models, we consider a class of esti-pE yt Ft y1 yt i and var yt Ft y1 2 yt2 i b2 .i 1i 1Thus, the conditional variance is a nonlinear function andhence the RCA model may be viewed as a non-lineartime series model. Nicholls and Quinn [7] studied linearas well as some nonlinear (proposed) forecast by fitting anonlinear (RCA) model for the classical lynx cycle data.Using heuristic reasoning they proposed a nonlinear1forecast and yˆ n 1 sgn 1 yˆ n 12 yˆ n2 2 2 theyshowedempirically that the forecast yn 1 is a better predictor(having smalller forecast errors when compared with theactual observations) than the linear forecast for the lynxdata. It is of interest to note that by defininght yt2 E yt2 Ft y1 , the optimal forecast for yn 1 canbe obtained as11yn* 1 E yt2 Ft y1 2 sgn yn2 12 n2 2 2 . i 1That is, the estimating function method can be used toobtain a nonlinear forecast for a nonlinear models byconsidering a class of elementary martingale estimatingfunctions generated by nonlinear functions of the observations. Using a similar argument we could also obtainforecasts for various class of GARCH models, see Thavaneswaran and Heyde [6] for details. The main messageis RCA models could be used to improve the forecastingperformance of stochastic volatility models.Lemma 2.1. When Zt is a standard normal randomvariable such that Zt N(0,1) then,Copyright 2012 SciRes.JMF2. Random Coefficient Volatility ModelsConsider the class of random coefficient autoregressive(RCA) models defined by allowing random additive perturbations of the autoregressive (AR) coefficients of ordinary AR models. That is, we assume that the process ytis given by,pyt i bi t yt i et(2.1)

S. S. APPADOO ET AL.161yt bt yt 1 t E Zt 1 2 . Now if Z t N 0, Zπ 2 22 2 2Z thenwhere Z denotes the set of integers and b 0 2 0 ,1) t , b2 t 0 0 2 1 E Zt . As a general case for evenπ 2 2 1 powers we have E Z 2 2 .2 π 2) 2 b2 1. The sequences {bt} and {εt} respectively, are the errorsin the model.Theorem 2.1. Let {yt} be a modified RCA (1) time series with an absolute value random coefficient satisfyingconditions (1) and (2). The modified RCA (1) model isgiven by2.1. RCA ModelsRandom coefficient autoregressive time series were introduced by Nicholls and Quinn [10] and some of theirproperties have been studied recently by Thavaneswaran[3]. RCA models exhibiting long memory propertieshave been considered in Leipus and Sugailis [8]. A sequence of random variables {yt} is called an RCA (1)time series if it satisfies the equations 1) E yt 0, E yt2 2) E yt43) K y t Z,yt bt yt 1 t(2.2) t N 0, 2 , bt N 0, b2 . Then we have the fol-lowing 2 1 2 2 b 2 b2 π 23 4 1 2 2 b b2 π , 3 8 2 b 2 224432 22 3 b 4 b 6 b 1 2 b b 1 πππ 2 222 b 3 1 2 b π . 8 2 b3 24432 2 6 b 3 b 4 b 1 ππ The autocovariance and the autocorrelation functions are given by 2 b2 k π Proof: 2 2 b2 1 k 1 and pk π 2 2k 1 . 2 E yt 0, E yt2 2 E yt2 1 2 E yt2 1 E bt E yt2 1 E bt2 E 2tThus, we have E yt2Copyright 2012 SciRes. 2 1 2 2 b 2 b2 π (2.3)JMF

S. S. APPADOO ET AL.162and we have3 4 E yt4 8 2 b3 2 4 3 b4 4 3 b 6 2 b2 1 ππ 26 4 2 2 b b2 π 3 8 2 b 22 4 3 b4 4 3 b 6 2 b2 1 2 2 b b2 1 πππ 23 4 1 2 2 b b2 π 3 8 2 b 22 6 2 b2 1 2 2 b b2 4 3 b4 4 3 b 1 πππ K E yt4 yE yt2 22 222 3 1 2 b b π 8 2 b3 24432 2 3 b 4 b 6 b 1 ππ When b2 0 , the kurtosis of the process yt convergeto K(y) 3. Thus, the autocorrelation function is given by k E bt k k 1 2 b2 π 2 1 2 k E y(2.6)and E y 2t 1 2b2π 3 6We have(2.5)where bt is an uncorrelated Gaussian process with zeromean and with variance b2 . εt is an uncorrelated Gaussian process with zero mean and with variance 2 .Then, we have the followingProof:4t 1 yt 1 t2 1 t 0E yt2 2 E yt2 1 2 E yt2 1 bwhere we use the fact that ρ0 1.Theorem 2.2. Suppose yt is an Random CoefficientMoving average process model of the formyt bt yt 1 t2 1 tE yt E bt(2.4) E yt2 63 1 2 2 b 2 b2 π (2.7)and 2 4 23 6 b2 3 4 2 6 2 3 4 2 E yt2 1 105 8 3 4 12 bπ 22432 23 3 b4 1 4 2 b 6 b 4 bππ 2 62 62 6212 36 b π 18 b 18 E yt 1 315 2 3242 23 3 b4 1 8 b 6 b 4 bππ Copyright 2012 SciRes.JMF

S. S. APPADOO ET AL.163Thus we have E yt4 1 2 62 62 6 36 b 18 b 18 6π3 2 32242 23 3 b4 1 2 2 b b2 1 8 b 6 b 4 bπππ 315 12 2 3242 23 3 b4 1 8 b 6 b 4 bππ The kurtosis of the process is given byK y 22 b2 36 b 6 18 b2 6 18 2 6 1 2 2 b ππ 2 3242 23 3 b4 3 6 1 8 b 6 b 4 b ππ 2 2 35 1 2 2 b b2 π 2 32 42 234 1 8 π b 6 b 4 b π 3 b When b2 0 , the kurtosis of the process yt converge2to K y 35 29 and when b2 0 , and 0 the1 2kurtosis of the process yt turns out to be 35.Theorem 2.3. Suppose yt is an Random CoefficientMoving average process model of the form (2.8)yt t bt t 1 (2.9)where bt is an uncorrelated noise process with zero meanand with variance b2 . εt is an uncorrelated noise process with zero mean and with variance 2 . Then, wehave the following relationships 2E yt2 2 1 2 2 b b2 ,π 2E yt4 4 3 4 3 b4 1 121 2 b2 2 b 6 b2 2 3 2 b2 ,π 3 4 3 b4 1 12 2 1 2 b2 2 b 6 b2 2 3 2 b2 yπK . 222 1 2 b π b and the autocorrelation functions are given by1k 0 2 b π k k 1 1 2 2 2 2 bb π 0k 2,3, Copyright 2012 SciRes.(2.10)JMF