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Quantitative Finance, Vol. 12, No. 1, January 2012, 89–105Extension of stochastic volatility equity models withthe Hull–White interest rate processLECH A. GRZELAK*yz, CORNELIS W. OOSTERLEEyx and SACHA VAN WEERENzDownloaded by [Lech A. Grzelak] at 10:55 24 January 20121. IntroductionyDelft Institute of Applied Mathematics, Delft University of Technology,Mekelweg 4, 2628 CD Delft, The NetherlandszDerivatives Research and Validation Group, Rabobank, Jaarbeursplein 22,3521 AP Utrecht, The NetherlandsxCWI, National Research Institute for Mathematics and Computer Science,Kruislaan 413, 1098 SJ Amsterdam, The Netherlands(Received 28 July 2008; in final form 6 July 2009)We present an extension of stochastic volatility equity models by a stochastic Hull–Whiteinterest rate component while assuming non-zero correlations between the underlyingprocesses. We place these systems of stochastic differential equations in the class of affinejump-diffusion–linear quadratic jump-diffusion processes so that the pricing of Europeanproducts can be efficiently performed within the Fourier cosine expansion pricing framework.We compare the new stochastic volatility Scho¨bel–Zhu–Hull–White hybrid model with aHeston–Hull–White model, and also apply the models to price hybrid structured derivativesthat combine the equity and interest rate asset classes.Keywords: Finance; Financial applications; Mathematical finance; Financial derivatives;Financial econometrics; Financial engineering; Mathematical models; Financial mathematicsIn recent years the financial world has focused on theaccurate pricing of exotic and hybrid products that arebased on a combination of underlyings from differentasset classes. In this paper we therefore present a flexiblemulti-factor stochastic volatility (SV) model that includesthe term structure of the stochastic interest rates (IR). Ouraim is to combine an arbitrage-free Hull–White IR modelin which the parameters are consistent with market pricesof caps and swaptions. In order to perform efficientoption valuation we fit this process in the class of affinejump-diffusion (AJD) processes (Duffie et al. 2000)(although jump processes are not included in this work).We specify under which conditions such a general modelcan fall in the class of AJD processes.A major step away from the assumption of constantvolatility in derivatives pricing was made by Hull andWhite (1990), Stein and Stein (1991) and Heston (1993),who defined the volatility as a diffusion process.This improved the pricing of derivatives under*Corresponding author. Email: l.a.grzelak@tudelft.nlheavy-tailed return distributions significantly and alloweda trader to quantify the uncertainty in the pricing.Stochastic volatility models have become popular forderivative pricing and hedging (see, for example, Fouqueet al. 2000), however financial engineers have developedmore complex exotic products that additionally requirethe modeling of a stochastic interest rate component. Aderivative pricing tool in which all these features areexplicitly modeled may have the potential of generatingmore accurate option prices for hybrid products. Theseproducts can be designed to provide capital or incomeprotection, diversification for portfolios and customizedsolutions for both institutional and retail markets.Fang and Oosterlee (2008a) developed a highly efficientalternative pricing method based on a Fourier-cosineexpansion of the density function and called it the COSmethod. This method can also determine a whole set ofoption prices in one computation. The COS algorithmrelies heavily on the availability of the characteristicfunction of the price process, which is guaranteed if westay within the AJD class (Duffie et al. 2000, Lewis 2001,Lee 2004). We examine the effect of correlated processesfor assets, stochastic volatility and interest rates on optionQuantitative FinanceISSN 1469–7688 print/ISSN 1469–7696 online ß 2012 Taylor & Francishttp://www.tandfonline.comhttp://dx.doi.org/10.1080/14697680903170809
90 L. A. Grzelak et al.Downloaded by [Lech A. Grzelak] at 10:55 24 January 2012prices through a comparison with, for example, the in which the squared volatility, v t ¼ t 2 , represents theHeston model.variance of the instantaneous stock return.pdS t ¼ rS t dt þffiffiffiffi v tStdW x t ,The dimension of the (complex-valued) ODEs for B(u, )dv t ¼ 2 v t þ t þ 2 dt þ 2ffiffiffifficorresponds to the dimension of the state vector, X t .pv t dWt2Typically, multi-factor models, such as the SZHW andthe HHW models, provide a better fit to the observedThe plan of the paper is as follows. In section 2 weperform an analysis of the Scho¨bel–Zhu–Hull–Whitemodel. In subsection 2.3 we show that the hybrid modelof interest admits a semi-closed form for the characteristicfunction. In the following subsection we derive theIt has already been reported by Heston (1993) andScho¨bel and Zhu (1999) that the plain Scho¨bel–Zhumodel is a particular case of the original Heston model.We can see that if ¼ 0, the Scho¨bel–Zhu model equalsthe Heston model in which H ¼ 2, H ¼ 2 =2, andHeston–Hull–White model with non-zero correlation H ¼ 2. This relation gives a direct connection betweenbetween the stock and the interest rate. In section 3 weshow how to efficiently price options with a Fouriercosinetheir discounted characteristic functions (Lord and Kahl2006). Finally, if we set r t constant and p ¼ 0 in the systemexpansion technique when the characteristic of equations (1), and zero correlations, the modelfunction with stochastic interest rate of the asset processis available. Further, in section 4 the two hybrid models,Scho¨bel–Zhu–Hull–White and Heston–Hull–White, andthe stochastic volatility Heston model are compared indetail with respect to calibration and hybrid productpricing. Section 5 concludes. The lengthy proofs of thecollapses to the standard Black–Scholes model.We will choose the parameters of equations (1) suchthat we deal with the Scho¨bel–Zhu–Hull–White (SZHW)model in subsection 2.3, and with the Heston–Hull–White(HHW) model in subsection 2.4. Gaspar (2004) andCheng and Scaillet (2007) showed that the so-calledlemmas are given in the appendices.linear-quadratic jump-diffusion (LQJD) models areequivalent to the AJD models with an augmented statevector.2. Extension of stochastic volatility equity models2.1. Affine jump-diffusion processesIn this section we present a hybrid stochastic volatilityThe AJD class refers to a fixed probability space (, F, P)equity model that includes a stochastic interest rateand a Markovian n-dimensional affine process X t in someprocess. In particular, we add to the SV model the wellknownHull–White stochastic interest rate process (Hullspace D R n . The model without jumps can be expressedby the following stochastic differential form:and White 1996), which is a generalization of the Vasˇicˇek(1977) model.dX t ¼ ðX t Þdt þ ðX t ÞdW t ,We consider a three-dimensional system of stochastic where W t is a F t -standard Brownian motion in R n , (X t ):differential equations of the formD ! R n , (X t ):D ! R nn . Moreover, for processes indS t ¼ r t S t dt þ t p S t dWt x ,the AJD class it is assumed that drift (X t ), volatilitydr t ¼ ð t r t Þdt þ dWt r (X t )(X t ) T and interest rate component r(X t ) are of theaffine form, i.e.d t ¼ ð t Þdt þ t1 p dW t , ð1Þ ðX t Þ¼a 0 þ a 1 X t , for any ða 0 , a 1 Þ2R n R nn , ð5Þwhere p is an exponent, and control the speed ofmean reversion, represents the interest rate volatility, ðX t ÞðX t Þ T ¼ðc 0 Þ ij þðc 1 Þ T ij X t, for arbitraryand 1 p determines the variance of the t process.Parameters and t are the long-run mean of theðc 0 , c 1 Þ2R nn R nnn , ð6Þvolatility and the interest rate processes, respectively.W i are correlated Wiener processes, also governed byrðX t Þ¼r 0 þ r T 1 X t, for ðr 0 , r 1 Þ2R R n : ð7Þan instantaneous covariance matrix,Then, for a state vector, X t , the discounted characteristic23function (CF) is of the following form:1 x, x,rR67T ¼ 4 ,x 1 ,r 5dt:ð2Þ ðu, X t , t, T Þ¼E Q rðes dsþiu T X Tt jF t Þ¼e Aðu, ÞþBT ðu, ÞX t, r,x r, 1where the expectation is taken under the risk-neutralIf we keep r t deterministic and p ¼ 1/2, we have the measure, Q. For a time lag, :¼ T t, the coefficientsHeston (1993) model,A(u, ) and B T (u, ) have to satisfy certain complex-valuedpdS t ¼ rS t dt þffiffiffiffi tStdW x t ,ordinary differential equations (ODEs) (Duffie et al.d t ¼ H ð H t Þdt þ H p ffiffiffiffi2000): t dWt : ð3ÞdFor p ¼ 1, our model is, in fact, the generalized Stein–d Aðu, Þ ¼ r 0 þ B T a 0 þ 1 2 BT c 0 B,Stein (Stein and Stein 1991) model, which is also calleddthe Scho¨bel–Zhu (Scho¨bel and Zhu 1999) model,d Bðu, Þ ¼ r 1 þ a T 1 B þ 1 ð8Þ2 BT c 1 B:
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