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A numerical method to price exotic path-dependent options on an underlying described by the Heston stochastic volatility model

A numerical method to price exotic path-dependent options on an underlying described by the Heston stochastic volatility model
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  A numerical method to price exotic path-dependentoptions on an underlying described by the Hestonstochastic volatility model  ∗ Luca Vincenzo Ballestra Dipartimento di Scienze Sociali “D. Serrani”,Universit`a Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy,Ph. N. +39-071-2207251, FAX N. +39-071-2207150, E-mail: Graziella Pacelli Dipartimento di Scienze Sociali “D. Serrani”,Universit`a Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy,Ph. N. +39-071-2207050, FAX N. +39-071-2207150, E-mail: Francesco Zirilli Dipartimento di Matematica “G. Castelnuovo”,Universit`a di Roma “La Sapienza”, Piazzale Aldo Moro 2, 00185 Roma, Italy,Ph. N. +39-06-49913282, FAX N. +39-06-44701007, E-mail: Abstract We consider the problem of pricing European exotic path-dependent derivatives onan underlying described by the Heston stochastic volatility model. Lipton has found aclosed form integral representation of the joint transition probability density functionof underlying price and variance in the Heston model. We give a convenient numer-ical approximation of this formula and we use the obtained approximated transitionprobability density function to price discrete path-dependent options as discounted ex-pectations. The expected value of the payoff is calculated evaluating an integral withthe Monte Carlo method using a variance reduction technique based on a suitableapproximation of the transition probability density function of the Heston model. Asa test case, we evaluate the price of a discrete arithmetic average Asian option, whenthe average over  n  = 12 prices is considered, that is when the integral to evaluate is a2 n  = 24 dimensional integral. We show that the method proposed is computationallyefficient and gives accurate results. JEL Classification Codes:  G13, C63. Key Words:  stochastic volatility, Heston model, path-dependent options, Monte Carlointegration. ∗ The numerical experience reported in this paper has been obtained using the computing grid of Enea(Roma, Italy). The support and sponsorship of Enea is gratefully acknowledged. 1  1 Introduction Let  R  and  R + be the set of real numbers and of the positive real numbers respectively, andlet  t  be a real variable that denotes time.Given  T >  0, let us consider the price of a stock as a function of time described as astochastic process  S  ( t ), 0  < t < T  . We assume that  S  ( t ) satisfies the stochastic differentialequation: dS  ( t ) =  µ d S  ( t ) dt  +  σ ( t ) S  ( t ) dW  (1) ( t ) ,  0  < t < T,  (1)with the initial condition: S  (0) =  S  0 ,  (2)where  S  0  ∈  R + ,  µ d  ∈  R  is an assigned parameter,  W  (1) t  is a standard Wiener process, dW  (1) ( t ) is its stochastic differential, and  σ ( t ) is a time-dependent volatility parameter, thatwe assume to be a stochastic process. In fact we suppose that the variance  V   ( t ) =  σ 2 ( t ),0  < t < T  , satisfies the stochastic differential equation: dV   ( t ) = − γ  ( V   ( t ) − θ ) dt  +  κ   V   ( t ) dW  (2) ( t ) ,  0  < t < T,  (3)with the initial condition: V   (0) =  v 0 ,  (4)where  v 0 ,  θ ,  γ  , and  κ  are positive constants,  W  (2) t  is a standard Wiener process and  dW  (2) t is its stochastic differential. Note that the initial condition (2) should be understood as S  (0) =  S  0  w.p.1 (with probability one) and that a similar statement holds for the initialcondition (4).Let  ρ  denote the correlation coefficient between  dW  (1) t  and  dW  (2) t  , we assume  ρ  to beconstant. To avoid unnecessary technicalities we assume  2 γθκ 2  >  1. This condition ensuresthat if   v 0  ∈ R + then  V   ( t )  >  0 w.p.1 for  t >  0 ([37]).In the financial literature, equation (3) is commonly known as the Cox-Ingersoll-Ross(CIR) square-root model, since it has been used by Cox, Ingersoll and Ross to model interestrates ([13]).The equations (1)-(4) represent a model of stock-price dynamics where the volatility isassumed to be a stochastic process, that is, they are a stochastic volatility model. Thismodel is known as Heston model ([33]).In the study of finance stochastic volatility models have been introduced in order togeneralize the log-normal dynamics of the underlying asset prices, that is usually employedwhen the volatility is assumed to be constant. Indeed, models with constant volatility, suchas, for instance, the Black and Scholes model ([6]), may fail to cope with the market reality,where often the asset price behaviour is affected by phenomena such as the “smile” effect2  ([48]), or by the presence of skewness and of kurtosis effects in the price probability densityfunction. A number of stochastic volatility models can be found in the literature, amongthem, the models of Ball and Roma ([5]), Heston ([33]), Hull and White ([35]), Melinoand Turnbull ([40]), Schobel and Zhu ([44]), Scott ([45]), Stein and Stein ([46]), Wiggins([47]), where the volatility is determined as the solution of a stochastic differential equation,and those of Fouque and Han ([27], [28]), Fouque, Papanicolaou, Sircar and Solna ([30]),where the volatility is modeled according to a multi-factor stochastic process, that is severalstochastic processes are used to model the volatility at different time scales. These laststochastic processes are defined as solutions of stochastic differential equations. Finallywe mention that Duan ([19]) and Heston and Nandi ([34]) developed stochastic volatilitymodels based on the discrete GARCH process ([32]).Among the stochastic volatility models present in the literature, the Heston model isparticularly interesting since it can give a satisfactory description of the price dynamics of the underlying asset ([8], [34], [42]), and admits a closed form representation that containsan one-dimensional integral of the associated joint transition probability density functionof the price of the underlying and of the variance. In particular, using the technique of thecharacteristic functions, Heston ([33]) was the first who derived a formula containing anone-dimensional integral for pricing European options with stochastic volatility, and used itto price bond and currency options. For the Heston model Dr˘ a gulesku and Yakovenko ([18])found closed-form expressions for the joint transition probability density function of pricesand variance and for the marginal probability distribution of the variance, which require theevaluation of a double integral and of a single integral, respectively. These formulae are basedon the use of Fourier and Laplace transforms, and are used to estimate the parameters of the Heston model by comparison of the theoretical transition probability density functionof prices with observed market price distributions ([18], [42]). Generalizing the formulaegiven in [33], Lipton has found a closed-form expression for the joint transition probabilitydensity function of prices and variance that contains an one-dimensional oscillatory integral([39], pag. 602-608). Lipton has used this formula to price forward starting (cliquet) options.Moreover, in the Heston model, Broadie and Kaya ([10]), using the Laplace transform, founda closed formula for the marginal (cumulate) probability distribution of the asset price, andused it in conjunction with Monte Carlo integration to price European and forward startingoptions. In addition, Broadie and Kaya ([11]) have exploited the closed formula of themarginal probability distribution of the asset price found to evaluate the Greeks of somevanilla and path-dependent options in the Heston model, using the so-called pathwise andlikelihood ratio methods (see [9] for a detailed description of these techniques).In this paper we show that the formula of the joint probability density function of pricesand variance for the Heston model derived in [39] can be used to develop a numerical methodto price efficiently discretely monitored exotic path-dependent options of European type.Exotic path-dependent options are options whose payoff at exercise depends non-triviallyon the path history of the underlying asset price. Depending on wether the underlying assetprice is continuously or discretely monitored, these options are said continuous or discrete,respectively. Moreover, a path-dependent option can be European or American, dependingon wether it can be exercised only at or also before maturity time, respectively. Path-dependent options include barrier options, that are active or expired depending on the fact3  that the price of the underlying asset reaches or does not reach one or more assigned prices(i.e. barriers) ([15]), lookback options, whose payoff at exercise depends on the extremalvalues taken by the underlying asset price over its path ([15]), and Asian options, whosepayoff at exercise depends on the (arithmetic or geometric) time average of the underlyingasset prices taken over a finite number of assigned time values or over an assigned timeinterval ([15]). Quite seldom path-dependent options admit explicit solutions. This happens,for instance, for some barrier options (see, e.g., [12] and references therein), for continuouslookbacks ([16], [31]), and for geometric Asian options ([17], [38]), in presence of constantvolatility and in the European case.In most cases when exotic options are considered, both for constant or stochastic volatil-ity models, the solution of the pricing problem is not available in closed-form, and numericalapproximations must be used. The most common techniques to price path-dependent deriva-tives are:1) the partial differential equation (PDE) approach, in this case a PDE with appropriateinitial and/or boundary conditions must be solved in order to price the option. The optionvalue is considered as a function of time, of the underlying price, when a stochastic volatil-ity model is used, of the volatility or of the variance, and, for certain kind of derivatives,of an additional independent variable that is introduced in order to take in account thepath-dependent payoff. This additional independent variable is introduced, for instance,for lookbacks and for the Asian options ([26], [36], [48]). The PDE approach allows tohandle both European and American style options ([25], [26]), but can be computationallyexpensive, especially in the presence of stochastic volatility, and/or when the additionalindependent variable is introduced, in fact in this case the PDE to be solved has three orfour independent variables;2) the lattice binomial (or multinomial) approach, which consists in simulating all the possi-ble (discrete-time) dynamics of asset prices and, eventually, of the stochastic volatility, overa lattice of nodes and then calculating the option value as the average of all the realizedpayoffs. This technique can be applied both to European and American style options, buthas a dramatic computational cost when a large number of lattice nodes or when a largenumber of possible realizations of the payoff must be considered, this last case is, for in-stance, the case of Asian options ([243) the Monte Carlo approach, where the option price is obtained as a statistical estimateof the interest rate discounted expected payoff. The Monte Carlo approach cannot han-dle easily American options, but has the advantage of being rather simple to implementand of being very well suited for parallel computing, and, in general, it allows to achieve asatisfactory level of accuracy with a moderate computational effort ([7]).In the literature there are several papers where the problem of pricing exotic path-dependent options in the presence of stochastic volatility is considered. Among them, thepapers of Fouque and Han ([27], [28]), where multi-factor stochastic volatility models areconsidered. In particular, in ([28]) the authors find an approximate solution for Europeanstyle continuous geometric Asian options, and use it as a control variate for the MonteCarlo simulation of continuous arithmetic Asian options. Moreover, in ([27]), Fouque andHan determine an asymptotic expansion formula for the price of European style continu-ous arithmetic Asian options. We also mention that in ([29]) Fouque and Han derive an4  asymptotic formula to price European style continuous arithmetic Asian options using asingle-scale mean-reverting stochastic volatility model. Furthermore, in the framework of Heston stochastic volatility model, Lipton ([39]) determines power series expressions for theprice of barrier options, and Clarke and Parrott ([14]), using the PDE approach, evaluatethe price of American continuous arithmetic Asian options using a finite difference scheme.In this paper we consider the problem of pricing European style discrete path-dependentoptions in the Heston stochastic volatility model using the Monte Carlo approach. Moreprecisely, let us consider a positive integer  f   ≥ 2, and  f   +1 time values: 0 =  t 0  < t 1  < ... <t f   =  T  . We are interested in valuing financial derivatives with maturity at time  t  =  T   onan asset described by the Heston model (1)-(4). It is known that the option price evaluatedat time  t  =  t 0  = 0 can be calculated using the risk-neutral formula: U  ( S  0 ,v 0 ,T  ) =  e − rT     + ∞ 0  dS  1   + ∞ 0  dS  2  ...   + ∞ 0  dS  f    + ∞ 0  dv 1   + ∞ 0  dv 2  ...   + ∞ 0  dv f  Payoff  ( S  0 ,S  1 ,...,S  f  ,v 0 ,v 1 ,...,v f  ) ·   p ( S  0 ,S  1 ,...,S  f  ,v 0 ,v 1 ,...,v f  ) ,  (5)where  r  is the risk-free interest rate,  Payoff  ( S  0 ,S  1 ,...,S  f  ,v 0 ,v 1 ,...,v f  ) is the payoff func-tion, that is the option value at time  T  ,    p ( S  0 ,S  1 ,...,S  f  ,v 0 ,v 1 ,...,v f  ) is the followingprobability density function associated to the process (1)-(4):   p ( S  0 ,S  1 ,...,S  f  ,v 0 ,v 1 ,...,v f  ) dS  1 dS  2  ...dS  f  dv 1 dv 2  ...dv f   = Prob { S  i  < S  ( t i )  < S  i  +  dS  i ,v i  < V   ( t i )  < v i  +  dv i , i  = 1 , 2 ,...,f  | S  ( t 0 ) =  S  0 ,V   ( t 0 ) =  v 0 } .  (6)Note that we wrote  Payoff  ( S  0 ,S  1 ,...,S  f  ,v 0 ,v 1 ,...,v f  ) even if, due to the fact that thevariance cannot be observed directly on the financial market, the payoff functions usuallyemployed in the financial industry depend only on the prices  S  0 ,S  1 ,...,S  f   of the underlyingat the time values  t 0 ,t 1 ,...,t f   respectively.Using the results of Lipton ([39]), the transition probability density function (6) canbe written as a product of   f   one-dimensional oscillatory integrals. We have approximatedthese oscillatory integrals using an adequate quadrature method, and we have exploited theresulting approximation of the probability density    p ( S  0 ,S  1 ,...,S  f  ,v 0 ,v 1 ,...,v f  ) to calculatethe option value (5) using the Monte Carlo integration method.It is well known that, in order to compute an accurate estimate of the option value, thatis in order to compute the integral that appears in (5), the Monte Carlo integration methodwith a variance reduction technique must be employed. In fact, this integral is usuallya high-dimensional integral and the quality of the result obtained with the Monte Carlosimulation increases very slowly with the size of the sample considered in the Monte Carlosimulation if the result is calculated as the average of a random variable, function of   S  ( t i )and  V   ( t i ),  i  = 1 , 2 ,...,f  , having a large variance. This usually occurs when dealing with alarge number of evaluation dates, that is when  f >  4 or 5. In order to solve this problem,we have used the so-called importance sampling technique ([43]), according to which therandom variables  S  ( t i ) and  V   ( t i ),  i  = 1 , 2 ,...,f  , are sampled from probability distributionsthat resembles their true ones, and that are easy to handle. These distributions can be5
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