A numerical method to price exotic pathdependentoptions 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. +390712207251, FAX N. +390712207150, Email: ballestra@posta.econ.unian.it
Graziella Pacelli
Dipartimento di Scienze Sociali “D. Serrani”,Universit`a Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy,Ph. N. +390712207050, FAX N. +390712207150, Email: g.pacelli@univpm.it
Francesco Zirilli
Dipartimento di Matematica “G. Castelnuovo”,Universit`a di Roma “La Sapienza”, Piazzale Aldo Moro 2, 00185 Roma, Italy,Ph. N. +390649913282, FAX N. +390644701007, Email: f.zirilli@caspur.it
Abstract
We consider the problem of pricing European exotic pathdependent 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 numerical approximation of this formula and we use the obtained approximated transitionprobability density function to price discrete pathdependent options as discounted expectations. 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, pathdependent 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 timedependent 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 CoxIngersollRoss(CIR) squareroot model, since it has been used by Cox, Ingersoll and Ross to model interestrates ([13]).The equations (1)(4) represent a model of stockprice 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 lognormal 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 multifactor 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 onedimensional 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 anonedimensional 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 closedform 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 closedform expression for the joint transition probabilitydensity function of prices and variance that contains an onedimensional oscillatory integral([39], pag. 602608). 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 pathdependent options in the Heston model, using the socalled 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 pathdependent options of European type.Exotic pathdependent options are options whose payoﬀ at exercise depends nontriviallyon 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 pathdependent option can be European or American, dependingon wether it can be exercised only at or also before maturity time, respectively. Pathdependent 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 pathdependent 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 volatility models, the solution of the pricing problem is not available in closedform, and numericalapproximations must be used. The most common techniques to price pathdependent derivatives 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 volatility 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 thepathdependent 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 possible (discretetime) 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 instance, 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 handle 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 pathdependent options in the presence of stochastic volatility is considered. Among them, thepapers of Fouque and Han ([27], [28]), where multifactor 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 continuous 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 asinglescale meanreverting 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 pathdependentoptions 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 riskneutral 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 riskfree interest rate,
Payoff
(
S
0
,S
1
,...,S
f
,v
0
,v
1
,...,v
f
) is the payoﬀ function, 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
onedimensional 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 highdimensional 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 socalled 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