American option pricing under two stochastic volatility processes

Applied Mathematics and Computation 224 (2013) 283–310
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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
American option pricing under two stochastic volatility
processes
Carl Chiarella a, Jonathan Ziveyi b,⇑
a
b
Finance Discipline Group, University of Technology Sydney, P.O. Box 123, Broadway, NSW 2007, Australia
Risk and Actuarial Studies, Australian School of Business, The University of New South Wales, Sydney, NSW 2052, Australia
a r t i c l e
i n f o
Keywords:
American options
Fourier transform
Laplace transform
Method of characteristics
a b s t r a c t
In this paper we consider the pricing of an American call option whose underlying asset
dynamics evolve under the inﬂuence of two independent stochastic volatility processes
as proposed in Christoffersen, Heston and Jacobs (2009) [13]. We consider the associated
partial differential equation (PDE) for the option price and its solution. An integral expression for the general solution of the PDE is presented by using Duhamel’s principle and this
is expressed in terms of the joint transition density function for the driving stochastic processes. For the particular form of the underlying dynamics we are able to solve the Kolmogorov PDE for the joint transition density function by ﬁrst transforming it to a
corresponding system of characteristic PDEs using a combination of Fourier and Laplace
transforms. The characteristic PDE system is solved by using the method of characteristics.
With the full price representation in place, numerical results are presented by ﬁrst approximating the early exercise surface with a bivariate log linear function. We perform numerical comparisons with results generated by the method of lines algorithm and note that our
approach provides quite good accuracy.
Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction
It has long been recognised that the original option pricing framework of Black and Scholes [5] suffers from the deﬁciency
that the Normal distribution is inadequate to pick up the skewness and kurtosis observed in real ﬁnancial data. There is a
mountain of empirical evidence from Mandelbrot [29] to Platen and Rendek [31] to this effect. Equally important has been
the impact on European option pricing when the underlying asset is driven by stochastic volatility processes. In this regard,
we refer the reader to Scott [32], Wiggins [36], Hull and White [19], Stein and Stein [34], and Heston [17], who all consider
European option pricing where the underlying asset dynamics evolve under the inﬂuence of various types of single factor
mean reverting stochastic volatility processes.
The standard option pricing framework of Black and Scholes [5] has been premised on a number of restrictive assumptions, one of which is constant volatility of asset returns. The constant volatility assumption is based on the early perception
that asset returns are characterized by the Normal distribution.
A variety of papers have also proved that multiple factor stochastic volatility models perform quite well relative to single
factor models. One such paper is the most recent work by Christoffersen et al. [13] who show empirically with the aid of
principal component analysis that a two factor model offers ﬂexibility in controlling the level and slope of the volatility smirk
⇑ Corresponding author.
E-mail addresses: [email protected] (C. Chiarella), [email protected] (J. Ziveyi).
0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.amc.2013.08.047
284
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
(a situation where in- and out-of-the-money options have higher implied volatilities as compared to Black–Scholes prices). It
has also been shown empirically by da Fonseca et al. [15] that multiple factor stochastic volatility models offer more consistent option prices as compared to single factor models. As demonstrated in da Fonseca et al. [15] and Christoffersen
et al. [13] the main advantages of a multiple volatility system is that they calibrate short-term and long-term volatility levels
better than a single process.
Whilst most of the initial work has focused on European style options, not much has been done on pricing American options under stochastic volatility. Touzi (1999) considers the pricing of American put options written on an underlying asset
whose dynamics evolve under the inﬂuence of stochastic volatility. Touzi describes the dependence of the early exercise
boundary of the American put option on the volatility parameter and proves that such a boundary is a decreasing function
of volatility implying that for a ﬁxed underlying asset price, as the volatility increases, the early exercise boundary decreases.
Tzavalis and Wang [35] derive the integral representation of an American call option price when the volatility process
evolves according to the square-root process proposed by Heston [17]. They derive the integral expressions, again using optimal stopping theory along the lines of Karatzas [25]. By appealing to the empirical ﬁndings by Broadie et al. [6] who suggest
that when variance evolves stochastically the early exercise boundary is a log-linear function of both time and instantaneous
variance, a Taylor series expansion is applied to the resulting early exercise surface around the long-run variance. The unknown functions resulting from the Taylor series expansion are then approximated by ﬁtting Chebyshev polynomials. Ikonen
and Toivanen [20] formulate and solve the linear complementarity problem of the American call option under stochastic volatility using componentwise splitting methods. The resulting subproblems from componentwise splitting are solved by
using standard partial differential equation methods.
Chiarella and Ziogas [10] derive the integral representation of the American call option pricing partial differential equation (PDE) when the dynamics of the underlying asset is inﬂuenced by a single stochastic volatility process by ﬁrst applying a
Fourier transform to the underlying asset domain and then guessing the general solution of the PDE along similar lines as
initially presented in [17] when pricing European call options under stochastic volatility. Adolfsson et al. [2] also derive
the integral representation of the American call1 option under stochastic volatility by formulating the pricing PDE as an inhomogeneous problem and then using Duhamel’s principle to represent the corresponding solution in terms of the joint transition
density function. The joint density function solves the associated backward Kolmogorov PDE and a systematic approach for solving such a PDE is developed. A combination of Fourier and Laplace transforms is used to transform the homogeneous PDE for the
density function to a characteristic PDE. The resulting system is then solved using ideas ﬁrst presented by Feller [16]. The early
exercise boundary is approximated by a log-linear function as proposed in [35]. Instead of using approximating polynomials as
in [35,2], Adolfsson et al. [2] derive an explicit characteristic function for the early exercise premium component and then use
numerical root ﬁnding techniques to ﬁnd the unknown functions from the log-linear approximation.
Motivated by the superior features of multifactor volatility processes as empirically proved in [15,13], we seek to extend the
American option pricing model of Adolfsson et al. [2] to the multifactor stochastic volatility case. We will assume that the
underlying asset is driven by two stochastic variance processes of Christoffersen et al. [13] type. Whilst da Fonseca [14,15] treat
the two stochastic variance processes to be effective during different periods of the maturity domain, in this work we model the
variance processes as independent risk factors inﬂuencing the dynamics of the underlying asset. As an example, we aim to devise semi-analytical solutions for American call options under this framework that are computationally tractable.
The PDE for the option price is solved subject to initial and boundary conditions that specify the type of option under
consideration. As is well known, the underlying asset of the American call option is bounded above by the early exercise
boundary and below by zero. We convert the upper bound of the underlying asset to an unbounded domain by using the
approach of Jamshidian [21]. The three stochastic processes; one for the underlying asset and the two variance processes
can also be used to derive the corresponding PDE for their joint transition probability density function which satisﬁes a backward Kolmogorov PDE. Coupled with this and the unbounded PDE for the option price, we derive the general solution for the
American option price by using Duhamel’s principle. The only unknown terms in the general solution are the transition density function which is the solution of the backward Kolmogorov PDE for the three driving processes and the free boundary,
for which an integral equation will be found from the initial and boundary conditions.
In solving the Kolmogorov PDE, we ﬁrst reduce it to a characteristic PDE by using a combination of Fourier and Laplace
transforms. The resulting equation is then solved by the method of characteristics. Once the solution is found, we revert back
to the original variables by applying the Fourier and Laplace inversion theorems. With the transition density in place, we can
readily obtain the full integral representation of the American option price. As implied by Duhamel’s principle, the American
option price is the sum of two components namely the European and the early exercise premium components. The European
option component can be readily reduced to the [17] form by using similar techniques to those in [2]. In dealing with the
early exercise premium component, we extend the idea of Tzavalis and Wang [35] and approximate the early exercise
boundary as a bivariate log-linear function. This approximation allows us to reduce the integral dimensions of the early exercise premium by simplifying the integrals with respect to the two variance processes. The reduction of the dimensionality
has the net effect of enhancing computational efﬁciency by reducing the computational time of the early exercise premium
component.
1
Their approach can be generalised to any option payoff function under the same framework.
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
285
The main aim of this paper is to obtain an analytical expression for the price of American call options when the underlying
asset is driven by more than one stochastic volatility process presented in [13]. The main use of our results is as a benchmark
for comparison with more general models for which only numerical solutions can be employed.
This paper is organized as follows, we present the problem statement in Section 2 followed by the corresponding general
solution for the American call option price in Section 3. We introduce key deﬁnitions of Fourier and Laplace transforms in
Section 4. A Fourier transform is applied to the underlying asset variable in the PDE for the density function in Section 5 followed by application of a bivariate Laplace transform to the variance variables in Section 6. Application of the Laplace transform yields the PDE which we solve by the method of characteristics, details of which are given in Section 7. Once this PDE is
solved the next step involves reverting back to the original underlying asset and variance variables. This is accomplished by
applying Laplace and Fourier inversion theorems as detailed in Sections 8 and 9 respectively. The resulting function is the
explicit representation of the transition density function. Section 10 nicely represents the integral form of the American call
option price. An approximation of the early exercise boundary is presented in Section 11. Having found a representation of
the American option price together with the early exercise boundary approximation, we then present details of how to
implement the pricing relationship in Section 12. Numerical results are then presented in Section 13 followed by concluding
remarks in Section 14. Lengthy derivations have been relegated to appendices.
2. Problem statement
We consider the evaluation of the American call option written on an underlying asset whose dynamics evolve under the
inﬂuence of two stochastic variance processes of the Heston [17] type. We represent the value of this option at the current
time, t, as Vðt; S; v 1 ; v 2 Þ where S is the price of the underlying asset paying a continuously compounded dividend yield at a
rate q in a market offering a risk-free rate of interest denoted by r, and v 1 and v 2 are the two variance processes driving S.
Under the real world probability measure, P, the underlying asset dynamics are governed by the stochastic differential equation (SDE) system
dS ¼ lSdt þ
dv 1
dv 2
pﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃ
v 1 SdW 1 þ v 2 SdW 2 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ
¼ j1 ðh1 v 1 Þdt þ q13 r1 v 1 dW 1 þ 1 q213 r1 v 1 dW 3 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃ
¼ j2 ðh2 v 2 Þdt þ q24 r2 v 2 dW 2 þ 1 q224 r2 v 2 dW 4 :
pﬃﬃﬃﬃﬃﬃ
ð2:1Þ
ð2:2Þ
ð2:3Þ
where l is the instantaneous return per unit time of the underlying asset, h1 and h2 are the long-run means of v 1 and v 2
respectively, j1 and j2 are the speeds of mean-reversion, while r1 and r2 are the instantaneous volatilities of v 1 and v 2
per unit time respectively. The processes, W j for j ¼ 1; . . . ; 4 are independent Wiener processes. We have chosen a special
correlation structure2 for the Eqs. (2.1)–(2.3) as this allows us to apply transform methods because we avoid the product term
pﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃ
v 1 v 2 which would make it impossible to apply transform based methods that we propose. Using the approach of Feller [16]
and Cheang et al. [7] show that for the Eqs. (2.2) and (2.3) to be positive processes, the conditions 2j1 h1 P r21 and 2j2 h2 P r22
need to be satisﬁed. The latter authors also show that in addition to these two inequalities the conditions
1 < q13 < min ðj1 =r1 ; 1Þ and 1 < q24 < min ðj2 =r2 ; 1Þ, also need to be satisﬁed for the two variances to be ﬁnite.
Cheang et al. [7] prove that the conditions on q13 and q14 ensure that the stochastic differential equation for the underlying asset presented in Eq. (2.1) can have an explicit solution under the real world probability measure. The conditions also
ensure that an equivalent solution to (2.1) under a suitable risk-neutral measure also exists and that the discounted stock
yield process under this measure is a martingale.
The system (2.1)–(2.3) contains four Wiener processes but only one traded asset S as of the two variance processes are
non-tradeable. This single asset is insufﬁcient to hedge away these four risk factors when combined in a portfolio with an
option dependent on the underlying asset, S. This situation leads to market incompleteness. In order to hedge away these
risk sources, the market needs to be completed in some way. The process of completing the market is usually done by placing
a sufﬁcient number of options of different maturities in the hedging portfolio,3 and then applying a change of measure4 to
preserve the original underlying asset dynamics. Under the risk neutral measure the dynamics become
dS ¼ ðr qÞSdt þ
dv 1
dv 2
pﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃ
f 1 þ v 2 Sd W
f2;
v 1 Sd W
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃ f
f3;
1 q213 r1 v 1 d W
¼ j1 ðh1 v 1 Þdt k3 ðtÞ 1 q213 r1 v 1 dt þ q13 r1 v 1 d W
1 þ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ
q
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃ f
pﬃﬃﬃﬃﬃﬃ f
1 q224 r2 v 2 d W
¼ j2 ðh2 v 2 Þdt k4 ðtÞ 1 q224 r2 v 2 dt þ q24 r2 v 2 d W
2 þ
4;
ð2:4Þ
pﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃ
In particular we note that the correlation between dS and dv 1 is q13 r1 v 1 whilst the correlation between dS and dv 2 is q24 r2 v 2 and the correlation
between dv 1 and dv 2 is zero.
3
After applying these hedging arguments, it turns out that the resulting option pricing PDE contains the two market prices of risk corresponding to the
number of non-traded factors under consideration.
4
f j ¼ kj ðtÞdt þ dW j , where
The change of measure is accomplished by applying the Girsanov’s Theorem for Wiener processes which states that there exist d W
f j are Wiener processes under the risk neutral measure Q and kj ðtÞ are the market prices of risk associated with the Wiener instantaneous shocks, dW j for
W
j ¼ 1; . . . ; 4.
2
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C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
where r is the risk-free interest rate and q is the continuously compounded dividend yield on the underlying asset, S. Note
that the processes for v 1 and v 2 contains two independent Wiener processes and cannot be combined into a single stochastic
pﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃ
volatility process because of the v 1 dt and v 2 dt terms that appear in the SDE for v 1 and v 2 respectively. Also the two indef 1 and d W
f 2 , are not additive. The key assumption we make on the market prices of volatility risk, k3 ðtÞ
pendent processes, d W
and k4 ðtÞ, is that both quantities are strictly positive to guarantee an investor a positive risk premium for holding the nontraded variance factors. In determining the market prices of the two variance risks, we use the same reasoning as in [17] with
a slight modiﬁcation such that
k3 ðtÞ ¼
pﬃﬃﬃﬃﬃﬃ
k1 v 1
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
r1 1 q213
and k4 ðtÞ ¼
pﬃﬃﬃﬃﬃﬃ
k2 v 2
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ;
r2 1 q224
ð2:5Þ
where k1 and k2 are constants. By substituting the two equations in (2.5) into the system (2.4) we obtain
dS ¼ ðr qÞSdt þ
dv 1
dv 2
pﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃ
f 1 þ v 2 Sd W
f 2;
v 1 Sd W
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃ f
f3;
¼ ½j1 h1 ðj1 þ k1 Þv 1 dt þ q13 r1 v 1 d W
1 q213 r1 v 1 d W
1 þ
q
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃ f
f4:
¼ ½j2 h2 ðj2 þ k2 Þv 2 dt þ q24 r2 v 2 d W 2 þ 1 q224 r2 v 2 d W
ð2:6Þ
ð2:7Þ
ð2:8Þ
Now with the system of Eqs. (2.6)–(2.8), the next step involves the derivation of the corresponding American option pricing
PDE for the option written on the underlying asset, S. The pricing PDE can be shown to be5
@V
ðs; S; v 1 ; v 2 Þ ¼ LVðs; S; v 1 ; v 2 Þ rV;
@s
ð2:9Þ
where L is the Dynkin operator associated with the SDE system (2.6)–(2.8) deﬁned as
L ¼ ðr qÞS
þ
@
@
@
1
@2
1
@2
1
@2
þ ½j1 ðh1 v 1 Þ k1 v 1 þ ½j2 ðh2 v 2 Þ k2 v 2 þ v 1 S2 2 þ r21 v 1 2 þ v 2 S2 2
@S
@v 1
@v 2 2
2
@v 1 2
@S
@S
1 2
@2
@2
@2
r2 v 2 2 þ q13 r1 v 1 S
þ q14 r2 v 2 S
:
2
@S@ v 1
@S@ v 2
@v 2
ð2:10Þ
The state variables are deﬁned in the domains 0 < v 1 ; v 2 < 1 and 0 6 S < bðs; v 1 ; v 2 Þ where S ¼ bðs; v 1 ; v 2 Þ, is the early exercise boundary of the American call option at time-to-maturity, s when the instantaneous variances are v 1 and v 2 respectively. For an American call option, the PDE (2.9) is to be solved subject to the initial and boundary conditions
Vð0; S; v 1 ; v 2 Þ ¼ ðS KÞþ ;
Vðs; 0; v 1 ; v 2 Þ ¼ 0;
0 < S < 1;
ð2:11Þ
s P 0;
Vðs; bðs; v 1 ; v 2 Þ; v 1 ; v 2 Þ ¼ bðs; v 1 ; v 2 Þ K;
@V
lim
ðs; S; v 1 ; v 2 Þ ¼ 1; s P 0:
S!bðs;v 1 ;v 2 Þ @S
ð2:12Þ
s P 0;
ð2:13Þ
ð2:14Þ
Condition (2.11) is the payoff of the option contract if it is held to maturity, while Eq. (2.12) is the absorbing state condition
which ensures that the option ceases to exist once the underlying asset price hits zero. Eq. (2.13) is the value matching condition that guarantees continuity of the option value function at the early exercise boundary, bðs; v 1 ; v 2 Þ. Eq. (2.14) is the
smooth pasting condition which together with the value matching condition are imposed to eliminate arbitrage opportunities. Boundary conditions at v 1 ¼ 0 and v 2 ¼ 0 are found by extrapolation techniques when numerically implementing the
resulting American call option pricing equation.
Also associated with the system of stochastic differential equations in (2.6)–(2.8) is the transition density function which
we denote here as Gðs; S; v 1 ; v 2 ; S0 ; v 1;0 ; v 2;0 Þ. This function represents the transition probability of passage from S; v 1 ; v 2 at
time-to-maturity s to S0 ; v 1;0 ; v 2;0 at maturity. It is well known that the transition density function satisﬁes the backward
Kolmogorov PDE associated with the stochastic differential equations in the system (2.6)–(2.8) (see for example [30]).
The Kolmogorov equation in the current situation can be shown to be of the form6
@G
¼ LG;
@s
ð2:15Þ
where 0 6 S < 1 and 0 6 v 1 ; v 2 < 1. Eq. (2.15) is solved subject to the initial condition
5
Here s ¼ T t is the time to maturity. Strictly speaking we should use different symbols to denote VðT s; S; v 1 ; v 2 Þ and Vðs; S; v 1 ; v 2 Þ, but for convenience
we use the same symbol. For completeness, we also assume that pricing function, Vðs; S; v 1 ; v 2 Þ is at least twice differentiable with respect to its underlying
state variables.
6
The function Gðs; S; v 1 ; v 2 Þ is also assumed to be at least twice differentiable with respect to its state variables.
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
Gð0; S; v 1 ; v 2 ; S0 ; v 1;0 ; v 2;0 Þ ¼ dðS S0 Þdðv 1 v 1;0 Þdðv 2 v 2;0 Þ;
287
ð2:16Þ
where dðÞ is the Dirac delta function.
3. Deriving the general solution of the pricing PDE
As noted in the PDE (2.9), the underlying asset domain is bounded above by the early exercise boundary, bðs; v 1 ; v 2 Þ. Jamshidian [21] shows that one can consider an unbounded domain for the underlying asset by introducing an indicator function
and transforming the PDE to
@V
ðs; S; v 1 ; v 2 Þ ¼ LVðs; S; v 1 ; v 2 Þ rV þ 1SPbðs;v 1 ;v 2 Þ ðqS rKÞ:
@s
ð3:1Þ
Here, 0 6 S < 1, 0 < v 1 ; v 2 < 1 and 1SPbðs;v 1 ;v 2 Þ is an indicator function which is equal to one if S P bðs; v 1 ; v 2 Þ and zero
otherwise. Now the PDE (3.1) is deﬁned on an unbounded domain for the underlying asset.
As an initial step to solving the PDE (3.1), we switch to log asset variables by letting S ¼ ex and setting
Cðs; x; v 1 ; v 2 Þ Vðs; ex ; v 1 ; v 2 Þ;
Uðs; x; v 1 ; v 2 ; x0 ; v 1;0 ; v 2;0 Þ Gðs; ex ; v 1 ; v 2 ; ex0 ; v 1;0 ; v 2;0 Þ;
ð3:2Þ
to obtain
@C
¼ MC rC þ 1xPln bðs;v 1 ;v 2 Þ ðqex rKÞ;
@s
ð3:3Þ
where the operator M is deﬁned by
1
1
@
@
@
1
@2
1
@2
@2
M ¼ r q v1 v2
þ ðU1 b1 v 1 Þ
þ ðU2 b2 v 2 Þ
þ v 1 2 þ v 2 2 þ q13 r1 v 1
2
2
@x
@v 1
@ v 2 2 @x
2 @x
@x@ v 1
þ q14 r2 v 2
U1 ¼ j1 h1 ;
@2
1
@2
1
@2
þ r21 v 1 2 þ r22 v 2 2 ;
@x@ v 2 2
@v 1 2
@v 2
U2 ¼ j2 h2 ;
b1 ¼ j1 þ k1 and b2 ¼ j2 þ k2 :
ð3:4Þ
ð3:5Þ
Eq. (3.3) is solved subject to the initial condition
þ
Cð0; x; v 1 ; v 2 Þ ¼ ðex KÞ ;
1 < x < 1:
ð3:6Þ
Likewise, the transition density PDE (2.15) is transformed to
@U
¼ MU:
@s
ð3:7Þ
Eq. (3.7) is to be solved subject to the initial condition
Uð0; x; v 1 ; v 2 ; x0 ; v 1;0 ; v 2;0 Þ ¼ dðx x0 Þdðv 1 v 1;0 Þdðv 2 v 2;0 Þ:
ð3:8Þ
The inhomogeneous PDE (3.3) is in a form whose general solution can be represented by use of Duhamel’s principle.7We
present the general solution of (3.3) in the proposition below.
7
Consider the one dimensional inhomogeneous parabolic PDE of the form
@C
¼ LC þ f ðs; xÞ;
@s
ð3:9Þ
where L is a parabolic partial differential operator and solved subject to the initial condition
Cð0; xÞ ¼ /ðxÞ:
ð3:10Þ
Let Uðs; xÞ be the transition probability density function which is the solution to
@U
¼ LU;
@s
ð3:11Þ
subject to the initial condition
Uð0; xÞ ¼ dðx x0 Þ;
ð3:12Þ
then the solution of Eq. (3.9) can be represented as
Cðs; xÞ ¼
Z
1
/ðyÞUðs; x yÞdy þ
1
Z sZ
0
1
f ðn; yÞUðs n; x yÞdydn:
1
More details about Duhamel’s principle can be found in [27]. This principle can easily be generalised to the multi-dimensional case which we present in this paper.
288
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
Proposition 3.1. The solution of the American call option pricing PDE (3.3) can be represented as8
Proof. Refer to Appendix A. h
The ﬁrst component of Eq. (3.16) is the European option component whilst the second component is the early exercise
premium. For us to operationalise the representation of Eq. (3.16), we need an explicit form of the density function,
Uðs; x; v 1 ; v 2 Þ which is the solution of the PDE in Eq. (3.7). In order to solve Eq. (3.7), we ﬁrst reduce it to a corresponding
system of characteristic PDEs which can then be readily solved using a vast array of methods for tackling such problems.
In this paper, we will apply a combination of Fourier and Laplace transforms to this PDE resulting in a system of characteristic PDEs. We will apply Fourier transforms to the underlying log asset variable as its domain matches that of the transform.
A bivariate Laplace transform will then be applied to the stochastic variance variables. We start by giving a brief review of
transform methods before applying them to the PDE (3.7).
4. Fourier and Laplace transforms
Fourier and Laplace transform techniques have been extensively applied in solving problems in the physical sciences. In
this section we provide key deﬁnitions of Fourier and Laplace transforms that we will utilise in transforming the PDE (3.7) to
a corresponding system of characteristic PDEs.
Deﬁnition 4.1. The Fourier transform of the function Uðs; x; v 1 ; v 2 Þ with respect to x is deﬁned as9
F fUðs; x; v 1 ; v 2 Þg ¼
Z
1
b s; g; v 1 ; v 2 Þ;
eigx Uðs; x; v 1 ; v 2 Þdx :¼ Uð
ð4:1Þ
1
where i ¼
pﬃﬃﬃﬃﬃﬃﬃ
1 is a complex number and 1 < g < 1.
b s; g; v 1 ; v 2 Þ is represented as,
Deﬁnition 4.2. The inverse of the Fourier transform of Uð
Z
1
2p
b s; g; v 1 ; v 2 Þg ¼
F 1 f Uð
1
b s; g; v 1 ; v 2 Þdg :¼ Uðs; x; v 1 ; v 2 Þ:
eigx Uð
b s; g; v 1 ; v 2 Þ with respect to
Deﬁnition 4.3. The bivariate Laplace transform of the function Uð
b s; g; v 1 ; v 2 Þg ¼
Lf Uð
Z
1
Z
0
ð4:2Þ
1
1
v1
and
v 2 is deﬁned as,
b s; g; v 1 ; v 2 Þdv 1 dv 2 :¼ Uð
e s; g; s1 ; s2 Þ;
es1 v 1 s2 v 2 Uð
ð4:3Þ
0
where s1 and s2 are complex variables whenever the improper integral exists.
8
Note that the American put option price can be represented as
Cðs; x; v 1 ; v 2 Þ ¼ C E ðs; x; v 1 ; v 2 Þ þ C P ðs; x; v 1 ; v 2 Þ;
ð3:13Þ
where
C E ðs; x; v 1 ; v 2 Þ ¼ ers
Z
1
Z
0
1
Z
0
1
þ
ðK eu Þ Uðs; x; v 1 ; v 2 ; u; w1 ; w2 Þdudw1 dw2 ;
ð3:14Þ
1
and
C P ðs; x; v 1 ; v 2 Þ ¼
Z s
erðsnÞ
Z
0
1
Z
0
1
Z
0
ln bðn;w1 ;w2 Þ
ðrK qeu Þ Uðs n; x; v 1 ; v 2 ; u; w1 ; w2 Þdudw1 dw2 dn:
ð3:15Þ
1
Cðs; x; v 1 ; v 2 Þ ¼ C E ðs; x; v 1 ; v 2 Þ þ C P ðs; x; v 1 ; v 2 Þ;
ð3:16Þ
where
C E ðs; x; v 1 ; v 2 Þ ¼ ers
Z
1
0
Z
1
Z
0
1
þ
ðeu KÞ Uðs; x; v 1 ; v 2 ; u; w1 ; w2 Þdudw1 dw2 ;
ð3:17Þ
1
and
C P ðs; x; v 1 ; v 2 Þ ¼
Z s
0
9
erðsnÞ
Z
1
0
Z
1
0
Z
1
ðqeu rKÞ Uðs n; x; v 1 ; v 2 ; u; w1 ; w2 Þdudw1 dw2 dn:
ln bðn;w1 ;w2 Þ
We assume that Uðs; x; v 1 ; v 2 Þ satisﬁes the Dirichlet’s conditions in the domain 1 < x < 1 and is absolutely integrable in this interval.
ð3:18Þ
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
289
e s; g; s1 ; s2 Þ with respect to s1 and s2 is deﬁned as,
Deﬁnition 4.4. The bivariate inverse Laplace transform10 of the function Uð
e s; g; s1 ; s2 Þg ¼
L1 f Uð
Z
1
ð2piÞ
2
c2 þi1
c2 i1
Z
c1 þi1
c1 i1
e s; g; s1 ; s2 Þds1 ds2 :¼ Uð
b s; g; v 1 ; v 2 Þ;
es1 v 1 þs2 v 2 Uð
ð4:4Þ
where the integration is done along the lines, Reðs1 Þ ¼ c1 and Reðs2 Þ ¼ c2 in the complex hyperplane such that c1 and c2 are
e s; g; s1 ; s2 Þ.
greater than the real part of all singularities of Uð
We will make the following assumptions about the density function:
lim Uðs; x; v 1 ; v 2 Þ ¼ lim
x!1
x!1
@U
ðs; x; v 1 ; v 2 Þ ¼ 0;
@x
@U
lim Uðs; x; v 1 ; v 2 Þ ¼ lim
v 1 !1 @ v 1
v 1 ;v 2 !1
ð4:5Þ
ðs; x; v 1 ; v 2 Þ ¼ lim
@U
v 2 !1 @ v 2
ðs; x; v 1 ; v 2 Þ ¼ 0;
lim Uðs; x; v 1 ; v 2 Þ ¼ 0:
ð4:6Þ
ð4:7Þ
v 1 ;v 2 !0
Such assumptions are made since by their nature, transition density functions and their derivatives converge to zero as their
underlying processes assume larger values.
5. Applying Fourier transforms
We ﬁrst apply the Fourier transform to the log asset price variable in the PDE (3.7).
b s; g; v 1 ; v 2 Þ of the function Uðs; x; v 1 ; v 2 Þ which is the solution of the homogeneous
Proposition 5.1. The Fourier transform, Uð
PDE (3.7) satisﬁes the PDE
2b
2b
b b
b
b
b
@U
1
1
b þ U1 @ U þ U2 @ U H1 v 1 @ U H2 v 2 @ U þ 1 r2 v 1 @ U þ 1 r2 v 2 @ U ;
¼ igðr qÞ þ KðgÞv 1 þ KðgÞv 2 U
1
2
@s
@v 1
@v 2
@v 1
@v 2 2
2
2
@ v 21 2
@ v 22
ð5:1Þ
where
H1 ¼ H1 ðgÞ b1 þ igq13 r1 ;
H2 ¼ H2 ðgÞ b2 þ igq24 r2 ;
and KðgÞ ¼ ig g2 :
ð5:2Þ
Eq. (5.1) is to be solved subject to the initial condition
b
Uð0;
g; v 1 ; v 2 Þ ¼ eigx0 dðv 1 v 1;0 Þdðv 2 v 2;0 Þ:
ð5:3Þ
Proof. By use of Eq. (4.1), the assumptions in (4.5) and using standard manipulations for Fourier transforms when applying
them to the PDE (3.7) we obtain the PDE in Eq. (5.1)
The Fourier transform of the initial condition in Eq. (3.8) is simpliﬁed as follows
F fUð0; x; v 1 ; v 2 Þg ¼
Z
1
eigx Uð0; x; v 1 ; v 2 Þdx ¼
1
Z
1
eigx dðx x0 Þdðv 1 v 1;0 Þdðv 2 v 2;0 Þdx
1
¼ eigx0 dðv 1 v 1;0 Þdðv 2 v 2;0 Þ;
which is the result presented in Eq. (5.3).
ð5:4Þ
h
6. Applying Laplace transforms
We have applied the Fourier transform to the log asset price variable. To successfully solve the resulting PDE (5.1), we
apply a bivariate Laplace transform to the variance variables and this is accomplished by the proposition below.
e s; g; s1 ; s2 Þ is found to satisfy the ﬁrst order
Proposition 6.1. By applying Deﬁnition 4.3 to the PDE (5.1) the Laplace transform, Uð
PDE
e e
e 1
@U
1
@U
1 2 2
1
@U
þ
r21 s21 H1 s1 þ KðgÞ
þ
r2 s2 H2 s2 þ KðgÞ
@s
@s1
@s2
2
2
2
2
e þ f1 ðs; s2 Þ þ f2 ðs; s1 Þ:
¼ ðU1 r2 Þs1 þ ðU2 r2 Þs2 igðr qÞ þ H1 þ H2 U
1
10
2
In this paper we will not directly use this inverse Laplace transform deﬁnition as we will make use of those tabulated in [1].
ð6:1Þ
290
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
Eq. (6.1) is to be solved subject to the initial condition
e
Uð0;
g; s1 ; s2 Þ ¼ eigx0 s1 v 1;0 s2 v 2;0 :
ð6:2Þ
The two functions, f1 and f2 are given by
f1 ðs; s2 Þ ¼
1 2
e s; g; 0; s2 Þ;
r1 U1 Uð
2
and f 2 ðs; s1 Þ ¼
1 2
e s; g; s1 ; 0Þ
r2 U2 Uð
2
and are determined such that
e s; g; s1 ; s2 Þ ¼ 0;
lim Uð
s1 !1
e s; g; s1 ; s2 Þ ¼ 0;
lim Uð
and
s2 !1
ð6:3Þ
respectively.
Proof. By applying Eq. (4.3) and the assumptions in (4.6) and (4.7) to the respective components of Eq. (5.1) and noting that
e s; g; 0; s2 Þ and Uð
e s; g; s1 ; 0Þ we obtain the result in Propf1 ðs; s2 Þ and f2 ðs; s1 Þ are terms involving the Laplace transforms of Uð
osition 6.1. Feller [16] has demonstrated that assumptions like those in the ﬁrst equation of (4.6) imply that
e s; g; s1 ; s2 Þ ¼ 0 and
lim Uð
s1 !1
e s; g; s1 ; s2 Þ ¼ 0;
lim Uð
s2 !1
which is Eq. (6.3) of Proposition 6.1.
ð6:4Þ
h
7. Solution of the characteristic equation
Eq. (6.1) is a ﬁrst order PDE known as a characteristic equation. We solve this equation using the method of characteristics. Similar techniques have been successfully used in [16,11] to solve PDEs like Eq. (6.1). Cheang et al. [7] use similar techniques when solving the American call option where the underlying asset is being driven by both stochastic volatility and
jumps. We present the solution of this PDE in the proposition below.
Proposition 7.1. The solution of Eq. (6.1) subject to the initial and boundary conditions in Eqs. (6.2) and (6.3) can be represented
as
e s; g; s1 ; s2 Þ ¼
Uð
22rU12 22U22
r
1
2X1
2X2
2
ðr21 s1 H1 þ X1 ÞðeX1 s 1Þ þ 2X1
ðr22 s2 H2 þ X2 ÞðeX2 s 1Þ þ 2X2
H1 X1
H X
exp v 1;0 2 2 2 v 2;0 þ igx0
2
r1
r2
ðU1 r21 ÞðH1 X1 Þ ðU2 r22 ÞðH2 X2 Þ
þ
igðr qÞ þ H1 þ H2 s
exp
2
2
r1
r2
2X1 v 1;0 ðr21 s1 H1 þ X1 ÞeX1 s
2X2 v 2;0 ðr22 s2 H2 þ X2 ÞeX2 s
exp
exp
r2 ½ðr2 s1 H1 þ X1 ÞðeX1 s 1Þ þ 2X1 r22 ½ðr22 s2 H2 þ X2 ÞðeX2 s 1Þ þ 2X2 1 1
2U1
2X1 v 1;0 eX1 s
2X1
1;
C
r21
r21 ðeX1 s 1Þ ðr21 s1 H1 þ X1 ÞðeX1 s 1Þ þ 2X1
2U2
2X2 v 2;0 eX2 s
2X2
1
;
1;
þC
r22
r22 ðeX2 s 1Þ ðr22 s2 H2 þ X2 ÞðeX2 s 1Þ þ 2X2
ð7:1Þ
where
X1 ¼
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
H21 KðgÞr21 and X2 ¼ H22 KðgÞr22 :
ð7:2Þ
The function Cðn; zÞ is the incomplete gamma function deﬁned as
Cðn; zÞ ¼
1
CðnÞ
Z
z
en nn1 dn;
ð7:3Þ
0
and
CðnÞ ¼
Z
1
en nn1 dn:
0
Proof. Refer to Appendix 4 of Chiarella and Ziveyi [12]. h
ð7:4Þ
291
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
8. Inverting the Laplace transform
Now that we have solved Eq. (6.1), the next step is to recover the original function, Uðs; x; v 1 ; v 2 Þ which is expressed in terms
of the original state variables. This process is accomplished by applying the Laplace and Fourier inversion theorems respectively.
In Proposition 8.1 below, we will ﬁrst present the inverse Laplace transform of the function given in Proposition 7.1.
e s; g; s1 ; s2 Þ in Eq. (7.1) is
Proposition 8.1. The inverse Laplace transform of Uð
H1 X1
b s; g; v 1 ; v 2 Þ ¼ exp
Uð
r
exp 2
1
H2 X2
ðv 1 v 1;0 þ U1 sÞ þ
ðv 2 v 2;0 þ U2 sÞ
2
2X1
r21 ðeX1 s 1Þ
2 1e
2 X1 s
1 ðe
X
r
I2U1 1
r2
1
X1 s
r
4X1
r21 ðeX1 s 1Þ
2X2
r22 ðeX2 s 1Þ
U1 1
U2 1
v 1;0 eX1 s r21 2 v 2;0 eX2 s r22 2
v 1;0 eX s þ v 1
1
X2 s
2 2e
2 X2 s
2 ðe
X
1Þ
r2
v1
1Þ
4 2
2 X2 s
ðe
2
X
r
2
2
v2
v 1 v 1;0 eX1 s I2U2 1
1
2
r2
v 2;0 eX s þ v 2
v 2 v 2;0 eX s
2
1Þ
1
2
eigx0 igðrqÞs
;
ð8:1Þ
where Ik ðzÞ is the modiﬁed Bessel function of the ﬁrst kind deﬁned as
Ik ðzÞ ¼
ð2z Þ2nþk
:
Cðn þ k þ 1Þn!
n¼0
1
X
ð8:2Þ
Proof. Refer to Appendix 5 of Chiarella and Ziveyi [12]. h
9. Inverting the Fourier transform
The next task is to ﬁnd the inverse Fourier transform of Eq. (8.1), this is accomplished by applying Deﬁnition 4.2 to Proposition 8.1 and we present the result in the proposition below.
Proposition 9.1. The inverse Fourier transform, Uðs; x; v 1 ; v 2 ; x0 ; v 1;0 ; v 2;0 Þ of Eq. (8.1) can be expressed as
Uðs; x; v 1 ; v 2 ; x0 ; v 1;0 ; v 2;0 Þ ¼
1
2p
where
Hðs; x; v 1 ; v 2 ; g; v 1;0 ; v 2;0 Þ ¼ exp
Z
1
eigx0 Hðs; x; v 1 ; v 2 ; g; v 1;0 ; v 2;0 Þdg
ð9:1Þ
1
H1 X1
r
exp 2
1
2X1
r21 ðeX1 s 1Þ
eigxþigðrqÞs
r2
1
X1 s
2 1e
2 X1 s
1 ðe
X
r2
2X2
X2 s
v
e
þ
v
2;0
2
r22 ðeX2 s 1Þ
U1 1 U2 1
v 1;0 eX1 s r21 2 v 2;0 eX2 s r22 2
v 1;0 eX s þ v 1
1
2 2e
2 X2 s
2 ðe
X
X2 s
v1
v2
1Þ r
1Þ
1
1
4X1
4
X
2
ðv 1 v 1;0 eX1 s Þ2 I2U2 1 2 X s
ðv 2 v 2;0 eX2 s Þ2 :
2 X1 s
r1 ðe 1Þ
r2 ðe 2 1Þ
r2
I2U1 1
H2 X2
ðv 1 v 1;0 þ U1 sÞ þ
ðv 2 v 2;0 þ U2 sÞ
2
r
ð9:2Þ
2
Proof. Refer to Appendix B. h
Now that we have managed to obtain Uðs; x; v 1 ; v 2 Þ, we revert back to the original density function Gðs; S; v 1 ; v 2 Þ which is
expressed in terms of the underlying asset variable, S.
Proposition 9.2. The transition density function expressed in terms of the original state variables can be represented as
Gðs; S; v 1 ; v 2 ; S0 ; v 1;0 ; v 2;0 Þ ¼
1
2p
Z
1
eig ln S0 Hðs; ln S; v 1 ; v 2 ; g; v 1;0 ; v 2;0 Þdg:
ð9:3Þ
1
Proof. Recall that S ex and Uðs; x; v 1 ; v 2 ; x0 ; v 1;0 ; v 2;0 Þ Gðs; ex ; v 1 ; v 2 ; ex0 ; v 1;0 ; v 2;0 Þ. Substituting these into Eq. (9.1) we
obtain the result in the above proposition. h
Having found the explicit form of the trivariate transition density function, we can now obtain the full representation of
the American call option presented in Proposition 3.1. As demonstrated in that proposition, the American option price is expressed in terms of the unknown early exercise boundary S ¼ bðs; v 1 ; v 2 Þ. We use the value matching condition to ﬁnd the
integral equation that determines this function, details of which are given in the next section.
292
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
Table 1
Parameters used for the American call option. The v 1 column contains parameters for the ﬁrst variance process whilst the v 2 column contains parameters for
the second variance process.
Parameter
Value
v 1 -Parameter
Value
v 2 -Parameter
Value
K
r
q
T
M
100
3%
5%
0.5
200
h1
6%
3
10%
0:5
0
20%
h2
8%
4
11%
0:5
0
20%
j1
r1
q13
k1
v max
1
j2
r2
q24
k2
v max
2
10. The American option price
As stated in the previous section, given the explicit transition probability density function in Proposition 9.2, we can derive the simpliﬁed version of the American call option written on the underlying asset, S whose dynamics evolve according to
the SDE system (2.6)–(2.8). By using the relationship Cðs; x; v 1 ; v 2 Þ Vðs; ex ; v 1 ; v 2 Þ, the value of the American call option can
be represented as
Vðs; S; v 1 ; v 2 Þ ¼ V E ðs; S; v 1 ; v 2 Þ þ V P ðs; S; v 1 ; v 2 Þ:
ð10:1Þ
The two terms on the RHS of Eq. (10.1) are presented in the two propositions below. As pointed out earlier, the ﬁrst component on the RHS of Eq. (10.1) is the European option component whilst the second is the early exercise premium.
Proposition 10.1. The European option component of the American call option can be written as
V E ðs; S; v 1 ; v 2 Þ ¼ eqs SP1 ðs; S; v 1 ; v 2 ; KÞ ers KP2 ðs; S; v 1 ; v 2 ; KÞ;
ð10:2Þ
where
1 1
Pj ðs; S; v 1 ; v 2 ; KÞ ¼ þ
2 p
Z
1
0
!
g j ðs; S; v 1 ; v 2 ; gÞeig ln K
Re
dg;
ig
ð10:3Þ
for j ¼ 1; 2 with
g j ðs; S; v 1 ; v 2 ; gÞ ¼ exp ig ln S þ Bj ðs; gÞ þ D1;j ðs; gÞv 1 þ D2;j ðs; gÞv 2 ;
1 Q 1;j eX1;j s
U1
Bj ðs; gÞ ¼ igðr qÞs þ 2 ðH1;j þ X1;j Þs 2 ln
1 Q 1;j
r1
X2;j s 1 Q 2;j e
U2
;
þ 2 ðH2;j þ X2;j Þs 2 ln
1 Q 2;j
r2
"
#
"
#
ðH1;j þ X1;j Þ
1 eX1;j s
ðH2;j þ X2;j Þ
1 eX2;j s
D1;j ðs; gÞ ¼
; D2;j ðs; gÞ ¼
:
r21
r22
1 Q 1;j eX1;j s
1 Q 2;j eX2;j s
ð10:4Þ
Here, Q m;j ¼ ðHm;j þ Xm;j Þ=ðHm;j Xm;j Þ for m ¼ 1; 2 and j ¼ 1; 2 where H1;1 ¼ H1 ði gÞ, H1;2 ¼ H1 ðgÞ, H2;1 ¼ H2 ði gÞ,
H2;2 ¼ H2 ðgÞ, X1;1 ¼ X1 ði gÞ, X1;2 ¼ X1 ðgÞ, X2;1 ¼ X2 ði gÞ and X2;2 ¼ X2 ðgÞ.
Proof. Refer to Appendix D. h
Remark 10.1. We recall that the deﬁnitions of H1 , H2 , X1 and X2 have been provided in Eq. (5.2). Also U1 and U2 have been
deﬁned in Eq. (3.5).
When numerically implementing this pricing function in Proposition 10.2, it is desirable to adopt the ideas proposed in
[23,4]. Such techniques prevent the possibilities of branch cuts11 and ensure that the density function is continuous in the
complex plane. Discontinuities are frequently more pronounced when pricing long maturity options. An example to this effect
can be found on Table 1 of Albrecher et al. [4].
Proposition 10.2. The early exercise premium component of the American call option can be represented as
V P ðs; S; v 1 ; v 2 Þ ¼
Z sZ
0
0
1
Z
0
1
½qeqðsnÞ SPA1 ðs n; S; v 1 ; v 2 ; w1 ; w2 ; bðn; w1 ; w2 ÞÞ rerðsnÞ KP A2
ðs n; S; v 1 ; v 2 ; w1 ; w2 ; bðn; w1 ; w2 ÞÞdw1 dw2 dn;
11
A branch cut is a curve in the complex plane across which a function is discontinuous.
ð10:5Þ
293
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
where
PAj ðs n; S; v 1 ; v 2 ; w1 ; w2 ; bðn; v 1 ; v 2 ÞÞ ¼
1 1
þ
2 p
Z
1
Re
g Aj ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þeig ln bðn;w1 ;w2 Þ
0
ig
!
dg;
ð10:6Þ
for j ¼ 1; 2 with
H2;j X2;j
ðv 1 w1 þ U1 ðs nÞÞ þ
ðv 2 w2 þ U2 ðs nÞÞ ð10:7Þ
2
r
r
2
2X1;j
2X2;j
w1 eX1;j ðsnÞ þ v 1 w2 eX2;j ðsnÞ þ v 2
exp 2 X ðsnÞ
X
ð
s
nÞ
2
r1 ðe 1;j
1Þ
r2 ðe 2;j
1Þ
U1 1
U2 1
X1;j ðsnÞ
2X2;j eX2;j ðsnÞ
w1 eX1;j ðsnÞ r21 2 w2 eX2;j ðsnÞ r22 2
ig ln SþigðrqÞðsnÞ 2X1;j e
e
v1
v2
r21 ðeX1 ðsnÞ 1Þ r22 ðeX2;j ðsnÞ 1Þ
1
1
4X1;j
4X2;j
ðv 1 w1 eX1;j ðsnÞ Þ2 I2U2 1 2 X ðsnÞ
ðv 2 w2 eX2;j ðsnÞ Þ2 :
I2U1 1 2 X ðsnÞ
r1 ðe 1;j
1Þ
r2 ðe 2;j
1Þ
r2
r2
1
2
g Aj ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þ ¼ exp
H1;j X1;j
2
1
The expressions for Hm;j and Xm;j are given in Proposition 10.1 above.
Proof. Refer to Appendix E. h
Eq. (10.1) is in terms of the early exercise boundary, bðs; v 1 ; v 2 Þ which is still unknown. This function needs to be determined for us to have the corresponding option price at each point in time. Also, the three integrals of the early exercise premium component cannot be integrated out as we do not know the functional form of this early exercise boundary. The only
knowledge we have is that it is a function of time and the two instantaneous variances. The early exercise boundary also
satisﬁes the value matching condition
bðs; v 1 ; v 2 Þ K ¼ Vðs; bðs; v 1 ; v 2 Þ; v 1 ; v 2 Þ;
ð10:8Þ
which, given the integral expression for Vðs; bðs; v 1 ; v 2 Þ; v 1 ; v 2 Þ is a non-linear Volterra integral equation. This can be solved
directly for the free-boundary but we seek some approximation techniques in order to reduce the computational burden
associated with solving the integral equation directly. We present one approximation technique for the early exercise
boundary in the next section.
The integral expression derived in this paper is applicable to any continuous payoff function, which is a powerful feature
of Fourier and Laplace transform based methods.
11. Approximating the early exercise surface
The idea of approximating early exercise boundaries has gained popularity in pricing standard12 American options; Ju [22]
uses multi-piece exponential functions to approximate the early exercise boundary of the American put option. Chiarella et al.
[8] approximate the value function using Fourier–Hermite series expansions. Ait-Sahlia and Lai [3] approximate Kim’s (1990)
early exercise boundary with a piecewise linear function. They ﬁrst discretise the time domain into equally spaced sub-intervals. Linear interpolation is then incorporated to ﬁt the early exercise boundary between two successive subintervals thereby
generating the entire free-boundary. Mallier [28] consider series solutions for the location of the early exercise boundary close
to expiry.
Approximation techniques have also been generalised to American options under stochastic volatility. Broadie et al. [6]
have shown empirically in the single stochastic volatility case that ln bðs; v Þ can well be approximated by a function that is
linear in v. Based on these empirical ﬁndings, Tzavalis and Wang [35] have expanded the logarithm of the early exercise
boundary using Taylor series around the long-run volatility. This expansion yields two unknown functions of time which
they later determine using Chebyshev polynomial expansion techniques. Instead of applying a Chebyshev approximation,
Adolfsson et al. [2] use numerical integration techniques and root ﬁnding methods to ﬁnd these unknown functions of time.
This method proves to be adequate enough in terms of accuracy and computational speed as compared to other valuation
methods that they consider. It is this approach that we employ in this paper.
We ﬁrst use a Taylor series expansion to expand the logarithm of the early exercise boundary, ln bðs; v 1 ; v 2 Þ around the
corresponding long-run variances such that
ln bðs; v 1 ; v 2 Þ b0 ðsÞ þ b1 ðsÞv 1 þ b2 ðsÞv 2 ;
12
Here, the term ‘‘standard’’ means an option pricing model that satisﬁes all Black and Scholes [5] assumptions.
ð11:1Þ
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C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
where b0 ðsÞ, b1 ðsÞ and b2 ðsÞ are functions of time to be determined. This approach allows us to simplify the two integrals
with respect to w1 and w2 in Eq. (10.5) before applying the numerical algorithm. Incorporating the expansion (11.1) into
Eq. (10.5) we obtain the results in the following proposition.
Proposition 11.1. By approximating the early exercise boundary with the expression
ln bðs; v 1 ; v 2 Þ b0 ðsÞ þ b1 ðsÞv 1 þ b2 ðsÞv 2 ;
ð11:2Þ
the value of the American call option in Eq. (10.1) can be re-expressed as
Vðs; S; v 1 ; v 2 Þ V E ðs; S; v 1 ; v 2 Þ þ V AP ðs; S; v 1 ; v 2 Þ;
ð11:3Þ
where V E ðs; S; v 1 ; v 2 Þ is as presented in Proposition 10.1 and the approximation to the early exercise premium is given by
V AP ðs; S; v 1 ; v 2 Þ ¼
Z s
0
b A ðs n; S; v 1 ; v 2 ; b0 ðnÞ; b1 ðnÞ; b2 ðnÞÞ rerðsnÞ K P
bA
½qeqðsnÞ S P
1
2
ðs n; S; v 1 ; v 2 ; b0 ðnÞ; b1 ðnÞ; b2 ðnÞÞdn;
ð11:4Þ
where
b A ðs n; S; v 1 ; v 2 ; b0 ðnÞ; b1 ðnÞ; b2 ðnÞÞ ¼ 1 þ 1
P
j
2 p
Z
1
Re
g^Aj ðs n; S; v 1 ; v 2 ; g; b1 ðnÞ; b2 ðnÞÞeigb0 ðnÞ
0
ig
!
dg;
ð11:5Þ
for j ¼ 1; 2 with
n
o
g^Aj ðs; S; v 1 ; v 2 ; g; b1 ; b2 Þ ¼ exp ig ln S þ BAj ðs; g; b1 ; b2 Þ þ DA1;j ðs; g; b1 Þv 1 þ DA2;j ðs; g; b2 Þv 2 ;
(
!)
1 Q A1;j eX1;j s
U1
A
Bj ðs; g; b1 ; b2 Þ ¼ igðr qÞs þ 2 ðH1;j þ X1;j Þs 2 ln
r1
1 Q A1;j
(
!)
1 Q A2;j eX2;j s
U2
;
þ 2 ðH2;j þ X2;j Þs 2 ln
r2
1 Q A2;j
"
#
ðH1;j þ X1;j Þ
1 eX1;j s
;
DA1;j ðs; g; b1 Þ ¼ igb1 þ
r21
1 Q A1;j eX1;j s
"
#
ðH2;j þ X2;j Þ
1 eX2;j s
A
:
D2;j ðs; g; b2 Þ ¼ igb2 þ
r22
1 Q A2;j eX2;j s
Here
Q Am;j ¼
Hm;j þ igr2m bm þ Xm;j
;
Hm;j þ igr2m bm Xm;j
Hm;j ¼ Hj ði gÞ;
and Xm;j ¼ Xj ði gÞ;
for m ¼ 1; 2 and j ¼ 1; 2.
Proof. Refer to Appendix F. h
By using the approximation (11.1), we have managed to reduce the number of integral dimensions from four to two as we
have explicitly simpliﬁed two integrals with respect to w1 and w2 appearing in Eq. (10.5). The simpliﬁed version of the early
exercise premium component enhances computational speed of our numerical scheme for ﬁnding both the early exercise
boundary and the corresponding option price as the resulting equation is now independent of the modiﬁed Bessel functions
which tends to consume much computational time. Given the approximation in Eq. (11.1), the value-matching condition can
also be expressed as
eb0 ðsÞþb1 ðsÞv 1 þb2 ðsÞv 2 K ¼ Vðs; eb0 ðsÞþb1 ðsÞv 1 þb2 ðsÞv 2 ; v 1 ; v 2 Þ:
ð11:6Þ
Eq. (11.6) is implicit in b0 ðsÞ, b1 ðsÞ and b2 ðsÞ, hence root ﬁnding techniques need to be employed for us to obtain explicit
forms of these functions. In determining these functions, we formulate three equations such that
b0 ðsÞ ¼ ln½Vðs; eb0 ðsÞþb1 ðsÞv 1 þb2 ðsÞv 2 ; v 1 ; v 2 Þ þ K b1 ðsÞv 1 b2 ðsÞv 2 ;
1
ln½Vðs; eb0 ðsÞþb1 ðsÞv 1 þb2 ðsÞv 2 ; v 1 ; v 2 Þ þ K b0 ðsÞ b2 ðsÞv 2 ;
b1 ðsÞ ¼
v1
b2 ðsÞ ¼
1
v2
ln½Vðs; eb0 ðsÞþb1 ðsÞv 1 þb2 ðsÞv 2 ; v 1 ; v 2 Þ þ K b0 ðsÞ b1 ðsÞv 1 :
These equations need to be solved iteratively at each instant, details of which are outlined in the next section.
ð11:7Þ
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C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
12. Numerical implementation
Having derived the integral expression for the American call option price in Eq. (11.3) and the corresponding system of
Eqs. (11.7) for approximating the early exercise boundary, we now present the numerical algorithm for the implementation
of these equations. A variety of techniques have been proposed in the literature for numerically solving equations like (11.7).
Huang et al. [18] use a numerical integration scheme to solve the [26] American put integral equation. Kallast and Kivinukk
[24] also use quadrature methods to approximate the price, delta, gamma and vega of both American call and put options.
Adolfsson et al. [2] use similar techniques to implement the integral expression for the American call option price when the
dynamics of the underlying asset evolve under the inﬂuence of a stochastic variance process of the Heston [17] type. In
implementing our pricing algorithm, we shall use quadrature techniques as applied in [24].
The European option component of Eq. (11.3) involves only one integral with respect to the Fourier transform variable, this
integration is easily handled by standard methods. However, the early exercise premium component has two integrals, one with
respect to the Fourier transform variable and the other with respect to running time-to-maturity, n. The integral with respect to
the Fourier transform variable is handled in a similar way as in the European component case. However, the integral with respect
to n requires the entire history of the three functions, b0 ðsÞ, b1 ðsÞ, and b2 ðsÞ up to and including the current time. We therefore
need to devise an algorithm to determine these three functions iteratively at each point in time.
In implementing Eq. (11.3) and the system (11.7), we treat the American option as a Bermudan option. The time interval
is partitioned into M-equally spaced subintervals of length h ¼ T=M. The algorithm is initiated at maturity and we then progress backwards in time. We denote the starting point as, s0 ¼ 0 which corresponds to maturity time. At maturity, it has
been shown in [26] that the early exercise boundary of the American call option takes the form
bð0; v 1 ; v 2 Þ ¼ max
r
K; K :
q
ð12:1Þ
By comparing coefﬁcients, we can readily deduce that
r
b0 ð0Þ ¼ max ln K; ln K ;
q
b1 ð0Þ ¼ 0;
and b2 ð0Þ ¼ 0:
ð12:2Þ
All other time steps are denoted as sm ¼ mh, for m ¼ 1; 2; . . . ; M. The discretised version of the American call option price is
thus
Vðmh; S; v 1 ; v 2 Þ V E ðmh; S; v 1 ; v 2 Þ þ V AP ðmh; S; v 1 ; v 2 Þ:
ð12:3Þ
m
b0
m
b1
At each subsequent time step we need to determine the three unknown boundary terms,
¼ b0 ðmhÞ,
¼ b1 ðmhÞ and
m
b2 ¼ b2 ðmhÞ each of which depends on the entire early exercise boundary history. We use iterative techniques to ﬁnd the
m
m
m
values of these three unknown functions at each time step, that is, when iterating for b0 , b1 and b2 , we use as initial
m
m1
m
m1
m
m1
guesses13 b0;0 ¼ b0 , b1;0 ¼ b1 and b2;0 ¼ b2 followed by solving the system of linked equations
h
i
m
m
m
m
m
m
b0;k ¼ ln V mh; eb0;k þb1;k1 v 1 þb2;k1 v 2 ; v 1 ; v 2 þ K b1;k1 v 1 b2;k1 v 2 ;
h
i
m
m
m
1
m
m
m
b1;k ¼
ln Vðmh; eb0;k þb1;k v 1 þb2;k1 v 2 ; v 1 ; v 2 Þ þ K b0;k b2;k1 v 2 ;
v1
m
b2;k
¼
1
v2
h
i
m
m
m
m
m
ln Vðmh; eb0;k þb1;k v 1 þb2;k v 2 ; v 1 ; v 2 Þ þ K b0;k b1;k v 1 :
m
m
m
m
ð12:4Þ
m
m
We ﬁnd the value of k such that jb0;k b0;k1 j < 0 , jb1;k b1;k1 j < 1 and jb2;k b2;k1 j < 2 , where 0 ; 1 and 2 are tolerance
values. Once the tolerance values are attained, we then proceed to the next time step. This algorithm is applicable to any root
ﬁnding method for determining the triplet, b0 , b1 and b2 . Adolfsson et al. [2] use Newton’s method while we prefer to use the
bisection method in this paper as it does not involve the computation of the ﬁrst derivative of the pricing function.
In the next section we present numerical and graphical results for the early exercise boundary and the corresponding
American call option prices obtained using the above approach. We also provide graphs for the joint probability density functions of the state variables. Such density functions are crucial as they give us a clue on how to handle the unbounded integral
domains of the state variables and address the convergence property of density functions.
13. Numerical results
Having presented the numerical algorithm as outlined above, we now provide some numerical examples in this section.
In what follows, we will dub our method the numerical integration scheme. We have also implemented the Method of Lines
(MOL) algorithm for the PDE (2.9) for comparison purposes. Details on how to implement the MOL algorithm can be found in
[9], who consider the valuation of American options under stochastic volatility and jump diffusion processes. In all the
13
m
m
m
The subscript k in the three functions b0;k , b1;k and b2;k represents the number of iterations required for convergence of the iterative process at time step m.
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C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
numerical experiments that follow unless otherwise stated, we will use the parameters provided in Table 1 where, v max
and
1
v max
are the maximum levels of the two instantaneous variances under consideration. We have discretised the two variance
2
domains into 30 equally spaced sub intervals and M ¼ 200 time steps. For the MOL algorithm a non-uniform grid is applied
to the underlying asset domain and a total number of 1,438 grid points has been used. The large number of grid points in the
underlying asset domain helps in stabilising the numerical scheme and enhancing the smoothness of the early exercise
boundary. For the numerical scheme to be stable, we have used 40 grid points in the interval 0 6 S 6 1, 198 points in the
interval 1 < S 6 100 and 1200 points within the interval 100 < S 6 500.
We start by presenting the joint probability density function of S and v 1 when v 2 is ﬁxed in Fig. 1 (that of S and v 2 when
v 1 is ﬁxed is similar in appearance). This surface is generated by implementing Eq. (9.3). The nature of these probability density functions guide us on how to truncate the inﬁnite domains of the state variables when performing numerical integration
experiments. From this ﬁgure we note that density functions are zero everywhere except near the origins of the state variables. For instance, instead of integrating the underlying asset domain from zero to inﬁnity in our case we have simply integrated from zero to 50 since beyond this point the density function is extremely close to zero. Such diagrams also provide a
natural way of analysing the distribution of asset returns under stochastic volatility.
Having established the integration domains for the state variables, we present in Fig. 2 the early exercise surface for the
American call option when v 2 if ﬁxed. This surface shows how an increase in v 1 affects the free-boundary of the American
Joint PDF of S and v when v =11%
1
2
5
x 10
10
1
pdf(S,v )
8
6
4
2
0
60
50
40
30
20
10
0.05
0.15
0.1
S
0.2
0.25
0.3
v1
Fig. 1. The probability density function of S and v 1 when v 2 is ﬁxed. We have considered the case when q13 ¼ 0:5 and q24 ¼ 0:5 with all other parameters as
provided in Table 1.
Free Surface of the American Call Option
170
160
b(τ,v 1,v 2)
150
140
130
120
110
100
0.5
0.4
0.2
0.3
0.15
0.2
τ
0.1
0.1
Fig. 2. Early exercise surface of the American Call option when
0.05
0
0
v
1
v 2 ¼ 0:67%, q13 ¼ 0:5 and q24 ¼ 0:5. All other parameters are as presented in Table 1.
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C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
call option. We note from this ﬁgure that the early exercise surface is an increasing function of v 1 and is of the form typical
for that of an American call option written on a single underlying asset whose dynamics evolve under the inﬂuence of a single stochastic variance process as presented in [9]. A similar surface can be obtained by ﬁxing v 1 and allowing v 2 to vary.
We can also compare the early exercise boundaries for the American call option when both v 1 and v 2 are ﬁxed. Fig. 3
shows these comparisons for varying correlation coefﬁcients. Note from this ﬁgure that for ﬁxed v 1 and v 2 , early exercise
boundaries typical for standard American call options are generated. We have also included the free-boundary generated
from the geometric Brownian motion (GBM) model to highlight the impact of stochastic volatility on the American call option free-boundary. Since the two instantaneous variance processes under consideration are mean reverting, we calculate
the corresponding GBM constant standard deviation as
rGBM ¼
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
h1 þ h2 ;
ð13:1Þ
where h1 and h2 are the long run variances of v 1 and v 2 respectively. From Fig. 3 we note that zero correlations almost correspond to the GBM case. The early exercise boundary generated when the correlations are negative lies above that of the
GBM model whilst that for positive correlations lies below as revealed in Fig. 3.
Comparing Early Exercise Boundaries of the American Call Option
170
160
1 2
b(τ,v ,v )
150
140
130
GBM
ρ = −0.5 & ρ
120
ρ
13
=0&ρ
ρ
13
= 0.5 & ρ
13
24
24
= −0.5
=0
24
= 0.5
110
100
0
0.1
0.2
0.3
0.4
0.5
τ
Fig. 3. Exploring the effects of stochastic volatility on the early exercise boundary of the American call option for varying correlation coefﬁcients when
rGBM ¼ 0:3742, v 1 ¼ 6% and v 2 ¼ 8%. All other parameters are provided in Table 1.
Early Exercise Boundary of the American Call Option
160
150
1
b(τ,v ,v2)
140
130
GBM
σ1 = 0.1 & σ2 = 0.11
120
σ1 = 0.15 & σ2 = 0.20
110
100
0
0.1
0.2
0.3
0.4
0.5
τ
Fig. 4. Exploring the effects of varying the volatilities of v 1 and v 2 on the early exercise boundary of the American call option. We have used the following
parameters, rGBM ¼ 0:3742, v 1 ¼ 6%, v 2 ¼ 8%, q13 ¼ 0:5 and q24 ¼ 0:5 with all other parameters as given in Table 1.
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Fig. 4 shows the effects of varying volatilities of v 1 and v 2 on the early exercise boundary. We note that increasing the
volatility of volatilities, r1 and r2 , has the effect of lowering the exercise boundary. We have considered the case when both
q13 and q24 are equal to 0.5 and the instantaneous variances equal to their long run means.
To justify the effectiveness of our approach in valuing American call options, we need to compare the results with other
pricing methods. In Fig. 5 we present the early exercise boundaries from the method of lines (MOL) algorithm and numerical
integration respectively. From this diagram we note that the early exercise boundary generated by the numerical integration
method is slightly lower than that from the MOL. This might be attributed to approximation and discretisation errors from
the numerical integration method. Discretisation errors can be reduced by making the grids ﬁner. Errors from early exercise
boundary approximation can be reduced by devising better approximating functions empirically or by any other suitable
approach. Similar comparisons can be made for different parameter combinations. We also present Fig. 6 which compares
the effects of different correlation coefﬁcients on the same process. For example when q13 ¼ 0:5 and q24 ¼ 0:5 we see that
the corresponding early exercise boundary is slightly below the q13 ¼ 0 and q24 ¼ 0 boundary. This might be due to cancelation effect of the inﬂuential stochastic terms of the variance processes.
Early Exercise boundary Comparison
160
150
1 2
b(τ,v ,v )
140
130
120
MOL
Numerical Integration
110
100
0
0.1
0.2
0.3
0.4
0.5
τ
Fig. 5. Comparing early exercise boundaries from the MOL and numerical integration approach when the two instantaneous variances are ﬁxed. Here,
v 1 ¼ 0:67%, v 2 ¼ 13:33%, q13 ¼ 0:5 and q24 ¼ 0:5 with all other parameters as given in Table 1.
Comparing Early Exercise Boundaries of the American Call Option
170
160
140
1 2
b(τ,v ,v )
150
130
ρ
120
13
= −0.5 & ρ
24
ρ
13
= −0.5 & ρ
24
ρ
13
=0&ρ
rho
13
24
= −0.5
= 0.5
=0
= 0.5 & ρ
24
= 0.5
110
100
0
0.1
0.2
0.3
0.4
0.5
τ
Fig. 6. Exploring the effects of mixed correlation coefﬁcients on the early exercise boundary of the American call option. We have used the following
parameters, v 1 ¼ 6%, v 2 ¼ 8% with all other parameters as given in Table 1.
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We now turn to an analysis of option prices using the two approaches. Fig. 7 shows the general American call option price
surface at a ﬁxed level of v 2 . A similar surface can be generated by ﬁxing v 1 and allowing S and v 2 to vary. We can also assess
the effects of stochastic volatility on the option prices for different correlation coefﬁcients by making comparisons with GBM
prices where we calculate the corresponding constant volatility using Eq. (13.1) which is the square-root of the average of
the two long run variances. Fig. 8 shows option price differences found by subtracting option prices from the numerical integration method from the corresponding GBM prices. As with the early exercise boundary comparisons, the zero correlation
price differences are not signiﬁcantly different from GBM prices. As documented in [17,9], higher price differences are noted
for far out-of-the-money and in-the-money options. Positive correlations yield option prices which are lower than GBM
prices for in-the-money options while generating prices which are higher for out-of-the-money options. The reverse effect
holds for negative correlations. Higher price differences of up to 0.1 are noted for both positive and negative correlations.
We also access the signiﬁcance of the two stochastic variance processes by comparing price differences between the GBM
prices with those from numerical integration for varying speeds of mean reversion. This is important for instance when pricing an option under two volatility regimes. Fig. 9 shows the effects on price changes when varying the speeds of mean reversion. We note that the price changes are very sensitive to the changing volatility factors. Detailed empirical studies on the
signiﬁcance of multiple factor stochastic volatility models have been presented in [13,15].
100
1 2
V(τ,S,v ,v )
80
60
40
20
0
200
160
0.2
120
0.15
80
0.1
40
S
Fig. 7. American call option price surface when
0.05
0
0
v
1
v 2 ¼ 13:33%, q13 ¼ 0 and q24 ¼ 0 with all other parameters provided in Table 1.
Price Differences for the American Call Option
0.1
0.08
GBM
ρ13 = −0.5 and ρ24 = −0.5
0.06
ρ
= 0.5 and ρ
ρ
= 0 and ρ
ρ13 = −0.5 and ρ24 = 0.5
13
13
24
24
= 0.5
=0
Price Differences
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
0
40
80
120
160
200
S
Fig. 8. Option prices from the geometric Brownian motion minus option prices from the Stochastic volatility model for varying correlation coefﬁcients.
Here, rGBM ¼ 0:3742, v 1 ¼ 6% and v 2 ¼ 8% with all other parameters provided in Table 1.
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Comparing Different Speeds of Mean Reversion
0.08
0.06
0.04
0.02
S
0
−0.02
κ =3&κ =4
−0.04
1
2
κ = 0.3 & κ = 8
1
2
−0.06
−0.08
−0.1
0
20
40
60
80
100
120
Price Differennces
140
160
180
200
Fig. 9. Option prices from the geometric Brownian motion minus option prices from the Stochastic volatility model for Speeds of Mean Reversion. Here,
rGBM ¼ 0:3742, v 1 ¼ 6% and v 2 ¼ 8% with all other parameters provided in Table 1.
Table 2
American call option price comparisons when v 1 ¼ 0:67%, v 2 ¼ 13:33%, q13 ¼ 0:5, q24 ¼ 0:5. We have taken
GBM volatility to be rGBM ¼ 0:3741657 and this is found by using Eq. (13.1).
S
Numerical integration
MOL
GBM
60
80
100
120
140
160
180
200
0.2036
2.4088
9.8082
23.1069
40.4756
60
80
100
0.2029
2.4000
9.7918
23.0920
40.4686
60
80
100
0.1850
2.4154
9.9452
23.3006
40.5922
60
80
100
Price Differences for the American Call Option
0.15
0.1
Price Differences
0.05
0
GBM
σ1 = 0.10 & σ2 = 0.11
−0.05
σ1 = 0.15 & σ2 = 0.20
−0.1
−0.15
−0.2
0
20
40
60
80
100
S
120
140
160
180
200
Fig. 10. Option prices from the geometric Brownian motion minus option prices from the Stochastic volatility model for varying volatilities of volatility.
Here, rGBM ¼ 0:3742, v 1 ¼ 6%, v 2 ¼ 8%, q13 ¼ 0:5 and q24 ¼ 0:5. All other parameters are provided in Table 1.
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Delta of the American Call Option
1
0.9
0.8
∂V(τ,S,v1,v2)/∂S
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
200
160
0.2
120
0.15
80
S
0.1
40
0.05
0
Fig. 11. Delta surface of the American call option when
0
v1
v 2 ¼ 0:67%, q13 ¼ 0:5 and q24 ¼ 0:5. All other parameters are as provided in Table 1.
We also present option prices obtained from the MOL and numerical integration methods together with the associated
GBM prices in Table 2 when q13 ¼ 0:5 and q24 ¼ 0:5. From this table we note that option prices obtained from the MOL
and numerical integration methods are not signiﬁcantly different from each other which shows that both methods are suitable for practical purposes in valuing American call options under stochastic volatility. We have included GBM prices to
highlight the impact of stochastic volatility on option prices. When we presented numerical results for the early exercise
boundaries we highlighted the effects of changes in the volatilities of v 1 and v 2 , we also provide graphical results on how
such changes affect option prices in Fig. 10. In this ﬁgure, we have used the case when q13 ¼ 0:5 and q24 ¼ 0:5. We can readily see that higher price differences occur for higher r1 and r2 with all other parameters as provided in Table 1. This implies
that higher volatilities of v 1 and v 2 have the effect of increasing the variances which then results in higher price differences
for in-and out-of-the-money options relative to GBM prices. Similar conclusions have been derived in [17] when considering
the European call option under stochastic volatility.
The most important feature of the MOL is that the option price, delta and the free-boundary are all generated simultaneously as part of the solution process at no added computational cost. Given such a tremendous convenience, we wrap
up this section by presenting the delta surface of the American call option in Fig. 11 for ﬁxed v 2 . A similar surface can be
obtained by holding v 1 constant. We also explore the effects of stochastic volatility on the delta by making comparisons with
the GBM delta in Fig. 11. From this ﬁgure, we note that the option delta is very sensitive to the changes in the variance.
14. Conclusion
In this paper we have presented a numerical integration technique for pricing an American call option written on an
underlying asset whose dynamics evolve under the inﬂuence of two stochastic variance processes of the [17] type. The approach involves the transformation of the pricing partial differential equation (PDE) to an inhomogeneous form by exploiting
Jamshidian’s [21] (1992) techniques. An integral expression has been presented as the general solution of the inhomogeneous PDE with the aid of Duhamel’s principle and is found to be a function of the transition density function. Fig. 12.
The transition density function is a solution of the associated Kolmogorov backward PDE for the three stochastic processes under consideration. We have assumed a special correlation structure that allows the use of transform techniques.
A systematic approach for solving the Kolmogorov PDE using a combination of Fourier and Laplace transforms has been presented and a means for numerically implementing the integral equation for the American call option has been provided. The
early exercise boundary approximation has allowed a simpliﬁcation of the double integrals with respect to the running variance variables. This reduces the computational burden when one proceeds to numerical implementation. The process yields
a semi-analytical formula for pricing American call options, which can be used as a benchmark method for comparison with
other pricing techniques.
Numerical results exploring the impact of stochastic volatility on both option prices and the free-boundary have been
provided and we have found that the correlations between the underlying asset and the two variance processes have a signiﬁcant effect on in-and out-of-the money options. The numerical results presented yield similar ﬁndings to Heston [17] and
Chiarella et al. [9] on the impact of stochastic volatility on option prices who they consider both European and American
option pricing under stochastic volatility respectively. We have also analysed the effects of varying the volatilities of instantaneous variances on both the early exercise boundary and the corresponding option prices. We note that an increase in the
302
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
Effects of Stochastic Volatility on the Delta
1
0.9
0.8
1 2
∂V(τ,S,v ,v )/∂S
0.7
0.6
v1 = 0.67%
0.5
v = 20%
1
GBM)
0.4
0.3
0.2
0.1
0
0
50
100
S
150
Fig. 12. Exploring the effects of Stochastic volatility on the Delta of the American call option when
parameters are as provided in Table 1.
200
v 2 ¼ 13:33%, q13 ¼ 0:5
and q24 ¼ 0:5. All other
volatility of the instantaneous variances increases the corresponding variance levels resulting in higher price differences for
in-and out-of-the-money options when compared with geometric Brownian motion prices.
We have assessed the accuracy of the numerical integration approach by making comparisons with numerical results
from the method of lines (MOL) algorithm. While the MOL is by nature computationally faster, both approaches provide
comparable results though there are slight differences in the early exercise boundary plots. Such differences are mainly
due to the early exercise boundary approximation and discretisation errors associated with both methods.
Appendix A. Proof of Proposition 3.1
Consider the PDE (3.3) whose initial condition is the payoff at maturity, Cð0; x; v 1 ; v 2 Þ ¼ ðex KÞþ and for convenience we
set f ðs; x; v 1 ; v 2 Þ ¼ 1xPln bðs;v 1 ;v 2 Þ ðqex rKÞ. The PDE (3.3) is to be solved in the region 0 6 s 6 T; 1 6 x < 1 and
0 6 v 1 ; v 2 < 1.
By use of Duhamel principle, the solution of the PDE (3.3) is given by
Cðs; x; v 1 ; v 2 Þ ¼ ers
Z
Z
0
1
1
Z
0
1
Z
1
þ
ðeu KÞ Uðs; x; v 1 ; v 2 ; u; w1 ; w2 Þdudw1 dw2 þ
1
Z s
0
erðsnÞ
Z
1
0
Z
1
0
f ðn; u; w1 ; w2 ÞUðs n; x; v 1 ; v 2 ; u; w1 ; w2 Þdudw1 dw2 dn:
1
C E ðs; x; v 1 ; v 2 Þ þ C P ðs; x; v 1 ; v 2 Þ:
ðA1:1Þ
We verify that this is the correct solution by showing that (A1.1) satisﬁes the PDE (3.3). Substituting
Cðs; x; v 1 ; v 2 Þ ¼ C E ðs; x; v 1 ; v 2 Þ þ C P ðs; x; v 1 ; v 2 Þ into (3.3) we compute
Z 1Z 1Z 1
@C
@U
þ rC MC f ðs; x; v 1 ; v 2 Þ ¼ ers
ðeu KÞ
MU dudw1 dw2 rC E þ rC E
@s
@s
Z 10 Z 10 Z 11
þ
f ðs; u; w1 ; w2 ÞUð0; x; v 1 ; v 2 ; u; w1 ; w2 Þdudw1 dw2
0
0
1
Z s
Z 1Z 1Z 1
@U
þ
erðsnÞ
f ðn; u; w1 ; w2 Þ
dudw1 dw2 dn rC P þ rC P
@s
0
0
0
1
Z s
Z 1Z 1Z 1
erðsnÞ
f ðn; u; w1 ; w2 ÞMU dudw1 dw2 dn f ðs; x; v 1 ; v 2 Þ
0
1
0
Z 1Z 1Z 10
f ðs; x; v 1 ; v 2 Þdðex eu Þdðv 1 w1 Þdðv 2 w2 Þdudw1 dw2
¼
0
0
1
Z s
Z 1Z 1Z 1
@U
þ
erðsnÞ
f ðn; x; v 1 ; v 2 Þ
MU dudw1 dw2 dn f ðs; x; v 1 ; v 2 Þ
@s
0
0
0
1
¼ f ðs; x; v 1 ; v 2 Þ þ 0 f ðs; x; v 1 ; v 2 Þ ¼ 0:
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
303
Appendix B. Proof of Proposition 9.1
b s; g; v 1 ; v 2 Þ from Eq. (8.1) we obtain
By applying Eq. (4.2) to Eq. (8.1) and substituting for Uð
Uðs; x; v 1 ; v 2 ; x0 ; v 1;0 ; v 2;0 Þ ¼
1
2p
Z
1
eigx0 Hðs; x; v 1 ; v 2 ; g; v 1;0 ; v 2;0 Þdg
ðA2:1Þ
1
where
Hðs; x; v 1 ; v 2 ; g; v 1;0 ; v 2;0 Þ ¼ exp
H1 X1
r21
exp 2X1
r21 ðeX1 s 1Þ
eigxþigðrqÞs
I2U1 1
1
r2
1
X1 s
2 1e
2 X1 s
1 ðe
X
2X2
X2 s
v
e
þ
v
2;0
2
r22 ðeX2 s 1Þ
U1 1 U2 1
v 1;0 eX1 s r21 2 v 2;0 eX2 s r22 2
v 1;0 eX s þ v 1
2 2e
2 X2 s
2 ðe
X
X2 s
v1
v2
1Þ r
1Þ
1
1
4X1
4
X
2
ðv 1 v 1;0 eX1 s Þ2 I2U2 1 2 X s
ðv 2 v 2;0 eX2 s Þ2 :
2 X1 s
r1 ðe 1Þ
r2 ðe 2 1Þ
r2
r2
H2 X2
ðv 1 v 1;0 þ U1 sÞ þ
ð
v
v
þ
U
s
Þ
2
2;0
2
2
r
ðA2:2Þ
2
which is the result given in Proposition 9.1.
Appendix C. Some useful complex integrals
In this appendix we reproduce the integral representation of complex functions given in [33] as they are required for the
calculations in Appendices D and E. We seek complex integral representations of expressions involving the function g which
satisﬁes the two conditions ðiÞ hð/Þ gð/ iÞ and ðiiÞ hð/Þ ¼ hð/Þ.
We need to consider integrals of the form
Q1 ¼
Z
1
2p
1
gð/Þ
Z
1
ey ei/y dyd/;
Q2 ¼
a
1
1
2p
Z
1
gð/Þ
Z
1
ei/y dyd/:
ðA3:1Þ
a
1
The ﬁrst equation in (A3.1) is simpliﬁed by letting n ¼ / i so that
Z 1
Z 1
1
gðn iÞ
einy dydn
2p 1
a
Z 1
Z 1
Z 1
1
1
eiga eigb
dg:
gðg iÞ
eigy dydg ¼
gðg iÞ lim
¼
b!1
2p 1
2p 1
ig
a
Q1 ¼
ðA3:2Þ
ðA3:3Þ
Eq. (A3.2) can be expressed as
Z 1
iga
iga
Z 1
1
e
eigb
e eigb
dg þ
dg
lim
gðg iÞ
gðg iÞ
2p b!1 0
ig
ig
0
Z 1
Z
1
1
gðg iÞeiga gðg iÞeiga
1
gðg iÞeigb gðg iÞeigb
¼
dg:
dg lim
2p 0
2p b!1 0
ig
ig
Q1 ¼
ðA3:4Þ
Now using the result in [33] that
FðxÞ ¼
where FðxÞð¼
1
b!1 2p
lim
Z
1
1
2 2p
Rx
1
Z
1
0
1
0
gðg iÞeigx gðg iÞeigx
dg;
ig
ðA3:5Þ
gðgÞdgÞ is a cumulative density function we can show that
gðg iÞeigb gðg iÞeigb
1
dg ¼ lim
b!1 2p
ig
Z
0
1
hðgÞeigb hðgÞeigb
1
1
dg ¼ lim FðbÞ ¼ :
b!1 2
2
ig
ðA3:6Þ
Using this result, Eq. (A3.4) can be represented as
Q1 ¼
1
1
þ
2 2p
Z
0
1
hðgÞeiga hðgÞeiga
dg:
ig
ðA3:7Þ
Also, if hðgÞ and hðgÞð¼ hðgÞÞ are complex conjugates then
hðigÞeiga
hðigÞeiga
:
¼
ig
ig
Using this result (A3.7) simpliﬁes to
ðA3:8Þ
304
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
Q1 ¼
1 1
þ
2 p
Z
1
0
hðgÞeiga
dg:
Re
ig
ðA3:9Þ
Performing similar operations to the second equation in (A3.1) we obtain
Q2 ¼
1 1
þ
2 p
Z
1
0
gðgÞeiga
Re
dg:
ig
ðA3:10Þ
Appendix D. Proof of Proposition 10.1
Let x ¼ logðSÞ and make use of the relation C E ðs; logðSÞ; v 1 ; v 2 Þ ¼ V E ðs; S; v 1 ; v 2 Þ introduced in Eq. (3.2). We substitute the
explicit density function presented in Proposition 9.1 into the European option component in Eq. (3.17) and obtain after
rearranging
V E ðs; S; v 1 ; v 2 Þ ¼ ers
Z
1
Z
0
1
Z
0
1
ðex KÞ
1
2p
ln K
Z
eigx u2 ðs; S; v 1 ; v 2 ; g; w1 ; w2 Þdg dxdw1 dw2 ;
1
ðA4:1Þ
1
where
u2 ðs; S; v 1 ; v 2 ; g; w1 ; w2 Þ ¼ exp
H1;2 X1;2
exp r21
H2;2 X2;2
ðv 1 w1 þ U1 sÞ þ
ð
v
w
þ
U
s
Þ
2
2
2
2
r2
2X2;2
X2;2 s
w
e
þ
v
2
2
r22 ðeX2;2 s 1Þ
U1 1
U2 1
2X1;2 eX1;2 s
2X2;2 eX2;2 s
w1 eX1;2 s r21 2 w2 eX2;2 s r22 2
eig ln SþigðrqÞs 2 X s
v1
v2
r1 ðe 1 1Þ r22 ðeX2;2 s 1Þ
1
1
4X1;2
4X2;2
I2U1 1 2 X s
ðv 1 w1 eX1;2 s Þ2 I2U2 1 2 X s
ðv 2 w2 eX2;2 s Þ2 ;
r1 ðe 1;2 1Þ
r2 ðe 2;2 1Þ
r2
r2
2X1;2
r21 ðeX1;2 s 1Þ
1
w1 e
X1;2 s
þ v1 ðA4:2Þ
2
with H1;1 ¼ H1 ði gÞ; H1;2 ¼ H1 ðgÞ; H2;1 ¼ H2 ði gÞ, H2;2 ¼ H2 ðgÞ; X1;1 ¼ X1 ði gÞ; X1;2 ¼ X1 ðgÞ, X2;1 ¼ X2 ði gÞ and
X2;2 ¼ X2 ðgÞ.
From Eq. (A4.2) we note that the payoff of the European call option is independent of the variance variables, namely w1
and w2 . This gives us the ﬂexibility to calculate the integrals with respect to w1 and w2 ﬁrst, thus Eq. (A4.1) can be written as
V E ðs; S; v 1 ; v 2 Þ ¼
ers
2p
Z
1
1
Z
1
Z
eigx ðex KÞ
ln K
0
1
Z
1
u2 ðs; S; v 1 ; v 2 ; g; w1 ; w2 Þdw1 dw2 dgdx:
ðA4:3Þ
0
In evaluating the double integral with respect to the running variance variables, we ﬁrst let14
A1 ¼
2X1;2 w1
r21 ð1 eX1;2 s Þ
;
A2 ¼
2X2;2 w2
r22 ð1 eX2;2 s Þ
;
ðA4:4Þ
and
y1 ¼
2X1;2 v 1
r21 ðeX1;2 s 1Þ
;
y2 ¼
2X2;2 v 2
r22 ðeX2;2 s 1Þ
:
ðA4:5Þ
Using g 2 ðs; S; v 1 ; v 2 Þ to denote the inner double integral in Eq. (A4.3) it can be written as
H2;2 X2;2
eig ln SþigðrqÞs
ð
v
þ
U
s
Þ
þ
ð
v
þ
U
s
Þ
1
1
2
2
r21
r22
2X1;2
2X2;2
exp 2 X s
v
v
1 r1 ðe 1;2 1Þ
r22 ðeX2;2 s 1Þ 2
U21 12
Z 1Z 1
ðH1;2 X1;2 Þð1 eX1;2 s Þ
A1 r1
exp þ 1 A1
2X1;2
y1
0
0
U22 12
X2;2 s
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðH2;2 X2;2 Þð1 e
Þ
A2 r2
exp þ 1 A2
I2U1 1 ð2 A1 y1 ÞI2U2 1
2X2;2
y2
r2
r2
1
2
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð2 A2 y2 ÞdA1 dA2 ;
g 2 ðs; S; v 1 ; v 2 Þ ¼ exp
14
H1;2 X1;2
Note that these functions were introduced before in the systems (A5.1) and (A5.11) of Chiarella and Ziveyi [12].
ðA4:6Þ
305
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
Now using the deﬁnition of the modiﬁed Bessel function, we can further simplify the above equation to15
H2;2 X2;2
eig ln SþigðrqÞs
ð
v
þ
U
s
Þ
þ
ð
v
þ
U
s
Þ
1
1
2
2
r21
r22
2X1;2
2X2;2
exp 2 X s
v
v
1 r1 ðe 1;2 1Þ
r22 ðeX2;2 s 1Þ 2
Z
Z
n
n
1 X
1
X
y11 y22 1 1
ðH1;2 X1;2 Þð1 eX1;2 s Þ þ 2X1;2
A1
exp n ! n2 ! 0
2X1;2
0
n ¼0n ¼0 1
g 2 ðs; S; v 1 ; v 2 Þ ¼ exp
H1;2 X1;2
1
2
2U 1
2U2
n1 þ 2 1
n2 þ 2 1
r
r
A1 1
A2 2
ðH2;2 X2;2 Þð1 eX2;2 s Þ þ 2X2;2
dA1 dA2
exp A2
2X2;2
C n1 þ 2rU21 C n2 þ 2rU22
1
2
H1;2 X1;2
H2;2 X2;2
ig ln SþigðrqÞs
¼ exp
ðv 1 þ U1 sÞ þ
ðv 2 þ U2 sÞ e
r21
r22
2X1;2
2X
exp 2 X s
v 2 X2;22;2
v
r1 ðe 1;2 1Þ 1
r2 ðe s 1Þ 2
n1 þ2U21
1 X
1
X
r
yn11
yn22
2X1;2
1
X1;2 s
2U1
2U2
ð
H
X
Þð1
e
Þ
þ
2
X
1;2
1;2
1;2
n
!
C
n
þ
n1 ¼0n2 ¼0 n1 !C n1 þ 2
2
2
2
r
r
1
2
2X2;2
ðH2;2 X2;2 Þð1 eX2;2 s Þ þ 2X2;2
n2 þ2U22 Z
1
r
2
0
Z
1
0
n1 þ
en1 n2 n1
2U1
1
r2
1
n2 þ
n2
2U2
1
r2
2
dn1 dn2 :
ðA4:7Þ
By noting that
Z
1
0
Z
0
n1 þ
1
en1 n2 n1
2U1
1
r2
1
2U 2
1
r2
2
n2 þ
n2
2U1
2U2
dn1 dn2 ¼ C n1 þ 2 C n2 þ 2 ;
r1
r2
ðA4:8Þ
we obtain
H2;2 X2;2
ð
Þ
ð
Þ
eig ln SþigðrqÞs
þ
v
þ
U
s
v
þ
U
s
1
1
2
2
r21
r22
2X1;2
2X2;2
exp 2 X s
v
v
1 r1 ðe 1;2 1Þ
r22 ðeX2;2 s 1Þ 2
2U21 2U22
r
r
2X1;2
2X2;2
1
2
ðH1;2 X1;2 Þð1 eX1;2 s Þ þ 2X1;2
ðH2;2 X2;2 Þð1 eX2;2 s Þ þ 2X2;2
n1
1 X
1
X
yn11 yn22
2X1;2
n ! n2 ! ðH1;2 X1;2 Þð1 eX1;2 s Þ þ 2X1;2
n1 ¼0n2 ¼0 1
n2
2X2;2
:
ðH2;2 X2;2 Þð1 eX2;2 s Þ þ 2X2;2
g 2 ðs; S; v 1 ; v 2 Þ ¼ exp
H1;2 X1;2
ðA4:9Þ
We apply a Taylor series expansion of the exponential function to the double summation to obtain
H2;2 X2;2
eig ln SþigðrqÞs
ð
v
þ
U
s
Þ
þ
ð
v
þ
U
s
Þ
1
1
2
2
r21
r22
2X1;2
2X2;2
exp 2 X s
v
v
1 r1 ðe 1;2 1Þ
r22 ðeX2;2 s 1Þ 2
2U21 2U22
r
r
2X1;2
2X2;2
1
2
X1;2 s
X2;2 s
ðH1;2 X1;2 Þð1 e
Þ þ 2X1;2
ðH2;2 X2;2 Þð1 e
Þ þ 2X2;2
2X1;2 y1
2X2;2 y2
:
exp
þ
ðH1;2 X1;2 Þð1 eX1;2 s Þ þ 2X1;2 ðH2;2 X2;2 Þð1 eX2;2 s Þ þ 2X2;2
g 2 ðs; S; v 1 ; v 2 Þ ¼ exp
Reverting to the
15
H1;2 X1;2
v 1 and v 2 variables from the Eq. (A4.5) we obtain
Note that we set n1 ¼
X1;2 s
ðH1;2 X1;2 Þð1e
2X1;2
Þþ2X1;2
A1 and n2 ¼
X2;2 s
ðH2;2 X2;2 Þð1e
2X2;2
Þþ2X2;2
A2 in the integrals.
ðA4:10Þ
306
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
g 2 ðs; S; v 1 ; v 2 Þ ¼ exp
H1;2 X1;2
r21
exp H2;2 X2;2
eig ln SþigðrqÞs
ðv 1 þ U1 sÞ þ
ð
v
þ
U
s
Þ
2
2
2
2 1;2
2 X1;2 s
ðe
1
X
r
v1 1Þ
r
r2
2 2;2
2 X2;2 s
ðe
2
X
1Þ
v2
2U22
r
2X1;2
2X2;2
2
ðH1;2 X1;2 Þð1 eX1;2 s Þ þ 2X1;2
ðH2;2 X2;2 Þð1 eX2;2 s Þ þ 2X2;2
2X1;2
2X1;2 v 1
exp
ðH1;2 X1;2 Þð1 eX1;2 s Þ þ 2X1;2 r21 ðeX1;2 s 1Þ
2X2;2
2X2;2 v 2
:
exp
ðH2;2 X2;2 Þð1 eX2;2 s Þ þ 2X2;2 r22 ðeX2;2 s 1Þ
2U 1
r2
1
ðA4:11Þ
For convenience, we now attempt to represent this density in the form presented in [17]. This is accomplished by adopting
the representation
g 2 ðs; S; v 1 ; v 2 ; gÞ ¼ exp ðig ln S þ B2 ðs; gÞ þ D1;2 ðs; gÞv 1 þ D2;2 ðs; gÞv 2 Þ;
ðA4:12Þ
where
ðH1;2 X1;2 Þð1 eX1;2 s Þ þ 2X1;2
ðH1;2 X1;2 Þs 2 ln
2X1;2
r
X2;2 s
U2
ðH2;2 X2;2 Þð1 e
Þ þ 2X2;2
;
þ 2 ðH2;2 X2;2 Þs 2 ln
2X2;2
r2
B2 ðs; gÞ ¼igðr qÞs þ
U1
2
1
ðH1;2 X1;2 Þ
D1;2 ðs; gÞ ¼
r21
ðH2;2 X2;2 Þ
D2;2 ðs; gÞ ¼
r22
2X1;2
2X1;2
þ
r21 ðeX1;2 s 1Þ ðH1;2 X1;2 Þð1 eX1;2 s Þ þ 2X1;2
2X2;2
2X2;2
þ
r22 ðeX2;2 s 1Þ ðH2;2 X2;2 Þð1 eX2;2 s Þ þ 2X2;2
2X1;2 v 1
r21 ðeX1;2 s 1Þ
2X2;2 v 2
r22 ðeX2;2 s 1Þ
;
:
By letting
Q 1;2 ¼
H1;2 þ X1;2
H1;2 X1;2
and Q 2;2 ¼
H2;2 þ X2;2
;
H2;2 X2;2
the above three functions reduce to
1 Q 1;2 eX1;2 s
1 Q 1;2
r21
1 Q 2;2 eX2;2 s
U2
;
þ 2 ðH2;2 þ X2;2 Þs 2 ln
1 Q 2;2
r2
ðH1;2 þ X1;2 Þ
1 eX1;2 s
D1;2 ðs; gÞ ¼
;
r21
1 Q 1;2 eX1;2 s
ðH2;2 þ X2;2 Þ
1 eX2;2 s
D2;2 ðs; gÞ ¼
:
r22
1 Q 2;2 eX2;2 s
B2 ðs; gÞ ¼igðr qÞs þ
U1
ðH1;2 þ X1;2 Þs 2 ln
ðA4:13Þ
ðA4:14Þ
ðA4:15Þ
Substituting Eq. (A4.12) into Eq. (A4.3) we obtain
V E ðs; S; v 1 ; v 2 Þ ¼
ers
2p
Z
1
1
g 2 ðs; S; v 1 ; v 2 ; gÞ
Z
1
ln K
ex eigx dxdg K
Z
1
g 2 ðs; S; v 1 ; v 2 ; gÞ
1
Z
1
eigx dxdg:
ðA4:16Þ
ln K
The two components on the RHS of the above equation have similar properties to the two respective equations in (A3.1) described in Appendix C. We can evaluate the integrals in Eq. (A4.16) using Eqs. (A3.9) and (A3.10) provided that
g 2 ðs; S; v 1 ; v 2 ; g iÞ satisﬁes appropriate assumptions. The ﬁrst assumption that we must verify is that g 2 ðs; S; v 1 ; v 2 ; g iÞ
can be expressed as a function of g. This assumption is satisﬁed since
g 2 ðs; S; v 1 ; v 2 ; g iÞ ¼ SeðrqÞs g 1 ðs; S; v 1 ; v 2 ; gÞ;
where
g 1 ðs; S; v 1 ; v 2 ; gÞ ¼ exp ðig ln S þ B2 ðs; gÞ þ D1;1 ðs; gÞv 1 þ D2;1 ðs; gÞv 2 Þ;
ðA4:17Þ
307
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
with
1 Q 1;2 eX1;1 s
ðH1;1 þ X1;1 Þs 2 ln
1 Q 1;1
r
1 Q 2;1 eX2;1 s
U2
;
þ 2 ðH2;1 þ X2;1 Þs 2 ln
1 Q 2;1
r2
ðH1;1 þ X1;1 Þ
1 eX1;1 s
;
D1;1 ðs; gÞ ¼
r21
1 Q 1;1 eX1;1 s
ðH2;1 þ X2;1 Þ
1 eX2;1 s
:
D2;1 ðs; gÞ ¼
2
r2
1 Q 2;1 eX2;1 s
B1 ðs; gÞ ¼igðr qÞs þ
U1
2
1
ðA4:18Þ
ðA4:19Þ
ðA4:20Þ
Furthermore, by using the same reasoning as in Eq. (A3.8) it can also be shown that
g j ðs; S; v 1 ; v 2 ; gÞ ¼ g j ðs; S; v 1 ; v 2 ; g; Þ;
for j ¼ 1; 2;
ðA4:21Þ
and hence all the assumptions required to carry out the calculations yielding (A3.9) and (A3.10) are satisﬁed. Thus Eq.
(A4.16) becomes
V E ðs; S; v 1 ; v 2 Þ ¼ eqs SP1 ðs; S; v 1 ; v 2 ; KÞ ers KP2 ðs; S; v 1 ; v 2 ; KÞ;
where
1 1
Pj ðs; S; v 1 ; v 2 ; KÞ ¼ þ
2 p
Z
ðA4:22Þ
!
g j ðs; S; v 1 ; v 2 ; gÞeig ln K
Re
dg;
ig
1
0
ðA4:23Þ
for j ¼ 1; 2 which is the result in Proposition 10.1.
Appendix E. Proof of Proposition 10.2
We proceed as we did in Appendix D. Substituting the density function presented in Proposition 9.1 into the early exercise
premium component in Eq. (3.18) we obtain after rearranging
V P ðs; S; v 1 ; v 2 Þ ¼
Z s
0
erðsnÞ
1
2p
Z
1
Z
0
Z
1
1
where
1
0
Z
1
½qey rK
ln bðn;w1 ;w2 Þ
eigy g A2 ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þdg dydw1 dw2 dn;
H2;2 X2;2
ð
v
w
þ
U
ð
s
nÞ
Þ
þ
ð
v
w
þ
U
ð
s
nÞ
Þ
1
1
1
2
2
2
r2
r22
1
2X1;2
2X2;2
X2;2 ðsnÞ
w1 eX1;2 ðsnÞ þ v 1 w
e
þ
v
exp 2 X ðsnÞ
2
2
r1 ðe 1;2
1Þ
r22 ðeX2;2 ðsnÞ 1Þ
U1 1
U2 1
2X1;2 eX1;2 ðsnÞ 2X2;2 eX2;2 ðsnÞ w1 eX1;2 ðsnÞ r21 2 w2 eX2;2 ðsnÞ r22 2
eig lnSþigðrqÞðsnÞ 2 X ðsnÞ
2
X
ð
s
nÞ
v1
v2
r1 ðe 1
1Þ r2 ðe 2;2
1Þ
1
1
4X1;2
4
X
2;2
I2U1 1 2 X ðsnÞ
ðv 1 w1 eX1;2 ðsnÞ Þ2 I2U2 1 2 X ðsnÞ
ðv 2 w2 eX2;2 ðsnÞ Þ2 :
r1 ðe 1;2
1Þ
r2 ðe 2;2
1Þ
r2
r2
g A2 ðs n;S; v 1 ; v 2 ; g;w1 ;w2 Þ ¼exp
ðA5:1Þ
H1;2 X1;2
1
2
Eq. (A5.1) is equivalent to
" Z
Z 1
1
1
V P ðs; S; v 1 ; v 2 Þ ¼
e
q
g A2 ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þ
ey eigy dydg
2p
0
0
1
ln bðn;w1 ;w2 Þ
0
#
Z 1
Z 1
A
igy
rK
g 2 ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þ
e dydg dw1 dw2 dn:
Z s
rðsnÞ
Z
1
Z
1
ðA5:2Þ
ln bðn;w1 ;w2 Þ
1
By proceeding in the same way that we did to handle Eq. (A4.16) when simplifying the complex integrals we can write
g A2 ðs n; S; v 1 ; v 2 ; g i; w1 ; w2 Þ ¼ eðrqÞðsnÞ Sg A1 ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þ:
ðA5:3Þ
so that Eq. (A5.2) reduces to
V P ðs; S; v 1 ; v 2 Þ ¼
Z sZ
0
1
0
Z
1h
0
i
qeqðsnÞ SP A1 ½s n; S; v 1 ; v 2 ; w1 ; w2 ; bðn; w1 ; w2 Þ rerðsnÞ KP A2 ½s n; S; v 1 ; v 2 ; w1 ; w2 ; bðn; w1 ; w2 Þ dw1 dw2 dn;
ðA5:4Þ
308
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
where
PAj ð
1 1
s n; S; v 1 ; v 2 ; w1 ; w2 ; bðn; w1 ; w2 ÞÞ ¼ þ
2 p
Z
1
Re
g Aj ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þeig ln bðn;w1 ;w2 Þ
!
ig
0
dg;
for j ¼ 1; 2 which is the result given in Proposition 10.2.
Appendix F. Proof of Proposition 11.1
Going back to Eq. (A5.1), we have expressed the early exercise premium value as
" Z
Z 1
1
1
s; S; v 1 ; v 2 Þ ¼
e
q
g A2 ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þ ey eigy dydg
2p
0
0
1
ln bðn;w1 ;w2 Þ
0
#
Z 1
Z 1
A
igy
g 2 ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þ
e dydg dw1 dw2 dn:
rK
Z s
V AP ð
rðsnÞ
Z
1
Z
1
ðA6:1Þ
ln bðn;w1 ;w2 Þ
1
With the approximation, ln bðs; v 1 ; v 2 Þ ¼ b0 ðsÞ þ b1 ðsÞv 1 þ b2 ðsÞv 2 , the above equation is transformed to
" Z
Z 1
1
1
s; S; v 1 ; v 2 Þ ¼
e
q
g A2 ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þ ey eigy dydg
2p
0
0
0
1
b0 ðnÞþb1 ðnÞw1 þb2 ðnÞw2
#
Z 1
Z 1
A
igy
g 2 ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þ
e dydg dw1 dw2 dn:
rK
Z s
V AP ð
rðsnÞ
Z
1
Z
1
ðA6:2Þ
b0 ðnÞþb1 ðnÞw1 þb2 ðnÞw2
1
By making the change of variables z ¼ y b1 ðnÞw1 b2 ðnÞw2 in Eq. (A6.2) we obtain
" Z
Z 1
1
1
s; S; v 1 ; v 2 Þ ¼
e
q
g A2 ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þ eð1þigÞðzþb1 ðnÞw1 þb2 ðnÞw2 Þ dzdg
2p
0
0
0
1
b0 ðnÞ
#
Z 1
Z 1
A
igðzþb1 ðnÞw1 þb2 ðnÞw2 Þ
g 2 ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þ
e
dzdg dw1 dw2 dn
rK
Z s
V AP ð
¼
Z s
0
rðsnÞ
1
erðsnÞ
Z
1
Z
1
Z
1
b0 ðnÞ
½qez J 1 ðn; zÞ rKJ2 ðn; zÞdzdn;
ðA6:3Þ
b0 ðnÞ
where we set
J 1 ðn; zÞ ¼
Z
1
Z
0
1
eb1 ðnÞw1 þb2 ðnÞw2 þigðzþb1 ðnÞw1 þb2 ðnÞw2 Þ
0
1
2p
Z
1
1
g A2 ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þdgdw1 dw2 ;
ðA6:4Þ
and
J 2 ðn; zÞ ¼
Z
1
Z
0
1
0
eig½b1 ðnÞw1 þb2 ðnÞw2 1
2p
Z
1
1
eigz g A2 ðs n; S; v 1 ; v 2 ; g; w1 ; w2 Þdgdw1 dw2 :
ðA6:5Þ
The two expressions for J 1 ðn; zÞ and J 2 ðn; zÞ are now in a form that can allow us to simplify the integrals with respect to w1 and
w2 . By making use of equations (A4.4) and (A4.5) and repeating similar steps to those from Eqs. (A4.6)–(A4.11) in Appendix D
we obtain
J 2 ðn; zÞ ¼
1
2p
Z
1
1
eigz g^A2 ðs n; S; v 1 ; v 2 ; g; b1 ; b2 Þdg;
ðA6:6Þ
where
n
o
g^A2 ðs;S; v 1 ; v 2 ; g;b1 ;b2 Þ ¼exp ig lnS þ BA2 ðs; g;b1 ;b2 Þ þ DA1;2 ðs; g;b1 Þv 1 þ DA2;2 ðs; g;b2 Þv 2 ;
U1
ðH1;2 þ igr21 b1 X12 Þð1 eX1;2 s Þ þ 2X1;2
BA2 ðs; gÞ ¼ igðr qÞs þ 2 ðH1;2 X1;2 Þs 2ln
2X1;2
r1
U2
ðH2;2 þ igr22 b2 X2;2 Þð1 eX2;2 s Þ þ 2X2;2
;
þ 2 ðH2;2 X2;2 Þs 2ln
2X2;2
r2
H1;2 X1;2
2X1;2
2X1;2
2X1;2
2 X s
DA1;2 ðs; gÞ ¼
þ
r21
r1 ðe 1;2 1Þ ðH1;2 þ igr21 b1 X1;2 Þð1 eX1;2 s Þ þ 2X1;2 r21 ðeX1;2 s 1Þ
H2;2 X2;2
2X2;2
2X2;2
2X2;2
þ
DA2;2 ðs; gÞ ¼
2 X s
:
r22
r2 ðe 2;2 1Þ ðH2;2 þ igr22 b2 X2;2 Þð1 eX2;2 s Þ þ 2X2;2 r22 ðeX2;2 s 1Þ
ðA6:7Þ
309
C. Chiarella, J. Ziveyi / Applied Mathematics and Computation 224 (2013) 283–310
The functions BA2 ðs; gÞ; DA1;2 ðs; gÞ and DA2;2 ðs; gÞ can be simpliﬁed further by letting
Q A1;2 ¼
H1;2 þ igr21 b1 þ X1;2
;
H1;2 þ igr21 b1 X1;2
and Q A2;2 ¼
H2;2 þ igr22 b2 þ X2;2
;
H2;2 þ igr22 b2 X2;2
ðA6:8Þ
such that
BA2 ðs; g; b1 ; b2 Þ ¼igðr qÞs þ
(
U1
1 Q A1;2 eX1;2 s
ðH1;2 þ X1;2 Þs 2 ln
r21
DA1;2 ð
s; g; b1 Þ ¼ igb1 þ
DA2;2 ðs; g; b2 Þ ¼ igb2 þ
ðH1;2 þ X1;2 Þ
r21
ðH2;2 þ X2;2 Þ
1 Q A1;2
"
"
r22
1 eX1;2 s
1 Q A1;2 eX1;2 s
1 eX2;2 s
1 Q A2;2 eX2;2 s
!)
þ
#
U2
(
r22
ðH2;2 þ X2;2 Þs 2 ln
1 Q A2;2 eX2;2 s
!)
1 Q A2;2
;
;
#
ðA6:9Þ
:
By using a similar transformation to that between Q 1 and Q 2 in Appendix C we can write
g^A2 ðs n; S; v 1 ; v 2 ; g i; b1 ; b2 Þ ¼ eðrqÞðsnÞ Sg^A1 ðs n; S; v 1 ; v 2 ; g; b1 ; b2 Þ;
ðA6:10Þ
where
n
o
g^A1 ðs; S; v 1 ; v 2 ; g; b1 ; b2 Þ ¼ exp ig ln S þ BA1 ðs; g; b1 ; b2 Þ þ DA1;1 ðs; g; b1 Þv 1 þ DA2;1 ðs; g; b2 Þv 2 ; BA1 ðs; g; b1 ; b2 Þ
(
!)
1 Q A1;1 eX1;2 s
U1
¼ igðr qÞs þ 2 ðH1;1 þ X1;1 Þs 2 ln
r1
1 Q A1;1
(
!)
1 Q A2;1 eX2;1 s
U2
þ 2 ðH2;1 þ X2;1 Þs 2 ln
; DA1;1 ðs; g; b1 Þ
r2
1 Q A2;1
"
#
"
#
ðH1;1 þ X1;1 Þ
1 eX1;1 s
ðH2;1 þ X2;1 Þ
1 eX2;1 s
A
;
D
:
¼ igb1 þ
ð
s
;
g
;
b
Þ
¼
i
g
b
þ
2
2
2;1
r21
r22
1 Q A1;1 eX1;1 s
1 Q A2;1 eX2;1 s
Incorporating these into Eq. (A6.3) we obtain
V AP ðs; S; v 1 ; v 2 Þ ¼
Z s
0
erðsnÞ
Z
1
qez
b0 ðnÞ
Z
1
rK
b0 ðnÞ
1
2p
Z
1
1
1
2p
Z
1
1
eigz eðrqÞðsnÞ Sg^A1 ðs n; S; v 1 ; v 2 ; g; b1 ðnÞ; b2 ðnÞÞdgdzdn Z s
erðsnÞ
0
eigz g^A2 ðs n; S; v 1 ; v 2 ; g; b1 ðnÞ; b2 ðnÞÞdgdzdn;
from which we obtain
V AP ðs; S; v 1 ; v 2 Þ ¼
Z s
0
erðsnÞ qSeðrqÞðsnÞ
Z s
0
erðsnÞ rK
1
2p
Z
1
2p
1
1
Z
1
g^A1 ðs n; S; v 1 ; v 2 ; g; b1 ðnÞ; b2 ðnÞÞ
1
g^A2 ðs n; S; v 1 ; v 2 ; g; b1 ðnÞ; b2 ðnÞÞ
Z
1
Z
1
ez eigz dzgdn
b0 ðnÞ
eigz dzdgdn:
ðA6:11Þ
b0 ðnÞ
Proceeding as we did in Eq. (A4.16) we obtain the result that
V P ðs; S; v 1 ; v 2 Þ ¼
Z s
0
b A ðs n; S; v 1 ; v 2 ; b0 ðnÞ; b1 ðnÞ; b2 ðnÞÞ rerðsnÞ K P
bA
½qeqðsnÞ S P
1
2
ðs n; S; v 1 ; v 2 ; b0 ðnÞ; b1 ðnÞ; b2 ðnÞÞdn;
ðA6:12Þ
where
b A ðs n; S; v 1 ; v 2 ; b0 ðnÞ; b1 ðnÞ; b2 ðnÞÞ ¼ 1 þ 1
P
j
2 p
Z
1
Re
g^Aj ðs n; S; v 1 ; v 2 ; g; b1 ðnÞ; b2 ðnÞÞeigb0 ðnÞ
0
ig
!
dg;
ðA6:13Þ
for j ¼ 1; 2, which is the result presented in Proposition 11.1.
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