The evaluation of early exercise exotic options
- Publication Type:
- Thesis
- Issue Date:
- 2011
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Research on the pricing of multifactor American options has been growing at a slow
pace due to the curse of dimensionality. If we start to consider the pricing of American
option contracts written on more than one underlying asset or relax the constant
volatility assumption of the Black and Scholes (1973) model, the computational burden
increases as more computing power is required to handle the increasing number of
dimensions.
This thesis deals with the problem of pricing multifactor American options under both
constant and stochastic volatility. The main focus of the thesis is to extend the representation
results of Kim (1990) and Carr, Jarrow and Myneni (1992) and to devise
higher dimensional numerical techniques for pricing multifactor American options. We
present numerical examples for two and three factor models. The pricing problems are
formulated using the well known hedging arguments. We adopt two main approaches;
the first involves deriving integral expressions for the American option prices with the
aid of Jamshidian’s (1992) transformation of the associated partial differential equation
from a homogeneous problem on a restricted domain to an inhomogeneous problem on
an unrestricted domain, Duhamel’s principle and integral transform methods. The
second technique involves implementing the method of lines algorithm for American
exotic options, with the spread call option under stochastic volatility being the main
example – this approach tackles directly the pricing partial differential equation. Chapter
1 contains an overview of the American option pricing problem from the viewpoint
of the applications in this thesis. The chapter concludes with some technical results
used in the rest of the thesis. The main contributions of the thesis are contained in
the subsequent chapters.
Chapter 2 extends the integral transform approach of McKean (1965) and Chiarella and
Ziogas (2005) to the pricing of American options written on two underlying assets under
Geometric Brownian motion. A bivariate transition density function of the two underlying
stochastic processes is derived by solving the associated backward Kolmogorov
partial differential equation. Fourier transform techniques are used to transform the
partial differential equation to a corresponding ordinary differential equation whose
solution can be readily found by using the integrating factor method. An integral expression
of the American option written on any two assets is then obtained by applying
Duhamel’s principle. A numerical algorithm for calculating American spread call option
prices is given as an example, with the corresponding early exercise boundaries
approximated by linear functions. Numerical results are presented and comparisons
made with other alternative approaches.
Chapter 3 considers the pricing of an American call option whose underlying asset
evolves under the influence of two independent stochastic variance processes of the Heston
(1993) type. We derive the associated partial differential equation (PDE) for the
option price using standard hedging arguments. An integral expression for the general
solution of the PDE is derived using Duhamel’s principle, which is expressed in terms
of the yet to be determined trivariate transition density function for the driving stochastic
processes. We solve the backward Kolmogorov PDE satisfied by the transition
density function by first transforming it to the corresponding characteristic PDE using
a combination of Fourier and Laplace transforms. The characteristic PDE is solved
by the method of characteristics. Having determined the density function, we provide
a full representation of the American call option price. By approximating the early
exercise surface with a bivariate log-linear function, we develop a numerical algorithm
to calculate the pricing function. Numerical results are compared with those from the
method of lines algorithm. The approach is generalised in Chapter 4 to the case when
the underlying asset evolves under the influence of more than two stochastic variance
processes by using a combination of induction proofs and some lengthy derivations.
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