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Affine processes in finance: Numerical approximation, simulation and model properties

Kyoung-Kuk Kim, 2008
Faculty Advisor: Paul Glasserman

Abstract

This thesis deals with theoretical and numerical questions related to affine jumpdiffusion

models used in finance. In more detail, we look at three different classes within

the affine jump-diffusion class.

The first is the Heston stochastic volatility model which has been used extensively since

its first introduction by Heston (1993). To price financial derivatives with complex payoff

structures, we have to resort to the Monte Carlo simulation. We propose new simulation

schemes for the Heston model based on the squared Bessel bridge decomposition. These

new methods perform well in different parameter settings and they are compared with two

other existing methods, first, the exact scheme of Broadie and Kaya (2006) and, second, the

QE method of Andersen (2005).

The second question is about the tail behavior of the canonical affine diffusion processes

which were introduced by Dai and Singleton (2000) in the context of financial econometrics

to study the term structure of interest rates. We show that the canonical models have light

tails or exponential bounded tails, and the explicit conditions that guarantee light tails are

given. Moreover, we prove that there exists a uniqte limiting stationary distribution for

each canonical model and the regions of finite exponential moments of such stationary

distributions are determined by the stability region of the dynamical system associated

with a given model.

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