This thesis deals with theoretical and numerical questions related to affine jump-diffusion 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 unique 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.
We further go into the detailed analysis of the dynamical system of a canonical affine diffusion process. We prove that the stability region of such a dynamical system can be represented by the union of stable sub-manifolds under some mild conditions, and also derive some partial differential equation of which solution is blow-up times of the dynamical system. Through an asymptotic analysis of those blow-up times, we calculate the implied volatility asymptotics for options with short maturities and extreme strikes based on Lee (2004).
The third and final question involves the general affine jump-diffusion models. It is computationally too expensive to apply numerical integration schemes to compute vanilla option prices in an affine jump-diffusion model which does not have an explicit Fourier transform formula. To extend the category of models that can be tested in financial econometrics, we apply the well known saddlepoint technique to affine jump-diffusion models. After we develop the basic idea and review some known saddlepoint techniques, we test them for the Heston model, the model of stochastic volatility with jumps (SVJ) and the Scott model. Implementation details and some modifications of existing methods are also given.
|School Location:||United States -- New York|
|Source:||DAI-B 69/10, Dissertation Abstracts International|
|Keywords:||Affine processes, Jump diffusion models, Saddlepoint approximation|
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