Jump Diffusion Processes in Financial Modeling

This thesis is a collection of papers focused on applications of jump diffusion processes in financial modeling. The aim of the first paper is to extend the analytical tractability of the Black-Scholes model to alternative models with jumps, no matter whether the jump sizes have exponential-type tails or power-type tails. More precisely, we study a jump diffusion model for asset prices whose jump sizes are hyper-exponentially distributed. The hyper-exponential distribution can approximate many heavy-tail distributions as closely as possible, including both power- and exponential-type distributions. We demonstrate the hyper-exponential jump diffusion model can lead to analytical solutions for popular path-dependent options such as lookback, barrier and perpetual American options. Numerical examples indicate that the formulae are easy to implement and accurate. These analytical solutions are made possible mainly because we solve several high-order integro-differential equations explicitly related to first passage time problems and optimal stopping problems. In the second paper, we propose a two-factor equilibrium model for the on-peak electricity spot price in the markets where the spot price exhibits large spikes and hence seems to have an infinite mean. More precisely, we model the logarithm of the scaled peak demand using two independent mean reverting affine (jump) diffusion processes, which describe different mean reversion structures of normal peak demands and spikes of demands separately, and then derive a model for the on-peak spot price based on the minimum Nash equilibrium for the demand-and-price function obtained from existing game theory results (see [3]). Not only does our model capture features of electricity spot prices such as spikes and seasonality, but it also has some other appealing properties. First, it can incorporate oligopoly. Second, the on-peak spot price has an infinite expectation, but the futures price has a finite expectation. Third, the on-peak spot price in our model has a closed form, and the analytical solutions to the futures prices and futures option prices can also be derived. In addition, our model can produce the following phenomenon relevant to the futures prices. Specifically, some plots are provided and indicate that futures prices are more stable than spot prices especially when significant spikes happen, that a short spike of demands can generate a much larger spike of the spot price due to the convexity of the cost function, and that an early spike has a much weaker effect on the futures prices than a spike which occurs when the time is close to the maturity. Furthermore, we conclude that our two-factor model seems better than the one-factor model in the literatures by comparing futures prices generated by these two models. The third paper presents an analytical solution of a double-Laplace transform for Asian options under the Black-Scholes model. The transform can be inverted numerically via a two-sided Laplace inversion algorithm. A theoretical error bound of the numerical inversion is also derived. Numerical results indicate that our pricing method is fast, stable, and accurate. A main theoretical contribution of the paper is that we prove the results in a straightforward way by solving an ordinary differential equation analytically rather than using Bessel processes and Lamperti's representation. We also point out that a well-known recursion used in pricing Asian options may not have a unique solution. Additionally, we make a conjecture for the pricing formula of Asian options under the double exponential jump diffusion model, which is analogous to the formula under the...