The main contribution of this thesis is to derive the properties and present a closed from solution of the exotic options under some specific types of Levy processes, such as American put options, American call options, British put options, British call options and American knockout put options under either double exponential jumpdiffusion processes or onesided exponential jumpdiffusion processes. Compared to the geometric Brownian motion, exponential jumpdiffusion processes can better incorporate the asymmetric leptokurtic features and the volatility smile observed from the market. Pricing the option with early exercise feature is the optimal stopping problem to determine the optimal stopping time to maximize the expected options payoff. Due to the Markovian structure of the underlying process, the optimal stopping problem is related to the freeboundary problem consisting of an integral differential equation and suitable boundary conditions. By the local timespace formula for semimartingales, the closed form solution for the options value can be derived from the freeboundary problem and we characterize the optimal stopping boundary as the unique solution to a nonlinear integral equation arising from the early exercise premium (EEP) representation. Chapter 2 and Chapter 3 discuss American put options and American call options respectively. When pricing options with early exercise feature under the double exponential jumpdiffusion processes, a nonlocal integral term will be found in the infinitesimal generator of the underlying process. By the local timespace formula for semimartingales, we show that the value function and the optimal stopping boundary are the unique solution pair to the system of two integral equations. The significant contributions of these two chapters are to prove the uniqueness of the value function and the optimal stopping boundary under less restrictive assumptions compared to previous literatures. In the degenerate case with only onesided jumps, we find that the results are in line with the geometric Brownian motion models, which extends the analytical tractability of the BlackScholes analysis to alternative models with jumps. In Chapter 4 and Chapter 5, we examine the British payoff mechanism under onesided exponential jumpdiffusion processes, which is the first analysis of British options for process with jumps. We show that the optimal stopping boundaries of British put options with only negative jumps or British call options with only positive jumps can also be characterized as the unique solution to a nonlinear integral equation arising from the early exercise premium representation. Chapter 6 provides the study of American knockout put options under negative exponential jumpdiffusion processes. The conditional memoryless property of the exponential distribution enables us to obtain an analytical form of the arbitragefree price for American knockout put options, which is usually more difficult for many other jumpdiffusion models.
Date of Award  1 Aug 2018 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Goran Peskir (Supervisor) & Kees Van Schaik (Supervisor) 

 Double Exponential JumpDiffusion Model
 ChangeofVariable Formula
 Optimal Stopping and FreeBoundary Problem
 Option Pricing
OPTION PRICING UNDER EXPONENTIAL JUMP DIFFUSION PROCESSES
Bu, T. (Author). 1 Aug 2018
Student thesis: Phd