Analysis of the Option Prices in Jump Diffusion Models.

Option Pricing;Jump Diffusions;Mathematics;Science;Mathematics

Xing, HaoYoung, Virginia R. ;

University of Michigan

关键词: Option Pricing;    Jump Diffusions;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/63868/haoxing_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

We study the option pricing problem in jump diffusion models from both probabilistic and PDE perspectives. This dissertation consists of the following four parts:(i) We study the regularity properties of the value function of an optimal stopping problem for a process with Levy jumps. Assuming the diffusion component of the process is non-degenerate and a mild assumption on the singularity of the Levy measure, we show that the value function is smooth in the continuation region for problems with either finite or infinite variation jumps. Moreover, the smooth-fit property is shown via the global regularity of the value function.(ii) We show that the optimal exercise boundary of the American put option for jump diffusions with compound Poisson jumps is continuously differentiable (except at the maturity). This differentiability result has been established by Yang et al. under the condition r ≥ q + λ int_R+(e^z − 1) ν(dz). We extend the result to the case where the condition fails via an unified approach that treats both cases simultaneously. We also show that the boundary is infinitely differentiable under a regularity assumption on the jump distribution.(iii), (iv) When the underlying asset price dynamics follows jump diffusions with compound Poisson jumps, we construct a sequence of functions that uniformly converge (on compact sets) to the American (Asian) option price exponentially fast. Each function in this sequence is the value function of a diffusion problem. This sequence gives us an efficient numerical algorithm to price options in jump diffusion models. We prove the convergence/stability of this numerical algorithm and apply it to price American and Asian options in Chapters IV and V respectively.