Small-time expansions for local jump-diffusion models

Yankeng Luo, Purdue University

Abstract

Markov processes have been widely used in physical science and finance to model stochastic phenomena. In particular, Markov processes with jumps can provide appealing model for many financial application. This dissertation focuses on the small-time asymptotic behavior of Markov processes that are solutions to stochastic differential equations driven by both diffusion processes and pure jump processes. We first consider a local jump-diffusion process {Xt}t≥0 whose paths exhibit infinitely jump activity (i.e., with probability 1, the paths of the process exhibit infinitely many jumps in every finite integral). We obtain a second order power series expansion in a small t of the tail probability of the process {X t}t≥0. As an application, we derived the leading term for out-of-the-money option prices in short maturity under an exponential local jump-diffusion model. Our proofs in this work are based on a regularizing technique, Malliavin calculus, flow of diffeomorphism for SDEs, and time reversibility. Then, we generalize our Markov model one step forward, to allow the intensity of jumps of the process to depend locally on the state of the process. In this case, we again obtain a small-time second-order polynomial expansion for the tail distribution of the process and apply this result to further obtain a second-order expansion for out-of-the-money option prices. In this work, we also generalize existing works on the regularity of diffeomorphism associated to stochastic differential equations. Besides the just described analytical estimates, we also an innovative diffusion approximation for the infinite jump activity of jump-diffusion processes, which is not seen in the literatures, to the best of our knowledge

Degree

Ph.D.

Advisors

Viens, Purdue University.

Subject Area

Mathematics

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