Date of Award

2019

Degree Type

Open Access Dissertation

Degree Name

Mathematics, PhD

Program

Institute of Mathematical Sciences

Advisor/Supervisor/Committee Chair

Henry Schellhorn

Dissertation or Thesis Committee Member

John Angus

Dissertation or Thesis Committee Member

Qidi Peng

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

Rights Information

© 2019 Zhengji Guo

Keywords

Exponential formula, Malliavin Calculus, option pricing, SABR model, stochastic volatility

Abstract

We develop two pricing formulae for European options in the SABR model with beta= 1 case by means of Malliavin Calculus. We follow the approach of Alòs et al (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation. We derive the Hull-White term, by using the conditional density of the average volatility, and write it as a two-dimensional integral. For the correction part, we use two different approaches. Both approaches rely on the pairing of the exponential formula developed by Jin, Peng, and Schellhorn (2016) with analytical calculations. The first approach, which we call "Dyson series on the return's idiosyncratic noise" yields a complete series expansion but necessitates the calculation of a 7-dimensional integral. Two of these dimensions come from the use of Yor's (1992) formula for the joint density of a Brownian motion and the time-integral of geometric Brownian motion.The second approach, which we call "Dyson series on the common noise" necessitates the calculation of only a one-dimensional integral, but the formula is more complex.

ISBN

9781085784054

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