NotesCarrRandomization

Abstract

Carr's randomization and new FFT techniques for fast and accurate pricing of barrier options

Dmitri Boyarchenko University of Chicago

I will explain how Carr's randomization approximation can be applied to the problem of pricing a knock-out option with one or two barriers in a wide class of models of stock prices used in mathematical finance. The approximation yields a backward induction procedure, each of whose steps can be implemented numerically with very high speed and precision. The resulting algorithms are significantly more efficient than the algorithms based on other approaches to the pricing of barrier options. In the first part of my talk I will focus on the classical Black-Scholes model and Kou's double-exponential jump-diffusion model, as well as a class of models that contains those two as special cases, namely, the hyper-exponential jump-diffusion (HEJD) models. For HEJD models, each step in the backward induction procedure for pricing a single or double barrier option can be made very explicit, so that the calculation of an option price using our method takes a small fraction of a second. In the second part of my talk I will discuss other prominent examples of models used in empirical studies of financial markets, including the Variance Gamma model and the CGMY model. In these examples, the aforementioned backward induction procedure can be reduced to computing a sequence of Fourier transforms and inverse Fourier transforms. However, the numerical calculation of Fourier transforms via FFT may lead to significant errors, which are often hard or impossible to control when standard FFT techniques are used. I will describe a new approach to implementing FFT techniques that allows one to control these errors without sacrificing the computational speed. The material I will present is based on joint works with Svetlana Boyarchenko (University of Texas at Austin) and Sergei Levendorskii (University of Leicester).

For more information about upcoming seminars, please visit: http://stevanovichcenter.uchicago.edu/seminars/ <http://stevanovichcenter.uchicago.edu/seminars/#boyarchenko>

Notes

Links

• http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1275666

-- RobGardner - 07 Nov 2008