Finance and Stochastics

, Volume 13, Issue 2, pp 269–303

Pricing options under stochastic volatility: a power series approach

Article

DOI: 10.1007/s00780-008-0086-4

Cite this article as:
Antonelli, F. & Scarlatti, S. Finance Stoch (2009) 13: 269. doi:10.1007/s00780-008-0086-4

Abstract

In this paper we present a new approach for solving the pricing equations (PDEs) of European call options for very general stochastic volatility models, including the Stein and Stein, the Hull and White, and the Heston models as particular cases. The main idea is to express the price in terms of a power series of the correlation parameter between the processes driving the dynamics of the price and of the volatility. The expansion is done around correlation zero and each term is identified via a probabilistic expression. It is shown that the power series converges with positive radius under some regularity conditions. Besides, we propose (as in Alós in Finance Stoch. 10:353–365, 2006) a further approximation to make the terms of the series easily computable and we estimate the error we commit. Finally we apply our methodology to some well-known financial models.

Keywords

Options Stochastic volatility SDEs PDEs Duhamel’s principle 

Mathematics Subject Classification (2000)

60H10 91B24 

JEL Classification

C02 G13 

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversità di L’AquilaL’AquilaItaly
  2. 2.School of EconomicsUniversità di Roma Tor VergataRomaItaly

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