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Volume 15,     Number 3,     Fall 2007

 

CHANGE OF TIME METHOD IN
MATHEMATICAL FINANCE
ANATOLIY SWISHCHUK

Abstract. In this paper, we consider applications of the change of time method to study the following three models in finance: the geometrical Brownian motion model for stock prices, the mean-reverting model for commodity asset prices and Heston's stochastic volatility model for stock prices. We apply the change of time method to derive the well-known Black-Scholes formula for European call options and to derive an explicit option pricing formula for a European call option for a mean-reverting model for commodity prices. We also derive explicit formulas for variance and volatility swaps for financial markets with a stochastic volatility following a Cox-Ingersoll-Ross process [21]. Two numerical examples are presented. One is the S&P60 Canada Index (January 1997- February 2002) to price variance and volatility swaps for the Heston model and the other is the AECO Natural Gas Index, (1 May 1998- 30 April 1999), to price a European call option for the mean-reverting asset model.

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© 2007, Canadian Applied Mathematics Quarterly (CAMQ)