A multiscale extension of the Margrabe formula under stochastic volatility. (April 2017)

• Record Type: Journal Article
• Authors: Kim, Jeong-Hoon; Park, Chang-Rae
• Is Part Of: Chaos, solitons and fractals. Volume 97(2017)
• Page Start: 59
• Page End: 65
• Publication Date: 2017-04
• Subjects: Margrabe's formula -- Exchange option -- Stochastic volatility -- Multiscale -- Greeks; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2017.02.006
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: Fast-mean-reverting stochastic volatility model is chosen to extend the classical Margrabe formula. The resultant formula is explicitly given by the greeks of Margrabe price itself. We show how the stochastic volatility corrects the Margrabe price behavior. Abstract: The pricing of financial derivatives based on stochastic volatility models has been a popular subject in computational finance. Although exact or approximate closed form formulas of the prices of many options under stochastic volatility have been obtained so that the option prices can be easily computed, such formulas for exchange options leave much to be desired. In this paper, we consider two different risky assets with two different scales of mean-reversion rate of volatility and use asymptotic analysis to extend the classical Margrabe formula, which corresponds to a geometric Brownian motion model, and obtain a pricing formula under a stochastic volatility. The resultant formula can be computed easily, simply by taking derivatives of the Margrabe price itself. Based on the formula, we show how the stochastic volatility corrects the Margrabe price behavior depending on the moneyness and the correlation coefficient between the two asset prices.