Pricing of Variance and Volatility Swaps with Semi-Markov Volatilities

25 Pages Posted: 21 May 2009  

Anatoliy V. Swishchuk

University of Calgary

Date Written: May 20, 2009


We consider a semi-Markov modulated market consisting of a riskless asset or bond, B, and a risky asset or stock, S, whose dynamics depend on a semi-Markov process x. Using the martingale characterization of semi-Markov processes, we note the incompleteness of semi-Markov modulated markets and find the minimal martingale measure. We price variance (Theorem 1) and volatility (Theorem 2) swaps for stochastic volatilities driven by the semi-Markov processes. We also discuss some extensions of the obtained results such as local semi-Markov volatility, Dupire formula for the local semi-Markov volatility and residual risk associated with the swap pricing.

Keywords: semi-Markov volatility, incomplete market, minimal martingale measure, variance and volatility swaps, Dupire formula, residual risk, risk-minimizing strategy

JEL Classification: G13, C61

Suggested Citation

Swishchuk, Anatoliy V., Pricing of Variance and Volatility Swaps with Semi-Markov Volatilities (May 20, 2009). Available at SSRN: or

Anatoliy V. Swishchuk (Contact Author)

University of Calgary ( email )

University Drive
Calgary, Alberta T2N 1N4

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