the Pricing and Hedging of Volatility Derivatives
Sam Howison, Avraam Rafailidis, Henrik Rasmussen
We consider the pricing of a range of volatility derivatives, including
volatility and variance swaps. Under risk-neutral valuation we provide
closed form formulae for volatility-average and variance swaps for a
variety of diffusion and jump-diffusion models for volatility. We describe
a general partial differential equation framework for derivatives that
have an extra dependence on an average of the volatility. We give
approximate solutions of this equation for volatility products written on
assets for which the volatility process fluctuates on a time-scale that is
fast compared with the lifetime of the contracts, analysing both the
``outer'' region and, by matched asymptotic expansions, the ``inner''
boundary layer near expiry.
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