A note on the pricing and hedging of volatility derivatives

We consider the pricing of volatility products and especially volatility and variance swaps. Under risk-neutral valuation we provide closed form formulae for volatility-average and variance swaps. Also we provide a general partial differential equation 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 timescale that is fast compared with the lifetime of the contracts.