Pricing Variance Swaps Under Fractional Stochastic Volatility Model

Variance swap is a kind of forward constracts whose value is based on the future volatility level of asset.This kind of volatility derivatives provides direct risk exposure to volatility.The article obtained the pricing formula of variance swap under the stochastic volatility model described by the fractional Ornstein-Uhlenbeck process(H ≥1/2)through the risk minimizing hedging strategy.In the article,we choose the fractional Ornstein-Uhlenbeck process(H ≥1/2)to build our stochastic volatility model to describe the value of underlying asset.The risk market described by our model is no arbitrage and incomplete.Regarding to the second fundamental theorem of asset pricing,we know that not all derivatives security can be perfectly hedged.we choose the risk minimizing hedging strategy under the minimal martingale measure to price variance swap.The strategy relaxes the requirements for hedging condition,i.e.we don’t require the strategy to be self-financing,but mean-self-financing and with minimal risk.Firstly,the consistency of deterministic function’s integrals with regarding to fractional brownian motion defined by several different ways is proved and the distribution property of the fractional Ornstein-Uhlenbeck process is obtained(the consistency of Pathwise Riemann-Stieljes integral and the integral defined by Pipiras et al.,as well as the distribution property of the process have been concluded,this paper only gives the proof).Secondly,we prove that the risk market described by our model is no arbitrage and incomplete.Thirdly,the existence of the risk minimizing hedging strategy under the minimal martingale measure of variance swap is proved.We also discussed the reasonability of using the strategy to price by the decomposition in the replicating strategy of variance swap.Finally,we obtained the pricing formula through the strategy.