In this paper, we consider price weighted-volatility swap and price weighted-variance swap. The underlying asset considered in this paper is assumed to follow a general stochastic differential equation and exhibits stochastic volatility. We obtain analytical pricing formulas for the weighted-variance swap and approximate expression for the weighted-volatility swap. Nice bounds for the arbitrage-free variance swap price are also found. The proposed pricing formulas are easy to implement in real time and can be applied efficiently for practical applications. We consider the problem of hedging volatility swap with variance swap and obtain analytical formula for the hedge ratio. We also consider a problem of hedging an asset with variance swap and option. We determined the optimal amount of the underlying asset that has to be held for minimizing the hedging error by taking positions in options and weighted-variance swap. A numerical example is also provided.
Variance Swap Premium under Stochastic Volatility and Self-Exciting Jumps