Market-based estimation of stochastic volatility models

We propose a method for estimating stochastic volatility models by adapting the HJM approach to the case of volatility derivatives. We characterize restrictions that observed variance swap dynamics have to satisfy to prevent arbitrage opportunities. When the drift of variance swap rates are affine under the pricing measure, we obtain closed form expressions for those restrictions and formulas for forward variance curves. Using data on the S&P500 index and variance swap rates on different time to maturities, we find that linear mean-reverting one factor models provide inaccurate representation of the dynamics of the variance swap rates while two-factor models significantly outperform the former both in and out of sample.