Variance Swaps with Deterministic and Stochastic Correlations

As market observations say that many financial quantities are correlated in a time dependent, nonlinear or unpredictable way, in this study, we present an approach to price discretely sampled variance swaps based on the Heston model extended by incorporating deterministic or stochastic correlation between an underlying asset and its variance. We obtain a closed form exact formula for the fair delivery prices under the deterministic correlation model and an affine approximation formula under the stochastic correlation model. A comparison with Monte–Carlo simulations supports the validity of the pricing formulas. Based on the analytic results, we find that the fair delivery price increases as time to maturity or leverage effect increases or sampling frequency decreases. On the other hand, the impact scale of the correlation volatility is so imperceptible that the time dependent deterministic correlation model can still be a good proxy of the stochastic correlation environment in the case of variance swap pricing, while the approximation formula with the stochastic correlation is better than the exact formula with the deterministic correlation in computing sense.