Closed-Form Formulae for Variance and Volatility Swaps Under Stochastic Volatility With Stochastic Liquidity Risks

We construct a stochastic volatility model considering stochastic liquidity risks when valuing variance and volatility swaps with discrete sampling. We base our model on Heston stochastic volatility, which is adopted for the modeling of stock prices when the market is perfectly liquid. Stock dynamics are further revised by discounting their prices through the employment of mean reverting market liquidity. We convert the stock dynamics under the physical measure into the one under a risk-neutral measure via measure transform, with which the analytical valuation of variance and volatility swaps is realized. By taking the limit of sampling frequency, we further consider how both swaps with continuous sampling can be priced. Numerical implementation is finally carried out, with which the capability of the constructed model in capturing the influence of the two common types of financial risks can be clear.