Short-maturity options on realized variance in local-stochastic volatility models

We derive the short-maturity asymptotics for prices of options on realized variance in local-stochastic volatility models. We consider separately the short-maturity asymptotics for out-of-the-money and in-the-money options cases. The analysis for the out-of-the-money case uses large deviations theory and the solution for the rate function involves solving a two-dimensional variational problem. In the special case when the Brownian noises in the asset price dynamics and the volatility process are uncorrelated, we solve this variational problem explicitly. For the correlated case, we obtain upper and lower bounds for the rate function, as well as an expansion around the at-the-money point. Numerical simulations of the prices of variance options in a local-stochastic volatility model with bounded local volatility are in good agreement with the asymptotic results for sufficiently small maturity. The leading-order asymptotics for at-the-money options on realized variance is dominated by fluctuations of the asset price around the spot value, and is computed in closed form.