Short-maturity asymptotics for VIX and European options in local-stochastic volatility models

We derive the short-maturity asymptotics for European and VIX option prices in local-stochastic volatility models where the volatility follows a continuous-path Markov process. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered. Using large deviations theory methods, the asymptotics for the OTM options are expressed as a two-dimensional variational problem, which is reduced to an extremal problem for a function of two real variables. This extremal problem is solved explicitly in an expansion in log-moneyness. We derive series expansions for the implied volatility for European and VIX options which should be useful for model calibration. We give explicit results for two classes of local-stochastic volatility models relevant in practice, with Heston-type and SABR-type stochastic volatility. The leading-order asymptotics for at-the-money options are computed in closed-form. The asymptotic results reproduce known results in the literature for the Heston and SABR models and for the uncorrelated local-stochastic volatility model. The asymptotic results are tested against numerical simulations for a local-stochastic volatility model with bounded local volatility.