Log-Normal Stochastic Volatility Model With Quadratic Drift

In this paper, we introduce the log-normal stochastic volatility (SV) model with a quadratic drift to allow arbitrage-free valuation of options on assets under money-market account and inverse martingale measures. We show that the proposed volatility process has a unique strong solution, despite non-Lipschitz quadratic drift, and we establish the corresponding Feynman–Kac partial differential equation (PDE) for computation of conditional expectations under this SV model. We derive conditions for arbitrage-free valuations when return–volatility correlation is positive to preclude the “loss of martingality†, which occurs in many traditional SV models. Importantly, we develop an analytic approach to compute an affine expansion for the moment generating function of the log-price, its quadratic variance (QV) and the instantaneous volatility. Our solution allows for semi-analytic valuation of vanilla options under log-normal SV models closing a gap in existing studies.We apply our approach for solving the joint valuation problem of vanilla and inverse options, which are popular in the cryptocurrency option markets. We demonstrate the accuracy of our solution for valuation of vanilla and inverse options.a By calibrating the model to time series of options on Bitcoin over the past four years, we show that the log-normal SV model can work efficiently in different market regimes. Our model can be well applied for modeling of implied volatilities of assets with positive return–volatility correlation.