Log-normal Stochastic Volatility Model with Quadratic Drift: Applications to Assets with Positive Return-Volatility Correlation and to Inverse Martingale Measures

Abstract

We show that the application of conventional stochastic volatility (SV) models may not be feasible for the arbitrage-free valuation of derivative securities on assets with positive return-volatility correlation under the money-market account (MMA) measure and under the inverse price measure. The existence of the inverse martingale measure, where an asset is used as the numeraire itself, is critical for valuation of options on cryptocurrencies, where most traded options have inverse paysoffs.

We introduce the log-normal SV model with a quadratic drift, which allows for the arbitrage-free valuation of options on assets with positive return-volatility correlation under both the MMA and the inverse measures. We show that the proposed volatility process has a strong solution. We then 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.

We finally demonstrate application of our model and the accuracy of our solution for valuation and calibration of listed options on assets with positive return-volatility correlation and for inverse options on Bitcoin cryptocurrency.

Keywords: Log-normal stochastic volatility, Non-affine models, Closed-form solution, Moment generating function, cryptocurrency derivatives, Quadratic variance

JEL Classification: G13, C63