Risk Adjustments of Option Prices under Time-changed Dynamics

We derive a closed-form expansion of option prices in terms of Black-Scholes prices and higher-order Greeks. We show how the true price of an option less its Black-Scholes price is given by a series of premiums on higher-order risks that are not priced under the Black-Scholes model assumptions. The expansion can be used for a broad class of option pricing models with dynamics governed by time-changed Brownian motions. Specifically, we study expansions for exponential Lévy models such as the Variance Gamma and the Normal Inverse Gaussian models as well as their stochastic volatility counterparts, e.g., the VGSV and NIGSV models. Moreover, we consider extensions of the expansion to a more general subclass of affine jump-diffusion models for which the pricing transform may not be known in closed form