Robust Dynamic Hedging

We consider a robust approach to price European and American options by modeling the market dynamic as a repeated game between the nature (adversary) and the investor. This model was recently studied by Abernethy et al. [AFW12], which asserted in the case that the payoff function is convex, the no-arbitrage upper bound of a European option price converges to the Black-Scholes model at the limit. In this paper, we offer an elementary analysis, which substantially simplifies the proofs in [AFW12] and settles two unsolved problems there: we give explicit trading strategies to replicate the options, and we quantify the capital budget necessitated from the trader. By leveraging the flexibility of our approach, moreover, we design approximation algorithm to price options with non-convex payoffs via a derandomization of the stochastic mesh method [Gla04,BG04]. We also characterize the limit for a large class of options as diffusion processes with controlled volatilities, which reduce to the Black-Scholes price for convex payoff and a degenerate price for the concave counterpart. Our framework can also be adapted to American options, as well as a semi-adversary model that incorporates rare shocks in asset prices. The latter extension serves as the analog in robust pricing theory to so-called “jump processes” in the stochastic studies on option pricing.