How to Hedge an Option Against an Adversary: Black-Scholes Pricing is Minimax Optimal

We consider a popular problem in finance, option pricing, through the lens of an online learning game between Nature and an Investor. In the Black-Scholes option pricing model from 1973, the Investor can continuously hedge the risk of an option by trading the underlying asset, assuming that the asset’s price fluctuates according to Geometric Brownian Motion (GBM). We consider a worst-case model, in which Nature chooses a sequence of price fluctuations under a cumulative quadratic volatility constraint, and the Investor can make a sequence of hedging decisions. Our main result is to show that the value of our proposed game, which is the “regret” of hedging strategy, converges to the Black-Scholes option price. We use significantly weaker assumptions than previous work—for instance, we allow large jumps in the asset price—and show that the Black-Scholes hedging strategy is near-optimal for the Investor even in this non-stochastic framework.