Abstract

Option contracts are a type of financial derivative that allow investors to hedge risk and speculate on an asset’s future market price. In 1973, Black and Scholes proposed a valuation model for options that essentially estimates the tail risk of the asset under the assumption that the price will fluctuate according to geometric Brownian motion. More recently, DeMarzo et al. proposed a more robust valuation scheme which has much weaker assumptions on the price path; indeed, in their model the asset’s price can even be chosen adversarially. This framework can be considered as a sequential two-player zero-sum regret game between the investor and Nature. We analyze the value of this game in the limit, where the investor can trade at smaller and smaller time intervals. Under weak assumptions on the actions of Nature (the adversary), we show that the minimax option price asymptotically approaches exactly the Black-Scholes valuation. The key piece of our analysis is showing that Nature’s minimax optimal dual strategy converges to geometric Brownian motion in the limit. Joint work with Jake Abernethy and Andre Wibisono.