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Option contracts are a type of financial derivative that allow investors tohedge risk and speculate on the variation of an asset's future market price. Inshort, an option has a particular payout that is based on the market price foran asset on a given date in the future. In 1973, Black and Scholes proposed avaluation model for options that essentially estimates the tail risk of theasset price under the assumption that the price will fluctuate according togeometric Brownian motion. More recently, DeMarzo et al., among others, haveproposed more robust valuation schemes, where we can even assume an adversarychooses the price fluctuations. This framework can be considered as asequential two-player zero-sum game between the investor and Nature. We analyzethe value of this game in the limit, where the investor can trade at smallerand smaller time intervals. Under weak assumptions on the actions of Nature (anadversary), we show that the minimax option price asymptotically approachesexactly the Black-Scholes valuation. The key piece of our analysis is showingthat Nature's minimax optimal dual strategy converges to geometric Brownianmotion in the limit.

Minimax Option Pricing Meets Black-Scholes in the Limit