  Option contracts are a type of financial derivative that allow investors to
hedge risk and speculate on the variation of an asset's future market price. In
short, an option has a particular payout that is based on the market price for
an asset on a given date in the future. In 1973, Black and Scholes proposed a
valuation model for options that essentially estimates the tail risk of the
asset price under the assumption that the price will fluctuate according to
geometric Brownian motion. More recently, DeMarzo et al., among others, have
proposed more robust valuation schemes, where we can even assume an adversary
chooses the price fluctuations. This framework can be considered as a
sequential two-player zero-sum 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 (an
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.
