The weighted majority algorithm
Information and Computation
A decision-theoretic generalization of on-line learning and an application to boosting
EuroCOLT '95 Proceedings of the Second European Conference on Computational Learning Theory
Online trading algorithms and robust option pricing
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Prediction, Learning, and Games
Prediction, Learning, and Games
Probability: Theory and Examples
Probability: Theory and Examples
Regret minimization algorithms for pricing lookback options
ALT'11 Proceedings of the 22nd international conference on Algorithmic learning theory
Pricing exotic derivatives using regret minimization
SAGT'11 Proceedings of the 4th international conference on Algorithmic game theory
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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. A key element of their analysis is that the investor can "hedge" the payout of the option by continuously buying and selling the asset depending on the price fluctuations. More recently, DeMarzo et al. proposed a more robust valuation scheme which does not require any assumption 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 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.