Regret Minimization Algorithms for Pricing Lookback Options

In this work, we extend the applicability of regret minimization to pricing financial instruments, following the work of [10]. More specifically, we consider pricing a type of exotic option called a fixed-strike lookback call option. A fixed-strike lookback call option has a known expiration time, at which the option holder has the right to receive the difference between the maximal price of a stock and some pre-agreed price. We derive upper bounds on the price of these options, assuming an arbitrage-free market, by developing two-way trading algorithms. We construct our trading algorithms by combining regret minimization algorithms and one-way trading algorithms. Our model assumes upper bounds on the absolute daily returns, overall quadratic variation, and stock price, otherwise allowing for fully adversarial market behavior.