Option Pricing & Partial Hedging: Theory of Polish Options

The twin problems of hedging and pricing of options in discrete-time markets are analyzed. We consider trading strategies consisting of one stock and one bond. The bond price rises deterministically over time while the stock price can change in several (more than two) ways at each instant of trading. Given such stock price movements, perfect hedging is not possible, and arbitrage arguments alone are not suucient. We determine hedging and bid and ask prices by balancing expected gain against risk. Using a recent approach of Bouchaud and Sornette, we work out in detail the case where the mean rate of return of the stock diiers from that of the bond. We identify a new kind of strategy open to operators that are suuciently insensitive to risk. We nd a candidate for market price of risky options, which reduces to the Black-Scholes prescription when risk can be eliminated. We report on data on stock price movements on the Warsaw Stock Exchange, and show that they are well described by a simple model where prices on each day can either increase, decrease or stay the same. We work out the details of the option pricing and hedging problems in this case.