Toward the Theory of Pricing of Options of Both European and American Types. II. Continuous Time

In the first part of the paper [29] the options pricing theory was developed under the assumption that a $(B,S)$-market is discrete (in space and in time). It is assumed in the present text that a $(B,S)$-market is operating continuously in time. The riskless bank accountB = (Bₜ )t 0 is evolving according to the “compound interests” formula (1.1), and a risky stock price S = (Sₜ )t 0 is governed by geometric Brownian motion (1.4). The “martingale” pricing theory is presented for fair (rational) option price, hedging strategies, and rational expiration times. The Black-Scholes formula for a standard European call option is derived. The paper considers a number of other particular examples of European as well as American options.