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

Abstract

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 account$B = (B_t )_{t \geqq 0} $ is evolving according to the “compound interests” formula (1.1), and a risky stock price $S = (S_t )_{t \geqq 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.