Geodesic
Toward the theory of pricing of options of both European and American types.~II. Continuous time
Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 1, pp. 80-129

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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 $B=(B_t)_{t\ge 0}$ is evolving according to the “compound interests” formula (1.1), and a risky stock price $S=(S_t)_{t\ge 0}$ 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.
Keywords: risky and riskless securities, hedging strategies, geometric (economic) Brownian motion, standard and exotic options, Black–Scholes formula, put-call parity, martingale and dual martingale measures.
Mots-clés : options 
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     title = {Toward the theory of pricing of options of both {European} and {American} {types.~II.} {Continuous} time},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     year = {1994},
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A. N. Shiryaev; Yu. M. Kabanov; D. O. Kramkov; A. V. Melnikov. Toward the theory of pricing of options of both European and American types.~II. Continuous time. Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 1, pp. 80-129. http://geodesic.mathdoc.fr/item/TVP_1994_39_1_a2/