Geodesic
Toward the theory of pricing of options of both European and American types.~I. Discrete time
Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 1, pp. 23-79

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This paper consisting of two parts (I — discrete time, II — continuous time [19]) considers the main concepts, statements of problems, and results of financial mathematics in connection with options and option contract pricing as a kind of derivative securities. In § 1 it is assumed that the contracts are exercised in discrete $(B,S)$-market. There are two assets: riskless bank account $B=(B_n )_{n\ge 0}$ and risky stock $S=(S_n )_{n\ge 0}$, European as well as American options are examined. Special attention is paid to the “martingale” methods of option pricing and hedging strategies in particular for call options and put options.
Keywords: security market, bonds and stocks, bank account, American and European options, rational cost (fair price), hedging strategies, Markov times, optimal stopping rules, market completeness.
Mots-clés : martingales, arbitrage 
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     title = {Toward the theory of pricing of options of both {European} and {American} {types.~I.} {Discrete} time},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     year = {1994},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1994_39_1_a1/}
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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.~I. Discrete time. Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 1, pp. 23-79. http://geodesic.mathdoc.fr/item/TVP_1994_39_1_a1/