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”, Teor. Veroyatnost. i Primenen., 39:1 (1994), 80–129; Theory Probab. Appl., 39:1 (1994), 61

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Teoriya Veroyatnostei i ee Primeneniya, 1994, Volume 39, Issue 1, Pages 80–129 (Mi tvp3763)

This article is cited in 62 scientific papers (total in 62 papers)

Toward the theory of pricing of options of both European and American types. II. Continuous time

A. N. Shiryaev^a, Yu. M. Kabanov^b, D. O. Kramkov^a, A. V. Melnikov^a

^a Steklov Mathematical Institute, Russian Academy of Sciences
^b Central Economics and Mathematics Institute, RAS

Full-text PDF (2229 kB) Citations (62)

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 $B=(B_t)_{t\ge 0}$ 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, options, hedging strategies, geometric (economic) Brownian motion, standard and exotic options, Black–Scholes formula, put-call parity, martingale and dual martingale measures.

Received: 05.07.1993

English version:
Theory of Probability and its Applications, 1994, Volume 39, Issue 1, Pages 61–102
DOI: https://doi.org/10.1137/1139003

Bibliographic databases:

Document Type: Article

Language: Russian

Citation: 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”, Teor. Veroyatnost. i Primenen., 39:1 (1994), 80–129; Theory Probab. Appl., 39:1 (1994), 61–102

Citation in format AMSBIB

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\transl

\jour Theory Probab. Appl.

\yr 1994

\vol 39

\issue 1

\pages 61--102

\crossref{https://doi.org/10.1137/1139003}

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Linking options:

https://www.mathnet.ru/eng/tvp3763

https://www.mathnet.ru/eng/tvp/v39/i1/p80

Cycle of papers

Toward the theory of pricing of options of both European and American types. I. Discrete time
A. N. Shiryaev, Yu. M. Kabanov, D. O. Kramkov, A. V. Melnikov
Teor. Veroyatnost. i Primenen., 1994, 39:1, 23–79
Toward the theory of pricing of options of both European and American types. II. Continuous time
A. N. Shiryaev, Yu. M. Kabanov, D. O. Kramkov, A. V. Melnikov
Teor. Veroyatnost. i Primenen., 1994, 39:1, 80–129

This publication is cited in the following 62 articles:

Tsvetelin Zaevski, “On the ϵ-optimality of American options”, CFRI, 2025
Tsvetelin S. Zaevski, “Quadratic American Strangle Options in Light of Two-Sided Optimal Stopping Problems”, Mathematics, 12:10 (2024), 1449
Tsvetelin S. Zaevski, “American strangle options with arbitrary strikes”, Journal of Futures Markets, 43:7 (2023), 880
Tsvetelin Zaevski, “Perpetual cancellable American options with convertible features”, Modern Stochastics: Theory and Applications, 2023, 367
A. A. Shishkova, “Raschet aziatskikh optsionov dlya modeli Bleka–Shoulsa”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2018, no. 51, 48–63
Anna Glazyrina, Alexander Melnikov, “Quadratic hedging of equity-linked life insurance contracts under the real-world measure in discrete time”, RDA, 6:2 (2017), 167
Finance Mathematics, 2016, 171
E. Daniliuk, S. Rozhkova, “Hedging of the Barrier Put Option in a Diffusion (B, S) – Market in case of Dividends Payment on a Risk Active”, IFAC-PapersOnLine, 48:25 (2015), 34
Elena Daniliuk, Communications in Computer and Information Science, 564, Information Technologies and Mathematical Modelling - Queueing Theory and Applications, 2015, 304
Patrick Gagliardini, Diego Ronchetti, “Semi-parametric estimation of American option prices”, Journal of Econometrics, 173:1 (2013), 57
Andreeva U.V., Danilyuk E.Yu., Demin N.S., Rozhkova S.V., Pakhomova E.G., “Evropeiskii optsion kupli lukbek s plavayuschim straikom”, Izvestiya tomskogo politekhnicheskogo universiteta, 321:6 (2012), 13–15
Andreeva U.V., Danilyuk E.Yu., Demin N.S., Rozhkova S.V., Pakhomova E.G., “Primenenie veroyatnostnykh metodov k issledovaniyu ekzoticheskikh optsionov kupli evropeiskogo tipa na osnove ekstremalnykh znachenii tseny riskovogo aktiva”, Izvestiya tomskogo politekhnicheskogo universiteta, 321:6 (2012), 5–12
Xiao-feng Yang, Jin-ping Yu, Wen-li Huang, Sheng-hong Li, “Pricing permanent convertible bonds in EVG model”, Appl. Math. J. Chin. Univ., 27:3 (2012), 268
Xiangling Hu, Charles L. Munson, Stergios B. Fotopoulos, “Purchasing decisions under stochastic prices: Approximate solutions for order time, order quantity and supplier selection”, Ann Oper Res, 201:1 (2012), 287
Jan Dhaene, Alexander Kukush, Daniël Linders, “The Multivariate Black & Scholes Market: Conditions for Completeness and No-Arbitrage”, SSRN Journal, 2012
R. V. Ivanov, “Optimal Stopping Problem in a Model with Compensated Refusal of Reward”, Math. Notes, 89:2 (2011), 238–244
N. S. Demin, U. V. Andreeva, “Ekzoticheskie optsiony kupli s ogranicheniem vyplat i garantirovannym dokhodom v modeli Bleka–Shoulsa”, Probl. upravl., 1 (2011), 33–39
Melnikova E.I., Shirshikova L.A., “Primenenie optsionnykh kontraktov dlya povysheniya konkurentosposobnosti promyshlennykh predpriyatii”, Vestnik yuzhno-uralskogo gosudarstvennogo universiteta. seriya: ekonomika i menedzhment, 2011, 131–137
Danilyuk E.Yu., Demin N.S., “Khedzhirovanie optsiona kupli s zadannoi veroyatnostyu na diffuzionnom (b, s)-rynke v sluchae vyplaty dividendov po riskovomu aktivu”, Vestnik tomskogo gosudarstvennogo universiteta. upravlenie, vychislitelnaya tekhnika i informatika, 2011, no. 1, 22–30 Quantile hedging call option in a diffusion (\it{b}, \it{s})-market in case of dividends payment on a risk active
R. V. Ivanov, “On the problem of optimal stopping for the composite Russian option”, Autom. Remote Control, 71:8 (2010), 1602–1607

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Theory of Probability and its Applications

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