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Teor. Veroyatnost. i Primenen., 1994, Volume 39, Issue 1, Pages 80–129 (Mi tvp3763)  

This article is cited in 27 scientific papers (total in 27 papers)

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

A. N. Shiryaeva, Yu. M. Kabanovb, D. O. Kramkova, A. V. Melnikova

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

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.

Full text: PDF file (2229 kB)

English version:
Theory of Probability and its Applications, 1994, 39:1, 61–102

Bibliographic databases:

Received: 05.07.1993

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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\by A.~N.~Shiryaev, Yu.~M.~Kabanov, D.~O.~Kramkov, A.~V.~Melnikov
\paper Toward the theory of pricing of options of both European and American types.~II. Continuous time
\jour Teor. Veroyatnost. i Primenen.
\yr 1994
\vol 39
\issue 1
\pages 80--129
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1348191}
\zmath{https://zbmath.org/?q=an:0833.60065}
\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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    2. E. V. Yagnyatinskii, “A problem of optimal sequential investing”, Russian Math. Surveys, 52:4 (1997), 850–851  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. S. S. Artemev, A. A. Nosikova, S. V. Soloboev, “Metod Monte-Karlo dlya modelirovaniya tseny aktsii”, Sib. zhurn. vychisl. matem., 3:1 (2000), 1–10  mathnet  zmath
    4. A. V. Melnikov, “On the Unity of Quantitative Methods of Pricing in Finance and Insurance”, Proc. Steklov Inst. Math., 237 (2002), 50–72  mathnet  mathscinet  zmath
    5. F. S. Nasyrov, “Symmetric Integrals and Their Application in Financial Mathematics”, Proc. Steklov Inst. Math., 237 (2002), 256–269  mathnet  mathscinet  zmath
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    7. Kisielewicz M., Michta M., Motyl J., “Set valued approach to stochastic control part II (viability and semimartingale issues)”, Dynamic Systems and Applications, 12:3–4 (2003), 433–466  mathscinet  zmath  isi
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    9. Asmussen S., Avram F., Pistorius M.R., “Russian and American put options under exponential phase–type Levy models”, Stochastic Processes and Their Applications, 109:1 (2004), 79–111  crossref  mathscinet  zmath  isi
    10. Duistermaat J.J., Kyprianou A.E., van Schaik K., “Finite expiry Russian options”, Stochastic Processes and Their Applications, 115:4 (2005), 609–638  crossref  mathscinet  zmath  isi
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    14. Fotopoulos S.B., Hu X., Munson C.L., “Flexible supply contracts under price uncertainty”, European Journal of Operational Research, 191:1 (2008), 253–263  crossref  mathscinet  zmath  isi
    15. Eberlein E., Papapantoleon A., Shiryaev A.N., “On the duality principle in option pricing: semimartingale setting”, Finance and Stochastics, 12:2 (2008), 265–292  crossref  mathscinet  zmath  isi
    16. Mykhailo Pupashenko, Alexander Kukush, “Reselling of european option if the implied volatility varies as Cox-Ingersoll-Ross process”, Theory Stoch. Process., 14(30):4 (2008), 114–128  mathnet
    17. Danilyuk E.Yu., Demin N.S., “Khedzhirovanie optsiona prodazhi s zadannoi veroyatnostyu v sluchae vyplaty dividendov po riskovomu aktivu”, Vestn. Tomskogo gos. un-ta. Upravlenie, vychislitelnaya tekhnika i informatika, 2009, no. 4, 32–42
    18. R. V. Ivanov, “On the problem of optimal stopping for the composite Russian option”, Autom. Remote Control, 71:8 (2010), 1602–1607  mathnet  crossref  mathscinet  zmath  isi
    19. U. V. Andreeva, N. S. Demin, A. V. Erlykova, E. A. Pan'shina, “Exotic European options with restrictions on the payoffs”, Autom. Remote Control, 71:9 (2010), 1864–1878  mathnet  crossref  mathscinet  zmath  isi
    20. Fotopoulos S.B., Jandhyala V.K., Khapalova E., “Exact Asymptotic Distribution of Change-Point Mle for Change in the Mean of Gaussian Sequences”, Annals of Applied Statistics, 4:2 (2010), 1081–1104  crossref  zmath  isi
    21. R. V. Ivanov, “Optimal Stopping Problem in a Model with Compensated Refusal of Reward”, Math. Notes, 89:2 (2011), 238–244  mathnet  crossref  crossref  mathscinet  isi
    22. 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  mathnet
    23. 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  elib
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