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Tr. Mat. Inst. Steklova, 2002, Volume 237, Pages 80–122 (Mi tm325)  

This article is cited in 16 scientific papers (total in 16 papers)

Bounds on Option Prices for Semimartingale Market Models

A. A. Gushchina, É. Mordeckib

a Steklov Mathematical Institute, Russian Academy of Sciences
b Facultad de Ciencias, Centro de Matemática

Abstract: We propose a methodology for determining the range of option prices of a European option with a convex payoff function in a general semimartingale market model. Prices are obtained as expectations with respect to the set of equivalent martingale measures. Since the set of prices is an interval on the real line, two main questions are considered: (i) how to find upper and lower estimates for the range of prices, and (ii) how to establish the attainability of these estimates. To solve the first question, we introduce a partial ordering in the set of distributions of discounted stock prices (adapted from the theory of statistical experiments), which allows us to find extremal distributions and, accordingly, the upper and lower bounds for the range of option prices. The weak convergence of probability measures is used to answer the second question, whether the bounds obtained at the first step are exact. Exploiting stochastic calculus, we give answers to both questions in terms (the most natural for this problem) of predictable characteristics of the stochastic logarithm of a discounted stock price process. Special attention is given to two examples: a discrete-time and a diffusion-with-jumps market models.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 237, 73–113

Bibliographic databases:
UDC: 519.2
Received in February 2001

Citation: A. A. Gushchin, É. Mordecki, “Bounds on Option Prices for Semimartingale Market Models”, Stochastic financial mathematics, Collected papers, Tr. Mat. Inst. Steklova, 237, Nauka, MAIK Nauka/Inteperiodika, M., 2002, 80–122; Proc. Steklov Inst. Math., 237 (2002), 73–113

Citation in format AMSBIB
\Bibitem{GusMor02}
\by A.~A.~Gushchin, \'E.~Mordecki
\paper Bounds on Option Prices for Semimartingale Market Models
\inbook Stochastic financial mathematics
\bookinfo Collected papers
\serial Tr. Mat. Inst. Steklova
\yr 2002
\vol 237
\pages 80--122
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm325}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1976509}
\zmath{https://zbmath.org/?q=an:1113.91319}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2002
\vol 237
\pages 73--113


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    2. Kirch M., Runggaldier W.J., “Efficient hedging when asset prices follow a geometric Poisson process with unknown intensities”, SIAM J. Control Optim., 43:4 (2004), 1174–1195  crossref  mathscinet  zmath  isi  scopus
    3. Møller Th., “Stochastic orders in dynamic reinsurance markets”, Finance Stoch., 8:4 (2004), 479–499  crossref  mathscinet  isi  scopus
    4. Bergenthum J., Rüschendorf L., “Comparison of option prices in semimartingale models”, Finance Stoch., 10:2 (2006), 222–249  crossref  mathscinet  zmath  isi  scopus
    5. Branger N., Mahayni A., “Tractable hedging: An implementation of robust hedging strategies”, Journal of Economic Dynamics & Control, 30:11 (2006), 1937–1962  crossref  mathscinet  zmath  isi  scopus
    6. A. A. Gushchin, “On extension of $f$-divergence”, Theory Probab. Appl., 52:3 (2008), 439–455  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Schied A., Stadje M., “Robustness of delta hedging for path-dependent options in local volatility models”, J. Appl. Probab., 44:4 (2007), 865–879  crossref  mathscinet  zmath  isi  scopus
    8. Bergenthum J., Rüdschendorf L., “Comparison of semimartingales and Levy processes”, Ann. Probab., 35:1 (2007), 228–254  crossref  mathscinet  zmath  adsnasa  isi  scopus
    9. Bergenthum J., Rueschendorf L., “Convex ordering criteria for Levy processes”, Advances in Data Analysis and Classification, 1:2 (2007), 143–173  crossref  mathscinet  zmath  isi  scopus
    10. Rudloff B., “Coherent hedging in incomplete markets”, Quant. Finance, 9:2 (2009), 197–206  crossref  mathscinet  zmath  isi  scopus
    11. S. A. Khihol, “Averaging local characteristics makes a semimartingale with independent increments closer to Lévy processes”, Russian Math. Surveys, 65:2 (2010), 386–387  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. D. B. Rokhlin, “Recurrence relations for price bounds of contingent claims in discrete time market models”, Theory Probab. Appl., 56:1 (2012), 72–95  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. S. A. Khihol, “Averaging the local characteristics brings a semimartingale with independent increments closer to Lévy processes”, Theory Probab. Appl., 58:3 (2014), 413–429  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    14. Bellini F., Pellerey F., Sgarra C., Sekeh S.Ya., “Comparison Results For Garch Processes”, J. Appl. Probab., 51:3 (2014), 685–698  crossref  mathscinet  zmath  isi
    15. Rueschendorf L., Schnurr A., Wolf V., “Comparison of time-inhomogeneous Markov processes”, Adv. Appl. Probab., 48:4 (2016), 1015–1044  crossref  mathscinet  zmath  isi  scopus
    16. Deaconu M., Lejay A., Salhi Kh., “Approximation of Cvar Minimization For Hedging Under Exponential-Levy Models”, J. Comput. Appl. Math., 326 (2017), 171–182  crossref  mathscinet  zmath  isi  scopus
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