Abstract
The purpose of this paper is to describe the appropriate mathematical framework for the study of the duality principle in option pricing. We consider models where prices evolve as general exponential semimartingales and provide a complete characterization of the dual process under the dual measure. Particular cases of these models are the ones driven by Brownian motions and by Lévy processes, which have been considered in several papers.
Generally speaking, the duality principle states that the calculation of the price of a call option for a model with price process S=eH (with respect to the measure P) is equivalent to the calculation of the price of a put option for a suitable dual model S′=eH′ (with respect to the dual measure P′). More sophisticated duality results are derived for a broad spectrum of exotic options.
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References
Andreasen, J.: The pricing of discretely sampled Asian and lookback options: a change of numeraire approach. J. Comput. Financ. 2(1), 5–30 (1998)
Barndorff-Nielsen, O.E.: Processes of normal inverse Gaussian type. Financ. Stoch. 2, 41–68 (1998)
Bates, D.S.: The skewness premium: option pricing under asymmetric processes. In: Ritchken, P., Boyle, P.P. (eds.) Advances in Futures and Options Research, vol. 9, pp. 51–82. Elsevier, Amsterdam (1997)
Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)
Carr, P., Bowie, J.: Static simplicity. Risk, pp. 44–50, August 1994. Reprinted In: Jarrow, R. (ed.) Over the Rainbow. Risk Publications, pp. 182–189 (1995). http://www.math.nyu.edu/research/carrp/papers/pdf/staticsimp.pdf
Carr, P., Chesney, M.: American put call symmetry. Preprint, H.E.C. (1996). http://www.math.nyu.edu/research/carrp/papers/pdf/apcs2.pdf
Carr, P., Ellis, K., Gupta, V.: Static hedging of exotic options. J. Financ. 53, 1165–1190 (1998)
Carr, P., Geman, H., Madan, D.B., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75, 305–332 (2002)
Carr, P., Geman, H., Madan, D.B., Yor, M.: Stochastic volatility for Lévy processes. Math. Financ. 13, 345–382 (2003)
Chesney, M., Gibson, R.: State space symmetry and two factor option pricing models. In: Boyle, P.P., Longstaff, F.A., Ritchken, P. (eds.) Advances in Futures and Options Research, vol. 8, pp. 85–112. Elsevier, Amsterdam (1995)
Detemple, J.: American options: symmetry properties. In: Cvitanić, J., Jouini, E., Musiela, M. (eds.) Option Pricing, Interest Rates and Risk Management, pp. 67–104. Cambridge University Press, Cambridge (2001)
Dupire, B.: Pricing with a smile. Risk 7, 18–20 (1994)
Eberlein, E.: Application of generalized hyperbolic Lévy motions to finance. In: Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S.I. (eds.) Lévy Processes: Theory and Applications, pp. 319–336. Birkhäuser, Basel (2001)
Eberlein, E., Keller, U.: Hyperbolic distributions in finance. Bernoulli 1, 281–299 (1995)
Eberlein, E., Kluge, W., Papapantoleon, A.: Symmetries in Lévy term structure models. Int. J. Theor. Appl. Financ. 9, 967–986 (2006)
Eberlein, E., Papapantoleon, A.: Equivalence of floating and fixed strike Asian and lookback options. Stoch. Process. Appl. 115, 31–40 (2005)
Eberlein, E., Papapantoleon, A.: Symmetries and pricing of exotic options in Lévy models. In: Kyprianou, A., Schoutens, W., Wilmott, P. (eds.) Exotic Option Pricing and Advanced Lévy Models, pp. 99–128. Wiley, New York (2005)
Eberlein, E., Prause, K.: The generalized hyperbolic model: financial derivatives and risk measures. In: Geman, H., Madan, D., Pliska, S., Vorst, T. (eds.) Mathematical Finance—Bachelier Congress 2000, pp. 245–267. Springer, New York (2002)
Eberlein, E., Hammerstein, E.A.v.: Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes. In: Dalang, R., Dozzi, M., Russo, F. (eds.) Seminar on Stochastic Analysis, Random Fields and Applications IV. Progress in Probability, vol. 58, pp. 221–264. Birkhäuser, Basel (2004)
Fajardo, J., Mordecki, E.: Duality and derivative pricing with Lévy processes. Preprint, unpublished (2003)
Fajardo, J., Mordecki, E.: Skewness premium with Lévy processes. Working paper, IBMEC. (2006) http://professores.ibmecrj.br/pepe/research/research.htm
Fajardo, J., Mordecki, E.: Symmetry and duality in Lévy markets. Quant. Financ. 6, 219–227 (2006)
Geman, H., El Karoui, N., Rochet, J.-C.: Changes of numéraire, changes of probability measures and option pricing. J. Appl. Probab. 32, 443–458 (1995)
Grabbe, J.O.: The pricing of call and put options on foreign exchange. J. Int. Money Financ. 2, 239–253 (1983)
Henderson, V., Wojakowski, R.: On the equivalence of floating- and fixed-strike Asian options. J. Appl. Probab. 39, 391–394 (2002)
Jacod, J.: Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math., vol. 714. Springer, New York (1979)
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, New York (2003)
Kallsen, J.: Optimal portfolios for exponential Lévy processes. Math. Method. Oper. Res. 51, 357–374 (2000)
Kallsen, J.: A didactic note on affine stochastic volatility models. In: Kabanov, Y., Lipster, R., Stoyanov, J. (eds.) From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, pp. 343–368. Springer, New York (2006)
Kallsen, J., Shiryaev, A.N.: The cumulant process and Esscher’s change of measure. Financ. Stoch. 6, 397–428 (2002)
Küchler, U., Tappe, S.: Bilateral gamma distributions and processes in financial mathematics. Stoch. Process. Appl. 118, 261–283 (2008)
Kyprianou, A.E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, New York (2006)
Madan, D.B., Seneta, E.: The variance gamma (VG) model for share market returns. J. Bus. 63, 511–524 (1990)
Papapantoleon, A.: Applications of semimartingales and Lévy processes in finance: duality and valuation. Ph.D. thesis, University of Freiburg (2006)
Peskir, G., Shiryaev, A.N.: A note on the call-put parity and a call-put duality. Theory Probab. Appl. 46, 167–170 (2002)
Protter, P.: Stochastic Integration and Differential Equations, 3rd edn. Springer, New York (2004)
Rachev, S.T. (ed.): Handbook of Heavy Tailed Distributions in Finance. Elsevier, Amsterdam (2003)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, New York (1999)
Samorodnitsky, G., Taqqu, M.: Stable Non-Gaussian Random Processes. Chapman and Hall, London (1994)
Schoutens, W.: The Meixner process: theory and applications in finance. In: Barndorff-Nielsen, O.E. (ed.) Mini-Proceedings of the 2nd MaPhySto Conference on Lévy Processes, pp. 237–241 (2002)
Schoutens, W.: Lévy Processes in Finance: Pricing Financial Derivatives. Wiley, New York (2003)
Schoutens, W., Teugels, J.L.: Lévy processes, polynomials and martingales. Commun. Stat. Stoch. Models 14, 335–349 (1998)
Schroder, M.: Changes of numeraire for pricing futures, forwards and options. Rev. Financ. Stud. 12, 1143–1163 (1999)
Shepp, L.A., Shiryaev, A.N.: A new look at pricing of the “Russian option”. Theory Probab. Appl. 39, 103–119 (1994)
Shiryaev, A.N.: Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific, Singapore (1999)
Shiryaev, A.N., Kabanov, Y.M., Kramkov, D.O., Mel’nikov, A.: Toward the theory of pricing of options of both European and American types. II. Continuous time. Theory Probab. Appl. 39, 61–102 (1994)
Skiadopoulos, G.: Volatility smile consistent option models: a survey. Int. J. Theor. Appl. Financ. 4, 403–437 (2001)
Vanmaele, M., Deelstra, G., Liinev, J., Dhaene, J., Goovaerts, M.J.: Bounds for the price of discrete arithmetic Asian options. J. Comput. Appl. Math. 185, 51–90 (2006)
Večeř, J.: Unified Asian pricing. Risk 15(6), 113–116 (2002)
Večeř, J., Xu, M.: Pricing Asian options in a semimartingale model. Quant. Financ. 4, 170–175 (2004)
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The second named author acknowledges the financial support from the Deutsche Forschungsgemeinschaft (DFG, Eb 66/9-2). This research was carried out while the third named author was supported by the Alexander von Humboldt foundation.
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Eberlein, E., Papapantoleon, A. & Shiryaev, A.N. On the duality principle in option pricing: semimartingale setting. Finance Stoch 12, 265–292 (2008). https://doi.org/10.1007/s00780-008-0061-0
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DOI: https://doi.org/10.1007/s00780-008-0061-0
Keywords
- Duality principle in option pricing
- Exponential semimartingale model
- Exponential Lévy model
- Call-put duality
- Exotic options