Abstract
We obtain the maximum entropy distribution for an asset from call and digital option prices. A rigorous mathematical proof of its existence and exponential form is given, which can also be applied to legitimise a formal derivation by Buchen and Kelly (J. Financ. Quant. Anal. 31:143–159, 1996). We give a simple and robust algorithm for our method and compare our results to theirs. We present numerical results which show that our approach implies very realistic volatility surfaces even when calibrating only to at-the-money options. Finally, we apply our approach to options on the S&P 500 index.
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Notes
In Csiszár’s paper, the minus sign in front of the definition of entropy is dropped and its minimisation (rather than maximisation) is studied.
References
Avellaneda, M., Friedman, C., Holmes, R., Samperi, D.: Calibrating volatility surfaces via relative-entropy minimization. Appl. Math. Finance 4, 37–64 (1997)
Borwein, J., Choksi, R., Maréchal, P.: Probability distributions of assets inferred from option prices via the principle of maximum entropy. SIAM J. Optim. 14, 464–478 (2003)
Breeden, D.T., Litzenberger, R.H.: Prices of state-contingent claims implicit in option prices. J. Bus. 51, 621–651 (1978)
Brody, D.C., Buckley, I.R.C., Meister, B.K.: Preposterior analysis for option pricing. Quant. Finance 4, 465–477 (2004)
Brody, D.C., Buckley, I.R.C., Constantinou, I., Meister, B.: Entropic calibration revisited. Phys. Lett. A 337, 257–264 (2005)
Brody, D., Buckley, I.R.C., Constantinou, I.: Option price calibration from Rényi entropy. Phys. Lett. A 366, 298–307 (2007)
Buchen, P.W., Kelly, M.: The maximum entropy distribution of an asset inferred from option prices. J. Financ. Quant. Anal. 31, 143–159 (1996)
Chicago Board Options Exchange Website: CBOE Binary Options, April 2010. www.cboe.com
Coval, J.D., Jakub, J., Stafford, E.: The economics of structured finance. J. Econ. Perspect. 23, 3–25 (2009)
Csiszár, I.: I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3, 146–158 (1975)
Dempster, M.A.H., Medova, E.A., Yang, S.W.: Empirical copulas for CDO tranche pricing using relative entropy. Int. J. Theor. Appl. Finance 10, 679–702 (2007)
Derman, E., Kani, I.: Riding on a smile. Risk 7, 32–39 (1994)
Dupire, B.: Pricing with a smile. Risk 7, 18–20 (1994)
Frittelli, M.: The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance 10, 39–52 (2000)
Gatheral, J.: The Volatility Surface—A Practitioner’s Guide, Wiley Finance. Wiley, New York (2006)
Gulko, L.: The entropic market hypothesis. Int. J. Theor. Appl. Finance 2, 293–329 (1999)
Gulko, L.: The entropy theory of bond option pricing. Int. J. Theor. Appl. Finance 5, 355–383 (2002)
Jäckel, P.: By Implication. Wilmott, pp. 60–66, November 2006
Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957)
Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948)
Acknowledgements
We thank David Chevance, Peter Jäckel, Yannick Malevergne and Wolfgang Scherer for helpful comments and suggestions. We also thank the organisers of the WBS 5th Fixed Income Conference in Budapest, where we had the opportunity to present some of our results.
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Neri, C., Schneider, L. Maximum entropy distributions inferred from option portfolios on an asset. Finance Stoch 16, 293–318 (2012). https://doi.org/10.1007/s00780-011-0167-7
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DOI: https://doi.org/10.1007/s00780-011-0167-7