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Limiting distributions for minimum relative entropy calibration

Part of: Mathematical economics Probabilistic methods, simulation and stochastic differential equations Communication, information Limit theorems

Published online by Cambridge University Press:  14 July 2016

Łukasz Kruk
Łukasz Kruk*
Affiliation:
Maria Curie-Skłodowska University, Lublin
*
∗ Postal address: Institute of Mathematics, Maria Curie-Skłodowska University, Lublin, Poland. Email address: lkruk@hektor.umcs.lublin.pl
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Abstract

We consider minimum relative entropy calibration of a given prior distribution to a finite set of moment constraints. We show that the calibration algorithm is stable (in the Prokhorov metric) under a perturbation of the prior and the calibrated distributions converge in variation to the measure from which the moments have been taken as more constraints are added. These facts are used to explain the limiting properties of the minimum relative entropy Monte Carlo calibration algorithm.

Keywords

Model calibrationentropyweak convergenceMonte Carlo

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 91B70: Stochastic models 94A17: Measures of information, entropy
Secondary: 65C05: Monte Carlo methods 65C20: Models, numerical methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 35 - 50
Copyright
Copyright © Applied Probability Trust 2004 

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References

Avellaneda, M. (1998). Minimum-entropy calibration of asset-pricing models. Internat. J. Theoret. Appl. Finance 1, 447–472.CrossRefGoogle Scholar
Avellaneda, M., Friedman, C., Holmes, R., and Samperi, D. (1997). Calibrating volatility surfaces via relative entropy minimization. Appl. Math. Finance 4, 37–64.Google Scholar
Avellaneda, M. et al. (2000). Weighted Monte Carlo: a new technique for calibrating asset-pricing models. Internat. J. Theoret. Appl. Finance 4, 91–119.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Buchen, P. W., and Kelly, M. (1996). The maximum entropy distribution of an asset inferred from option prices. J. Financial Quant. Anal. 31, 143–159.Google Scholar
Cover, T. M., and Thomas, J. A. (1991). Elements of Information Theory. John Wiley, New York.Google Scholar
Csiszár, I. (1975). I-divergence geometry of probability distributions and minimization problems. Ann. Prob. 3, 146–158.Google Scholar
Dunford, N., and Schwartz, J. T. (1958). Linear Operators. Part I. John Wiley, New York.Google Scholar
Ethier, S. N., and Kurtz, T. G. (1985). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
Gihman, I. I., and Skorohod, A. V. (1979). Controlled Stochastic Processes. Springer, New York.CrossRefGoogle Scholar
Gulko, L. (1998). The entropy pricing theory. Doctoral Thesis, Yale School of Management, Yale University.Google Scholar
Hubalek, F., and Hudetz, T. (2000). Convergence of minimum entropy option prices for weakly converging incomplete market models. Internat. J. Theoret. Appl. Finance 3, 559–560.CrossRefGoogle Scholar
Hubalek, F., and Schachermayer, W. (1998). When does convergence of asset price processes imply convergence of option prices? Math. Finance 8, 385–403.Google Scholar
Karatzas, I., and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.CrossRefGoogle Scholar
Kloeden, P. E., and Platen, E. (1992). Numerical Solutions of Stochastic Differential Equations. Springer, New York.Google Scholar
Kullback, S. (1967). A lower bound for discrimination information in terms of variation. IEEE Trans. Inf. Theory 13, 126–127.CrossRefGoogle Scholar
Kushner, H. J. (1984). Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory. MIT Press, Cambridge, MA.Google Scholar
Kushner, H. J., and Dupuis, P. G. (1992). Numerical Methods for Stochastic Control Problems in Continuous Time. Springer, New York.Google Scholar
Newman, J. (1999). Model calibration in mathematical finance. Doctoral Thesis, New York University.Google Scholar
Platen, E., and Rebolledo, R. (1996). Principles for modelling financial markets. J. Appl. Prob. 33, 601–613.CrossRefGoogle Scholar
Rümelin, W. (1982). Numerical treatment of stochastic differential equations. SIAM J. Numer. Anal. 19, 604–613.Google Scholar
Samperi, D. (2000). Model calibration using entropy and geometry. Preprint.Google Scholar
Stroock, D. W., and Varadhan, S. R. S. (1969). Diffusion processes with continuous coefficients. II. Commun. Pure Appl. Math. 22, 479–530.Google Scholar