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Optimal importance sampling for the Laplace transform of exponential Brownian functionals

Part of: Limit theorems Mathematical finance

Published online by Cambridge University Press:  21 June 2016

Je Guk Kim
Je Guk Kim*
Affiliation:
The University of Tennessee
*
* Current address: Department of mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul, South Korea, 03722. Email address: jkim74@vols.utk.edu
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Abstract

We present an asymptotically optimal importance sampling for Monte Carlo simulation of the Laplace transform of exponential Brownian functionals which plays a prominent role in many disciplines. To this end we utilize the theory of large deviations to reduce finding an asymptotically optimal importance sampling measure to solving a calculus of variations problem. Closed-form solutions are obtained. In addition we also present a path to the test of regularity of optimal drift which is an issue in implementing the proposed method. The performance analysis of the method is provided through the Dothan bond pricing model.

Keywords

Exponential Brownian functionalMonte Carlo methodimportance samplingcalculus of variationlarge deviationsDothan model

MSC classification

Primary: 91G60: Numerical methods (including Monte Carlo methods)
Secondary: 60F10: Large deviations 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 531 - 542
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1]Clarke, F. (2013).Functional Analysis, Calculus of Variations and Optimal Control.Springer, London.Google Scholar
[2]Comtet, A., Monthus, C. and Yor, M. (1998).Exponential functionals of Brownian motion and disordered systems.J. Appl. Prob. 35, 255–271.Google Scholar
[3]Dembo, A. and Zeitouni, O. (1998).Large Deviations Techniques and Applications, 2nd edn.Springer, New York.CrossRefGoogle Scholar
[4]Dufresne, D. (2001).The integral of geometric Brownian motion.Adv. Appl. Prob. 33, 223–241.CrossRefGoogle Scholar
[5]Higham, D. J. (2001).An algorithmic introduction to numerical simulation of stochastic differential equations.SIAM Rev. 43, 525–546.Google Scholar
[6]Glasserman, P. (2004).Monte Carlo Methods in Financial Engineering.Springer, New York.Google Scholar
[7]Glasserman, P., Heidelberger, P. and Shahabuddin, P. (1999).Asymptotically optimal importance sampling and stratification for pricing path-dependent options.Math. Finance 9, 117–152.Google Scholar
[8]Guasoni, P. and Robertson, S. (2008).Optimal importance sampling with explicit formulas in continuous time.Finance Stoch. 12, 1–19.CrossRefGoogle Scholar
[9]Kallenberg, O. (2002).Foundations of Modern Probability, 2nd edn.Springer, New York.Google Scholar
[10]Linetsky, V. (2004).Spectral expansions for Asian (average price) options.Operat. Res. 52, 856–867.Google Scholar
[11]Morters, P. and Peres, Y. (2010).Brownian Motion.Cambridge University Press.Google Scholar
[12]Pintoux, C. and Privault, N. (2011).The Dothan pricing model revisited.Math. Finance 21, 355–363.Google Scholar
[13]Privault, N. and Uy, W. I. (2013).Monte Carlo computation of the Laplace transform of exponential Brownian functionals.Methodol. Comput. Appl. Prob. 15, 511–524.Google Scholar
[14]Schilder, M. (1966).Some asymptotic formulas for Wiener integrals.Trans. Amer. Math. Soc. 125, 63–85.CrossRefGoogle Scholar
[15]Varadhan, S. R. S. (1966).Asymptotic probabilities and differential equations.Commun. Pure Appl. Math. 19, 261–286.Google Scholar