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Number of paths versus number of basis functions in American option pricing. (English) Zbl 1062.60041
Summary: An American option grants the holder the right to select the time at which to exercise the option, so pricing an American option entails solving an optimal stopping problem. Difficulties in applying standard numerical methods to complex pricing problems have motivated the development of techniques that combine Monte Carlo simulation with dynamic programming. One class of methods approximates the option value at each time using a linear combination of basis functions, and combines Monte Carlo with backward induction to estimate optimal coefficients in each approximation. We analyze the convergence of such a method as both the number of basis functions and the number of simulated paths increase. We get explicit results when the basis functions are polynomials and the underlying process is either Brownian motion or geometric Brownian motion. We show that the number of paths required for worst-case convergence grows exponentially in the degree of the approximating polynomials in the case of Brownian motion and faster in the case of geometric Brownian motion.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
65C05 Monte Carlo methods
65C50 Other computational problems in probability (MSC2010)
60G35 Signal detection and filtering (aspects of stochastic processes)
91G60 Numerical methods (including Monte Carlo methods)
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References:
[1] Arvanitis, A. and Gregory, J. (2001). Credit : The Complete Guide to Pricing , Hedging and Risk Management . Risk Books, London.
[2] Becherer, D. (2001). Rational hedging and valuation with utility-based preferences. Ph.D. thesis, TU Berlin. Available at edocs.tu-berlin.de/diss/2001/becherer_dirk.htm.
[3] Becherer, D. (2004). Utility-indifference hedging and valuation via reaction–diffusion systems. Proc. Roy. Soc. London Ser. A 460 27–51. · Zbl 1087.91019
[4] Bélanger, A., Shreve, S. and Wong, D. (2004). A general framework for pricing credit risk. Math. Finance 14 317–350. · Zbl 1134.91395
[5] Bielecki, T. and Rutkowski, M. (2002). Credit Risk : Modelling , Valuation and Hedging . Springer, Berlin. · Zbl 0979.91050
[6] Blanchet-Scalliet, C. and Jeanblanc, M. (2004). Hazard rate for credit risk and hedging defaultable contingent claims. Finance and Stochastics 8 145–159. · Zbl 1052.91036
[7] Brémaud, P. (1981). Point Processes and Queues . Springer, Berlin.
[8] Davis, M. and Lischka, F. (2002). Convertible bonds with market risk and credit risk. In Applied Probability. Studies in Advanced Mathematics (D. Chan, Y.-K. Kwok, D. Yao and Q. Zhang, eds.) 45–58. Amer. Math. Soc., Providence, RI. · Zbl 1029.91033
[9] Davis, M. and Lo, V. (2001). Modelling default correlation in bond portfolios. In Mastering Risk 2 . Applications (C. Alexander, ed.) 141–151. Financial Times/Prentice Hall, Englewood Cliffs, NJ.
[10] Di Masi, G., Kabanov, Y. and Runggaldier, W. (1994). Mean-variance hedging of options on stocks with Markov volatilities. Theory Probab. Appl. 39 172–182. · Zbl 0836.60075
[11] Duffie, D. and Epstein, L. (1992). Stochastic differential utility. Econometrica 60 353–394. · Zbl 0763.90005
[12] Duffie, D., Schroder, M. and Skiadas, C. (1996). Recursive valuation of defaultable securities and the timing of resolution of uncertainty. Ann. Appl. Probab. 6 1075–1090. · Zbl 0868.90008
[13] Föllmer, H. and Schweizer, M. (1991). Hedging of contingent claims under incomplete information. In Applied Stochastic Analysis. Stochastics Monographs 5 (M. H. A. Davis and R. J. Elliott, eds.) 389–414. Gordon and Breach, New York. · Zbl 0738.90007
[14] Föllmer, H. and Sondermann, D. (1986). Hedging of non-redundant contingent claims. In Contributions to Mathematical Economics in Honor of Gerard Debreu (W. Hildenbrand and A. Mas-Colell, eds.) 205–223. North-Holland, Amsterdam. · Zbl 0663.90006
[15] Freidlin, M. (1985). Functional Integration and Partial Differential Equations. Princeton, New Jersey. · Zbl 0568.60057
[16] Friedman, A. (1975). Stochastic Differential Equations and Applications 1 . Academic Press, New York. · Zbl 0323.60056
[17] Heath, D. and Schweizer, M. (2001). Martingales versus PDEs in finance: An equivalence result with examples. J. Appl. Probab. 37 947–957. · Zbl 0996.91069
[18] Jacod, J. and Protter, P. (1982). Quelques remarques sur un nouveau type d’équations différentielles stochastiques. Séminaire de Probabilités XVI. Lecture Notes in Math. 920 447–458. Springer, Berlin. · Zbl 0482.60056
[19] Jarrow, R. and Yu, F. (2001). Counterparty risk and the pricing of derivative securities. J. Finance 56 1765–1799.
[20] Jeanblanc, M. and Rutkowski, M. (2003). Modelling and hedging of credit risk. In Credit Derivatives : The Definitive Guide (J. Gregory, ed.) 385–416. Risk Books, London.
[21] Krylov, N. (1987). Nonlinear Elliptic and Parabolic Equations of the Second Order. Reidel Publishing and Kluver Academic Publishers, Dordrecht. · Zbl 0619.35004
[22] Kunita, H. (1984). Stochastic differential equations and stochastic flows of diffeomorphisms. École d ’ Été de Probabilités de Saint-Flour XII. Lecture Notes in Math. 1097 142–303. Springer, Berlin. · Zbl 0554.60066
[23] Kusuoka, S. (1999). A remark on default risk models. Advances in Mathematical Economics 1 69–82. · Zbl 0939.60023
[24] Lando, D. (1998). On Cox processes and credit risky securities. Review of Derivatives Research 2 99–120. · Zbl 1274.91459
[25] Madan, D. and Unal, H. (1998). Pricing the risk of default. Review of Derivatives Research 2 121–160. · Zbl 1274.91426
[26] Møller, T. (2001). Risk-minimizing hedging strategies for insurance payment processes. Finance and Stochastics 5 419–446. · Zbl 0983.62076
[27] Møller, T. (2003). Indifference pricing of insurance contracts in a product space model. Finance and Stochastics 7 197–217. · Zbl 1038.62097
[28] Pardoux, E. (1999). BSDEs, weak convergence and homogenization of semilinear PDEs. In Nonlinear Analysis , Differential Equations and Control (F. H. Clarke and R. J. Stern, eds.) 503–549. Kluwer Academic, Dordrecht. · Zbl 0959.60049
[29] Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin. · Zbl 0516.47023
[30] Protter, P. (2004). Stochastic Integration and Differential Equations . Springer, Berlin. · Zbl 1041.60005
[31] Ramlau-Hansen, H. (1990). Thiele’s differential equation as a tool in product development in life insurance. Scand. Actuar. J. 97–104. · Zbl 0746.62101
[32] Schönbucher, P. J. (2003). Credit Derivatives Pricing Models . Wiley, Chichester.
[33] Schweizer, M. (1991). Option hedging for semimartingales. Stochastic Process. Appl. 37 339–363. · Zbl 0735.90028
[34] Schweizer, M. (2001). A guided tour through quadratic hedging approaches. In Option Pricing , Interest Rates and Risk Management (E. Jouini, J. Cvitanić and M. Musiela, eds.) 538–574. Cambridge Univ. Press. · Zbl 0992.91036
[35] Smoller, J. (1994). Shock Waves and Reaction-Diffusion Equations . Springer, Berlin. · Zbl 0807.35002
[36] Steffensen, M. (2000). A no arbitrage approach to Thiele’s differential equation. Insurance Math. Econom. 27 201–214. · Zbl 0994.91032
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