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Monte Carlo methods for security pricing. (English) Zbl 0901.90007
Summary: The Monte Carlo approach has proved to be a valuable and flexible computational tool in modern finance. This paper discusses some of the recent applications of the Monte Carlo method to security pricing problems, with emphasis on improvements in efficiency. We first review some variance reduction methods that have proved useful in finance. Then we describe the use of deterministic low-discrepancy sequences, also known as quasi-Monte Carlo methods, for the valuation of complex derivative securities. We summarize some recent applications of the Monte Carlo method to the estimation of partial derivatives or risk sensitivities and to the valuation of American options. We conclude by mentioning other applications.

MSC:
91G60 Numerical methods (including Monte Carlo methods)
65C05 Monte Carlo methods
91G20 Derivative securities (option pricing, hedging, etc.)
Software:
TOMS659
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[1] Aiworth, P.; Broadie, M.; Glasserman, P., A comparison of some Monte Carlo and quasi Monte Carlo methods for option pricing, () · Zbl 0888.90010
[2] Andersen, L., Efficient techniques for simulation of interest rate models involving non-linear stochastic differential equations, ()
[3] Andersen, L.; Brotherton-Ratcliffe, R., Exact exotics, Risk, 9, 85-89, (1996), October
[4] Barlow, R.E.; Proschan, F., Statistical theory of reliability and life testing, (1975), Holt, Reinhart and Winston New York · Zbl 0379.62080
[5] Barraquand, J., Numerical valuation of high dimensional multivariate European securities, Management science, 41, 1882-1891, (1995) · Zbl 0852.90021
[6] Barraquand, J.; Martineau, D., Numerical valuation of high dimensional multivariate American securities, Journal of financial and quantitative analysis, 30, 383-405, (1995)
[7] Beaglehole, D.; Dybvig, P.; Zhou, G., Going to extremes: correcting simulation bias in exotic option valuation, Financial analysts journal, 62-68, (1997), (Jan/Feb)
[8] Beckström, R.; Campbell, A., An introduction to VAR, (1995), CATS Software Palo Alto, CA
[9] Berman, L., Comparison of path generation methods for Monte Carlo valuation of single underlying derivative securities, ()
[10] Birge, J.R., Quasi-Monte Carlo approaches to option pricing, ()
[11] Bossaerts, P., Simulation estimators of optimal early exercise, ()
[12] Boyle, P., Options: a Monte Carlo approach, Journal of financial economics, 4, 323-338, (1977)
[13] Boyle, P.; Emanuel, D., The pricing of options on the generalized Mean, ()
[14] Bratley, P.; Fox, B., ALGORITHM 659: implementing Sobol’s quasirandom sequence generator, ACM transactions on mathematical software, 14, 88-100, (1988) · Zbl 0642.65003
[15] Bratley, P.; Fox, B.L.; Niederreiter, H., Implementation and tests of low-discrepancy sequences, ACM transactions on modelling and computer simulation, 2, 195-213, (1992) · Zbl 0846.11044
[16] Broadie, M.; Detemple, J., The valuation of American options on multiple assets, (), to appear in Mathematical Finance · Zbl 0882.90005
[17] Broadie, M.; Detemple, J., American option valuation: new bounds, approximations, and a comparison of existing methods, Review of financial studies, 9, 1211-1250, (1996)
[18] Broadie, M.; Glasserman, P., Estimating security price derivatives by simulation, Management science, 42, 269-285, (1996) · Zbl 0881.90018
[19] Broadie, M.; Glasserman, P., Pricing American-style securities using simulation, (), this issue · Zbl 0901.90009
[20] Caflisch, R.E.; Morokoff, W.; Owen, A., Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension, ()
[21] Carr, P., Deriving derivatives of derivative securities, ()
[22] Carriere, J.F., Valuation of the early-exercise price for derivative securities using simulations and splines, Insurance: mathematics and economics, 19, 19-30, (1996) · Zbl 0894.62109
[23] Carverhill, A.; Pang, K., Efficient and flexible bond option valuation in the heath, jarrow and morton framework, Journal of fixed income, 5, 70-77, (1995)
[24] Cheyette, O., Term structure dynamics and mortgage valuation, Journal of fixed income, 2, 28-41, (1992), (March)
[25] Clewlow, L.; Carverhill, A., On the simulation of contingent claims, Journal of derivatives, 2, 66-74, (1994)
[26] Devroye, L., Non-uniform random variate generation, (1986), Springer New York · Zbl 0593.65005
[27] Dixit, A.; Pindyck, R., Investment under uncertainty, (1994), Princeton University Press Princeton
[28] Duan, J.-C., The GARCH option pricing model, Mathematical finance, 5, 13-32, (1995) · Zbl 0866.90031
[29] Duan, J.-C.; Simonato, J.-G., Empirical martingale simulation for asset prices, () · Zbl 0989.91533
[30] Duffie, D., Dynamic asset pricing theory, (1996), Princeton University Press Princeton, NJ
[31] Duffie, D.; Glynn, P., Efficient Monte Carlo simulation of security prices, Annals of applied probability, 5, 897-905, (1995) · Zbl 0877.65099
[32] Faure, H., Discrépance de suites associées à un système de nuḿeration (en dimension s), Acta arithmetica, 41, 337-351, (1982) · Zbl 0442.10035
[33] Fox, B.L., ALGORITHM 647: implementation and relative efficiency of quasi-random sequence generators, ACM transactions on mathematical software, 12, 362-376, (1986) · Zbl 0615.65003
[34] Fu, M.; Hu, J.Q., Sensitivity analysis for Monte Carlo simulation of option pricing, Probability in the engineering and information sciences, 9, 417-446, (1995) · Zbl 1335.91101
[35] Fu, M.; Madan, D.; Wong, T., Pricing continuous time Asian options: a comparison of analytical and Monte Carlo methods, ()
[36] Geske, R.; Johnson, H.E., The American put options valued analytically, Journal of finance, 39, 1511-1524, (1984)
[37] Glasserman, P., Gradient estimation via perturbation analysis, (1991), Kluwer Academic Publishers Norwell, MA · Zbl 0746.90024
[38] Glasserman, P., Filtered Monte Carlo, Mathematics of operations research, 18, 610-634, (1991) · Zbl 0780.65084
[39] Glasserman, P.; Yao, D.D., Some guidelines and guarantees for common random numbers, Management science, 38, 884-908, (1992) · Zbl 0758.65091
[40] Glynn, P.W., Likelihood ratio gradient estimation: an overview, (), 366-374
[41] Glynn, P.W., Optimization of stochastic systems via simulation, (), 90-105
[42] Glynn, P.W.; Iglehart, D.L., Simulation methods for queues: an overview, Queueing systems, 3, 221-255, (1988) · Zbl 0665.60101
[43] Glynn, P.W.; Whitt, W., Indirect estimation via L = λW, Operations research, 37, 82-103, (1989) · Zbl 0719.62098
[44] Glynn, P.W.; Whitt, W., The asymptotic efficiency of simulation estimators, Operations research, 40, 505-520, (1992) · Zbl 0760.62009
[45] Grant, D.; Vora, G.; Weeks, D., Path-dependent options: extending the Monte Carlo simulation approach, () · Zbl 0902.90006
[46] Halton, J.H., On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numerische Mathematik, 2, 84-90, (1960) · Zbl 0090.34505
[47] Hammersley, J.M.; Handscomb, D.C., Monte Carlo methods, (1964), Chapman & Hall London · Zbl 0121.35503
[48] Haselgrove, C.B., A method for numerical integration, Mathematics of computation, 15, 323-337, (1961) · Zbl 0209.46902
[49] Hlawka, E., Discrepancy and Riemann integration, () · Zbl 0218.10064
[50] Hull, J., Options, futures, and other derivative securities, (1997), Prentice-Hall Englewood Cliffs, NJ
[51] Hull, J.; White, A., The pricing of options on assets with stochastic volatilities, Journal of finance, 42, 281-300, (1987)
[52] Iben, B.; Brotherton-Ratcliffe, R., Credit loss distributions and required capital for derivatives portfolios, Journal of fixed income, 4, 6-14, (1994)
[53] Johnson, H., Options on the maximum or the minimum of several assets, Journal of financial and quantitative analysis, 22, 227-283, (1987)
[54] Johnson, H.; Shanno, D., Option pricing when the variance is changing, Journal of financial and quantitative analysis, 22, 143-151, (1987)
[55] Joy, C.; Boyle, P.P.; Tan, K.S., Quasi-Monte Carlo methods in numerical finance, Management science, 42, 926-938, (1996) · Zbl 0880.90006
[56] Kemna, A.G.Z.; Vorst, A.C.F., A pricing method for options based on average asset values, Journal of banking and finance, 14, 113-129, (1990) · Zbl 0638.90013
[57] Kloeden, P.; Platen, E., Numerical solution of stochastic differential equations, (1992), Springer New York · Zbl 0752.60043
[58] L’Ecuyer, P.; Perron, G., On the convergence rates of IPA and FDC derivative estimators, Operations research, 42, 643-656, (1994) · Zbl 0863.65096
[59] Lavenberg, S.S.; Welch, P.D., A perspective on the use of control variables to increase the efficiency of Monte Carlo simulations, Management science, 27, 322-3356, (1981) · Zbl 0452.65004
[60] Lawrence, D., Aggregating credit exposures: the simulation approach, ()
[61] Marchuk, G.; V. Shaidurov, V., Difference methods and their extrapolations, (1983), Springer New York · Zbl 0511.65076
[62] McKay, M.D.; Conover, W.J.; Beckman, R.J., A comparison of three methods for selecting input variables in the analysis of output from a computer code, Technometrics, 21, 239-245, (1979) · Zbl 0415.62011
[63] Morokoff, W.J.; Caflisch, R.E., Quasi-Monte Carlo integration, Journal of computational physics, 122, 218-230, (1995) · Zbl 0863.65005
[64] Moskowitz, B.; Caflisch, R.E., Smoothness and dimension reduction in quasi-Monte Carlo methods, (), to appear · Zbl 0858.65023
[65] Niederreiter, H., Low discrepancy and low dispersion sequences, Journal of number theory, 30, 51-70, (1988) · Zbl 0651.10034
[66] Niederreiter, H., On the distribution of pseudo-random numbers generated by the linear congruential method. III, Mathematics of computation, 30, 571-597, (1976) · Zbl 0342.65002
[67] Niederreiter, H., Random number generation and quasi-Monte Carlo methods, () · Zbl 0761.65002
[68] Niederreiter, H.; Xing, C., Low-discrepancy sequences and global function fields with many rational places, () · Zbl 0893.11029
[69] Nielsen, S., Important sampling in lattice pricing models, () · Zbl 0881.90015
[70] Ninomiya, S.; Tezuka, S., Toward real-time pricing of complex financial derivatives, Applied mathematical finance, 3, 1-20, (1996) · Zbl 1097.91530
[71] Owen, A., Monte Carlo variance of scrambled equidistribution quadrature, () · Zbl 0890.65023
[72] Owen, A., Randomly permuted (t, m, s)-nets and (t, s)-sequences, () · Zbl 0831.65024
[73] Paskov, S.; Traub, J., Faster valuation of financial derivatives, Journal of portfolio management, 22, 113-120, (1995), Fall
[74] Pollard, D., Convergence of stochastic processes, (1984), Springer New York · Zbl 0544.60045
[75] Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P., Numerical recipes in C: the art of scientific computing, (1992), Cambridge University Press Cambridge · Zbl 0845.65001
[76] Raymar, S.; Zwecher, M., A Monte Carlo valuation of American call options on the maximum of several stocks, ()
[77] Reider, R., ()
[78] Rubinstein, R.; Shapiro, A., Discrete event systems, (1993), Wiley New York
[79] Rust, J., Using randomization to break the curse of dimensionality, () · Zbl 0872.90107
[80] Schwartz, E.S.; Torous, W.N., Prepayment and the valuation of mortgage-backed securities, Journal of finance, 44, 375-392, (1989)
[81] Scott, L.O., Option pricing when the variance changes randomly: theory, estimation, and an application, Journal of financial and quantitative analysis, 22, 419-438, (1987)
[82] Shaw, J., Beyond VAR and stress testing, ()
[83] Sobol, I.M., On the distribution of points in a cube and the approximate evaluation of integrals, USSR computational mathematics and mathematical physics, 7, 86-112, (1967) · Zbl 0185.41103
[84] Spanier, J.; Maize, E.H., Quasi-random methods for estimating integrals using relatively small samples, SIAM review, 36, 18-44, (1994) · Zbl 0824.65009
[85] Stein, M., Large sample properties of simulations using Latin hypercube sampling, Technometrics, 29, 143-151, (1987) · Zbl 0627.62010
[86] Stulz, R.M., Options on the minimum or the maximum of two risky assets, Journal of financial economics, 10, 161-185, (1982)
[87] Tezuka, S., A generalization of faure sequences and its efficient implementation, ()
[88] Tezuka, S., Uniform random numbers: theory and practice, (1995), Kluwer Academic Publishers Boston · Zbl 0841.65004
[89] Tilley, J.A., Valuing American options in a path simulation model, Transactions of the society of actuaries, 45, 83-104, (1993)
[90] Turnbull, S.M.; Wakeman, L.M., A quick algorithm for pricing European average options, Journal of financial and quantitative analysis, 26, 377-389, (1991)
[91] Van Rensberg, J.; Torrie, G.M., Estimation of multidimensional integrals: is Monte Carlo the best method?, Journal of physics A: mathematical and general, 26, 943-953, (1993) · Zbl 0781.65015
[92] Wiggins, J.B., Option values under stochastic volatility: theory and empirical evidence, Journal of financial economics, 19, 351-372, (1987)
[93] Willard, G.A., Calculating prices and sensitivities for path-dependent derivative securities in multifactor models, ()
[94] Worzel, K.J.; Vassiadou-Zeniou, C.; Zenois, S.A., Integrated simulation and optimization models for tracking indices of fixed-income securities, Operations research, 42, 223-233, (1994) · Zbl 0925.90026
[95] Zaremba, S.K., The mathematical basis of Monte Carlo and quasi-Monte Carlo methods, SIAM review, 10, 310-314, (1968) · Zbl 0034.35902
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