Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-27T02:08:11.242Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Valuation of High Dimensional Multivariate American Securities

Published online by Cambridge University Press:  06 April 2009

Jérôme Barraquand  and
Didier Martineau
Jérôme Barraquand
Affiliation:
Salomon Brothers International Ltd., Victoria Plaza, 111 Buckingham Palace Road, London SW1W 0SB, UK
Didier Martineau
Affiliation:
Long-Term Capital Management, 61 Conduit Street, London W1R 9FD, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Several efficient numerical lattice-based techniques exist for pricing American securities depending on one or few (up to three) risk sources. However, these methods cannot be used for high dimensional problems, since their memory requirement is exponential in the number of risk sources. We present an efficient numerical technique that combines Monte Carlo simulation with a particular partitioning method of the underlying assets space, which we call Stratified State Aggregation (SSA). Using this technique, we can compute accurate approximations of prices of American securities with an arbitrary number of underlying assets. Our numerical experiments show that the method is practical for pricing American claims depending on up to 400 risk sources.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 30 , Issue 3 , September 1995 , pp. 383 - 405
Copyright
Copyright © School of Business Administration, University of Washington 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barone-Adesi, G., and Elliott, R.. “Approximations for the Values of American Options.” Stochastic Analysis and Applications, 9 (1991), 115–131.CrossRefGoogle Scholar
Barone-Adesi, G., and Whaley, R. E.. “Efficient Analytic Approximation of American Option Values.” Journal of Finance, 42 (1987), 301–320.CrossRefGoogle Scholar
Barraquand, J. “Numerical Valuation of High Dimensional Multivariate European Securities.” Management Science (forthcoming 1995).CrossRefGoogle Scholar
Barraquand, J., and Pudet, T.. “Pricing of American Path-Dependent Contingent Claims.” Mathematical Finance (forthcoming 1995).CrossRefGoogle Scholar
Bellman, R.Dynamic Programming. Princeton, NJ: Princeton Univ. Press (1957).Google ScholarPubMed
Bensoussan, A. “On the Theory of Option Pricing.” Acta Applicandae Mathematicae, 2 (1984), 139–158.CrossRefGoogle Scholar
Bertsekas, D. P.Dynamic Programming, Deterministic and Stochastic Models. Englewood Cliffs, NJ: Prentice Hall, Inc. (1987).Google Scholar
Bossaerts, P. “Simulation Estimators of Optimal Early Exercise.” Unpubl. Manuscript, Graduate School of Industrial Administration, Carnegie Mellon Univ. (1989).Google Scholar
Boyle, P. P. “Options: A Monte Carlo Approach.” Journal of Financial Economics, 4 (1977), 323–338.CrossRefGoogle Scholar
Boyle, P. P. “The Quality Option and the Timing Option in Futures Contracts.” Journal of Finance, 44 (1989), 101–113.CrossRefGoogle Scholar
Boyle, P. P.; Evnine, J.; and Gibbs, S.. “Numerical Evaluation of Multivariate Contingent Claims.” Review of Financial Studies, 2 (1989), 241–250.CrossRefGoogle Scholar
Boyle, P. P., and Tse, Y.. “An Algorithm for Computing Values of Options on the Maximum or Minimum of Several Assets.” Journal of Financial and Quantitative Analysis, 25 (1990), 215–227.CrossRefGoogle Scholar
Brennan, M., and Schwartz, E.. “Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis.” Journal of Financial and Quantitative Analysis, 13 (1978), 461–474.CrossRefGoogle Scholar
Brennan, M., and Schwartz, E.. “A Continuous Time Approach to the Pricing of Bonds.” Journal of Banking and Finance, 3 (1979), 133–155.CrossRefGoogle Scholar
Cheng, S. “Pricing Models for Multiple Currency Option Bonds.” Unpubl. Manuscript, Graduate School of Business, Stanford Univ. (1987).Google Scholar
Cox, J., and Rubinstein, M.. Options Markets. Englewood Cliffs, NJ: Prentice Hall (1985).Google Scholar
Cox, J. C.; Ross, S. A.; and Rubinstein, M.. “Option Pricing: A Simplified Approach.” Journal of Financial Economics, 7 (1979), 229–263.CrossRefGoogle Scholar
Duffle, D.Security Markets: Stochastic Models. Boston, MA.: Academic Press (1988).Google Scholar
Duffle, D.Dynamic Asset Pricing Theory. Princeton, NJ: Princeton Univ. Press (1992).Google Scholar
Fabozzi, F. J., and Pollack, I. M.. The Handbook of Fixed Income Securities. Homewood, IL.: Dow Jones-Irwin, Second Edition (1987).Google Scholar
Geske, R., and Johnson, H.. “The American Put Option Valued Analytically.” Journal of Finance, 39 (1984), 1511–1524.CrossRefGoogle Scholar
Harrison, J. M., and Kreps, D. M.. “Martingales and Arbitrage in Multiperiod Securities Markets.” Journal of Economic Theory, 20 (1979), 381–408.CrossRefGoogle Scholar
Harrison, J. M., and Pliska, S. R.. “Martingales and Stochastic Integrals in the Theory of Continuous Trading.” Stochastic Processes and Their Applications, 11 (1981), 261–271.CrossRefGoogle Scholar
Heath, D.; Jarrow, R.; and Morton, A.. “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation.” Econometrica, 30 (1992), 77–107.CrossRefGoogle Scholar
Hutchinson, J. M., and Zenios, S. A.. “Financial Simulations on a Massively Parallel Connection Machine.” International Journal of Supercomputer Applications, 5 (1991), 27–45.Google Scholar
Jaillet, P.; Lamberton, D.; and Lapeyre, B.. “Inéquations Variationelles et Théorie des Options.” Comptes Rendus de l'Académie des Sciences de Paris, 307 (1988), 961–965.Google Scholar
Johnson, H. “Options on the Maximum or the Minimum of Several Assets.” Journal of Financial and Quantitative Analysis, 22 (1987), 277–282.CrossRefGoogle Scholar
Karatzas, I., and Shreve, S. E.. Brownian Motion and Stochastic Calculus. Springer-Verlag (1988).CrossRefGoogle Scholar
Karatzas, M. “On the Pricing of American Options.” Applied Mathematics and Optimization, 17 (1988), 37–60.CrossRefGoogle Scholar
Kushner, H. J., and Dupuis, P. G.. Numerical Methods for Stochastic Control Problems in Continuous Time. Springer Verlag (1992).CrossRefGoogle Scholar
McKean, H. “Appendix: Free Boundary Problem for the Heat Equation Arising from a Problem in Mathematical Economics.” Industrial Management Review, 6 (1965), 32–39.Google Scholar
Merton, R. “The Theory of Rational Option Pricing.” Bell Journal of Economics and Management Science, 4 (1973), 141–183.Google Scholar
Myneni, R. “The Pricing of American Options.” Annals of Applied Probability, 2 (1992), 1–23.CrossRefGoogle Scholar
Schwartz, E. S., and Torous, W. N.. “Prepayment and the Valuation of Mortgage-Backed Securities.” Journal of Finance, 44 (1989).Google Scholar
Stulz, R. M. “Options on the Minimum or the Maximum of Two Risky Assets: Analysis and Applications.” Journal of Financial Economics, 10 (1982), 161–185.CrossRefGoogle Scholar