Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T17:02:52.274Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact Monte Carlo simulation of killed diffusions

Published online by Cambridge University Press:  01 July 2016

Bruno Casella  and
Gareth O. Roberts
Bruno Casella*
Affiliation:
University of Warwick
Gareth O. Roberts*
Affiliation:
University of Warwick
*
∗ Current address: via novara, 31, Milan, 20147, Italy.
∗∗ Postal address: Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We describe and implement a novel methodology for Monte Carlo simulation of one-dimensional killed diffusions. The proposed estimators represent an unbiased and efficient alternative to current Monte Carlo estimators based on discretization methods for the cases when the finite-dimensional distributions of the process are unknown. For barrier option pricing in finance, we design a suitable Monte Carlo algorithm both for the single barrier case and the double barrier case. Results from numerical investigations are in excellent agreement with the theoretical predictions.

Keywords

Monte Carlodiffusionsexact algorithm
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 1 , March 2008 , pp. 273 - 291
Copyright
Copyright © Applied Probability Trust 2008 

References

Anderson, T. W. (1960). A modification of the sequential probability ratio test to reduce the sample size. Ann. Math. Statist. 31, 165–197.Google Scholar
Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.Google Scholar
Bertoin, J. and Pitman, J. (1994). Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118, 147–166.Google Scholar
Beskos, A. and Roberts, G. O. (2005). Exact simulation of diffusions. Ann. Appl. Prob. 15, 2422–2444.Google Scholar
Beskos, A. and Papaspiliopoulos, O. and Roberts, G. O. (2006a). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12, 1077–1098.Google Scholar
Beskos, A. and Papaspiliopoulos, O. and Roberts, G. O. (2008). A new factorisation of diffusion measure and sample path reconstruction. To appear in Methodology Comput. Appl. Prob. Google Scholar
Beskos, A. and Papaspiliopoulos, O. and Roberts, G. O. and Fearnhead, P. (2006b). Exact and efficient likelihood based inference for discretely observed diffusions (with discussion). J. R. Statist. Soc. B 68, 333–382.Google Scholar
Black, F. and Cox, J. C. (1976). Valuing corporate securities: some effects of bond indenture provisions. J. Finance 31, 351–367.Google Scholar
Bossy, M., Gobet, E. and Talay, D. (2004). A symmetrized Euler scheme for an efficient approximation of reflected diffusions. J. Appl. Prob. 41, 877–889.CrossRefGoogle Scholar
Davydov, D. and Linetsky, V. (2001). The valuation and hedging of barrier and lookback options under the CEV process. Manag. Sci. 47, 949–965.CrossRefGoogle Scholar
Devroye, L. (1986). Nonuniform Random Variate Generation. Springer, New York.Google Scholar
Doob, J. L. (1949). Heuristic approach to the Kolmogorov–Smirnov theorems. Ann. Math. Statist. 20, 393–403.CrossRefGoogle Scholar
Fearnhead, P., Papaspiliopoulos, O. and Roberts, G. O. (2008). Particle filters for partially observed diffusions. To appear in J. R. Statist. Soc. B.Google Scholar
Gobet, E. (2000). Weak approximation of killed diffusion using Euler schemes. Stoch. Process. Appl. 87, 167–197.Google Scholar
Gobet, E. (2001). Euler schemes and half-space approximation for the simulation of diffusion in a domain. ESAIM Prob. Statist. 5, 261–297.CrossRefGoogle Scholar
Lépingle, D. (1995). Euler scheme for reflected stochastic differential equations. Math. Comput. Simul. 38, 119–126.CrossRefGoogle Scholar
Lerche, H. R. (1986). Boundary Crossing of Brownian Motion. (Lecture Notes Statist. 40). Springer, Berlin.Google Scholar
Longstaff, F. A. and Schwarz, E. S. (1995). A simple approach to valuing risky and floating rate debt. J. Finance 50, 789–819.Google Scholar
Merton, R. C. (1971). Theory of rational option pricing. Bell J. Econom. Manag. 4, 141–183.Google Scholar
Øksendal, B. K. (1998). Stochastic Differential Equations. An Introduction with Applications, 5th edn. Springer, Berlin.Google Scholar
Pötzelberger, K. and Wang, L. (2001). Boundary crossing probability for Brownian motion. J. Appl. Prob. 38, 152–164.CrossRefGoogle Scholar
Reiner, E. and Rubinstein, M. (1991). Breaking down the barriers. Risk 4, 28–35.Google Scholar