Exact Simulation of Jump-Diffusion Processes with Monte Carlo Applications

Casella, Bruno and Roberts, Gareth O. (2011): Exact Simulation of Jump-Diffusion Processes with Monte Carlo Applications. Published in: Methodology and Computing in Applied Probability , Vol. 13, No. 3 (9 January 2010): pp. 449-473.

Abstract

We introduce a novel algorithm (JEA) to simulate exactly from a class of one-dimensional jump-diffusion processes with state-dependent intensity. The simulation of the continuous component builds on the recent Exact Algorithm (Beskos et al., Bernoulli 12(6):1077–1098, 2006a). The simulation of the jump component instead employs a thinning algorithm with stochastic acceptance probabilities in the spirit of Glasserman and Merener (Proc R Soc Lond Ser A Math Phys Eng Sci 460(2041):111–127, 2004). In turn JEA allows unbiased Monte Carlo simulation of a wide class of functionals of the process’ trajectory, including discrete averages, max/min, crossing events, hitting times. Our numerical experiments show that the method outperforms Monte Carlo methods based on the Euler discretization.

• Item Type: MPRA Paper
• Language: English
• Keywords: Jump diffusion, Simulation, Exact Algorithms, Barrier option pricing
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C18 - Methodological Issues: General; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling

References

Beskos A, Roberts GO (2005) Exact simulation of diffusions. Ann Appl Probab 15(4):2422–2444zbMATHCrossRefMathSciNetGoogle Scholar

Beskos A, Papaspiliopoulos O, Roberts GO (2006a) Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12(6):1077–1098zbMATHCrossRefMathSciNetGoogle Scholar

Beskos A, Papaspiliopoulos O, Roberts GO, Fearnhead P (2006b) Exact and efficient likelihood based inference for discretely observed diffusions (with discussion). J R Stat Soc Ser B Stat Methodol 68(3):333–382zbMATHCrossRefMathSciNetGoogle Scholar

Beskos A, Papaspiliopoulos O, Roberts GO (2008) A new factorisation of diffusion measure and sample path reconstruction. Methodology and Computing in Applied Probability 10(1):85–104zbMATHCrossRefMathSciNetGoogle Scholar

Caines PE, Zhang JF (1995) On adaptive control for jump parameter systems via nonlinear filtering. SIAM J Control Optim 33:1758–1777zbMATHCrossRefMathSciNetGoogle Scholar

Cariboni J, Schoutens W (2004) Pricing credit default swaps under lévy models. www.defaultrisk.com

Casella B (2006) Exact monte carlo simulation of diffusion and jump-diffusion processes with financial applications. PhD thesis, Università Commerciale L. Bocconi, Istituto Metodi QuantitativiGoogle Scholar

Casella B, Roberts GO (2007) Exact monte carlo simulation of killed diffusions. Adv Appl Probab 41:273–291Google Scholar

Das S (2002) The surprise element: jumps in interest rates. J Econom 106:27–65zbMATHCrossRefGoogle Scholar

Devroye L (1986) Nonuniform random variate generation. Springer, New YorkGoogle Scholar

Doob JL (1949) Heuristic approach to the Kolmogorov-Smirnov theorems. Ann Math Stat 20:393–403zbMATHCrossRefMathSciNetGoogle Scholar

Giesecke K (2004) Credit risk modeling and evaluation: an introduction. In: Credit risk: models and management. Risk books, vol 2. D. ShimkoGoogle Scholar

Giraudo M, Sacerdote L (1997) Jump-diffusion processes as models for neuronal activity. Biosystems 40:75–82CrossRefGoogle Scholar

Glasserman P, Merener N (2004) Convergence of a discretization scheme for jump-diffusion processes with state-dependent intensities. Proc R Soc Lond Ser A Math Phys Eng Sci 460(2041):111–127. Stochastic analysis with applications to mathematical financezbMATHCrossRefMathSciNetGoogle Scholar

Gobet E (2000) Weak approximation of killed diffusion using Euler schemes. Stoch Process Appl 87(2):167–197zbMATHCrossRefMathSciNetGoogle Scholar

Ikeda N, Watanabe S (1989) Stochastic differential equations and diffusion processes. North-Holland mathematical library, vol 24, 2nd edn. North-Holland, AmsterdamzbMATHGoogle Scholar

Johannes M (2004) The statistical and economic role of jumps in continuous-time interest rate models. J Finance 59(1):227–260CrossRefGoogle Scholar

Johannes M, Kumar R, Polson N (1999) State dependent jump models: how do us equity indices jump? Working paper. University of ChicagoGoogle Scholar

Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus. In: Graduate texts in mathematics, vol 113. 2nd edn. Springer, New YorkGoogle Scholar

Merton RC (1976) Option pricing when undrlying stock returns are discontinuous. J Financ Econ 3:125–144zbMATHCrossRefGoogle Scholar

Metwally S, Atiya A (2002) Using brownian bridge for fast simulation of jump-diffusion processes and barrier options. J Deriv 10:43–54CrossRefGoogle Scholar

Pelucchetti S (2008) An analysis of the efficiency of the exact algorithm. PhD thesis, Università Commerciale L. Bocconi, Istituto Metodi QuantitativiGoogle Scholar

Platen E, Rebolledo R (1985) Weak convergence of semimartingales and discretisation methods. Stoch Process Appl 20(1):41–58zbMATHCrossRefMathSciNetGoogle Scholar

Revuz D, Yor M (1991) Continuous Martingales and Brownian motion. Springer, New YorkzbMATHGoogle Scholar

Zhou G (2001) The term structure of credit spreads with jump risk. J Bank Financ 25:504–531CrossRefGoogle Scholar

Zhu SC (1999) Stochastic jump-diffusion process for computing medial axes in markov random fields. IEEE Trans Pattern Anal Mach Intell 21:1158–1169CrossRefGoogle Scholar