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Approximation of jump diffusions in finance and economics

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Abstract

In finance and economics the key dynamics are often specified via stochastic differential equations (SDEs) of jump-diffusion type. The class of jump-diffusion SDEs that admits explicit solutions is rather limited. Consequently, discrete time approximations are required. In this paper we give a survey of strong and weak numerical schemes for SDEs with jumps. Strong schemes provide pathwise approximations and therefore can be employed in scenario analysis, filtering or hedge simulation. Weak schemes are appropriate for problems such as derivative pricing or the evaluation of risk measures and expected utilities. Here only an approximation of the probability distribution of the jump-diffusion process is needed. As a framework for applications of these methods in finance and economics we use the benchmark approach. Strong approximation methods are illustrated by scenario simulations. Numerical results on the pricing of options on an index are presented using weak approximation methods.

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References

  • Bally V., Talay D. (1996a). The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function. Probability Theory and Related Fields 104(1): 43–60

    Article  Google Scholar 

  • Bally V., Talay D. (1996b). The law of the Euler scheme for stochastic differential equations II. Convergence rate of the density. Monte Carlo Methods and Applications 2(2): 93–128

    Article  Google Scholar 

  • Bruti-Liberati, N., & Platen, E. (2004). On the efficiency of simplified weak Taylor schemes for Monte Carlo simulation in finance. In Computational science—ICCS 2004, Vol. 3039, of lecture notes in computer science, (pp. 771–778). Berlin: Springer.

  • Bruti-Liberati N., Platen E. (2005). On the strong approximation of jump-diffusion processes. University of Technology Sydney, Sydney, Technical report, Quantitative Finance Research Papers 157

    Google Scholar 

  • Bruti-Liberati N., Platen E. (2006).On the weak approximation of jump-diffusion processes. University of Technology Sydney, Sydney, Technical report

    Google Scholar 

  • Bruti-Liberati, N., Platen, E., Martini, F., & Piccardi, M. (2005). A multi-point distributed random variable accelerator for Monte Carlo simulation in finance. In Proceedings of the Fifth International Conference on Intelligent Systems Design and Applications (pp. 532–537). Los Alamitos, California: IEEE Computer Society Press.

  • Carr P., Madan D. (1999). Option pricing and the fast Fourier transform. Journal of Computational Finance 2(4): 61–73

    Google Scholar 

  • Cont R., Tankov P. (2004). Financial modelling with jump processes. Financial mathematics series. Chapman and Hall/CRC, London, Boca Raton

    Google Scholar 

  • Cont R., Voltchkova E. (2005). Finite difference methods for option pricing in jump-diffusion and exponential Lévy models. SIAM Journal of Numerical Analysis 43(4): 1596–1626

    Article  Google Scholar 

  • Craddock M., Heath D., Platen E. (2000). Numerical inversion of Laplace transforms: A survey with applications to derivative pricing. Journal of Computational Finance 4(1): 57–81

    Google Scholar 

  • Davis M.H.A. (1997). Option pricing in incomplete markets. In: Dempster M.A.H., Pliska S.R. (eds). Mathematics of derivative securities. Cambridge University Press, Cambridge, pp. 227–254

    Google Scholar 

  • D’Halluin Y., Forsyth P.A., Vetzal K.R. (2005). Robust numerical methods for contingent claims under jump diffusion processes. IMA Journal of Numerical Analysis 25, 87–112

    Article  Google Scholar 

  • Duffie D., Pan J., Singleton K. (2000). Transform analysis and option pricing for affine jump diffusions. Econometrica 68, 1343–1376

    Article  Google Scholar 

  • Föllmer H., Schweizer M. (1991). Hedging of contingent claims under incomplete information. In: Davis M., Elliott R. (eds). Applied stochastic analysis, stochastics monograph, vol5. Gordon and Breach, London, New York, pp. 389–414

    Google Scholar 

  • Gardoǹ A. (2004). The order of approximations for solutions of Itô-type stochastic differential equations with jumps. Stochastic Analysis and Applications 22(3): 679–699

    Article  Google Scholar 

  • Glasserman, P., & Merener, N. (2003). Convergence of a discretization scheme for jump-diffusion processes with state-dependent intensities. In Proceedings of the Royal Society, Vol. 460 (pp. 111–127).

  • Guyon, J. (2006). Euler scheme and tempered distributions. Stochastic Processes and their Applications. Forthcoming.

  • Higham D., Kloeden P. (2005). Numerical methods for nonlinear stochastic differential equations with jumps. Numerische Matematik, 110(1): 101–119

    Article  Google Scholar 

  • Higham D., Kloeden P. (2006). Convergence and stability of implicit methods for jump-diffusion systems. International Journal of Numerical Analysis and Modeling 3(2): 125–140

    Google Scholar 

  • Hofmann N., Platen E. (1996). Stability of superimplicit numerical methods for stochastic differential equations. Fields Institute Communications 9, 93–104

    Google Scholar 

  • Ikeda N., Watanabe, S. (1989). Stochastic differential equations and diffusion processes (2nd ed.). Amsterdan: North-Holland.(1st ed., 1981).

  • Jacod J. (2004). The Euler scheme for Lévy driven stochastic differential equations: limit theorems. Annals of Probability 32(3A), 1830–1872

    Article  Google Scholar 

  • Jacod J., Kurtz T., Méléard S., Protter P. (2005). The approximate Euler method for Lévy driven stochastic differential equations. Annals de l’Institut Henri Poincaré (B) Probabiliteś et Statistiques 41(3): 523–558

    Article  Google Scholar 

  • Jacod J., Protter P. (1998). Asymptotic error distribution for the Euler method for stochastic differential equations. Annals of Probability 26(1): 267–307

    Article  Google Scholar 

  • Karatzas I., Shreve S.E. (1998). Methods of mathematical finance, Applied Mathematics. vol. 39 Springer, Berlin

    Google Scholar 

  • Kelly J.R. (1956). A new interpretation of information rate. Bell System Technical Journal 35, 917–926

    Google Scholar 

  • Kloeden P.E., Platen E. (1999). Numerical solution of stochastic differential equations, Applied Mathematics (Vol. 23). Springer, Berlin, Third corrected printing

    Google Scholar 

  • Kohatsu-Higa A., Protter P. (1994). The Euler scheme for SDEs driven by semimartingales. In: Kunita H., Kuo H.H. (eds). Stochastic analysis on infinite dimensional spaces. Pitman, London, pp. 141–151

    Google Scholar 

  • Kubilius K., Platen E.(2002). Rate of weak convergence of the Euler approximation for diffusion processes with jumps. Monte Carlo Methods and Applications 8(1): 83–96

    Article  Google Scholar 

  • Li, C. W. (1995). Almost sure convergence of stochastic differential equations of jump-diffusion type. In Seminar on stochastic analysis, random fields and applications, Programme Probability (pp. 187–197). Basel: Birkhäuser Verlag.

  • Liu X.Q., Li C.W. (2000a). Almost sure convergence of the numerical discretisation of stochastic jump diffusions. Acta Applicandae Mathmaticae 62, 225–244

    Article  Google Scholar 

  • Liu X.Q., Li C.W. (2000b). Weak approximations and extrapolations of stochastic differential equations with jumps. SIAM Journal of Numerical Analysis 37(6): 1747–1767

    Article  Google Scholar 

  • Long J.B. (1990). The numeraire portfolio. Journal of Financial Economics 26, 29–69

    Article  Google Scholar 

  • Maghsoodi Y. (1996). Mean-square efficient numerical solution of jump-diffusion stochastic differential equations. SANKHYA A 58(1): 25–47

    Google Scholar 

  • Maghsoodi Y. (1998). Exact solutions and doubly efficient approximations of jump-diffusion Itô equations. Stochastic Analysis and Applications 16(6): 1049–1072

    Google Scholar 

  • Maghsoodi Y., Harris C.J. (1987). In-probability approximation and simulation of nonlinear jump-diffusion stochastic differential equations. IMA Journal of Mathematics Control Information 4(1): 65–92

    Article  Google Scholar 

  • Merton R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 2, 125–144

    Article  Google Scholar 

  • Mikulevicius R., Platen E. (1988). Time discrete Taylor approximations for Ito processes with jump component. Mathematische Nachrichten 138, 93–104

    Google Scholar 

  • Øksendal B., Sulem A. (2005). Applied stochastic control of jump-duffusions. Universitext of Berlin: Springer.

  • Platen E. (1982a). An approximation method for a class of Itô processes with jump component. Lietuvos Mathematikos Rinkinys 22(2): 124–136

    Google Scholar 

  • Platen E. (1982b). A generalized Taylor formula for solutions of stochastic differential equations. SANKHYA A, 44(2): 163–172

    Google Scholar 

  • Platen, E., & Heath, D. (2006). Introduction to quantitative finance: A Benchmark approach. Berlin: Springer Finance. (Springer, Forthcoming).

  • Protter P. (2004). Stochastic integration and differential equations (2nd ed). Springer, Berlin

    Google Scholar 

  • Protter P., Talay D. (1997). The Euler scheme for Lévy driven stochastic differential equations. Annals of Probability 25(1): 393–423

    Article  Google Scholar 

  • Runggaldier W.J. (2003). Jump-diffusion models. In: Rachev S.T. (eds). Handbook of heavy tailed distributions in finance, of Handbooks in Finance vol.1. Elsevier, Amsterdam, pp. 170–209

    Google Scholar 

  • Schönbucher P.J. (2004). A measure of survival. Risk Magazine 17(8): 79–85

    Google Scholar 

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  1. School of Finance and Economics, University of Technology Sydney, PO Box 123, Broadway, NSW, 2007, Australia

    Nicola Bruti-Liberati & Eckhard Platen

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  1. Nicola Bruti-Liberati
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  2. Eckhard Platen
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Correspondence to Nicola Bruti-Liberati.

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Bruti-Liberati, N., Platen, E. Approximation of jump diffusions in finance and economics. Comput Econ 29, 283–312 (2007). https://doi.org/10.1007/s10614-006-9066-y

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