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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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  • Nicola Bruti-Liberati & Eckhard Platen, 2006. "Approximation of Jump Diffusions in Finance and Economics," Research Paper Series 176, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:176
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    1. N. Hofmann & Eckhard Platen, 1994. "Stability of weak numerical schemes for stochastic differential equations," Published Paper Series 1994-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    2. Nicola Bruti Liberati & Eckhard Platen, 2004. "On the Efficiency of Simplified Weak Taylor Schemes for Monte Carlo Simulation in Finance," Research Paper Series 114, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Remigijus Mikulevicius & Eckhard Platen, 1988. "Time Discrete Taylor Approximations for Ito Processes with Jump Component," Published Paper Series 1988-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    4. Kubilius Kestutis & Platen Eckhard, 2002. "Rate of Weak Convergence of the Euler Approximation for Diffusion Processes with Jumps," Monte Carlo Methods and Applications, De Gruyter, vol. 8(1), pages 83-96, December.
    5. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151, January.
    6. Rama Cont & Ekaterina Voltchkova, 2005. "A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models," Post-Print halshs-00445645, HAL.
    7. Long, John Jr., 1990. "The numeraire portfolio," Journal of Financial Economics, Elsevier, vol. 26(1), pages 29-69, July.
    8. Nicola Bruti-Liberati & Eckhard Platen, 2005. "On the Strong Approximation of Jump-Diffusion Processes," Research Paper Series 157, Quantitative Finance Research Centre, University of Technology, Sydney.
    9. Mark Craddock & David Heath & Eckhard Platen, 1999. "Numerical Inversion of Laplace Transforms: A Survey of Techniques with Applications to Derivative Pricing," Research Paper Series 27, Quantitative Finance Research Centre, University of Technology, Sydney.
    10. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    11. Guyon, Julien, 2006. "Euler scheme and tempered distributions," Stochastic Processes and their Applications, Elsevier, vol. 116(6), pages 877-904, June.
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    Cited by:

    1. Jozef Barunik & Cathy Yi-Hsuan Chen & Jan Vecer, 2019. "Sentiment-Driven Stochastic Volatility Model: A High-Frequency Textual Tool for Economists," Papers 1906.00059, arXiv.org.
    2. Mao, Wei & Hu, Liangjian & Mao, Xuerong, 2015. "The existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and Lévy jumps," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 883-896.
    3. Björn Lutz, 2010. "Pricing of Derivatives on Mean-Reverting Assets," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-642-02909-7, December.
    4. Kristensen, Dennis & Shin, Yongseok, 2012. "Estimation of dynamic models with nonparametric simulated maximum likelihood," Journal of Econometrics, Elsevier, vol. 167(1), pages 76-94.
    5. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1, July-Dece.
    6. Benella, Simone & Consolini, Giuseppe & Stumpo, Mirko & Alberti, Tommaso & Gjerloev, Jesper W., 2022. "Markov property of the Super-MAG auroral electrojet indices," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    7. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2007.

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    More about this item

    Keywords

    jump-diffusion processes; discrete time approximation; simulation; strong covergence; weak convergence; benchmark approach; growth optimal portfolio;
    All these keywords.

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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