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Multiplicative noise, fast convolution, and pricing

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  • Giacomo Bormetti
  • Sofia Cazzaniga
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    Abstract

    In this work we detail the application of a fast convolution algorithm computing high dimensional integrals to the context of multiplicative noise stochastic processes. The algorithm provides a numerical solution to the problem of characterizing conditional probability density functions at arbitrary time, and we applied it successfully to quadratic and piecewise linear diffusion processes. The ability in reproducing statistical features of financial return time series, such as thickness of the tails and scaling properties, makes this processes appealing for option pricing. Since exact analytical results are missing, we exploit the fast convolution as a numerical method alternative to the Monte Carlo simulation both in objective and risk neutral settings. In numerical sections we document how fast convolution outperforms Monte Carlo both in velocity and efficiency terms.

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    File URL: http://arxiv.org/pdf/1107.1451
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1107.1451.

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    Date of creation: Jul 2011
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    Publication status: Published in Quantitative Finance, 2012, 1-14 iFirst
    Handle: RePEc:arx:papers:1107.1451

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    Web page: http://arxiv.org/

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    1. Eydeland, A, 1994. "A Fast Algorithm for Computing Integrals in Function Spaces: Financial Applications," Computational Economics, Society for Computational Economics, vol. 7(4), pages 277-85.
    2. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    3. G. Bormetti & V. Cazzola & G. Livan & G. Montagna & O. Nicrosini, 2009. "A Generalized Fourier Transform Approach to Risk Measures," Papers 0909.3978, arXiv.org, revised May 2012.
    4. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
    5. J.L. McCauley & G.h. Gunaratne, 2002. "An empirical model of volatility of returns and option pricing," Computing in Economics and Finance 2002 186, Society for Computational Economics.
    6. Carl Chiarella & Nadima El-Hassan, 1997. "Evaluation of Derivative Security Prices in the Heath-Jarrow-Morton Framework as Path Integrals Using Fast Fourier Transform Techniques," Working Paper Series 72, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    7. G. Bormetti & G. Montagna & N. Moreni & O. Nicrosini, 2004. "Pricing Exotic Options in a Path Integral Approach," Papers cond-mat/0407321, arXiv.org, revised May 2006.
    8. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
    9. Alejandro-Quiñones, Ángel L. & Bassler, Kevin E. & Field, Michael & McCauley, Joseph L. & Nicol, Matthew & Timofeyev, Ilya & Török, Andrew & Gunaratne, Gemunu H., 2006. "A theory of fluctuations in stock prices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 363(2), pages 383-392.
    10. Evert Wipplinger, 2007. "Philippe Jorion: Value at Risk – The New Benchmark for Managing Financial Risk," Financial Markets and Portfolio Management, Springer, vol. 21(3), pages 397-398, September.
    11. McCauley, Joseph L. & Gunaratne, Gemunu H., 2003. "An empirical model of volatility of returns and option pricing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 329(1), pages 178-198.
    12. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    13. Danilo Delpini & Giacomo Bormetti, 2010. "Minimal model of financial stylized facts," Papers 1011.5983, arXiv.org, revised Mar 2011.
    14. Tyler McMillen, . "Simulation and Inference for Stochastic Differential Equations: With R Examples," Journal of Statistical Software, American Statistical Association, vol. 26(b02).
    15. Montagna, Guido & Nicrosini, Oreste & Moreni, Nicola, 2002. "A path integral way to option pricing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 310(3), pages 450-466.
    16. Michel Vellekoop & Hans Nieuwenhuis, 2007. "On option pricing models in the presence of heavy tails," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 563-573.
    17. G. Bormetti & G. Montagna & N. Moreni & O. Nicrosini, 2006. "Pricing exotic options in a path integral approach," Quantitative Finance, Taylor & Francis Journals, vol. 6(1), pages 55-66.
    18. Bormetti, Giacomo & Cisana, Enrica & Montagna, Guido & Nicrosini, Oreste, 2007. "A non-Gaussian approach to risk measures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 376(C), pages 532-542.
    19. McCauley, Joseph L. & Gunaratne, Gemunu H. & Bassler, Kevin E., 2007. "Hurst exponents, Markov processes, and fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(1), pages 1-9.
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