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The adaptive mesh model: a new approach to efficient option pricing

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  • Figlewski, Stephen
  • Gao, Bin
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Abstract

Exact closed-form valuation equations for traded derivative securities are rare. Numerical approximation, most commonly with Binomial and Trinomial lattice models, gives exact valuation in the limit, but convergence is non-monotonic and often slow, due to 'distribution error' and 'truncation error.' This paper explains how truncation error arises and describes the Adaptive Mesh Model (AMM), a new approach that sharply reduces it by grafting one or more small sections of fine high-resolution lattice onto a tree with coarser time and price steps. Three different AMM structures are presented, one for pricing ordinary options, one for barrier options, and one for computing delta and gamma efficiently. The AMM approach can be adapted to a wide variety of contingent claims, yielding significant improvement in efficiency with very little increase in computational effort. For some common problems, including calculating delta, accuracy increases by several orders of magnitude relative to the standard models with no measurable increase in execution time at all.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Financial Economics.

Volume (Year): 53 (1999)
Issue (Month): 3 (September)
Pages: 313-351

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Handle: RePEc:eee:jfinec:v:53:y:1999:i:3:p:313-351

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Web page: http://www.elsevier.com/locate/inca/505576

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References

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  1. Hull, John & White, Alan, 1990. "Valuing Derivative Securities Using the Explicit Finite Difference Method," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 25(01), pages 87-100, March.
  2. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
  3. 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.
  4. Canina, Linda & Figlewski, Stephen, 1993. "The Informational Content of Implied Volatility," Review of Financial Studies, Society for Financial Studies, vol. 6(3), pages 659-81.
  5. Mark Rubinstein., 1991. "Exotic Options," Research Program in Finance Working Papers RPF-220, University of California at Berkeley.
  6. Brennan, Michael J & Schwartz, Eduardo S, 1977. "The Valuation of American Put Options," Journal of Finance, American Finance Association, vol. 32(2), pages 449-62, May.
  7. Geske, Robert & Johnson, Herb E, 1984. " The American Put Option Valued Analytically," Journal of Finance, American Finance Association, vol. 39(5), pages 1511-24, December.
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Citations

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Cited by:
  1. Zvan, R. & Vetzal, K. R. & Forsyth, P. A., 2000. "PDE methods for pricing barrier options," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1563-1590, October.
  2. Chang, Chuang-Chang & Lin, Jun-Biao, 2010. "The valuation of contingent claims using alternative numerical methods," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 20(5), pages 490-508, December.
  3. Kristensen, Dennis & Mele, Antonio, 2011. "Adding and subtracting Black-Scholes: A new approach to approximating derivative prices in continuous-time models," Journal of Financial Economics, Elsevier, vol. 102(2), pages 390-415.
  4. Doobae Jun & Hyejin Ku, 2013. "Valuation of American partial barrier options," Review of Derivatives Research, Springer, vol. 16(2), pages 167-191, July.
  5. Tian-Shyr Dai & Jr-Yan Wang & Hui-Shan Wei, 2008. "Adaptive placement method on pricing arithmetic average options," Review of Derivatives Research, Springer, vol. 11(1), pages 83-118, March.
  6. Benjamin Jourdain & Antonino Zanette, 2008. "A moments and strike matching binomial algorithm for pricing American Put options," Decisions in Economics and Finance, Springer, vol. 31(1), pages 33-49, May.
  7. Simona Sanfelici, 2004. "Galerkin infinite element approximation for pricing barrier options and options with discontinuous payoff," Decisions in Economics and Finance, Springer, vol. 27(2), pages 125-151, December.
  8. D. Andricopoulos, Ari & Widdicks, Martin & Newton, David P. & Duck, Peter W., 2007. "Extending quadrature methods to value multi-asset and complex path dependent options," Journal of Financial Economics, Elsevier, vol. 83(2), pages 471-499, February.
  9. Li, Minqiang, 2009. "A Quasi-analytical Interpolation Method for Pricing American Options under General Multi-dimensional Diffusion Processes," MPRA Paper 17348, University Library of Munich, Germany.
  10. N. Hilber & N. Reich & C. Schwab & C. Winter, 2009. "Numerical methods for Lévy processes," Finance and Stochastics, Springer, vol. 13(4), pages 471-500, September.
  11. Ben R. Craig & Joachim G. Keller, 2003. "The empirical performance of option-based densities of foreign exchange," Working Paper 0313, Federal Reserve Bank of Cleveland.
  12. Chung, San-Lin & Shih, Pai-Ta & Tsai, Wei-Che, 2013. "Static hedging and pricing American knock-in put options," Journal of Banking & Finance, Elsevier, vol. 37(1), pages 191-205.
  13. Fusai, Gianluca & Recchioni, Maria Cristina, 2007. "Analysis of quadrature methods for pricing discrete barrier options," Journal of Economic Dynamics and Control, Elsevier, vol. 31(3), pages 826-860, March.
  14. Yang, Sharon S. & Dai, Tian-Shyr, 2013. "A flexible tree for evaluating guaranteed minimum withdrawal benefits under deferred life annuity contracts with various provisions," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 231-242.
  15. Chung, San-Lin & Shih, Pai-Ta, 2009. "Static hedging and pricing American options," Journal of Banking & Finance, Elsevier, vol. 33(11), pages 2140-2149, November.

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