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Article

Option Pricing via QUAD: From Black–Scholes–Merton to Heston with Jumps

Haozhe Su, Ding Chen and David P. Newton
The Journal of Derivatives Spring 2017, 24 (3) 9-27; DOI: https://doi.org/10.3905/jod.2017.24.3.009
Haozhe Su
is a doctoral student at Nottingham University Business School, Jubilee Campus, in Nottingham U.K.
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  • For correspondence: haozhe.su@nottingham.ac.uk
Ding Chen
is a lecturer in finance at the School of Business, Management and Economics at the University of Sussex Falmer in Brighton, U.K.
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  • For correspondence: ding.chen@sussex.ac.uk
David P. Newton
is a professor of finance at the University of Bath School of Management in Bath, U.K.
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  • For correspondence: d.p.newton@bath.ac.uk
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Abstract

The Black–Scholes model is the rare closed-form formula for pricing options, but its shortcomings are well known. Adding stochastic volatility, American or Bermudan early exercise, non-diffusive jumps in the returns process, and/or other “exotic” payoff features quickly takes one into the realm where approximate answers must by computed by numerical methods. Binomial and other lattice models, Monte Carlo simulation, and numerical solution of the valuation partial differential equation are the standard techniques. They are well known to eventually arrive at as close an approximation to the exact model value as one wants but can require a large amount of calculation and execution time to get there. In this article, the authors present what amounts to the culmination of a series of papers on using quadrature methods to solve option problems of increasing complexity. Quadrature entails modeling and approximating the transition densities between critical points in the option’s lifetime. Depending on the specifics of the option’s payoff, looking at only a few points in time can greatly reduce the amount of calculation needed to price it. For example, a Bermudan option that can be exercised at three possible early dates needs quadrature calculations for only those three dates plus expiration, while other numerical methods require calculations for every date and possible asset price. This article summarizes how to apply quadrature to standard option problems, from plain-vanilla Black–Scholes up to Bermudan and American options under a stochastic volatility jump-diffusion returns process. The procedure can be speeded up even more by use of Richardson extrapolation and caching techniques that allow reuse of results calculated in intermediate steps. The range of derivatives models that can be valued extremely efficiently using quadrature is very broad.

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The Journal of Derivatives: 24 (3)
The Journal of Derivatives
Vol. 24, Issue 3
Spring 2017
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Option Pricing via QUAD: From Black–Scholes–Merton to Heston with Jumps
Haozhe Su, Ding Chen, David P. Newton
The Journal of Derivatives Feb 2017, 24 (3) 9-27; DOI: 10.3905/jod.2017.24.3.009

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Option Pricing via QUAD: From Black–Scholes–Merton to Heston with Jumps
Haozhe Su, Ding Chen, David P. Newton
The Journal of Derivatives Feb 2017, 24 (3) 9-27; DOI: 10.3905/jod.2017.24.3.009
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  • Article
    • Abstract
    • THE QUAD METHOD
    • ACCELERATION TECHNIQUES
    • USING QUAD TO PRICE MERTON’S JUMP-DIFFUSION MODEL
    • USING QUAD TO PRICE THE STOCHASTIC VOLATILITY JUMP-DIFFUSION MODEL
    • CONCLUSION
    • APPENDIX A
    • APPENDIX B
    • ENDNOTE
    • REFERENCES
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