Option Pricing via QUAD

From Black-Scholes-Merton to Heston with Jumps

Haozhe Su, Ding Chen, David P Newton

Research output: Contribution to journalArticle

1 Citation (Scopus)

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.
Original languageEnglish
Pages (from-to)9-27
Number of pages19
JournalJournal of Derivatives
Volume24
Issue number3
DOIs
Publication statusPublished - 2017

Fingerprint

Jump
Quadrature
Black-Scholes
Option pricing
Heston
Numerical methods
Stochastic volatility
Bermudan options
Numerical solution
Jump diffusion
Lattice model
Reuse
Early exercise
Derivatives
Partial differential equations
Richardson extrapolation
Critical point
Black-Scholes model
American options
Monte Carlo simulation

Cite this

Option Pricing via QUAD : From Black-Scholes-Merton to Heston with Jumps. / Su, Haozhe; Chen, Ding; Newton, David P.

In: Journal of Derivatives, Vol. 24, No. 3, 2017, p. 9-27.

Research output: Contribution to journalArticle

@article{584a73e765a547078cd5b9786c607ae0,
title = "Option Pricing via QUAD: From Black-Scholes-Merton to Heston with Jumps",
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.",
author = "Haozhe Su and Ding Chen and Newton, {David P}",
year = "2017",
doi = "10.3905/jod.2017.24.3.009",
language = "English",
volume = "24",
pages = "9--27",
journal = "Journal of Derivatives",
issn = "1074-1240",
publisher = "Institutional Investor, Inc",
number = "3",

}

TY - JOUR

T1 - Option Pricing via QUAD

T2 - From Black-Scholes-Merton to Heston with Jumps

AU - Su, Haozhe

AU - Chen, Ding

AU - Newton, David P

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

UR - https://doi.org/10.3905/jod.2017.24.3.009

U2 - 10.3905/jod.2017.24.3.009

DO - 10.3905/jod.2017.24.3.009

M3 - Article

VL - 24

SP - 9

EP - 27

JO - Journal of Derivatives

JF - Journal of Derivatives

SN - 1074-1240

IS - 3

ER -