Abstract
The Black–Scholes model is the rare closedform formula for pricing options, but its shortcomings are well known. Adding stochastic volatility, American or Bermudan early exercise, nondiffusive 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 plainvanilla Black–Scholes up to Bermudan and American options under a stochastic volatility jumpdiffusion 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 language  English 

Pages (fromto)  927 
Number of pages  19 
Journal  Journal of Derivatives 
Volume  24 
Issue number  3 
DOIs  
Publication status  Published  2017 
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David Newton
 Management  Head of Division
 Accounting, Finance & Law
 Centre for Governance, Regulation and Industrial Strategy
Person: Research & Teaching