Premium
On the Rate of Convergence of Discrete‐Time Contingent Claims[Note 1. We are grateful to Mark Broadie, Phil Dybvig, Raymond ...]
Author(s) -
Heston Steve,
Zhou Guofu
Publication year - 2000
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/1467-9965.00080
Subject(s) - stochastic game , mathematics , rate of convergence , smoothness , smoothing , discrete time and continuous time , multinomial distribution , mathematical economics , differentiable function , limit (mathematics) , convergence (economics) , mathematical optimization , econometrics , economics , computer science , statistics , mathematical analysis , key (lock) , computer security , economic growth
This paper characterizes the rate of convergence of discrete‐time multinomial option prices. We show that the rate of convergence depends on the smoothness of option payoff functions, and is much lower than commonly believed because option payoff functions are often of all‐or‐nothing type and are not continuously differentiable. To improve the accuracy, we propose two simple methods, an adjustment of the discrete‐time solution prior to maturity and smoothing of the payoff function, which yield solutions that converge to their continuous‐time limit at the maximum possible rate enjoyed by smooth payoff functions. We also propose an intuitive approach that systematically derives multinomial models by matching the moments of a normal distribution. A highly accurate trinomial model also is provided for interest rate derivatives. Numerical examples are carried out to show that the proposed methods yield fast and accurate results.