Binomial models for option valuation - examining and improving convergence

Binomial models, which describe the asset price dynamics of the continuous-time model in the limit, serve for approximate valuation of options, especially where formulas cannot be derived analytically due to properties of the considered option type. To evaluate results, one inevitably must understand the convergence properties. In the literature we find various contributions proving convergence of option prices. We examine convergence behaviour and convergence speed. Unfortunately, even in the case of European call options, distorted results occur when calculating prices along the iteration of tree refinements. These convergence patterns are examined and order of convergence one is proven for the Cox-Ross-Rubinstein model as well as for two alternative tree parameter selections from the literature. Furthermore, we define new binomial models, where the calculated option prices converge smoothly to the Black-Scholes solution, and we achieve order of convergence two with much smaller initial error. Notably, ...

[1]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[2]  Kaushik I. Amin.  On the Computation of Continuous Time Option Prices Using Discrete Approximations , 1991, Journal of Financial and Quantitative Analysis.

[3]  Yisong S. Tian.  A modified lattice approach to option pricing , 1993 .

[4]  Kaushik I. Amin,et al.  CONVERGENCE OF AMERICAN OPTION VALUES FROM DISCRETE‐ TO CONTINUOUS‐TIME FINANCIAL MODELS1 , 1994 .

[5]  Burton H. Camp.  Approximation to the Point Binomial , 1951 .

[6]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[7]  Kuldeep Shastri,et al.  Valuation by Approximation: A Comparison of Alternative Option Valuation Techniques , 1985, Journal of Financial and Quantitative Analysis.

[8]  On the Accuracy of Gaussian Approximation to the Distribution Functions of Sums of Independent Variables , 1966 .

[9]  Richard J. Rendleman,et al.  Two-State Option Pricing , 1979 .

[10]  E. C. Fieller.  THE DISTRIBUTION OF THE INDEX IN A NORMAL BIVARIATE POPULATION , 1932 .

[11]  C. Esseen.  Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law , 1945 .

[12]  S. Ross,et al.  Option pricing: A simplified approach☆ , 1979 .

[13]  H. Trotter.  An elementary proof of the central limit theorem , 1959 .

[14]  M. S. Raff.  On Approximating the Point Binomial , 1956 .

[15]  Lenos Trigeorgis,et al.  A Log-Transformed Binomial Numerical Analysis Method for Valuing Complex Multi-Option Investments , 1991, Journal of Financial and Quantitative Analysis.

[16]  Maurice G. Kendall,et al.  The advanced theory of statistics , 1945 .

[17]  Alan G. White,et al.  Efficient Procedures for Valuing European and American Path-Dependent Options , 1993 .

[18]  J. Ingersoll.  Theory of Financial Decision Making , 1987 .

[19]  John W. Pratt,et al.  A Normal Approximation for Binomial, F, Beta, and other Common, Related Tail Probabilities, II , 1968 .

[20]  P. Boyle.  Option Valuation Using a Three Jump Process , 1986 .

[21]  P. Boyle,et al.  Numerical Evaluation of Multivariate Contingent Claims , 1989 .

[22]  Daniel B. Nelson,et al.  Simple Binomial Processes as Diffusion Approximations in Financial Models , 1990 .

[23]  J. Hull.  Options, futures, and other derivative securities , 1989 .

[24]  Richard Breen,et al.  The Accelerated Binomial Option Pricing Model , 1991 .

[25]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[26]  Edward Omberg,et al.  Efficient Discrete Time Jump Process Models in Option Pricing , 1988, Journal of Financial and Quantitative Analysis.

[27]  E. B. Wilson,et al.  The Distribution of Chi-Square. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[28]  Dilip B. Madan,et al.  The multinomial option pricing model and its Brownian and poisson limits , 1989 .

[29]  P. Wilmott,et al.  Option pricing: Mathematical models and computation , 1994 .

[30]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[31]  R. Fisher.  The Advanced Theory of Statistics , 1943, Nature.

[32]  E. Paulson,et al.  An Approximate Normalization of the Analysis of Variance Distribution , 1942 .

[33]  S. Ross,et al.  The valuation of options for alternative stochastic processes , 1976 .

[34]  A. C. Berry.  The accuracy of the Gaussian approximation to the sum of independent variates , 1941 .

[35]  Paul L. Butzer,et al.  On the rate of approximation in the central limit theorem , 1975 .

[36]  R. C. Geary.  The Frequency Distribution of the Quotient of Two Normal Variates , 2022 .

[37]  M. Broadie,et al.  American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods , 1996 .

[38]  A. K. Basu,et al.  On the rate of approximation in the central limit theorem for dependent random variables and random vectors , 1980 .

[39]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

[40]  Some Numerical Comparisons of Several Approximations to the Binomial Distribution , 1969 .