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The rate of convergence of the binomial tree scheme. (English) Zbl 1062.91027

The binomial tree scheme introduced by J. C. Cox, S. A. Ross and M. Rubinstein [J. Financ. Econ. 7, No. 3, 229–263 (1979; Zbl 1131.91333)] as a simplification of the Black-Scholes model for valuing options is a popular and practical way to evaluate various contingent claims. Its usefulness comes from the fact that it reflects the real-time development of the stochastic price and it is easy to adapt it to the computation of American and other options. From another point of view it is simply a numerical solution method for the initial value problem for certain partial differential equations and it is known to be of the first order, that is, the error varies inversely to the number of time steps. The author studies the convergence closely. The exact rate of convergence is determined. An expression for constants of the rate of convergence is found. Knowing the form of the error allows the author to modify the Richardson extrapolation method to get a method of order 3/2. The delta, which determine the hedging strategy, can also be determined from the tree scheme, and converges with the same rate. The procedure called Skorokhod embedding is used to embed the Markov chain in the Black-Scholes diffusion model. This makes it possible to evaluate the error. This was done in a slightly different way by L. C. G. Rogers and E. J. Stapleton [Finance Stoch. 2, No. 1, 3–17 (1998; Zbl 0894.90025)] who used it to speed up the convergence of the binomial tree scheme.
The proposed technique is applicable to much more general cases.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
60G40 Stopping times; optimal stopping problems; gambling theory
60G44 Martingales with continuous parameter
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