Asymptotics of the price oscillations of a European call option in a tree model

It is well known that the price of a European vanilla option computed in a binomial tree model converges toward the Black‐Scholes price when the time step tends to zero. Moreover, it has been observed that this convergence is of order 1/n in usual models and that it is oscillatory. In this paper, we...

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Bibliographic Details
Published in:Mathematical finance 2004-04, Vol.14 (2), p.271-293
Main Authors: Diener, Francine, Diener, MARC
Format: Article
Language:English
Subjects:
asymptotic expansions
binomial trees
Decision trees
Derivatives
Economic models
Europe
European options
Finance
Laplace integral
Mathematical analysis
Mathematical models
Model testing
Monte Carlo simulation
Options markets
Options on stocks
oscillations
Prices
Studies
Value
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Summary:It is well known that the price of a European vanilla option computed in a binomial tree model converges toward the Black‐Scholes price when the time step tends to zero. Moreover, it has been observed that this convergence is of order 1/n in usual models and that it is oscillatory. In this paper, we compute this oscillatory behavior using asymptotics of Laplace integrals, giving explicitly the first terms of the asymptotics. This allows us to show that there is no asymptotic expansion in the usual sense, but that the rate of convergence is indeed of order 1/n in the case of usual binomial models since the second term (in ) vanishes. The next term is of type C2(n)/n, with C2(n) some explicit bounded function of n that has no limit when n tends to infinity.
ISSN:0960-1627
1467-9965
DOI:10.1111/j.0960-1627.2004.00192.x