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Asymptotics of the price oscillations of a European call option in a tree model
Author(s) -
Diener Francine,
Diener MARC
Publication year - 2004
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/j.0960-1627.2004.00192.x
Subject(s) - mathematics , limit (mathematics) , infinity , bounded function , term (time) , laplace transform , type (biology) , order (exchange) , zero (linguistics) , convergence (economics) , binomial options pricing model , call option , rate of convergence , binomial (polynomial) , tree (set theory) , valuation of options , mathematical analysis , econometrics , economics , physics , finance , statistics , ecology , linguistics , philosophy , channel (broadcasting) , engineering , quantum mechanics , electrical engineering , biology , economic growth
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 C 2 ( n )/ n , with C 2 ( n ) some explicit bounded function of n that has no limit when n tends to infinity.