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

Abstract

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 1/ √ n) 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.

Improving Convergence of Binomial Schemes and the Edgeworth Expansion

“…The asymptotics for Equation (5) has already been provided in [9] and later in [11] based on an integral representation of binomial sums (see e.g. [16]).…”

Random dynamics and finance: constructing implied binomial trees from a predetermined stationary density

“…Hubalek and Schachermayer (1998) explores the conditions needed to ensure that convergence of a general binomial prices imply convergence of option prices for general binomial models. And Diener and Diener (2004) explores the nature of the convergence of binomial models.…”

A European option general first-order error formula

“…Motivation. The problematic of describing and controlling the error for options evaluated under random walk approximations { () } of a geometric Brownian motion  has attracted the attention of several researchers, such as for instance [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. Knowledge and control of the error is of obvious interest when evaluating options through random walk approximations.…”

“…In Walsh [25] such first order error formula is given for general piecewise  (2) payoffs, but only in the specific case where the binomial scheme is the Cox Ross and Rubinstein scheme applied to the discounted process. In Diener and Diener [4], a first order error formula is provided for General Binomial Schemes, but only in the specific case where the payoff is a call option. In Diener and Diener [5], this first order error formula is obtained for digital options.…”

“…Since error formulae for digital and call options are already known, thanks to [4], [5] and [2], the contribution of this paper is to find the error formula for the  (1) part of . For the sake of simplicity, we will restrict our exposition to continuous payoff functions .…”