Error estimates for the binomial approximation of American put options
type: Item
instance of: scholarly article
title: Error estimates for the binomial approximation of American put options
zbMATH Open document ID: 0939.60022
author: Damien Lamberton
published in: The Annals of Applied Probability
publication date: 2000-07-05T00:00:00Z
review text: As is known the value of American put option in the Black-Scholes model can be written as a function \(P(t,\ln S_t)\) of time \(t\) and the current stock price \(S_t\) with \[ P(t,x)= \sup_{\tau\in F_{0,T-t}}Ee^{-r\tau} \psi (x+\mu\tau+ \sigma B_\tau),\tag{1} \] where \(\psi(x)= (K-e^x)^+\), \(B=(B_t)_{0\leq t\leq T}\) is the standard Brownian motion, \(F_{0,T-t}\) denotes the set of all stopping times of the natural filtration of \(B\) and \(\mu=r-\sigma^2/2\) \((r\) is the interest rate, \(\sigma\) is the so-called volatility). On the other side one of the natural numerical methods to compute the function \(P\) defined by (1) is to approximate the underlying Brownian motion \(B\) by normed random walk \(B_t^{(n)}= \sqrt{T/n}\sum^{[nt/T]}_{k=1}\varepsilon_k\) and to use the function \[ P^{(n)}(t,x)= \sup_{\tau\in F^{(n)}_{0,T-t}} Ee^{-r\tau} \psi(x+ \mu\tau+ \sigma B_\tau^{(n)}) \] with corresponding set of stopping times \(F^{(n)}_{0,T-t}\), \(((\varepsilon_k)_{k\geq 1})\) is a sequence of i.i.d. r.v. satisfying \(P( \varepsilon_k=1) =P(\varepsilon_k=0) =1/2\). The main result of this paper is the following statement: For any real number \(x\), there exists positive constants \(c\) and \(C\) such that \[ \forall n\in N :-{c\over n^{2/3}}\leq P^{(n)} (0,x)-P(0,x) \leq{C\over n^{3/4}}; \] moreover, if \(\mu\leq 0\) (i.e., \(r\leq \sigma^2/2)\), \[ \forall x\in R,\;\exists \widetilde C>0,\;\forall n\in N : P^{(n)}(0,x)- P(0,x)\leq\widetilde C(\sqrt{\ln n}/n)^{4/5}. \]
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 91B28
zbMATH DE Number: 1319851
zbMATH Keywords: American put option
zbMATH Keywords: Black-Scholes model
zbMATH Keywords: put prices's binomial approximation
reviewed by: Serguey Victorovich Zhulenev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1027961041
OpenAlex ID: W2025218455
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DOI: 10.1214/AOAP/1027961041
description: scientific article; zbMATH DE number 1319851