Error estimates for the binomial approximation of American put options
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., \(r\leq \sigma^2/2)\), \[ \forall x\in R,\;\exists \widetilde C>0,\;\forall n\in N : P^{(n)}(0,x)- P(0,x)\leq\widetilde C(\sqrt{\ln n}/n)^{4/5}. \]
scientific article; zbMATH DE number 1319851
scientific article