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On the early exercise boundary of the American put option. (English) Zbl 1029.91028

Let us consider the Black-Scholes model for the value of an American put option, \(f(S,t)\), where \(S\) is the price of the stock underlying the option and \(t\) is the time remaining until the option expires. The American put option may be exercised at any time for a yield of \(K-S\), where \(K\) is the strike price of the option. This leads to splitting the domain \(\{S\in(-\infty,\infty),t\in[0,\infty)\}\) into two regions which are separated by the early exercise boundary, \(S(t)=b(t)\): if \(S\leq b(t)\), then the option should be immediately exercised and, therefore, \(f(S,t)=K-S\); if \(S>b(t)\), then the option should not be exercised, and \(f(S,t)\) is governed by the Black-Scholes equation \(f_{t}={\sigma^2S^2\over 2} f_{SS}+rSf_{S}-rf\), where \(\sigma\) is the stock’s volatility and \(r\) is the risk-free interest rate. The authors obtain an integral equation for \(b(t)\): \[ \hat G\left(\ln\left({b(t)\over K}\right),t\right)=-{2r\over\sigma^2}\int_{0}^{t}{\dot b(s)\over b(s)}\hat G\left(\ln\left({b(t)\over b(s)}\right),t-s\right) ds, \] where \(\hat G(X,T)={K\sigma\over \sqrt{8\pi T}}\exp\left\{-{\left(X-\left(r-{\sigma^2\over 2}\right)T\right)^2\over 2\sigma^2T}-rT\right\}\). For the short time bounds \[ b(t)=K(1-\sigma\sqrt{-t\ln(Ctv^2(t))}),\;\text{where} C={8\pi r^2\over\sigma^2} \text{and} \lim_{t\to 0}v^2(t)\geq {1\over 4},\;\lim_{t\to 0}{v^2(t)\over-\ln(Ct)}\leq 1. \] The authors show that \(v\approx 1+1/\ln(Ct)+O((1/\ln(Ct))^2)\).

MSC:

91B28 Finance etc. (MSC2000)
35R35 Free boundary problems for PDEs
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