## 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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