According to Kim (1990, p.560) in "The Analytic Valuation of American Options".
I understand the first minimum condition where K sets the lower bound of the optimal exercise boundary at expiry, but the second one is unclear to me,
Update: $\delta$ = divdend rate, risk-free interest rate = $r$, optimal exercise boundary as a function of time $B(s)$ , exercise price $K$, in addition i found the following explanation,
An additional explanation was provided by Huang 1996
[1]: https://i.stack.imgur.com/1vjKt.png
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Re: Optimal exercise boundary at expiration...
there's a standard explanation obtained by substituting the pay-off (S-K) at expiration into the PDE and looking at the sign of [$]\frac{\partial V}{\partial t}[$] at expiration
[$]\frac{\partial V}{\partial t}=-\frac{S^{2}\sigma^{2}}{2}\frac{\partial^{2}V}{\partial S^{2}}-(r-D)S\frac{\partial V}{\partial S}+rV[$]
substitute [$]V=S-K[$] in the right-hand side and you get
[$]\left.\frac{\partial V}{\partial t}\right|_{t=T}=-(r-D)S+r(S-K)=DS-rK[$]
and that term [$]DS-rK[$] changes sign at [$]S=rK/D[$]
I think I saw that in Paul's books
[$]\frac{\partial V}{\partial t}=-\frac{S^{2}\sigma^{2}}{2}\frac{\partial^{2}V}{\partial S^{2}}-(r-D)S\frac{\partial V}{\partial S}+rV[$]
substitute [$]V=S-K[$] in the right-hand side and you get
[$]\left.\frac{\partial V}{\partial t}\right|_{t=T}=-(r-D)S+r(S-K)=DS-rK[$]
and that term [$]DS-rK[$] changes sign at [$]S=rK/D[$]
I think I saw that in Paul's books
Re: Optimal exercise boundary at expiration...
Book name?there's a standard explanation obtained by substituting the pay-off (S-K) at expiration into the PDE and looking at the sign of [$]\frac{\partial V}{\partial t}[$] at expiration
[$]\frac{\partial V}{\partial t}=-\frac{S^{2}\sigma^{2}}{2}\frac{\partial^{2}V}{\partial S^{2}}-(r-D)S\frac{\partial V}{\partial S}+rV[$]
substitute [$]V=S-K[$] in the right-hand side and you get
[$]\left.\frac{\partial V}{\partial t}\right|_{t=T}=-(r-D)S+r(S-K)=DS-rK[$]
and that term [$]DS-rK[$] changes sign at [$]S=rK/D[$]
I think I saw that in Paul's books
Re: Optimal exercise boundary at expiration...
Paul = Wilmott
Option Pricing: Mathematical Models and Computation
Wilmott, dewynne, howison
Option Pricing: Mathematical Models and Computation
Wilmott, dewynne, howison
Re: Optimal exercise boundary at expiration...
I attach a one-page excerpt from my 2016 book that may be of interest here.
This particular book chapter (Ch18) is actually a reprint of an article I wrote for Paul's magazine: "American Options under Jump-diffusions: an Introduction". It begins with the Black-Scholes case.
This particular book chapter (Ch18) is actually a reprint of an article I wrote for Paul's magazine: "American Options under Jump-diffusions: an Introduction". It begins with the Black-Scholes case.
Lewis.Ch18.Excerpt.zip- (186.45 KiB) Downloaded 130 times