In order to deduce the pricing of permanent options, we should first introduce a very classic thinking method: δ hedging technology. That is to say, by constructing a portfolio of first-class assets (stocks) and options ∏ = V-δ S, after a period of time, the income of this portfolio is equivalent to buying government bonds with the same value after making up the dividend loss of short selling stocks. That is to say, dv-δ ds = r (v-δ s) dt+δ sqdt. Because it is a permanent option, V is only related to S. According to the research of option theory, it is always assumed that S satisfies ds=r(S, t)dt+σ(S, t)dWt. By fully differentiating this formula with ito formula, δ = DV/DS can be obtained. (If you don't understand this part, you can refer to relevant reference books, because this part belongs to BS theorem and is generally taught before American options. I don't think you asked.
Substituting δ = dv/ds back into dv-δ ds = r (v-δ s) dt+δ sqdt, we can get the equation.
[(σs)^2*d^2v/(ds)^2]/2+(r-q)sdv/ds-rp=0
At the same time, there are initial conditions V(S0)=S0-K (strike price) and V(0)=0 (of course, when the stock price falls to 0, the option becomes a piece of paper).
This is actually a very basic work, because it is different from the option equation with duration and needs to solve the heat conduction equation. Now the equation [(σ s) 2 * d2v/(ds) 2]/2+(r-q) sdv/ds-RP = 0 is a simple second-order ordinary differential equation, which can be easily obtained by using the method of characteristic roots and two definite solution conditions.
The following step should be the core part of the permanent beauty solution.
The option pricing obtained above is a function of S0, in other words, the option holder needs to decide how big S0 is, and he can get the maximum income when exercising his right. This is the so-called free boundary. V(S) is closely related to the size of S0 and determines each other.
What I have to do now is to take V as a function of S0 and take the derivative of S0. When dV/dS0=0, the determined S0 is the S0 that optimizes V(S, S0). The result I made is S0=[a/(a- 1)]*K, and the option pricing is V (s) = {
I don't know if you can understand. I think this question should be a graduate student. It is not difficult to calm down and perform well.