Solution of a PDE using price of a european put option

Question

I'm reading some articles about PDE and I found the following PDE, with $q_1,A >0$:
$g_t(t,y)+ \beta^2yg_y(t,y)+\frac{1}{2}\beta^2y^2g_{yy}(t,y)-q_1 g(t,y)=0 \quad (t,y) \in [0,T), \times (0,+\infty)$.
with boundary conditions $\begin{cases} g(t,0)=F, t \in [0,T] \\ g(T,y)= \max\{F-Ay,S\} \end{cases}$
The solution proposed is the following:
$g(t,y)=Ae^{(\beta^2-q_1)(T-t)}p_{put}(t,y)+Se^{-q_1(T-t)}$
where $p_{put}$ is the price of a European put option with strike price $\frac{1}{A} (F-S)$ in a BS market with volatility of risky asset $\beta$ and rate of riskless asset $\beta^2$.
My question is the following. I cannot understand actually the solution becasue if I substitute in the PDE it solves but it doesn't satisfy the condition $g(t,0)=F$ becasue I obtain $g(t,0)=Fe^{-q_1(T-t)}$. Can anyone help me?

Answer 1

At expiration, the put option has value $p(T,y)=\max\left(A^{-1}(F-S) - y,0 \right)$ where $y$ is the underlying price at that time. When the underlying price is $0$ the payoff is $p(T,0) = A^{-1}(F-S)$. As is usual for equities when bankruptcy occurs, the state $y=0$ is assumed to be an absorbing barrier. If $y = 0$ is attained at an earlier time $t < T$, then the value of the put must be the present value of the payoff at expiration to avoid arbitrage. Hence,

$$p(t,0) = A^{-1}(F-S)e^{-\beta^2(T-t)}$$

With $g(t,y)$ as given, the corresponding condition for $g(y,0)$ is

$$g(t,0) = Ae^{(\beta^2-q_1)(T-t)}p(t,0)+Se^{-q_1(T-t)} = Ae^{(\beta^2-q_1)(T-t)}A^{-1}(F-S)e^{-\beta^2(T-t)}+Se^{-q_1(T-t)} \\ = (F-S)e^{-q_1(T-t)}+ Se^{-q_1(T-t)}= Fe^{-q_1(T-t)}$$

It appears you are correct and the condition $g(t,0) = F$ in the article is a typographical error.