derivate discounted payoff

Question

Let $S(t)$ be the stock price at time $t\geq 0$ with $S(0)=s_0$ and let $\Pi(S_t)=\max\{K-S_t,0\}=(K-S_t)_+$ the payoff of an american put with strike price $K$. How can I calculate the derivate $\frac{d}{d \psi} \mathbb E[e^{-\delta t} \Pi(S(t))\chi_{\{S(t)<\psi\}} | S(0)=s]$ for a stochastic $t$ and fixed $s$? I dont know how to start here, thanks for a hint.

Answer 1

I'd reason like this:

first we can notice that in order for the argument to be different to zero, i must be that both $S < K$ and $S < \psi$. Hence we have two cases:

1. $\psi > K$: in this case it must be $S< K$ and then the function does not depend on $\psi$ which means its derivative is zero
2. $\psi < K$: in this case it must be $S< \psi$ and then the expectation becomes

$$ A:= \int_{-\infty}^{\psi}e^{-\delta t}(K-S)f_S(S) dS \Rightarrow A_{\psi} = e^{-\delta t}(K-\psi)f_S(\psi) $$

Putting both together, I'd say the derivative is $ e^{-\delta t}(K-\psi)^{\text{+}}f_S(\psi)$