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Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends. Consider the perpetual American put option with payoff $(K-S_\tau)^+$ when exercised at time $\tau >0$. Given that $0<L<K$, consider the stopping time $\tau_L=inf(u\geq0:S_u\leq L)$.

For any $\lambda \in \Re$, $$Y_{\lambda,t} = (S_t/S_0)^\lambda e^{-(r\lambda-\lambda(1-\lambda)\sigma^2/2)t}$$.

For $\lambda=\lambda_-$, $$Y_{\lambda,t} = (S_t/S_0)^\lambda e^{-rt}$$

If $S_0\geq L$, then prove that: $E^Q[e^{-r\tau L}] = (S_0/L)^\lambda$

Was thinking to make use of the process $Y_{\lambda,t}$ in the formulation of the expectation. However, I can't seem to link it to the part regarding $\tau_L$.

I would really appreciate all the help I can get! Thank you!

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