Using a Stopped Martingale to Value a Perpetual American Put

Summary

The document asks how to evaluate the discounted probability that a Black–Scholes stock price, starting at or above a lower barrier L, ever reaches that barrier. It introduces a family of exponential stock-price processes and notes that, for a particular negative root, the process takes the form of the stock raised to that power and discounted at the risk-free rate. The proposed connection is to stop this process at the first passage time and use its martingale property to derive the expectation.

The post does not provide the proof or specify the root equation and assumptions needed to justify optional stopping. In particular, the displayed expectation appears to contain a typographical ambiguity: the intended quantity is likely the discounted indicator or exponential of the hitting time, rather than discounting the barrier level itself. The result also depends on the stock dynamics, parameter conditions, and treatment of paths that never hit the barrier.

Key ideas

• A first-passage time to a lower stock-price barrier can be analyzed by stopping a suitable martingale.
• The negative characteristic root links the stock-price power to risk-free discounting.
• A complete proof requires conditions that justify stopping the martingale at the hitting time.
• The stated expectation is ambiguous and may contain a notation error.

Tags

# Proving an Expectation


# Proving an Expectation












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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