Valuing a Perpetual American Put Option

This question focuses on valuing a perpetual American put option written on a risky asset following a standard continuous-time diffusion. With no fixed maturity date, the holder can choose any exercise time, which turns the problem into a classic optimal stopping exercise in option pricing. The setup is closely related to textbook treatments of real options and is common in more theoretical interviews for derivatives quant and exotics pricing roles, where candidates are expected to move comfortably between probabilistic and differential equation viewpoints for valuation problems.

Solving it leans heavily on transforming the usual time-dependent pricing equation into a time-homogeneous ordinary differential equation, then identifying a value function that satisfies appropriate boundary conditions. The key ingredients include risk-neutral pricing, free-boundary problems, and the smooth pasting condition at the optimal exercise threshold. Interviewers look for a clean derivation of the functional form, correct use of discounting and drift adjustments, and a clear explanation of how the critical exercise boundary is determined and interpreted economically.

What it tests

The core structure of perpetual American option problems is that, with infinite time to expiry, the value function becomes time-independent and the problem reduces from a partial differential equation to an ordinary differential equation in the underlying asset price. The solution relies on finding a function that satisfies the ODE everywhere except at the optimal exercise boundary, where the value must match the immediate exercise payoff and be smooth (the smooth pasting condition). The perpetual nature means the optimal stopping problem is governed by balancing the instantaneous benefits of exercise against the expected discounted future payoff, leading to a free boundary problem. The solution is characterized by power-law behavior, with exponents determined by the roots of the characteristic equation from the ODE, and the critical threshold where early exercise becomes optimal is found by matching value and slope at the boundary. This pattern holds because, without time decay, the only tradeoff is between the option's intrinsic value and the present value of waiting to exercise, so the solution is fully determined by the asset's dynamics and the discount rate.