For a perpetual American put with strike 100, a 5% continuously compounded rate, 20% volatility and no dividends, find the spot level at which exercise becomes optimal and the option's value when spot is 100.

For a perpetual American put with strike 100, a 5% continuously compounded rate, 20% volatility and no dividends, find the spot level at which exercise becomes optimal and the option's value when spot is 100.

Approach: With no maturity the value is a function of spot alone, so solve the ordinary differential equation, keep the decaying power solution, and impose value matching and smooth pasting at the free boundary.

71.43. The perpetual put value satisfies 0.5*sigma^2*S^2*V'' + r*S*V' - r*V = 0, whose decaying solution is V = A*S^{-gamma} with gamma = 2r/sigma^2 = 2*0.05/0.04 = 2.5. Value matching at the exercise boundary requires A*B^{-gamma} = K - B and smooth pasting requires the derivatives to agree, and solving the pair gives B = K*gamma/(1 + gamma) = 100*2.5/3.5 = 71.43. Substituting back, the option is worth (K - B)*(B/S)^gamma, which at spot 100 is 28.57*(0.71429)^{2.5} = 28.57*0.4312 = 12.32. Smooth pasting is what makes the answer unique: value matching alone admits every boundary, and requiring the derivative to be continuous picks the one that maximises the value, which is the standard optimal stopping condition. Raising the rate lowers gamma and pushes the boundary up, since waiting to exercise costs more interest on the strike.

Follow-up: Add a continuous dividend yield of 3% and say what happens to the boundary and to the perpetual American call?

Key concepts: exercise boundary, smooth pasting, perpetual put, optimal stopping.