Question

Consider the American put option with value P(S,t), where S is the asset price and t is time. Let E be the strike price, T be the expiry date, r be the interest rate, σ be the volatility, and Sf(t) be an optimal exercise boundary. Write down the partial differential equation on P(S,t) when S > Sf(t), the formula for P(S,t) when S < Sf(t), the inequality which P(S,t) must satisfy for S < Sf(t), the two conditions at S = Sf(t), the condition at t = T, the boundary condition at S = 0, and the boundary condition at S → ∞

Answer 1

S > Sf(t):

$∂P +\frac{1}{ 2} σ^2S^2∂^2sP + rS∂sP − rP = 0$

S < Sf(t):

$P (S, T) = max(E − S, 0)$

for S = Sf (t) and S = ∞:

$P (Sf (t), t) = E − Sf (t)$

$∂SP (Sf (t), t) = −1$

$P (+∞, t) = 0$