Smooth Pasting of the American Put Suppose a portfolio consists of an American put and the corresponding underlying. Hence the value of the portfolio is Π := V Am P + S, where S satisfies the SDE (1.33). Sf is the value for which we have high contact, compare

(4.22).

a) Show that dΠ =

⎧

⎨

⎩

0 for



S

∂V Am P

∂S + 1

σS dW + O(dt) for S>Sf .

b) Use this to argue

∂V Am P

∂S (Sf(t),t) = −1 .