# Does price of american (put) option exhibit smooth pasting in time direction under B-S model?

Let us consider the BS model and let $f(s,t)$ denote the price of an American put option with $t$ to expiry, then it is known the solution of the optimal stopping (when it is risk neutral) related to this American put option can be characterised by a curve continuous, monotonically decreasing, convex curve $c(t)$ such that $c(0)=K$ and $c(\infty)$ is a known limit from the perpetual problem.

Let us denote $C$ as the continuation region of this problem, that is $\{(s,t):s>C(t)\}$ and $D=C^c$ is the stopping region. It is well established that smooth pasting is exhibited at the boundary, that is to say

$$\lim_{(t,s)\rightarrow(T,C(T))}\partial_sf(s,t)= -1$$

This condition can be proved in many different ways via classical theory as well as arguments using viscosity solutions. My question is that: is anything known about time derivative when we approach the boundary?

For example, does

$$\lim_{(t,s)\rightarrow(T,C(T))}\partial_tf(s,t)= 0$$

hold?

## 1 Answer

The Black-Scholes PDE holds in the continuation region : $$u_t = - \frac{1}{2} \sigma^2 u_{ss}$$ (ignoring interest rate). This says that theta is as smooth as gamma. The "smooth pasting" literature you mention shows that delta is continuous at the exercise boundary, but gamma has a jump. So theta has a jump as well. In other words, in the time direction price is continuous but its derivative is not.

• I do not quite understand why you can ignore the other terms. if you have 0 interest rate, an american put is a European put (by convexity of the pay off) then your continution is everywhere and this problem is meaningless. – Lost1 Jan 29 '16 at 21:49
• Anyhow, i see your point, i think it is valid if you put the other terms back - do you know any source which says the second derivative is not continous? (I know this is true for a perpetual option as explicit expression exists, but seen nothing explicit written in the finite horizon put option case) – Lost1 Jan 29 '16 at 21:54