Why an American Put’s Theta Can Jump at the Exercise Boundary
Summary
The note asks whether an American put’s time derivative, or theta, is continuous where the optimal exercise boundary meets the continuation region. It distinguishes this question from the familiar smooth-pasting condition, which says the option’s delta matches the derivative of its exercise payoff at the boundary.
The answer uses the Black–Scholes PDE in the continuation region, with interest rates omitted, to relate theta to gamma. Since smooth pasting ensures delta continuity but does not ensure gamma continuity, gamma may jump across the boundary; theta can consequently jump as well. The price itself remains continuous, but its time derivative need not be. This is a concise qualitative explanation rather than a derivation of the boundary behavior, and the stated PDE relation is presented under the simplified assumption of no interest rate.
Key ideas
- Smooth pasting at an American put’s exercise boundary establishes continuity of delta.
- It does not guarantee that gamma is continuous across the boundary.
- In the continuation region, the Black–Scholes PDE links theta to gamma when interest is ignored.
- A jump in gamma can therefore produce a jump in theta even though option value remains continuous.
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Full text
# Does price of american (put) option exhibit smooth pasting in time direction under B-S model?
# 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?
## Answer by q.t.f. (score 1, accepted)
https://quant.stackexchange.com/a/23007
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.