Why American Put Pricing Uses an Inequality Near Exercise
Summary
The document discusses why the differential equation for an American put may be written as an inequality in the continuation region above the optimal exercise boundary. One response links the inequality to the option’s delta at the boundary and the condition that a portfolio’s return cannot exceed the return on a bank deposit. Another response argues that equality holds strictly above the boundary, and that the inequality is a generalized statement across both exercise and continuation regions.
The second explanation connects this distinction to numerical pricing. In tree or grid methods, continuation values are computed and then compared with immediate exercise values; near the exercise boundary, inserting an exercise value into a finite-difference stencil can disrupt the equality. The document presents both a theoretical interpretation and a practical discretization issue, but does not fully reconcile the two answers. Its treatment is therefore a concise explanation of the complementarity and numerical context, rather than a complete derivation.
Key ideas
- The American put has an optimal exercise boundary separating exercise and continuation regions.
- The response associates the boundary condition with a delta of minus one at the exercise boundary.
- Equality may hold in the continuation region, while an inequality expresses the broader early-exercise condition.
- Numerical schemes compare continuation values with immediate exercise payoffs.
- Grid updates near the boundary can affect finite-difference equality.
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Full text
# Question regarding the derivation of American Put option
# Question regarding the derivation of American Put option
When we derive the boundary conditions for the American put options, if we let $S_f(t)$ be the optimal exercise boundary, for $S \gt S_f(t)$ we get $$\frac{1}{2}\sigma^2S^2\frac{\partial^2P}{\partial S^2}+\frac{\partial P}{\partial t}-rP+rS\frac{\partial P}{\partial S} \le 0$$
Why is it $\le$ and not $=$ like the derivation for the European put option?
## Answer by user16651 (score 1)
https://quant.stackexchange.com/a/31197
Hint
The slop of option , delta, at $S=S_f(t)$ is $-1$, indeed $$\frac{\partial P}{\partial S}=-1$$ thus we can say that the return from the portfolio can not be greater than the return from a bank deposit, therefore
$$\frac{1}{2}\sigma^2S^2\frac{\partial^2P}{\partial S^2}+\frac{\partial P}{\partial t}-rP+rS\frac{\partial P}{\partial S} \le 0$$
Reference
- The Mathematics of Financial Derivatives: A Student Introduction, Paul Wilmott , Sam Howison and Jeff Dewynn (1995)
## Answer by Brian B (score 0)
https://quant.stackexchange.com/a/31194
Since we are specifying $S > S_f(t)$ you are correct, and equality holds. The author was probably trying to generalize to an equation where no restriction on being above the exercise boundary holds.
Mathematically, this is clumsy and useless but it often motivates the early exercise updates when we are using finite difference schemes (like trees or grids) to price the option.
Recall that in these schemes we discretize underling prices to a grid $S_n$ and backwardate from time $t^{m}$ to $t^{m+1}$ by applying a matrix operator to grid prices $P^m$.
If we use an explicit scheme like a tree this consists of computing $$ \tilde{P}^{m+1} = C \cdot P^m $$ for some matrix $C$ and then handling early exercise by setting $$ {P}^{m+1} = \max\left(\tilde{P}^{m+1} , X\right) $$ where $X$ are exercise values.
For nodes on the grid with no neighbors above exercise value, equality holds exactly. But right near exercise value the finite difference formula for $\frac{\partial^2 P}{\partial S^2}$ term gets an updated term inside, destroying the equality.
$$ \frac{\partial^2 P_n}{\partial S^2} \approx \frac{P_{n+1} - 2P_n + {\color{red} {{X}_{n-1}}}}{\Delta S^2} $$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.