Handling Discrete Dividends in American Option Finite Differences
Summary
The note explains how a finite-difference valuation of an American option should handle a discrete dividend at an ex-dividend date. The stock-price jump is represented by mapping the option value just after the dividend to the value just before it, evaluating the post-dividend value at the pre-dividend stock price less the dividend. Interpolation can implement this mapping on a discrete spot grid.
For an American option, the suggested procedure checks for exercise on both sides of the dividend event: first compare continuation value with intrinsic value just after the payment, then apply the dividend mapping, and then make the exercise comparison just before payment. This accommodates possible exercise opportunities on either side of the date. The advice is framed for a single discrete dividend and a finite-difference scheme; grid design and interpolation accuracy are not examined. The note does not establish a universal ordering for all numerical methods.
Key ideas
- A discrete dividend creates a jump between pre-dividend and post-dividend stock prices.
- Map values across the dividend date by evaluating the post-dividend option value at the reduced stock price.
- For American exercise, compare intrinsic and continuation values both after and before the dividend adjustment.
- Interpolation may be needed to apply the stock-price mapping on a finite-difference grid.
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Full text
# pricing american put option with fdm
# pricing american put option with fdm
Assume I use some finite difference solver to solve for American type of exercise in BS framework where stock pays dividend discretely. Then at every time iteration, for call option, I firstly adjust for dividends and then account for early exercise by taking a max between intrinsic value and "value to hold".
Should I swap the order in a case of a put? Should I first adjust for early exercise and after for a dividend?
The reason I am in doubt is because with a call I better exercise right before the dividend and with a put right after. Is that correct?
## Answer by Quantuple (score 1, accepted)
https://quant.stackexchange.com/a/29564
I would recommend to do both.
Consider the situation where a single discrete dividend is paid at $t$. You use a Finite Difference (FD) scheme to price a European option. Starting from the terminal condition at $T$, by backward induction you manage to obtain the solution $$V(t^+, \mathcal{S})$$ for a discrete grid of spot levels $\mathcal{S}$ at time $t^+$. For the moment, it is as if you did not consider dividends.
Now, because the stock goes ex at $t$, the no-jump condition writes: $$V(t^-,\mathcal{S}) = V(t^+,\mathcal{S}-D) \tag{1}$$
When you talk about "accounting for the dividend payment", I assume you talk about the transformation $V(t^+,\mathcal{S}) \to V(t^-,\mathcal{S})$ you need to take care of before being able to resume your FD backward stepping up to time $t=0$. From $(1)$ you know that $V(t^+,\mathcal{S}) \to V(t^-,\mathcal{S})=V(t^+,\mathcal{S}-D)$. This means you can (for instance) perform an interpolation to find $V(t^+,\mathcal{S}-D)$ from $V(t^+,\mathcal{S})$ and set $V(t^-,\mathcal{S})$ equal to the result.
For American options I would advise to do, when you reach time step $t$
- Once you get $V(t^+,\mathcal{S})$, do $V(t^+,\mathcal{S}) = \max( V(t^+,\mathcal{S}), (\phi(\mathcal{S}-K))^+)$ = check for optimal exercise opportunity after a dividend payment
- Account for the dividend payment $V(t^+,\mathcal{S}) \to V(t^-,\mathcal{S})=V(t^+,\mathcal{S}-D)$ = account for dividend payment
- Once you get $V(t^-,\mathcal{S})$, do $V(t^-,\mathcal{S}) = \max( V(t^-,\mathcal{S}), (\phi(\mathcal{S}-K))^+)$ = check for optimal exercise opportunity before a dividend payment
- Move backward to previous time step $t-\Delta t$... and repeat.
with $\phi=\pm1$ for call/put respectively.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.