Early Exercise Premiums for American Options with Continuous Dividends
Summary
The response addresses early exercise of American calls and puts on a stock that pays a continuous dividend yield. It presents an early exercise premium decomposition: each American option’s value is expressed as its European value plus terms involving rates, the dividend yield, and probabilities that the stock price lies beyond an optimal exercise boundary. The cited derivations use an optimal exercise curve and, for some terms, a stock-based probability measure.
The key limiting case given is that when the dividend yield is zero, early exercise of a call is not optimal, while a put can be exercised early. The document does not give a simple threshold rule for when to exercise with a positive continuous yield, nor does it work through parameter values or a numerical example. Its formulas assume a continuous diffusion framework; it notes that jump-diffusion models require additional terms to account for crossing the exercise boundary by jumps.
Key ideas
- An American option’s value can be decomposed into its European value and an early exercise premium.
- The premium depends on the dividend yield, interest rate, and probabilities relative to an optimal exercise boundary.
- With no dividends, early exercise is not optimal for a call under the stated framework.
- A put may be optimal to exercise early even when the dividend yield is zero.
- Jump-diffusion settings add terms for jumps across the exercise boundary.
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# Early exercise American Options with Dividend
# Early exercise American Options with Dividend
this is a basic question but I have not fully understood it.
Let's say we have dividend paying stock (continuous dividend yield), when would we exercise the Option early? Since the Dividend yield is continuous I am a bit confused. If it was discrete I would say, in the case of a call Option, it could make sense to exercise shortly before a Dividend payout.
Now the main questions. Would we always exercise the American Call Option early(cont. div)? Would we ever exercise the American Put Option early (cont. div)?
Thanks a lot!
## Answer by Kevin (score 2)
https://quant.stackexchange.com/a/66663
Suppose the stock price $(S_t)$ follows a continuous diffusion which pays dividends with yield $q$. Using different levels of rigour, Kim (1990), Jacka (1991), Myneni (1992) and Carr et al. (1992) derive the following decomposition of the fair values of American options \begin{align} C_A(S_0;K,T) &= C_E(S_0;K,T)+ \underbrace{q S_0\int_0^T e^{-q t} \mathbb{Q}_S[\{S_t\geq B_t\}]\text{d}t - rK \int_0^T e^{-rt} \mathbb{Q}[\{S_t\geq B_t\}]\text{d}t}_\text{Early Exercise Premium}, \\ P_A(S_0;K,T) &= P_E(S_0;K,T) + \underbrace{rK \int_0^T e^{-rt} \mathbb{Q}[\{S_t\leq B_t\}]\text{d}t - q S_0\int_0^T e^{-q t} \mathbb{Q}_S[\{S_t\leq B_t\}]\text{d}t}_\text{Early Exercise Premium}, \end{align} where $B_t$ is the optimal exercise curve and $\mathbb{Q}_S$ is the probability measure which uses $S_te^{qt}$ as numéraire. Mathematically, this decomposition resembles Riesz' decomposition or Doob-Meyer's decomposition.
If the asset pays no dividends, $q=0$, the early exercise premium for a call option would be negative and early exercise is never optimal. A put option, on the other hand, can very well be early exercised if $q=0$.
Pham (1997) and Gukhal (2001) generalise the decomposition to finite-active jump diffusions (you get extra terms capturing the possibility jumping above/below $B_t$).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.