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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!

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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 $B_t$).

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