Early Exercise of American Calls with Continuous Dividends
Summary
The document explains how dividends affect early exercise of American call options. With a continuous dividend yield, the answer describes an adjusted rate equal to the risk-free rate minus the dividend yield. When this adjusted rate is positive, it says early exercise is not strongly optimal; when negative, exercise boundaries can arise for some strikes, making the problem more like pricing an American put. A second explanation frames the dividend yield as interest on a foreign currency account, connecting the problem to FX options and relative interest rates.
For discrete dividends, the discussion says companies commonly announce the amount and payment date in advance. Before announcement, dividends must instead be estimated with a model. The document provides conceptual guidance rather than derivations or numerical examples, and its exercise conclusions depend on the stated rate setup and option assumptions. It does not detail market frictions, taxes, or the full mechanics of dividend forecasts.
Key ideas
- A continuous dividend yield can be incorporated by adjusting the risk-free rate downward by the yield.
- When the adjusted rate is negative, early exercise may be optimal for some strikes and times.
- The FX analogy treats continuous dividends as interest earned on a foreign currency account.
- Dividend dates and amounts are often announced in advance, while earlier valuation requires modeled estimates.
- The discussion gives intuition but no numerical examples or detailed treatment of market frictions.
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Full text
# Exercise on American call option and dividends # Exercise on American call option and dividends Consider an americal Call option on an underlying paying dividends. Then it is often argued that it is only optimal to exercise right before the dividend is paid out, otherwise one will not exercise. Now what if the dividend is continuous -can one then always see exercise? Furthermore, is a reasonable assumption that the above strategy is possible? I have little experience with how things actually works, but is it, in practice, known beforehand when lump sum dividends are paid out? ## Answer by Ulysses (score 4, accepted) https://quant.stackexchange.com/a/16013 When dividends are continuous, they are essentially negative interest rates, so you should price options w.r.t. new interest rate $\hat r := r-d$ where $r$ is the original interest rate and $d$ is the continuous dividend yield. If $\hat r>0$ then the price of the call is still a submartingale, so early exercise is not (strongly) optimal, however in a more realistic setting $\hat r<0$ so that pricing of the call becomes as intricate as that of the put: at each moment of time some strikes are early exercise and some are not. In practice, it is often known how often does the company pays/decides on the dividends. They always anounce the dividend amount and date in advance, before that you have to price the dividends using some model. ## Answer by Mark Joshi (score 1) https://quant.stackexchange.com/a/18834 in the continuous case, you can regard the dividend rate as the interest on a foreign bank account if we invest it so the number of shares grows at the rate $d.$ So we can think it as a call option on a foreign exchange rates. Now calls and puts are the same thing in foreign exchange just by changing viewpoint. So the pricing is just as hard for calls as for puts and exercise can happen any time depending on the relative interest rates.
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