Implied Volatility for an American Call Around a Known Dividend

Summary

The document poses an implied-volatility problem for an American call when a known dividend is paid before expiration and interest rates are zero. Because early exercise may be optimal immediately before the ex-dividend date, the contract can be viewed as having exercise opportunities at the dividend date and at expiration. The question asks how to infer volatility from observed option prices and whether the zero-rate setting permits a better approximation than Black’s approximation.

No solution, pricing method, or numerical evidence is supplied. The useful concept is the interaction between a discrete dividend and early exercise: an American option may require accounting for exercise at the dividend date rather than treating it as a standard European-style option. The prompt’s inequality and proposed simplification are not resolved, so the document does not support a specific implied-volatility procedure or approximation.

Key ideas

• A known dividend before expiration can make early exercise of an American call relevant.
• The setup can be represented with exercise opportunities at the dividend date and at expiration.
• The question seeks implied volatility from observed prices when the option has discrete exercise opportunities.
• No pricing solution or improvement over Black’s approximation is provided.

Tags

# American options -- doing better than Black's approximation when $r = 0$


# American options -- doing better than Black's approximation when $r = 0$












I am trying to find the implied volatility smile for an American call option with a known dividend during the option tenor. For the sake of argument, let's say today is Jan 1, the dividend $D$ is paid out in 30 days at time $T_1$, and the option expires in 60 days at time $T_2$. Since $r = 0$, $$D < K(1-e^{-r(T-T_1)}) = 0$$ isn't satisfied, so it may be optimal to exercise at $T_1$. In effect, we have a Bermudan option with possible exercise times $T_1$ and $T_2$. If I have price data for the option, how can I back out the implied volatility? Is there an additional simplification or approximation we can make since $r = 0$ that does better than Black's approximation?

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