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Explain the effect dividends have on whether American calls and puts should be exercised early.

FRM® Part I · Valuation and Risk Models · The Black-Scholes-Merton Model · FRM1.VR.15.f

The idea

An American option can be exercised at any time, so the question is when exercising beats holding or selling. A call on a stock paying no dividends should never be exercised early, because exercise yields only S − K, while the call is worth at least S minus the present value of K, which is more. The call also keeps its insurance against a price fall. With dividends, early exercise of a call can be optimal only just before an ex-dividend date. It is ruled out when the dividend D is no more than the interest on the strike, given by Saved = K × (1 − e^(−r×t)). Here t is the time left, from the ex-dividend date to expiry or to the next ex-dividend date. American puts differ. Even with no dividends, a deep in-the-money put may be exercised early because the strike received earns interest. Dividends make waiting through the ex-dividend date more attractive for a put, because the price drop raises the put's intrinsic value.

Formulas

Interest on the strike, the test for exercising a call before an ex-dividend date

I = K × (1 − e^(−r × τ))

K
the strike price
r
the continuously compounded risk-free rate
τ
the time left from the ex-dividend date, in years

Early exercise just before the ex-date is ruled out if the dividend is no more than I. Comparing the dividend with interest on the dividend itself is the trap.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement correctly explains why it is never optimal to exercise an American call on a non-dividend-paying stock before expiry?

  1. American calls cannot be exercised before expiry, so the question does not arise.Why not: American options can be exercised at any time; it is European options that cannot.
  2. The call's time value is zero whenever the stock pays no dividends, so there is nothing to lose by waiting.Why not: Time value is positive for a non-dividend stock; early exercise gives it up, which is why it is not optimal.
  3. Exercising gives only S - K, while the call is worth more than that and keeps its insurance value, so selling it beats exercising it. - correct
  4. The intrinsic value of a call, S - K, is always zero before expiry.Why not: Intrinsic value is positive whenever S exceeds K; the argument is that the call is worth more than it.

The call's value is at least S - K x e^(-rT), which exceeds S - K, and early exercise also gives up protection against a price fall. Selling is better than exercising.

A harder one

An American call has strike 60. The last ex-dividend date is three months before expiry, the dividend is 0.45 per share, and the continuously compounded risk-free rate is 4%. Which statement about exercising just before that ex-dividend date is correct?

  1. It is not optimal, because the dividend of 0.45 is less than the interest on the strike over the remaining time, about 0.60. - correct
  2. It is optimal, because the call is in the money and a dividend is about to be paid.Why not: Being in the money with a dividend due is not enough; the dividend must exceed the interest on the strike.
  3. It is optimal, because the dividend of 0.45 exceeds the interest on the dividend itself over three months, about 0.0045.Why not: Compares the dividend with interest on the dividend; the right comparison is with interest on the strike.
  4. It is not optimal, because early exercise of a call is never optimal when the stock pays dividends.Why not: The 'never exercise early' rule applies to stocks paying no dividends; with dividends it depends on their size.

The condition is D versus K x (1 - e^(-r x time left)) = 60 x (1 - e^(-0.04 x 0.25)) = 0.60. Since 0.45 < 0.60, the dividend does not compensate for paying the strike early.

The hardest tier

A dealer values an American call with nine months to expiry on a stock that goes ex-dividend once, in six months. Black's approximation takes the larger of two European calls: one maturing in nine months on the stock price less the present value of the dividend, and one maturing just before the ex-date. Values: nine-month European call on the adjusted price 4.10; call maturing just before the ex-date 4.35; nine-month European call on the unadjusted price 4.80. The approximate American call value is:

  1. 4.80Why not: Uses the nine-month call on the unadjusted price, which ignores the dividend that the holder will not receive.
  2. 4.10Why not: Takes only the dividend-adjusted European call, ignoring the value of exercising just before the ex-date.
  3. 4.35 - correct
  4. 4.23Why not: Averages the two eligible values instead of taking the larger.

Black's approximation is max(4.10, 4.35) = 4.35: exercising just before the ex-dividend date is worth more than holding to expiry on a dividend-adjusted price.

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