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Explain and assess the possible reasons for exercising American call and put options early.

FRM® Part I · Financial Markets and Products · Properties of Options · FRM1.FM.13.d

The idea

An American call on a stock paying no dividends should never be exercised early. Exercising gives up the insurance that the holder keeps if the stock falls, and it pays the strike sooner, losing interest on K. The call is worth at least S − K × exp(−rT), which exceeds S − K, so selling beats exercising. With dividends, early exercise can be optimal only just before an ex-dividend date, when the dividend exceeds the interest saved on the strike over the remaining life. An American put can be worth exercising early when it is deep in the money, because the gain is capped at K and interest on the cash received outweighs the remaining insurance value. High rates and low volatility favour early put exercise. The exam trap is applying call logic to puts.

Formulas

Dividend test for exercising an American call before the last ex-dividend date

D ≤ K × (1 − e^(−r × (T − t)))

D
the dividend paid on the last ex-dividend date
K
the strike price
r
the continuously compounded risk-free rate
T
the option's expiry, in years from now
t
the time of the ex-dividend date, in years from now

When this holds, the dividend is no more than the interest saved by paying the strike later, so early exercise is not optimal and the call is held.

A practice question

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

Why is it never optimal to exercise an American call early on a stock that pays no dividends?

  1. Exercising is penalised by the exchange because early exercise is a breach of the contractWhy not: Invents an exchange penalty; American options may be exercised at any time and the reason is economic.
  2. The call's value is always equal to the stock price, so exercising gains nothingWhy not: Confuses the upper bound S with the call's value; the call is worth less than S.
  3. Because the holder is obliged to hold a call until expiry if the stock pays no dividendWhy not: Misstates the contract; the holder may exercise at any time, but it is simply not profitable.
  4. Exercising forfeits the insurance value and time value, and pays the strike earlier; the call is worth more than S - K, so selling is better - correct

The American call is worth at least S - K x exp(-rT), which exceeds the exercise value S - K. Exercising early gives up the insurance against a price fall and pays the strike sooner, so holding or selling dominates.

A harder one

Which combination of conditions makes early exercise of an American put on a non-dividend stock more attractive?

  1. The put is out of the money, interest rates are high and volatility is highWhy not: Out-of-the-money options have no intrinsic value to realise, and high volatility raises the insurance value of waiting.
  2. The put is deep in the money, interest rates are high and volatility is low - correct
  3. The put is deep in the money, interest rates are low and volatility is highWhy not: Low rates lessen the interest earned on the strike, and high volatility makes waiting more valuable.
  4. The put is at the money, interest rates are low and volatility is lowWhy not: At-the-money puts have little intrinsic value, and low rates give little incentive to take the cash early.

Early exercise gives the strike now, earning interest, but gives up insurance. The trade is best when the put is deep in the money (little insurance left), rates are high (more interest earned) and volatility is low (less value in waiting).

The hardest tier

A trader holds an American call on a non-dividend stock currently at USD 80, with strike USD 50 and two years to expiry. The continuously compounded rate is 5%. The trader wants to realise value now. What is the lower bound on the market price at which the call could be sold, versus the exercise value of USD 30?

  1. USD 30.00, so selling and exercising are equivalentWhy not: Uses intrinsic value as the bound; the call also has the interest benefit of deferring payment of the strike.
  2. USD 34.76, so selling beats exercising - correct
  3. USD 24.74, so exercising beats sellingWhy not: Compounds the strike forward instead of discounting it, getting a bound below intrinsic value.
  4. USD 80.00, so the call can be sold for the stock priceWhy not: Reports the upper bound S as the sale price; the lower bound is S - K x exp(-rT).

The call is worth at least S - K x exp(-rT) = 80 - 50 x exp(-0.10) = 80 - 45.24 = 34.76, which exceeds the exercise value of 30. Selling captures at least 4.76 more than exercising.

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