Difference between the values of European and American options

CFA level I / Derivatives / Basics of Derivative Pricing and Valuation / Difference between the values of European and American options

Minimum values of European call and European put options: The minimum value of the European call and European put options is zero.

For calculating the minimum values at a particular strike price, we need to consider the put-call parity equation:

p₀ + S₀ = c₀ + X(1+r)^-T

From here, the value of call option, c₀ = p₀ + S₀ - X(1+r)^-T

The value of put option cannot be lower than zero. Therefore, the minimum value of call option equals S₀ - X(1+r)^-T

Similarly, the minimum value of put option equals X(1+r)^-T - S₀

So, we can say that the minimum value of European call option = Max(0, S₀ - X(1+r)^-T)

The minimum value of European put option = Max(0, X(1+r)^-T - S₀)

Minimum value of American options: Since the American options can be exercised at any time, they are more valuable than the European options due to that additional feature. Using the uppercase letters for the American call and American put prices: C₀ and P₀, we can say that:

C₀ ≥ c₀ and P₀ ≥ p₀

The options can be exercised anytime. Therefore the minimum value of American call option and American put option can be written as:

C₀ = Max(0,S₀ - X)

P₀ = Max(0, X - S₀)

But since these values have to be greater than their European counterparts, we deduce:

C₀ = Max(0, S₀ - X(1+r)^-T) because S₀ - X(1+r)^-T is greater than S₀ - X.

The minimum value of American put option, however, will be different than the European put option and is given by P₀ = Max(0, X- S₀) because X - S₀ is greater than X(1+r)^-T - S₀.

Minimum value	American option	European option
Call option	Max(0, S₀ - X(1+r)^-T)	Max(0, S₀ - X(1+r)^-T)
Put option	X - S₀	X(1+r)^-T - S₀

Early exercise or not: The above result leads to a conclusion that an American call option is always worth more in the market than exercised before the expiry date. Therefore, an American call option will never be exercised early.

The only scenario where the early exercising of American call option is beneficial is when the underlying contain interim cash flows like dividends or coupons. When the stock goes ex-dividend, then there would be a fall in the stock price. So, an early exercise could be advantageous just before the ex-dividend date.

The motivation of early exercise is weakened with high carrying costs. Storage costs lend a preference of owning the option over owning the underlying.

The minimum value of American put option exceeds the minimum value of European put option. So, there is a much stronger motivation for early exercise especially when the option is deep-in-the-money and the discount factor dwarfs the volatility factor. A deep-in-the-money put option has a limit to its ultimate value as the underlying cannot fall below zero. However, there is no limit to its moneyness for the call option as there is no limit for the price rise.

The dividends and coupon encourage early exercise of American call option, and they discourage the early exercise of American put options. The value of put option should rise even more post ex-dividend date because the stock will fall in the price.

Check your concepts:

(58.27) In which of the following events, the early exercise of American call options can be advisable?

(a) Presence of dividends or other cash payment
(b) If the call option is deep-in-the-money
(c) If the call option is deep-out-of-money

(58.28) When there is a dividend for the underlying stock, then at what time the early exercise of American call option is advisable?

(a) Just before the ex-dividend date
(b) Just after the ex-dividend date
(c) Presence of dividends discourages the early exercise of American call option

Solutions:

(58.27) Correct Answer is A: The presence of dividends or cash flows encourages the early exercise of American call option.

(58.28) Correct Answer is A: The early exercise of American call option is advisable just before the ex-dividend date if the option is in-the-money as the call option price would fall after the ex-dividend date.

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Basics of Derivative Pricing and Valuation: Chapter Test