### European Options

European options only allow for the exercise of options at the expiry date while American options allow for early exercise. The terms do not convey any information about where, geographically, the options are traded. The right to exercise is a complex feature of an option and as such, European options with a fixed exercise date are dealt with first.

## Standard Notation

The following notion is used:

• cT = Price of European call at expiration
• ST = Price of the underlying at time T
• X = Exercise price
• pT = Price of a European put option at expiration

## Value at Expiration

### European Call Options

The value of a European call option at expiration is the exercise value, which is the greater of zero or the underlying price at expiration minus the exercise price:

$$c_T = max(0, S_T – X)$$

### European Put Options

The value of a European put option at expiration is simply the opposite – the greater of zero or the exercise price minus the underlying price at expiration:

$$p_T = max(0, X – S_T)$$

## Question

There is a European call option that gives the holder the right to buy 100 shares of company XYZ at $100 in 3 months’ time. What is the value of this call option if the spot price at expiration is$110.15?

A. $1,015 B.$0

C. -$1,015 Solution The correct answer is A. CT = 100 × max($0, $110.15-$100) = $100 ×$10.15 = $1,015 Since we have a call option with a strike price of$100, it makes sense that if the price at expiration is higher than the strike price, our option has gained in value.

Explain how the value of a European option is determined at expiration

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