European Option

Contents (hide)

  1. 1. Definitions
  2. 2. The payoff structure
  3. 3. Option premium
  4. 4. Examples

1.  Definitions

European options give the right (not the obligation) to the holder to buy or sell the underlying asset on a predetermined date, at a predetermined price. The holder of the option will only exercise this right if it is profitable. They are very important examples of financial derivatives.

2.  The payoff structure

A European call gives the right to buy the asset, while a European put gives the right to sell. We can buy or sell these options in the marketplace. Thus, we have four different investment alternatives:

  • Long european call: we have bought the right to buy the asset
  • Short european call: we have sold the right to buy the asset
  • Long european put: we have bought the right to sell the asset
  • Short european put: we have sold the right to sell the asset
Since the buyer of the option has the right to exercise, and will only do so if her cashflows are positive. For example, the figure on the right gives the payoffs of a European call option with strike price \fs4 X=\$100. They payoffs are shown on maturity. The holder of the option will make the following decisions, based on the price of the underlying asset on maturity, denoted with \fs4 S_T:
  • If the current price is above the exercise price, that is if \fs4 S_T>X, she will exercise the option which means
    • Buying the asset at the exercise price, \fs4 X=\$100
    • Making an instant profit by selling the asset in the market. Apparently, the profit will be \fs4 C_T = S_T-X.
  • If the current price is below the exercise price, \fs4 S_T\le X, the holder will leave the option to expire worthless. It is cheaper to but the asset in the market, rather than exercising. The payoffs will be zero, \fs4 C_T=0

One can compactly write the payoffs of the European call option on maturity as \fs4 C_T = \max\{S_T-X, 0\} = (S_T-X)^+. They are zero up to the strike price, and increase linearly (at the \fs4 45^o line) after that point.

The payoffs for the European put can be retrieved using similar arguments. Now, the option will be exercised if \fs4 S_T<X, since the holder will buy at the market price \fs4 S_T and sell at \fs4 X. The payoffs are now compactly written as \fs4 P_T = \max\{X-S_T, 0\} = (X-S_T)^+. Short positions give the corresponding losses, relative to the long positions. All payoff graphs are given below.

3.  Option premium

Since the writer of the option faces only potential losses (while the holder faces only potential profits) an incentive is needed for the writer to engage in the contract. This incentive is called the option price or option premium. It is an amount that the long party (buyer) pays the short party (seller), at time zero. Thus, if the option is not exercised, the short party at least keeps the premium. These premiums are quoted in the marketplace for most underlying assets. For example at http://finance.yahoo.com/q/op?s=MSFT one can find the quoted prices for options written on the Microsoft stock price.

4.  Examples

As an example, on Jun-16 2005, the closing price for Microsoft was approximately $25. A European call option that expires on Oct-21 2005, with strike price $22.50 was quoted at $2.85-$2.95 (bid-ask). This means that

  • If we wanted to buy this option, we should pay $2.95 today. Then, after approximately four months, we have the right to buy a Microsoft share at $22.50. If we don't exercise this right, we will lose the $2.95 premium. This will happen if the Microsoft share price after four months is below $22.50 (today it is $25). Nevertheless, we cannot lose more than the premium.
  • If we wanted to sell this option, we will receive $2.85 today. We give the right to someone else to exercise, and buy the share from us at $22.50 in four months time. If the other party chooses not to exercise, we are left with a profit of $2.85. This will happen if the price is below $22.50. Notice that we cannot make more than the premium; in fact, if the price increases, we are facing unlimited losses.

The task of determining a fair option price is one of the most challenging topics in finance today. Loosely speaking, to determine the price of a call or a put one will have to

The seminal Black-Scholes model revolutionized option pricing theory since its introduction in 1973, and is still widely used today in one form or another.

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