Quantitative Finance for Physicists

delivers the asset to the buyer and the buyer pays the cash amount at

the delivery price. At maturity, the current (spot) asset price, S

T

, may

differ from the delivery price, K. Then the payoff from the long

position is S

T

 K and the payoff from the short position is K  S

T

.

Future contracts are the forward contracts that are traded on

organized exchanges, such as the Chicago Board of Trade (CBOT)

and the Chicago Mercantile Exchange (CME). The exchanges deter-

mine the standardized amounts of traded assets, delivery dates, and

the transaction protocols.

In contrast to the forward and future contracts, options give an

option holder the right to trade an underlying asset rather than the

obligation to do this. In particular, the call option gives its holder the

right to buy the underlying asset at a specific price (so-called exercise

price or strike price) by a certain date (expiration date or maturity).

The put option gives its holder the right to sell the underlying asset at a

strike price by an expiration date. Two basic option types are the

European options and the American options.

2

The European options

can be exercised only on the expiration date while the American

options can be exercised any time up to the expiration date. Most of

the current trading options are American. Yet, it is often easier to

analyze the European options and use the results for deriving proper-

ties of the corresponding American options.

The option pricing theory has been an object of intensive research

since the pioneering works of Black, Merton, and Scholes in the

1970s. Still, as we shall see, it poses many challenges.

9.2 GENERAL PROPERTIES OF STOCK OPTIONS

The stock option price is determined with six factors:

. Current stock price, S

. Strike price, K

. Time to maturity, T

. Stock price volatility, s

. Risk-free interest rate,

3

r

. Dividends paid during the life of the option, D.

Let us discuss how each of these factors affects the option price

providing all other factors are fixed. Longer maturity time increases

94 Option Pricing

the value of an American option since its holders have more time to

exercise it with profit. Note that this is not true for a European option

that can be exercised only at maturity date. All other factors, how-

ever, affect the American and European options in similar ways.

The effects of the stock price and the strike price are opposite for

call options and put options. Namely, payoff of a call option increases

while payoff of a put option decreases with rising difference between

the stock price and the strike price.

Growing volatility increases the value of both call options and put

options: it yields better chances to exercise them with higher payoff.

In the mean time, potential losses cannot exceed the option price.

The effect of the risk-free rate is not straightforward. At a fixed

stock price, the rising risk-free rate increases the value of the call

option. Indeed, the option holder may defer paying for shares and

invest this payment into the risk-free assets until the option matures.

On the contrary, the value of the put option decreases with the risk-

free rate since the option holder defers receiving payment from selling

shares and therefore cannot invest them into the risk-free assets.

However, rising interest rates often lead to falling stock prices,

which may change the resulting effect of the risk-free rate.

Dividends effectively reduce the stock prices. Therefore, dividends

decrease value of call options and increase value of put options.

Now, let us consider the payoffs at maturity for four possible

European option positions. The long call option means that the in-

vestor buys the right to buy an underlying asset. Obviously, it makes

sense to exercise the option only if S > K. Therefore, its payoff is

P

LC

¼ max [S  K, 0] (9:2:1)

The short call option means that the investor sells the right to buy an

underlying asset. This option is exercised if S > K, and its payoff is

P

SC

¼ min [K  S, 0] (9:2:2)

The long put option means that the investor buys the right to sell an

underlying asset. This option is exercised when K > S, and its payoff

is

P

LP

¼ max [K  S, 0] (9:2:3)

The short put option means that the investor sells the right to sell an

underlying asset. This option is exercised when K > S, and its payoff is

Option Pricing 95