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
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
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
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