Options, Futures, and other Derivatives
By John C. Hull
CHAPTER 10 Properties of Stock Options
We use a number of different arbitrage arguments to explore the relationships between
European option prices, American option prices, and the underlying stock price. The
most important of these relationships is put–call parity, which is a relationship between
the price of a European call option, the price of a European put option, and the
underlying stock price. The chapter examines whether American options should be
exercised early. It shows that it is never optimal to exercise an American call option on a
non-dividend-paying stock prior to the option’s expiration, but that under some
circumstances the early exercise of an American put option on such a stock is optimal.
When there are dividends, it can be optimal to exercise either calls or puts early.
Factors affecting the price of a stock option
There are six factors affecting the price of a stock option:
• The current stock price, S0
• The strike price, K
• The time to expiration, T
• The volatility of the stock price, σ
• The risk-free interest rate, r
• The dividends that are expected to be paid
The table below summarizes the effect on the price of a stock option of increasing one
variable while keeping all other fixed.
Variable European call European put American call American put
S0 + - + -
K - + - +
T ? ? + +
σ + + + +
R + - + -
Div - + - +
The payoff of a call options is the amount by which the stock price exceeds the
strike price. For a put option the payoff is the amount by which the strike price
exceeds the stock price, and thus behave in the opposite way from call options.
The time to expiration has a positive effect on American options, because the
owner of a long-life option has all the exercise opportunities open to the owner of
the short-life option – and more.
A dividend causes the stock price to decline.
The volatility of a stock price is a measure of how uncertain we are about future
stock price movements. The owner of a call benefits from price increases but has
limited downside risk in the event of price decreases because the most the owner
can lose is the price of the option. Similarly, the owner of a put benefits from price
decreases, but has limited downside risk in the event of price increases.
As interest rates in the economy increase, the expected return required by
By John C. Hull
CHAPTER 10 Properties of Stock Options
We use a number of different arbitrage arguments to explore the relationships between
European option prices, American option prices, and the underlying stock price. The
most important of these relationships is put–call parity, which is a relationship between
the price of a European call option, the price of a European put option, and the
underlying stock price. The chapter examines whether American options should be
exercised early. It shows that it is never optimal to exercise an American call option on a
non-dividend-paying stock prior to the option’s expiration, but that under some
circumstances the early exercise of an American put option on such a stock is optimal.
When there are dividends, it can be optimal to exercise either calls or puts early.
Factors affecting the price of a stock option
There are six factors affecting the price of a stock option:
• The current stock price, S0
• The strike price, K
• The time to expiration, T
• The volatility of the stock price, σ
• The risk-free interest rate, r
• The dividends that are expected to be paid
The table below summarizes the effect on the price of a stock option of increasing one
variable while keeping all other fixed.
Variable European call European put American call American put
S0 + - + -
K - + - +
T ? ? + +
σ + + + +
R + - + -
Div - + - +
The payoff of a call options is the amount by which the stock price exceeds the
strike price. For a put option the payoff is the amount by which the strike price
exceeds the stock price, and thus behave in the opposite way from call options.
The time to expiration has a positive effect on American options, because the
owner of a long-life option has all the exercise opportunities open to the owner of
the short-life option – and more.
A dividend causes the stock price to decline.
The volatility of a stock price is a measure of how uncertain we are about future
stock price movements. The owner of a call benefits from price increases but has
limited downside risk in the event of price decreases because the most the owner
can lose is the price of the option. Similarly, the owner of a put benefits from price
decreases, but has limited downside risk in the event of price increases.
As interest rates in the economy increase, the expected return required by
