Steven R.

Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

Stochastic Processes and

Advanced Mathematical Finance
Options and Derivatives

Rating
Student: contains scenes of mild algebra or calculus that may require guidance.

Section Starter Question

Suppose your rich neighbor offered an agreement to you today to sell his classic Jaguar sports-car to you (and only you) a year from today at a reasonable
price agreed upon today. (You and your neighbor will exchange cash and car
a year from today.) What would be the advantages and disadvantages to you
of such an agreement? Would that agreement be valuable? How would you
determine how valuable that agreement is?

Key Concepts
1. A call option is the right to buy an asset at an established price at a
certain time.
2. A put option is the right to sell an asset at an established price at a
certain time.
3. A European option may only be exercised at the end of its life on the
expiry date, an American option may be exercised at any time during
its life up to the expiry date.
4. Six factors affect the price of a stock option:
the current stock price S;
the strike price K;
the time to expiration T t where T is the expiration time and t
is the current time;
the volatility of the stock price ;
the risk-free interest rate r; and
the dividends expected during the life of the option.

Vocabulary
1. A call option is the right to buy an asset at an established price at a
certain time.
2. A put option is the right to sell an asset at an established price at a
certain time.
3. A future is a contract to buy (or sell) an asset at an established price
at a certain time.
4. Volatility is a measure of the variability and therefore the risk of a
price, usually the price of a security.

Mathematical Ideas
Definitions
A call option is the right to buy an asset at an established price at a certain
time. A put option is the right to sell an asset at an established price at a
certain time. Another slightly simpler financial instrument is a future which
is a contract to buy or sell an asset at an established price at a certain time.
More fully, a call option is an agreement or contract by which at a definite time in the future, known as the expiry date, the holder of the option
may purchase from the option writer an asset known as the underlying
asset for a definite amount known as the exercise price or strike price.
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A put option is an agreement or contract by which at a definite time in

the future, known as the expiry date, the holder of the option may sell to
the option writer an asset known as the underlying asset for a definite
amount known as the exercise price or strike price. The holder of a European option may only exercise it at the end of its life on the expiry date.
The holder of an American option may exercise it at any time during its
life up to the expiry date. For comparison, in a futures contract the writer
must buy (or sell) the asset to the holder at the agreed price at the prescribed
time. The underlying assets commonly traded on options exchanges include
stocks, foreign currencies, and stock indices. For futures, in addition to these
kinds of assets the common assets are commodities such as minerals and
agricultural products. In this text we will usually refer to options based on
stocks, since stock options are easily described, commonly traded and prices
are easily found.
Jarrow and Protter [2, page 7] tell a story about the origin of the names
European options and American options. While writing his important 1965
article on modeling stock price movements as a geometric Brownian motion,
Paul Samuelson went to Wall Street to discuss options with financial professionals. Samuelsons Wall Street contact informed him that there were two
kinds of options, one more complex that could be exercised at any time, the
other more simple that could be exercised only at the maturity date. The
contact said that only the more sophisticated European mind (as opposed
to the American mind) could understand the former more complex option.
In response, when Samuelson wrote his paper, he used these prefixes and
reversed the ordering! Now in a further play on words, financial markets
offer many more kinds of options with geographic labels but no relation to
that place name. For example; two common types are Asian options and
Bermuda options.

The Markets for Options

In the United States, some exchanges trading options are the Chicago Board
Options Exchange (CBOE), the American Stock Exchange (AMEX), and the
New York Stock Exchange (NYSE) among others. Not all options trade on
exchanges. Over-the-counter options markets where financial institutions and
corporations trade directly with each other are increasingly popular. Trading
is particularly active in options on foreign exchange and interest rates. The
main advantage of an over-the-counter option is that a financial institution
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Figure 1: This is not the market for options!

can tailor it to meet the needs of a particular client. For example, the strike
price and maturity do not have to correspond to the set standards of the
exchanges. The parties to the option can incorporate other nonstandard
features into the option. A disadvantage of over-the-counter options is that
the terms of the contract need not be open to inspection by others and the
contract may be so different from standard derivatives that it is hard to
evaluate in terms of risk and value.
A European put option allows the holder to sell the asset on a certain
date for a prescribed amount. The put option writer is obligated to buy the
asset from the option holder. If the underlying asset price goes below the
strike price, the holder makes a profit because the holder can buy the asset
at the current low price and sell it at the agreed higher price instead of the
current price. If the underlying asset price goes above the strike price, the
holder exercises the right not to sell. The put option has payoff properties
that are the opposite to those of a call. The holder of a call option wants the
asset price to rise, the higher the asset price, the higher the immediate profit.
The holder of a put option wants the asset price to fall as low as possible.
The further below the strike price, the more valuable is the put option.
The expiry date specifies the month in which the European option ends.
The precise expiration date of exchange traded options is 10:59 PM Central
Time on the Saturday immediately following the third Friday of the expiration month. The last day on which options trade is the third Friday of
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the expiration month. Exchange traded options are typically offered with
lifetimes of 1, 2, 3, and 6 months.
Another item used to describe an option is the strike price, the price
at which the asset can be bought or sold. For exchange traded options on
stocks, the exchange typically chooses strike prices spaced $2.50, $5, or $10
apart. The usual rule followed by exchanges is to use a $2.50 spacing if the
stock price is below $25, $5 spacing when it is between $25 and $200, and
$10 spacing when it is above $200. For example, if Corporation XYZ has
a current stock price of $12.25, options traded on it may have strike prices
of $10, $12.50, $15, $17.50 and $20. A stock trading at $99.88 may have
options traded at the strike prices of $90, $95, $100, $105, $110 and $115.
Options can be in the money, at the money or out of the money.
An in-the-money option would lead to a positive cash flow to the holder if it
were exercised immediately. Similarly, an at-the-money option would lead to
zero cash flow if exercised immediately, and an out-of-the-money would lead
to negative cash flow if it were exercised immediately. If S is the stock price
and K is the strike price, a call option is in the money when S > K, at the
money when S = K and out of the money when S < K. Clearly, an option
will be exercised only when it is in the money.

Characteristics of Options
The intrinsic value of an option is the maximum of zero and the value it
would have if exercised immediately. For a call option, the intrinsic value
is therefore max(S K, 0). Often it might be optimal for the holder of an
American option to wait rather than exercise immediately. The option is then
said to have time value. Note that the intrinsic value does not consider the
transaction costs or fees associated with buying or selling an asset.
The word may in the description of options, and the name option
itself implies that for the holder of the option or contract, the contract is a
right, and not an obligation. The other party of the contract, known as the
writer does have a potential obligation, since the writer must sell (or buy)
the asset if the holder chooses to buy (or sell) it. Since the writer confers on
the holder a right with no obligation an option has some value. The holder
must pay for the right at the time of opening the contract. Conversely, the
writer of the option must be compensated for the obligation taken on. Our
main goal is to answer the following questions:

Option Intinsic Value

Stock Price

Figure 2: Intrinsic value of a call option.

How much should one pay for that right? That is, what is the
value of an option? How does that value vary in time? How does
that value depend on the underlying asset?
Note that the value of the option contract depends essentially on the characteristics of the underlying asset. If the asset has relatively large variations
in price, then we might believe that the option contract would be relatively
high-priced since with some probability the option will be in the money. The
option contract value is derived from the asset price, and so we call it a
derivative.
Six factors affect the price of a stock option:
the current stock price S;
the strike price K;
the time to expiration T t where T is the expiration time and t is the
current time;
the volatility of the stock price;
the risk-free interest rate; and
the dividends expected during the life of the option.
Consider what happens to option prices when one of these factors changes
while all the others remain fixed. Table 1 summarizes the results. The
changes regarding the stock price, the strike price, the time to expiration
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Increase in
Variable
Stock Price
Strike Price
Time to Expiration
Volatility
Risk-free Rate
Dividends

European
Call
Increase
Decrease
?
Increase
Increase
Decrease

European
Put
Decrease
Increase
?
Increase
Decrease
Increase

American
Call
Increase
Decrease
Increase
Increase
Increase
Decrease

American
Put
Decrease
Increase
Increase
Increase
Decrease
Increase

Table 1: Effect on price of increases in the variables influencing option prices.

and the volatility are easy to explain; the other variables are less important
for our considerations.
Upon exercising it at some time in the future, the payoff from a call option
will be the amount by which the stock price exceeds the strike price. Call
options therefore become more valuable as the stock price increases and less
valuable as the strike price increases. For a put option, the payoff on exercise
is the amount by which the strike price exceeds the stock price. Put options
therefore behave in the opposite way to call options. Put options become less
valuable as stock price increases and more valuable as strike price increases.
Consider next the effect of the expiration date. Both put and call American options become more valuable as the time to expiration increases. The
owner of a long-life option has all the exercise options open to the shortlife option and more. The long-life option must therefore, be worth at
least as much as the short-life option. European put and call options do not
necessarily become more valuable as the time to expiration increases. The
owner of a long-life European option can only exercise at the maturity of the
option.
Roughly speaking, the volatility of a stock price is a measure of how
much future stock price movements may vary relative to the current price.
As volatility increases, the chance that the stock price will either increase or
decrease greatly relative to the present price also increases. For the owner of
a stock, these two outcomes tend to offset each other. However, this is not
so for the owner of a put or call option. The owner of a call benefits from
price increases, but has limited downside risk in the event of price decrease
since the most that he or she can lose is the price of the option. Similarly,
the owner of a put benefits from price decreases but has limited upside risk
in the event of price increases. The values of puts and calls therefore increase
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as volatility increases.
The reader will observe that the language about option prices in this
section has been qualitative and imprecise:
an option is a contract to buy or sell an asset at an established price
without specifying how the price is obtained;
. . . the option contract would be relatively high-priced . . . ;
Call options therefore become more valuable as the stock price increases . . . without specifiying the rate of change; and
As volatility increases, the chance that the stock price will either increase or decrease greatly . . . increases.
The goal in following sections is to develop a mathematical model which
gives quantitative and precise statements about options prices and to judge
the validity and reliability of the model.

Sources
The ideas in this section are adapted from Options, Futures and other Derivative Securities by J. C. Hull, Prentice-Hall, Englewood Cliffs, New Jersey,
1993 and The Mathematics of Financial Derivatives by P. Wilmott, S. Howison, J. Dewynne, Cambridge University Press, 195, Section 1.4, What are
options for?, Page 13 and R. Jarrow and P. Protter, A short history of
stochastic integration and mathematical finance the early years, 18801970,
IMS Lecture Notes, Volume 45, 2004, pages 7591.

Problems to Work for Understanding

1. (a) Find and write the definition of a future, also called a futures
contract. Graph the intrinsic value of a futures contract at its
contract date, or expiration date, as in 2.
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(b) Explain why holding a call option and writing a put option with
the same strike price K on the same asset is the same as having a
futures contract on the asset with strike price K. Draw a graph of
the value of the option combination and the value of the futures
contract on the same axes.
2. Puts and calls are not the only option contracts available, just the most
fundamental and the simplest. Puts and calls eliminate the risk of up
or down price movements in the underlying asset. Some other option
contracts designed to eliminate other risks are combinations of puts
and calls.
(a) Draw the graph of the value of the option contract composed of
holding a put option with strike price K1 and holding a call option
with strike price K2 where K1 < K2 . (Assume both the put and
the call have the same expiration date.) The holder profits only
if the underlier moves dramatically in either direction. This is
known as a long strangle.
(b) Draw the graph of the value of an option contract composed of
holding a put option with strike price K and holding a call option
with the same strike price K. (Assume both the put and the call
have the same expiration date.) This is called an long straddle,
and also called a bull straddle.
(c) Draw the graph of the value of an option contract composed of
holding one call option with strike price K1 and the simultaneous
writing of a call option with strike price K2 with K1 < K2 . (Assume both the options have the same expiration date.) This is
known as a bull call spread.
(d) Draw the graph of the value of an option contract created by simultaneously holding one call option with strike price K1 , holding
another call option with strike price K2 where K1 < K2 , and writing two call options at strike price (K1 + K2 )/2. This is known as
a butterfly spread.
(e) Draw the graph of the value of an option contract created by
holding one put option with strike price K and holding two call
options on the same underlying security, strike price, and maturity
date. This is known as a triple option or strap
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Reading Suggestion:
References
[1] John C. Hull. Options, Futures, and other Derivative Securities. PrenticeHall, second edition, 1993. economics, finance, HG 6024 A3H85.
[2] R. Jarrow and P. Protter. A short history of stochastic integration and
mathematical finance: The early years, 1880-1970. In IMS Lecture Notes,
volume 45 of IMS Lecture Notes, pages 7591. IMS, 2004. popular history.
[3] Staff. Over the counter, out of sight. The Economist, pages 9396, November 14 2009.
[4] Paul Wilmott, S. Howison, and J. Dewynne. The Mathematics of Financial Derivatives. Cambridge University Press, 1995.

Outside Readings and Links:

1. What are stock options? An explanation from youtube.com

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