MA3667 Computing assignmentBackground information for MA3667 Computing assignment.

EUROPEAN CALL AND PUT OPTIONS

Large institutions will often trade securities known as

derivatives

. They are financial instru-

ments whose value depends upon the value of more basic underlying stocks or assets. We will

meet a few types of derivative in this course, but among the most prominent will be

options

. An

option

is a contract that gives the option holder the right to buy or sell a particular asset either

up to or on a specified date in the future, for a price that is agreed upon now.

Some terminology and symbols:

Call Option

. A call option gives the holder the right to

buy

the underlying asset for a

specified price known as the

exercise

or

strike

price.

Put Option

. A put option gives the holder the right to

sell

the underlying asset during a

certain time window for a specified price known as the

exercise

or

strike

price.

Exercising an option

. If an option holder executes their right to buy or sell the underlying

asset then they are said to

exercise

the option.

Note that an option holder is not forced

to exercise their right to buy/ sell the asset

.

European Option

. In European options the option holder is only allowed to exercise the

option at one specified time, known as the

expiration date

.

American Option

. In American options the option holder can exercise the right to buy/sell

at

any time until expiration date

.

Maturity

,

Expiration date

,

Exercise date

. These all mean the same thing – the date in

the future on which (European), or by which (American), the option to buy/sell the asset

must be exercised.

Some symbols:

T

:= the expiration date.

S

t

:= the price of the asset at time

t

, so in

particular

S

T

:= the asset price (

not

the option price!!) at expiration date

T

.

K

:= the

exercise price.

2

Option price

. The buyer of the option gains protection against risk, whereas the seller is

exposing themselves to it. The buyer of the option gains protection against unfavourable

movements in the underlying asset price: the holder of a put option knows that they can

always get at least

K

for it whatever happens to the price of the asset in the market, whereas

the holder of a call option knows that they will have to spend at most

K

to buy the asset

at time

T

, however high the price of the asset may be in the market at time

T

. In order to

gain this protection against risk the holder of the option has to pay the seller of the option

a fee – the

option price

.

More symbols: in

this document

we will use:

P

t

(

T; K

) := the price at time

t

of a European

put option with expiration date

T

and strike price

K

. We will also use:

C

t

(

T; K

) := the

price at time

t

of a European call option with expiration date

T

and strike price

K

.

Payoff

. Suppose that you are the holder of a European call option to buy a stock for strike

price

K

at time

T

. Then you would only bother to exercise the option if it the market price

of the stock at time

T

turned out to be more than

K

, in which case the option would give

you a payoff of

S

T