MA3667 Computing assignment

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