Brief Summary of Math Finance
1
Fundamentals
Stock markets around the world sell a great variety of different products. These include shares
in companies, commodities, futures, currencies and options.
We will consider a simplified
market without commodities or futures and only one currency.
A dominating factor in markets is the risk aversion of investors. This means that a key
feature of any market model is the
riskfree interest rate
. We will always assume that cash
H.2
can earn interest at some rate
r
without risk (many texts describe this by the purchase or
sale of
bonds
). For any sequence of cash flows
x
i
at time
t
i
the
net present value
of the cash
flow is
v
(
x, t
) =
summationdisplay
i
x
i
(1 +
r
)
−
t
i
when the times
t
i
are measured in multiples of the
compounding period
.
This becomes
v
(
x, t
) =
∑
i
x
i
e
−
rt
i
if interest is compounded continuously.
The observed behaviour of stocks.
Price changes of shares are largely unpredictable
and the study of many sequences for many different shares has lead to the observation that
the quantities
S
(
t
+
δt
)
−
S
(
t
)
S
(
t
)
are approximately Normally distributed for a wide range of values of
∂t
and that changes
over nonoverlapping time intervals appear to be independent.
Portfolios
We will describe our assets at any time as a
portfolio
. It consists of the shares,
options and cash we have at that time and we suppose that any of these can be negative. For
shares and options this can be achieved by
selling short
i.e. selling things you haven’t got!
When you do this you will be required to buy the relevant stocks or shares at some future
date at the market price to honour your short sale.
Arbitrage
When it is possible to assemble a portfolio that
with certainty
returns more
than the risk free interest rate then we say that an
arbitrage opportunity
exists.
Definition 1.1
A European call option (
put option
) gives its
holder
the right to buy
H.1
from (
sell to
) the
writer
an asset at a nominated price at a specified future time (but the
holder need not exercise the option). The nominated price is often called the
exercise price
or
strike price
while the specified time is the
expiry date
.
A European call option returns
max(
S
(
T
)
−
K,
0)
≡
(
S
(
T
)
−
K
)
+
at the expiry date
T .
The main aim of the course this term is to explore the pricing of European call options in
terms of the strike price
K
, expiry date
T
, the interest rate and any components of the stock
price model which might be relevant.
The essential aspect of pricing options is that they
should not permit arbitrage opportunities.
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 Spring '10
 DrI.M.MacPhee
 Math

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