Fundamentals

1
Fundamentals
The trouble with first chapters is that nearly everyone jumps over them and goes straight to
the meat. So, assuming the reader gets this far before jumping, let me say what will be missed
and why it might be worth coming back sometime.
Section 1.1 is truly jumpable, so long as you really understand continuous as opposed to
discrete interest and dividends, sign conventions for long and short securities positions and
conventions for designating the passing of time. Section 1.2 gives a first description of the
concept of arbitrage, which is of course central to the subject of this book. This description is
rather robust and intuitive, as opposed to the fancy definition couched in heavy mathematics
which is given much later in the book; it is a practical working-man’s view of arbitrage, but it
yields most of the results of modern option theory.
Forward contracts are really only common in the foreign exchange markets; but the concept
of a forward rate is embedded within the analysis of more complex derivatives such as options,
in all financial markets. We look at forward contracts in Section 1.3 and introduce one of the
central mysteries of option theory: risk neutrality.
Finally, Section 1.4 gives a brief description of the nature of a futures contract and its
relationship with a forward contract.
1.1 CONVENTIONS
(i) Continuous Interest: If we invest $100 for a year at an annual rate of 10%, we get $110
after a year; at a semi-annual rate of 10%, we get $100 × 1.05
2
= $110.25 after a year, and
at a quarterly rate, $100 × 1.025
4
= $110.38. In the limit, if the interest is compounded each
second, we get
$100 × lim
n→∞

1 +

0.1
n

n
= $100 × e
0.1
= $110.52
The factor by which the principal sum is multiplied when we have continuous compounding
is e
r
c
T
, where T is the time to maturity and r
c
is the continuously compounding rate.
In commercial contracts, interest payments are usually specified with a stated compounding
period, but in option theory we always use continuous compounding for two reasons: first, the
exponential function is analytically simpler to handle; and second, the compounding period
does not have to be specified.
When actual rates quoted in the market need to be used, it is a simple matter to convert
between continuous and discrete rates:
Annual Compounding: e
r
c
= 1 + r
1
⇒ r
c
= ln(1 + r
1

)
Semi-annual Compounding: e
r
c
=

1 +
r
1/2
2

2
⇒ r
c
= 2ln

1 +
r
1/2
2

Quarterly Compounding: e
r
c
=

1 +
r
1/4
4


4
⇒ r
c
= 4ln

1 +
r
1/4
4

1 Fundamentals
(ii) Stock Prices: This book deals with the mathematical treatment of options on a variety of
different underlying instruments. It is not of course practical to describe some theory for
foreign exchange options and then repeat the same material for equities, commodities, indices,
etc. We therefore follow the practice of most authors and take equities as our primary example,
unless there is some compelling pedagogical reason for using another market (as there is in
the next section).
The price of an equity stock is a stochastic variable, i.e. it is a random variable whose
value changes over time. It is usually assumed that the stock has an expected financial return
which is exponential, but superimposed on this is a random fluctuation. This may be expressed
mathematically as follows:
S
t
= S
0
e
µt
+ RV
where S

0
and S
t
are the stock price now and at time t, µ is the return on the stock and RV is a
random variable (we could of course assume that the random fluctuations are multiplicative,
and later in the book we will see that this is indeed a better representation; but we keep things
simple for the moment). It is further assumed that the random fluctuations, which cause the
stock price to deviate from its smooth path, are equally likely to be upwards or downwards:
we assume the expected value E[ RV] = 0.
It follows that
E[S
t
] = S e
µt
which is illustrated in Figure 1.1.
S
t
t
t
Se
t
RV
t
0
=+
µ
Figure 1.1 Stock price movement
A word is in order on the subject of the stock return µ. This is the increase in wealth which
comes from investing in the stock and should not be confused with the dividend which is
merely the cash throw-off from the stock.

(iii) Discrete Dividends: Anyone who owns a stock on its ex-dividend date is entitled to receive the
dividend. Clearly, the only difference between the stock one second before and one second after
its ex-dividend date is the right to receive a sum of money $d on the dividend payment date.
Market prices of equities therefore drop by the present value of the dividend on ex-dividend
date. The declaration of a dividend has no effect on the wealth of the stockholder but is just
a transfer of value from stock price to cash. This suggests that before an ex-dividend date, a
stock price may be considered as made up of two parts: d e
−rT
, which is the present value of
the known future dividend payment; and the variable “pure stock” part, which may be written
S
0
− d e
−rT
. In terms of today’s stock price S
0
, the future value of the stock may then be written
S
t
= (S
0
− d e
−rT
)e
µt
+ RV
4
1.1 CONVENTIONS
We could handle several dividends into the future in this way, with the dividend term in the
last equation being replaced by the sum of the present values of the dividends to be paid before

time t; but it is rare to know the precise value of the dividends more than a couple of dividend
payment dates ahead.
Finally, the reader is reminded that in this imperfect world, tax is payable on dividends. The
above reasoning is easily adapted to stock prices which are made up of three parts: the pure
stock part, the future cash part and the government’s part.
(iv) Continuous Dividends: As in the case of interest rates, the mathematical analysis is much sim-
plified if it is assumed that the dividend is paid continuously (Figure 1.2), and proportionately
t
0
Se
E
t
S
()t
0
Se
0
S
m
t
m
-q
Figure 1.2 Continuous dividends
to the stock price. The assumption is
that in a small interval of time δt,
the stock will lose dividend equal to
qS
t
δt, where q is the dividend rate.
If we were to assume that µ = 0,

this would merely be an example
of exponential decay, with ES
t
=
S
0
e
−qt
. Taking into account the un-
derlying stock return (growth rate)
E[S
t
] = S
0
e
(µ−q) t
The non-random part of the stock
price can be imagined as trying to
grow at a constant exponential rate
of µ, but with this growth attenuated by a constant exponential rate of “evaporation” of value
due to the continuous dividend.
It has been seen that for a stockholder, dividends do not represent a change in wealth but only
a transfer from stock value to cash. However, there are certain contracts such as forwards and
options in which the holder of the contract suffers from the drop in stock price, but does not ben-
efit from the dividends. In pricing such contracts we must adjust for the stock price as follows:
S
0
→ S
0
− PV[expected dividends] (discrete)

S
0
→ S
0
e
−qt
(continuous)
(v) Time: As the theory is developed in this book, it will be important to be consistent in the use of
the concept of time. When readers cross refer between various books and papers on options,
they might find mysterious inconsistencies occurring in the signs of some terms in equations;
these are most usually traceable to the conventions used in defining time.
The time variable “T ” will refer to a length of time until some event, such as the maturity of a
deposit or forward contract. The most common use of T in this course will be the length of time
to the maturity of an option, and every model we look at (except one!) will contain this variable.
Time is also used to describe the concept of date, designated by t. Thus when a week elapses,
t increases by 1/52 years. “Now” is designated by t = 0 and the maturity date of one of the
above contracts is t = T .
This all looks completely straightforward; t and T describe two different, although inter-
related concepts. But it is this inter-relationship which requires care, especially when we
come to deal with differentials with respect to time. Suppose we consider the price to-
day (t = 0) of an option expiring in T years; if we now switch our attention to the value
5
1 Fundamentals
of the same option a day later, we would say that δt = 1 day; but the time to maturity
of the option has decreased by a day, i.e. δT =−1 day. The transformation between in-
crements in “date” and “time to maturity” is simply δt ↔−δT ; a differential with re-
spect to t is therefore equal to minus the differential with respect to T, or symbolically
∂/∂t ⇒−∂/∂T .
(vi) Long and Short Positions: In the following chapters, the concepts of long and short positions
are used so frequently that the reader must be completely familiar with what this means in

practice. We take again our example of an equity stock: if we are long a share of stock today,
this simply means that we own the share. The value of this is designated as S
0
, and as the price
goes up and down, so does the value of the shareholding. In addition, we receive any dividend
that is paid.
If we are short of a share of stock, it means that we have sold the stock without owning it.
After the sale, the purchaser comes looking for his share certificate, which we do not possess.
Our remedy is to give him stock which we borrow from someone who does own it.
Such stock borrowing facilities are freely available in most developed stock markets. Even-
tually we will have to return the stock to the lender, and since the original shares have gone to
the purchaser, we have no recourse but to buy the stock in the market. The value of our short
stock position is designated as −S
0
, since S
0
is the amount of money we must pay to buy in
the required stock.
The lender of stock would expect to receive the dividend paid while he lent it; but if the
borrower had already sold the stock (i.e. taken a short position), he would not have received any
dividends but would nonetheless have to compensate the stock lender. While the short position
is maintained, we must therefore pay the dividend to the stock lender from his own resources.
The stock lender will also expect a fee for lending the stock; for equities this is usually in the
region of 0.2% to 1.0% of the value of the stock per annum. The effect of this stock borrowing
cost when we are shorting the stock is similar to that of dividends, i.e. we have to pay out some
periodic amount that is proportional to the amount of stock being borrowed. In our pricing
models we therefore usually just add the stock lending rate to the dividend rate if our hedge
requires us to borrow stock.
The market for borrowing stocks is usually known as the repo market. In this market the
stock borrower has to put up the cash value of the stock which he borrows, but since he receives

the market interest rate on his cash (more or less), this leg of the repo has no economic effect
on hedging cost.
A long position in a derivative is straightforward. If we own a forward contract or an option,
its value is simply designated as f
0
. This value may be a market value (if the instrument is
traded) or the fair price estimated by a model. A short position implies different mechanics
depending on the type of instrument: take, for example, a call option on the stock of a company.
Some call options (warrants) are traded securities and the method of shorting these may be
similar to that for stock. Other call options are non-traded, bilateral contracts (over-the-counter
options). A short position here would consist of our writing a call giving someone the right to
buy stock from us at a fixed price. But in either case we have incurred a liability which can be
designated as − f
0
.
Cash can similarly be given this mirror image treatment. A long position is written B
0
.It
is always assumed that this is invested in some risk-free instrument such as a bank deposit
or treasury bill, to yield the interest rate. A short cash position, designated −B
0
, is simply a
borrowing on which interest has to be paid.
6
1.2 ARBITRAGE
1.2 ARBITRAGE
Having stated in the last section that most examples will be taken from the world of equities,
we will illustrate this key topic with a single example from the world of foreign exchange; it
just fits better.
Most readers have at least a notion that arbitrage means buying something one place and

selling it for a profit somewhere else, all without taking a risk. They probably also know that op-
portunities for arbitrage are very short-lived, as everyone piles into the opportunity and in doing
so moves the market to a point where the opportunity no longer exists. When analyzing financial
markets, it is therefore reasonable to assume that all prices are such that no arbitrage is possible.
Let us be a little more precise: if we have cash, we can clearly make money simply by
depositing it in a bank and earning interest; this is the so-called risk-free return. Alternatively,
we may make rather more money by investing in a stock; but this carries the risk of the stock
price going down rather than up. What is assumed to be impossible is to borrow money from
the bank and invest in some risk-free scheme which is bound to make a profit. This assumption
is usually known as the no-arbitrage or no-free-lunch principle. It is instructive to state this
principle in three different but mathematically equivalent ways.
(i) Equilibrium prices are such that it is impossible to make a risk-free profit.
Consider the following sequence of transactions in the foreign exchange market:
(A) We borrow $100 for a year from an American bank at an interest rate r
$
. At the end of the
year we have to return $100 (1 + r
$
) to the bank. Using the conventions of the last section,
its value in one year will be −$100 (1 + r
$
).
(B) Take the $100 and immediately do the following three things:
r
Convert it to pounds sterling at the spot rate S
now
to give £
100
S
now

;
r
Put the sterling on deposit with a British bank for a year at an interest rate of r
£
.Ina
year we will receive back £
100
S
now
(
1 + r
£
)
;
r
Take out a forward contract at a rate F
1 year
to exchange £
100
S
now
(
1 + r
£
)
for
$
100
S
now

(
1 + r
£
)
F
1 year
at the end of the year.
(C) In one year we receive $
100
S
now
(
1 + r
£
)
F
1 year
from this sequence of transactions and return
$100
(
1 + r
$
)
to the American bank. But the no-arbitrage principle states that these two
taken together must equal zero. Therefore
F
1 year
= S
now
(

1 + r
$
)
(
1 + r
£
)
(1.1)
(ii) If we know with certainty that two portfolios will have precisely the same value at some time
in the future, they must have precisely the same value now.
We use the same example as before. Consider two portfolios, each of which is worth $100 in
one year:
(A) The first portfolio is an interest-bearing cash account at an American bank. The amount
of cash in the account today must be $
100
(
1+r
$
)
.
(B) The second portfolio consists of two items:
r
A deposit of £
100
(
1+r
£
)
F
1 year

with a British bank;
r
A forward contract to sell £
100
F
1 year
for $100 in one year.
7