Principles of Option Pricing

4
Principles of Option Pricing
This is the most important chapter in the book and needs to be mastered if the reader is to
get a firm grasp of option theory. We start with a simple, stylized example. These examples
are often irritating to new students of derivatives who regard them as toy models with little
relevance to real-life financial problems. However, the reader is strongly advised not to dis-
miss them. Firstly, they allow concepts such as risk neutrality or pseudo-probabilities to be
introduced in a relatively painless way; introducing such concepts for the first time in a more
generalized or continuous context is definitely harder on the reader – trust me. Secondly, as will
be demonstrated in a few chapters, simple models which allow only two outcomes can easily
be generalized into powerful computational tools which accurately represent real financial
markets.
4.1 SIMPLE EXAMPLE
(i) Suppose a company is awaiting a crucially important yes/no decision from a government
regulator, to be announced in one month. The outcome will radically alter the company’s
future in a way which is predictable, once we know which way the decision goes. If the
decision is “yes”, the stock price will rise to S
high
but for a “no” the price will fall to S
low
.
Obviously, S
high
and S
low
must be above and below the present stock price S
0
(if they were
both above, S
0
would rise immediately). Let us further assume that everyone knows that given

the political climate, the yes probability is 70% and the no probability 30%.
We are equity derivatives investors and are holding an unquoted option on this company’s
stock which matures immediately after the announcement. The payoff of the option is f
1 month
,
which takes values f
high
or f
low
depending on whether the stock price becomes f
high
or S
low
.
How would we go about working out today’s value for this option?
(ii) Considering first the stock price itself, the expected value in one month and the expected growth
rate over that month µ are defined by
E[S
1 month
] = 0.7S
high
+ 0.3S
low
= (1 + µ)S
0
(4.1)
At the risk of emphasizing the obvious, let us be clear on this point: µ is definitely not the
rate by which S
0
will grow, since the final stock price will be either S

high
or S
low
.Itisthe
mathematical expectation of the stock price growth. In this example we can work out µ from
our knowledge of the probabilities of yes and no; alternatively, if we knew µ at the beginning,
we could work out the probabilities.
The expected value for f
1 month
is similarly given by
E[ f
1 month
] = 0.7 f
high
+ 0.3 f
low
which we can evaluate since we know the payoff values. It should not be too hard to calculate
the present value, but how? The simplest way might be just to discount back by the interest
4 Principles of Option Pricing
rate, but remember that this is only valid for finding the present value of some certain future
amount; for a risky asset, we must discount back by the rate of return (growth rate) of the
particular asset. This is clear from the slightly rewritten equation (4.1):
S
0
=
E[S
1 month
]
(1 + µ)
Maybe the answer is to use (1 + µ) as the discount factor; but µ is the growth rate of

the underlying equity stock, not the option. There is nothing to suggest that the expected
growth rate of the stock µ should equal the expected growth rate of the option λ. Nor is
there any simple general way of deriving λ from µ. This was the point at which option
theory remained stuck for many years. At this point, we enter the world of modern option
theory.
(iii) Instead of trying to value the option, let us switch our attention to another problem. We could
lose a lot of money on the derivative in one month if the stock price moves against us. Is it
possible to hedge the option against all risk of loss?
Suppose there were some quantity of stock  that we can short, such that the value of the
option plus the short stock position is the same in one month, whether the stock price goes
up or down. Today’s value of this little portfolio consisting of option plus short stock position
is written f
0
− S
0
. Note the convention whereby f
0
is the price of an option on one share
of stock, and  is some negative or positive number which will probably not be an integer.
Obviously you cannot buy or short fractions of an equity stock, but the arguments would
be exactly the same if we multiplied everything by some number large enough that we only
consider integral amounts of stock and derivatives; it is simply easier to accept the convention
of fractional .
If this little portfolio is to achieve its stated aim of having the same value in one month
whichever outcome occurs, we must have
f
high
− S
high
 = f

low
− S
low

or rearranging
 =
f
high
− f
low
S
high
− S
low
We have not yet managed to calculate a value for f
0
, but we have devised a method of hedging
the position. Note that this makes no reference to λ or µ, the growth rates of the derivative and
the underlying stock.
(iv) Saying that the derivative is hedged is precisely the same as saying that the value of the portfolio
of derivative plus stock is certain and predictable. Its value today is f
0
− S
0
 and its value
in one month is f
1 month
− S
1 month
, which is the same whether the stock goes up or down.

In Section 1.2 we saw that the return on a perfectly hedged portfolio must be the risk-free
rate
f
1 month
− S
1 month

f
0
− S
0

= 1 + r
36
4.1 SIMPLE EXAMPLE
or
( f
1 month
− f
0
) + rS
0
 − (S
1 month
− S
0
) = rf
0
(4.2)
This is expressed in terms of the general quantities f

1 month
and S
1 month
; more specifically, we
can write
f
high
− S
high
 = f
low
− S
low
 = (1 + r )( f
0
− S
0
)
A little algebra yields
(1 + r) f
0
= pf
high
+ (1 − p) f
low
(4.3)
where
p =
(1 + r)S
0

− S
low
S
high
− S
low
or alternatively
(1 + r)S
0
= pS
high
+ (1 − p)S
low
(4.4)
(v) Let us take a moment to contemplate the last couple of equations. The parameter p is defined
by equation (4.4). This is just a number which is made up of a combination of the observable
quantities S
0
, S
high
, S
low
and r . As was pointed out previously, S
high
and S
low
lie above and below
S
0
so that p takes values between 0 and 1. Compare equations (4.1) and (4.4): the first illustrates

the connection between the expected return and the probabilities of the stock price moving to
S
high
or S
low
. The second is rather similar in form, but in place of the expected growth rate µ
for the stock, it has the risk-free interest rate r; and in place of the probabilities 0.7 and 0.3 it
has the numbers p and (1 − p), which have values between 0 and 1. These numbers are called
pseudo-probabilities, but are not of course the real probabilities of any outcome. Suppose there
exists some fantasy world where people are all insensitive to risk. In such a risk-neutral world,
everybody would be content to receive the risk-free rate r on all their investments. Equations
(4.3) and (4.4) would then be equations which connect r, the expected return on both the stock
and the derivative, to the probabilities of S
high
or S
low
being achieved. But remember, this is
only a fantasy world and does not describe what is going on in the real world. As the reader
becomes more familiar with option theory, he will find that the concept of risk neutrality is a
very useful tool in working out option prices; but he must remember that this is only an intellec-
tual construction which is a useful way of remembering computational rules. He must not drift
into the common trap of forgetting precisely where the real world ends and the fantasy world
begins.
(vi) These distinctions are best illustrated with a step-by-step comparison of a derivative pricing
in the real world and in a risk-neutral world.
37
4 Principles of Option Pricing
REAL WORLD RISK-NEUTRAL WORLD
1. We start with a knowledge of the true prob-
abilities (0.7 and 0.3 in our example). Al-

ternatively, if we only know the expected
growth rate we use equation (4.1):
(1 + µ)S
0
= 0.7S
high
+ 0.3S
low
2. The probabilities of achieving S
high
and
S
low
are just the same as achieving f
high
and
f
low
. The true expected value of f
1 month
is
E[ f
1 month
]
real world
= 0.7 f
high
+ 0.3 f
low
3. The present expected value of the deriva-

tive is given by discounting the future ex-
pected value by λ, the expected growth rate
of the derivative:
f
0
=
1
(1 + λ)
E[ f
1 month
]
real world
4. Unfortunately, neither µ nor λ are known
in most circumstances so this method is
useless.
Calculate the pseudo-probabilities from equa-
tion (4.4) :
(1 + r)S
0
= pS
high
+ (1 − p)S
low
Pretend that the probabilities of achiev-
ing S
high
and S
low
(and therefore also f
high

and f
low
) are the pseudo-probabilities. The
pseudo-expectation is then
E
[
f
1 month
]
pseudo
= pf
high
+ (1 − p) f
low
Equation (4.3) shows that f
0
is just
E[ f
1 month
]
pseudo
discounted back at the inter-
est rate:
f
0
=
1
(1 + r)
E[ f
1 month

]
pseudo
This allows us to obtain f
0
entirely from ob-
servable quantities.
Astonishingly, we have suddenly found a way of calculating f
0
in terms of known or
observable quantities, yet only a page or two back, it looked as though the problem was
insoluble since we had no way of calculating the returns µ and λ. The log-jam was broken by
an arbitrage argument which hypothesized that an option could be hedged by a certain quantity
of underlying stock. The principle is exactly the same as for a forward contract, explained in
Section 1.3. Remember, this approach can only be used if the underlying commodity can be
stored, otherwise the hedge cannot be set up: equities, foreign exchange and gold work fine,
but tomatoes and electricity need a different approach; this book deals only with the former
category.
4.2 CONTINUOUS TIME ANALYSIS
(i) The simple “high–low” example of the last section has wider applicability than a reader might
expect at this point. However this remains to be developed in Chapter 7, and for the moment
we will extend the theory in a way that describes real financial markets in a more credible way.
Following the reasoning of the last section, we assume that we can construct a little portfolio
in such a way that a derivative and − units of stock hedge each other in the short term. Only
short-term moves are considered since it is reasonable to assume that the  units of short stock
position needed to hedge one derivative will vary with the stock price and the time to maturity.
Therefore the hedge will only work over small ranges before  needs to be changed in order
to maintain the perfect hedge.
38
4.2 CONTINUOUS TIME ANALYSIS
The value of the portfolio at time t may be written f

t
− S
t
. The increase in value of this
portfolio over a small time interval δt, during which S
t
changes by δS
t
, may be written
δ f
t
− S
t
 − S
t
qδt
The first two terms are obvious while the last term is just the amount of dividend which we must
pay to the stock lender from whom we have borrowed stock in the time interval δt, assuming
a continuous dividend proportional to the stock price.
The quantity  is chosen so that the short stock position exactly hedges the derivative over
a small time interval δt; this is the same as saying that the outcome of the portfolio is certain.
The arbitrage arguments again lead us to the conclusion that the return of this portfolio must
equal the interest rate:
δ f
t
− δS
t
 − S
t
qδt

f
t
− S
t

= r δt
or
δ f
t
− δS
t
 + (r − q)S
t
δt = rf
t
δt (4.5)
These equations are the exact analogue of equations (4.2) for the simple high–low model of
the last section.
(ii) As they stand, equations (4.5) are not particularly useful. However, it is assumed that S
t
follows
a Wiener process so that small movements are described by the equation
δS
t
S
t
= (µ − q)δt + σ δW
t
We can now invoke Ito’s lemma in the form of equation (3.12) and substitute for δ f
t

and δS
t
into the first of equations (4.5) to give

∂ f
1
∂t
+ (µ − q)S
t
∂ f
t
∂ S
t
+
1
2
σ
2
S
2
t
∂
2
f
t
∂ S
2
t

δt + σ S

t
∂ f
t
∂ S
t
δW
t
−S
t
[(µ − q)δt + σ δW
t
] − S
t
qδt = ( f
t
− S
t
)rδt (4.6)
Recall that the left-hand side of this equation is the amount by which the portfolio increases in
value in an interval δt; but by definition, this amount cannot be uncertain in any way because
the derivative is hedged by the stock. Therefore it cannot be a function of the stochastic
variable δW
t
, which means that the coefficient of this factor must be equal to zero. This
gives
∂ f
t
∂ S
t
=  (4.7)

We return to an examination of the exact significance of this in subsection (vi) below.
(iii) Black Scholes Differential Equation: Setting the coefficient of δW
t
to zero in equation (4.6)
leaves us with the most important equation of option theory, known as the Black Scholes
equation:
∂ f
t
∂t
+ (r − q)S
t
∂ f
t
∂ S
t
+
1
2
σ
2
S
2
t
∂
2
f
t
∂ S
2
t

= rf
t
(4.8)
39
4 Principles of Option Pricing
Any derivative for which a neutral hedge can be constructed is governed by this equation;
and all formulas for the prices of derivatives are solutions of this equation, with boundary
conditions depending on the specific type of derivative being considered. The immediately
remarkable feature about this equation is the absence of µ, the expected return on the stock,
and indeed the expected return on the derivative itself. This is of course the continuous time
equivalent of the risk-neutrality result that was described in Section 4.1(iv).
When the Black Scholes equation is used for calculating option prices, it is normally pre-
sented in a more directly usable form. Generally we want to derive a formula for the price of
an option at time t = 0, where the option matures at time t = T . Using the conventions of
Section 1.1(v), we write ∂ f
0
/∂t ⇒−∂ f
0
/∂T so that the Black Scholes equation becomes
∂ f
t
∂T
= (r − q)S
t
∂ f
t
∂ S
t
+
1

2
σ
2
S
2
t
∂
2
f
t
∂ S
2
t
− rf
t
(4.9)
(iv) Differentiability: For what is a cornerstone of option theory, the Black Scholes differential
equation has been derived in a rather minimalist way, so we will go back and examine some
issues in greater detail. First, we need to look at some of the mathematical conditions that must
be met.
It is clear from any graph of stock price against time that S
t
is not a smoothly varying function
of time. It is really not the type of function that can be differentiated with respect to time. So
just how valid is the analysis leading up to the derivation of the Black Scholes equation? This
is really not a simple issue and is given thorough treatment in Part 4 of the book; but for the
moment we content ourselves with the following commonsense observations:
r
S
t

and the derivative price f
S
t
t
are both stochastic variables. In this subsection we explicitly
show the dependence of f
S
t
t
on S
t
for emphasis.
r
Both S
t
and f
S
t
t
are much too jagged for dS
t
/dt or for d f
S
t
t
/dt to have any meaning at all, i.e.
in the infinitesimal time interval dt, the movements of δS
t
and δS
S

t
t
are quite unpredictable.
r
However, partial derivatives are another matter. If you know the time to maturity and the
underlying stock price, there is a unique value for a given partial derivative. These values
might be determined either by working out a formula or by devising a calculation procedure;
but you will be able to plot a unique smooth curve of f
S
t
t
vs. S
t
for a given constant t, and
also a unique curve for f
S
t
t
vs. t for a given constant S
t
.
r
The derivation of the Black Scholes equation ultimately depends on Ito’s lemma which
in turn depends on a Taylor expansion of f
S
t
t
to first order in t and second order in S
t
.

Underlying this is the assumption that the curves for f
S
t
t
against t and S
t
are at least once
differentiable with respect to t and twice differentiable with respect to S
t
.
r
A partial derivative is a derivative taken while holding all other variables constant. d f
S
t
t
/dt
and ∂ f
S
t
t
/∂t mean quite different things. Consider the following standard result of differential
calculus:
d f
S
t
t
dt
≡
∂ f
S

t
t
∂t
+
∂ f
S
t
t
∂ S
t
∂ S
t
∂t
We have already seen that the first two partial derivatives on the right-hand side of this
identity are well defined. However ∂ S
t
/∂t is just a measure of the rate at which the stock
price changes with time, which is random and undefined; thus d f
S
t
t
/dt is also undefined.
r
In pragmatic terms, this is summed up as follows: we know that the stock price jumps around
in a random way and therefore cannot be differentiated with respect to time; the same is
40