Simple generally accepted economic assumptions are insufficient
to
develop a rational option pricing theory. Assuming a perfect
financial market in Section 2.1 lead to elementary
arbitrage relations which options have to fulfill. While these
relations can be used as a verification tool for sophisticated
mathematical models, they do not provide an explicit option
pricing function depending on parameters such as time and the
stock price as well as the options underlying parameters
To obtain such a pricing function the value of the underlying
financial instrument (stock, currency, ...) has to be modelled. In
general, the underlying instrument is assumed to follow a
stochastic process either in discrete or in continuous time. While
the latter are analytically easier to handle, the former, which we
will consider as approximations of continuous time processes for
the time being, are particularly useful for numerical
computations. In the second part of this text, the discrete time
version will be discussed as financial time series models.
A model for stock prices which is frequently used and is also the
basis of the classical Black-Scholes approach, is the so-called
geometric Brownian motion. In this model the stock price
is a solution of the
stochastic differential equation
 |
(7.1) |
Equivalently, the process of stock price returns can be assumed to
follow a standard Brownian motion, i.e.
 |
(7.2) |
The drift
is the expected return on the stock in the time
interval
The volatility
is a measure of the return
variability around its expectation
Both parameters
and
are dependent on each other and are important factors
of the investors' risk preferences involved in the investment
decision: The higher the expected return
the higher, in
general, the risk quantified by
Modelling the underlying as geometric Brownian motion provides a
useful approximation to stock prices accepted by practitioners for
short and medium maturity. In real practice, numerous model
departures are known: in some situations the volatility function
of the general model (5.8) is different from
the linear specification
of geometric Brownian
motion. The Black-Scholes' approach is still used to approximate
option prices. The basic idea to derive option prices can be
applied to more general stock price models.
Black-Scholes' approach relies on the idea introduced in Chapter
2, i.e. duplicating the portfolio consisting of
the option by means of a second portfolio consisting exclusively
of financial instruments whose values are known. The duplicating
portfolio is chosen such that both
portfolios have equal values at options maturity
Then, it
follows from the assumption of a perfect financial market, and in
particular of no-arbitrage opportunities, that both portfolios
must have equal values at any time prior to time
The
duplicating portfolio can be created in two equivalent ways which
we illustrate in an example of a call option on a stock with price
:
1. Consider a portfolio consisting of one call of which the price
is to be computed. The duplicating portfolio is composed of stocks
and risk-less zero bonds of which the quantity adapts
continuously to changes in the stock price. Without loss of
generality, the zero bond's face value can be set equal to one
since the number of zero bond's in the duplicating portfolio is
free parameter. At time
the two portfolios consist of:
- Portfolio
:
- One call option (long position) with delivery
price
and maturity date
- Portfolio
:
-
stocks and
zero bonds with face value
and maturity date
2. Consider a perfect hedge-portfolio, which consists of stocks and written calls (by means
of short selling). Due to a dynamic hedge-strategy the portfolio
bears no risk at any time, i.e. profits due to the calls are
neutralized by losses due to the stocks. Correspondingly, the
duplicating portfolio is also risk-less and consists exclusively
of zero bonds. Again, the positions are adjusted continuously to
changes in the stock price. At time
the two portfolios are
composed of:
- Portfolio
:
- One stock and
(by means of
short selling) written call options on the stock with delivery
price
and maturity date
- Portfolio
:
-
zero bonds with face value
and maturity dates
Let
be the time when the call option expires worthless,
and otherwise let
be the time at which the option is
exercised. While for a European call option it holds
at
any time, an American option can be exercised prior to maturity.
We will see that both in 1. the call value is equal to the value
of the duplicating portfolio, and in 2. the hedge-portfolio's
value equals the value of the risk-less zero bond portfolio at
any time
and thus the same partial differential
equation for the call value results, which is called Black-Scholes equation.
The Black-Scholes approach can be applied to any financial
instrument
contingent on an underlying with price
if the latter price follows a geometric Brownian motion, and if
the derivatives price
is a function only of the price
and time:
Then, according to the theorem below,
a portfolio duplicating the financial instrument exists, and the
approach illustrated in 1. can be applied to price the instrument.
Pricing an arbitrary derivative the duplicating portfolio must
have not only the same value as the derivative at exercising time
but also the same cash flow pattern,
i.e. the duplicating portfolio has to generate equal amounts of
withdrawal profits or contributing costs as the derivative. The
existence of a perfect hedge-portfolio of approach 2. can be
shown analogously.
Theorem 7.1
Let the value

of an object be a geometric Brownian motion
(
6.1). Let

be a derivative contingent on the
object and maturing in

Let

be the time at which
the derivative is exercised, or rather

if it is not. Let
the derivative's value at any time

be given by a
function

of the object's price and time.
- a)
- It exists a portfolio consisting of the underlying
object and risk-less bonds which duplicates the derivative
in the sense that it generates up to time
the same cash
flow pattern as
and that it has the same time
value as
- b)
- The derivatives value function
satisfies
Black-Scholes partial differential equation
 |
(7.3) |
Proof:
To simplify we proceed from the assumption that the object is a
stock paying a continuous dividend yield
and thus involving
costs of carry
with
the continuous compounded
risk-free interest rate. Furthermore, we consider only the case
where
is a derivative on the stock, and that
does not generate any payoff before time
We construct a portfolio consisting of
shares of
the stock and
zero bonds with maturity date
and a face value of
Let
be the zero bond's value discounted to time
We denote the
time
portfolio value by
It is to show that
and
can be chosen such that at
exercise time respectively at maturity of
the portfolio
value is equal to the derivative's value, i.e.
Furthermore, it is shown that the portfolio does
not generate any cash flow prior to
i.e. it is neither
allowed to withdraw nor to add any money before time
All
changes in the positions must be realized by buying or selling
stocks or bonds, or by means of dividend yields.
First of all, we investigate how the portfolio value
changes
in a small period of time
By doing this, we use the notation
etc.
and thus
 |
(7.4) |
Since the stochastic process
is a geometric Brownian motion
and therefore an Itô-process (5.8) with
and
it follows from the generalized
Itô lemma (5.10) and equation (6.1)
 |
(7.5) |
and an analogous relation for
Using

and
and neglecting terms of size smaller than
it follows:
 |
(7.6) |
 |
(7.7) |
The fact that neither the derivative nor the duplicating portfolio
generates any cash flow before time
means that the terms
and
of
in equation
(6.4) which correspond to purchases and sales of
stocks and bonds have to be financed by the dividend yields. Since
one share of the stock pays in a small time interval
a
dividend amount of
we have
Substituting equations (6.6) and (6.7) in
the latter equation, it holds:
Using equation (6.1) and summarizing the stochastic terms
with differential
as well as the deterministic terms with
differential
containing the drift parameter
and all
other deterministic terms gives:
This is only possible if the stochastic terms disappear, i.e.
 |
(7.9) |
Thus the first term in (6.8) is neutralized as well.
Hence the middle term must also be zero:
 |
|
|
|
 |
|
|
(7.10) |
To further simplify we compute the partial derivative of equation
(6.9) with respect to
 |
(7.11) |
and substitute this in equation (6.10). We then obtain
 |
(7.12) |
Since the stock price
does not depend explicitly on time,
i.e.
the derivative of the
portfolio value
with respect to time
gives:
This implies
Substituting this equation in equation (6.12) we
eliminate
and obtain
 |
(7.13) |
Since the zero bond value
is independent of the stock price
i.e.
the derivative of the
portfolio value
with respect to the stock
price gives (using equation (6.9))
and thus
 |
(7.14) |
That is,
is equal to the so-called delta or
hedge-ratio of the portfolio (see Section 6.3.1).
Since
we can construct a duplicating
portfolio if we know
We can obtain this function of stock price and time as
a solution of the Black-Scholes differential equation
 |
(7.15) |
which results from substituting equation (6.14) in
equation (6.13). To determine
we have to take into
account a boundary condition which is obtained from the fact that
the cash flows at exercising time respectively maturity, i.e. at
time
of the duplicating portfolio and the derivative are
equal:
 |
(7.16) |
Since the derivative has at any time the same cash flow as the
duplicating portfolio,
also satisfies the Black-Scholes
differential equation, and at any time
it holds
Black-Scholes' differential equation fundamentally relies on the
assumption that the stock price can be modelled by a geometric
Brownian motion. This assumption is justified, however, if the
theory building on it reproduces the arbitrage relations derived
in Chapter 2. Considering an example we verify
this feature. Let
be the value of a future contract
with delivery price
and maturity date
. The underlying
object involves costs of carry at a continuous rate
. Since
depends only on the price of the underlying and time it
satisfies the conditions of Theorem 6.1. From Theorem
2.1 and substituting
for the time to
maturity it follows
Substituting the above equation into equation (6.3) it can
be easily seen that it is the unique solution of Black-Scholes'
differential equation with boundary condition
Hence, Black-Scholes' approach gives the same price for the
future contract as the model free no-arbitrage approach.
Finally, we point out that modelling stock prices by geometric
Brownian motion gives reasonable solutions for short and medium
terms. Applying the model to other underlyings such as currencies
or bonds is more difficult. Bond options typically have
significant longer time to maturity than stock options. Their
value does not only depend on the bond price but also on interest
rates which have to be considered stochastic. Modelling interest
rates reasonably involves other stochastic process, which we will
discuss in later chapters.
Generally exchange rates cannot be modelled by geometric Brownian
motion. Empirical studies show that the performance of this model
depends on the currency and on the time to maturity. Hence,
applying Black-Scholes' approach to currency options has to be
verified in each case. If the model is used, the foreign currency
has to be understood as the option underlying with a continuous
foreign interest rate
corresponding to the continuous dividend
yield of a stock. Thus, continuous costs of carry with rate
equal the interest rate differential between the domestic
and the foreign market. If the investor buys the foreign currency
early, then he cannot invest his capital at home any more, and
thus he looses the domestic interest rate
. However, he can
invest his capital abroad and gain the foreign interest rate
.
The value of the currency option results from solving
Black-Scholes' differential equation (6.3) respecting the
boundary condition implied by the option type.