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Subsections
The pioneering work of Black and Scholes[7] started the
serious study of the theory of option pricing. All further advances in
this field have been extensions and refinements of the original idea
expressed in that paper. Considering its immense importance in the
field of option pricing, we will study the theory behind the
Black-Scholes equation in some detail. We will also introduce the
quantum mechanical formulation of the Black-Scholes theory which will
be of great use in the following chapters as we consider stochastic
volatility.
There are several assumptions involved in the derivation of the
Black-Scholes equation. It is important to understand these properly
so as to see the limitations of the theory. These assumptions are
summarised below :
- The efficient market hypothesis is assumed to be satisfied. In
other words, the markets are assumed to be liquid, have
price-continuity, be fair and provide all players with equal access to
available information. This implies that zero transaction costs are
assumed in the Black-Scholes analysis.
- It is assumed that the underlying security is perfectly
divisible and that short selling with full use of proceeds is possible.
- Constant risk-free interest rates are assumed. In other words,
we assume that there exists a risk-free security which returns $1 at
time
when $
is invested at time
.
- As we will see later, the Black-Scholes analysis requires
continuous trading. This is, of course, not possible in practice as
the more frequently one trades, the larger the transaction costs.
- The prinicple of no arbitrage is assumed to be satisfied.
- The price of the underlying security is assumed to follow a
geometric Brownian process of the form
where
is white noise.
It should be obvious that none of these principles can be perfectly
satisfied. Transaction costs exist in all markets, all securities come
in discrete units, short selling with full use of proceeds is very
rare, interest rates vary with time and we will later see that there
is evidence that the price of most stocks do not precisely follow a
geometric Brownian process.
Considering the above, it may seem strange that we pay so much
importance to the Black-Scholes theory. There are several reasons for
this. The most important one is that the concepts behind the
Black-Scholes analysis provide the framework for thinking about option
pricing. All the research in option pricing since the Black-Scholes
analysis has been done either to extend it or to generalize
it. Another important reason for studying the Black-Scholes theory is
that the financial world uses it as a standard. In fact, traders quote
Black-Scholes volatility to each other, not the actual price of the
options! Further, Black-Scholes prices still give very good
approximations to the prices of options.
One common way of pricing a derivative is to form a self-financing,
replicating hedging strategy for it. (A strategy is basically an
algorithm which allows us to form a portfolio with certain
properties.) To understand what this means, we will have to look at
each term in turn. Self-financing means that the portfolio produced by
the strategy must not itself take up any money apart from a possible
initial investment. As we will soon see, this initial investment will
be the price of the security that the strategy is replicating. The
term replicating means that the strategy must replicate the payoff of
the security we are trying to price. Further, the portfolio produced
by the strategy should always produce the same result regardless of
price changes in the underlying security. In other words, the value of
the portfolio generated by the strategy should be deterministic and
cannot have a stochastic component (except for the stochastic
components of the underlying securities of the derivative). This
explains the term ``hedging'' used for the strategy.
For example, suppose we have a 2-year coupon bond with payoffs of $1
every half a year and $11 at the maturity. We can replicate this bond
using the following portfolio : one zero-coupon bond (we assume that all
zero-coupon bonds have a payoff of $1 at maturity) each of maturity
0.5 year, 1 year, 1.5 years and 11 zero-coupon bonds of maturity 2
years. As each of the zero-coupon bonds mature, they replicate the
payoff of the coupon-bearing bond. The portfolio in this example is
perfectly hedged since the underlying price of the bonds is known at
maturity. The present value of the bonds is not important for our
purposes since our aim is only to replicate the payoff of the
coupon-bearing bond.
By the principle of no arbitrage, any portfolio that can replicate the
payoff of an option (or, for that matter, any derivative) must have
the same value as that of the option. In the above example, the price
of the coupon bond must be the same as the combination of zero-coupon
bonds. The Black-Scholes analysis makes a portfolio which replicates
the payoff of an option, hence solving the valuation problem.
This is the most important principle in derivative valuation. It
states that the value of a derivative is its expected future value
discounted at the risk-free interest rate. This is exactly the same
result that we would obtain if we assumed that the world was
risk-neutral. In such a world, investors would require no compensation
for risk. This means that the expected return on all securities would
be the risk-free interest rate. This is a very useful principle as it
states that we can assume that the world is risk-neutral when
calculating option prices. The result would still be correct in the
real world even if (as is most probably the case) it is not
risk-neutral.
We derive the put-call parity relation (1.4) using this
principle. To do so, use the fact that the sum of the payoffs of a
long call and a short put option with the same strike price and
maturity (and, of course, on the same underlying security)is given by
. Hence, the value of the call and put option is given by
, the
last equality coming from risk-neutral valuation. This gives us
or
which is the same as
(1.4).
We need one more piece of information before we can derive the
Black-Scholes equation. This is Ito's lemma which is an important
result in the theory of stochastic processes. We will also provide a
simple though non-rigorous derivation of the lemma.
Ito's lemma states that if a variable
follows a stochastic
process of the form
 |
(43) |
where
is white noise, then any smooth function
follows
the process
 |
(44) |
We now present a non-rigorous definition of Ito's lemma using the
Taylor series formula. For a smooth function
, the normal
Taylor series expansion goes as
 |
(45) |
For a non-stochastic process, when
, the above equation becomes
 |
(46) |
However, as
follows the process
, the
discretised form of which is
where
is a standard normal
variable, we see that
 |
(47) |
Now, since
is a standard normal variable, we know that
. Thus,
which is a
first order term! Further, the variance of
is
. Hence, as
,
becomes
deterministic and equal to its expected value which is
. Hence, we get (3.2) which is Ito's lemma. While this is far
from a rigorous proof which can be found in any advanced book on
stochastic processes such as Ross[4], it gives us a good idea
of why (3.2) is correct.
We are now in a position to present a derivation of the Black-Scholes
equation. We consider a general derivative
whose value is a
function of the value of the underlying security
.
is assumed
to follow the stochastic process
 |
(48) |
where
(the average growth rate of the underlying security) and
(the volatility) are constants. Using Ito's lemma, we see
that
 |
(49) |
We cannot value this directly as there is a stochastic term. To
eliminate the stochastic term, we consider the portfolio
. We see that
 |
(50) |
with the last equality following from the no-arbitrage condition
(since there is no stochastic term,
is a risk-free investment
and hence must offer the same return as any other risk-free
investment). Simplifying the above equation, we obtain the
Black-Scholes equation
 |
(51) |
The initial (or, in finance, usually final) conditions determine the
kind of derivative that we are pricing. For a call option, the final
condition we have to use is
. We see that the
principle of risk-neutral valuation is clearly satisfied in this case
since the Black-Scholes equation is independent of
, the
expected rate of growth of the underlying security price.
It is important to note that the portfolio
represents a
self-financing, replicating, hedging strategy. It replicates a
risk-free investment and it is hedged since it has no stochastic
component.
There are several ways of solving the Black-Scholes equation. I will
present two solutions, one using the principle of risk-neutral
valuation and the other using the quantum mechanical
formulation. These are two of the most elegant solutions using minimal
mathematics.
We can try to solve equation (3.9) directly but a simpler way
exists to get the solution. This involves analysing the assumed
process for the stock prices (3.6) using Ito's lemma and
applying the principle of risk-neutral valuation to the
result. Applying Ito's lemma to (3.6) gives us
 |
(52) |
Now, the time integral of the white noise
will give us a random
walk whose distribution we know to be normal. In fact, it can be
easily seen from the above equation that
![\begin{displaymath}
\ln S - \ln S_0 \sim N\left[\left(\phi -
\frac{\sigma^2}{2}\right)(T-t), \sigma\sqrt{T-t}\right]
\end{displaymath}](img320.png) |
(53) |
(where
and
are the prices of the underlying security at time
and
respectively) or
![\begin{displaymath}
\ln S \sim N\left[\ln S_0 + \left(\phi -
\frac{\sigma^2}{2}\right)(T-t), \sigma\sqrt{T-t}\right]
\end{displaymath}](img325.png) |
(54) |
which shows that
follows a lognormal distribution.
We are now almost done. The principle of risk-neutral valuation
implies that the present value of the option is the expected final
value
of the option discounted at the risk-free
interest rate. So, we have
![\begin{displaymath}
c = e^{-r(T-t)}E[\max(S-K, 0)] = e^{-r(T-t)}\int_K^\infty (S-K)g(S)dS
\end{displaymath}](img328.png) |
(55) |
where
, the probability density function of
is given by
(3.12) which can be explicitly written as
 |
(56) |
where
has been replaced by
in accordance with the
principle of risk-neutral valuation. We can easily verify that this
solution satisfies the principle of risk-neutral valuation by
evaluating
.
The value of the integral (3.13) can be found with a bit of
algebraic manipulation and is
 |
(57) |
where
 |
(58) |
and
is the cumulative standard normal distribution.
While the result looks very complicated, it has an intuitive
interpretation. Equation (3.15) can be written as
![\begin{displaymath}
c = e^{-r(T-t)}[e^{r(T-t)}SN(d_1) - KN(d_2)].
\end{displaymath}](img338.png) |
(59) |
is the probability that the final stock price will be above
(in other words, that the option will be exercised) in a
risk-neutral world so that
is the strike price times the
probability that the strike price will be paid. The expression
is the expected value of a variable that equals
if
and 0 otherwise in a risk-neutral world. In other words,
is the expected value of the option at
maturity. The above result is therefore just an expression of the
principle of risk-neutral valuation.
We can also form a self-financing, replicating hedging strategy for
the call option. The portfolio in this case is made up of
units of the underlying security and
zero-coupon bonds with
the same maturity as the option. This approach is discussed in great
detail in Chriss[1]. We do not consider this approach here as
it is unintuitive since the result must be known in advance.
The following solution is adapted from Baaquie [8].
We can recast the Black-Scholes equation (3.9) into a
Schrödinger-like equation. We identify
, the value of the option
as a wave function dependent on time and the price of the underlying
security. Then, the Schrödinger equation becomes
 |
(60) |
where
is the Hamiltonian. From (3.9), we see that
the Hamiltonian is given by
 |
(61) |
For subsequent simplification, we introduce the variable
. The Hamiltonian is then given by
 |
(62) |
The price of the European call option is then given by
 |
(63) |
where
is the final wave function (note that
time runs backward in this formulation).
We now change the basis to the momentum basis where
is
diagonal. The transformation from the ``position'' basis to the
``momentum'' basis is defined by
 |
(64) |
The Hamiltonian
in the momentum basis is given by
![\begin{displaymath}
\matel{p}{\hat{H}_{BS}}{p'} = \left[\frac{\sigma^2p^2}{2} +
ip\left(\frac{\sigma^2}{2} - r\right)\right]\delta(p-p')
\end{displaymath}](img359.png) |
(65) |
Hence,
where
. After changing variables from
to
and
noting that time runs backward here, we see that this distribution is
the same as (3.14) as it should be. Hence, the solution is the
same as the one using the risk-neutral valuation method as it should
be.
While the advantages of the quantum mechanical method may not be very
obvious here, they will become obvious when we consider stochastic
volatility (i.e., when we allow
to follow a stochastic
process).
Next: Stochastic Volatility
Up: thesis
Previous: Stochastic Processes : An
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Marakani Srikant
2000-08-15
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