Appendix C Chapter 5 Technical Topic:
Limiting Results
Appendix C Chapter 5 Technical Topic:
Limiting Results
In this topic, we describe some technical results that look
at what happens to the binomial model when the number of periods goes to
infinity. You will see how the
binomial model converges with the Black-Scholes model of option pricing.
Suppose that the option expires at time t, and n is some large
number of periods into which we divide the time interval between now (which is
time zero) and t.
We are interested in what happens when n goes to infinity.
If you think of the binomial tree, then this corresponds to being able to
trade continuously (rather than at a discrete number of periods).
Once n is fixed, we
need to rescale r, u, and d
to account for the fact that a period can now be a very small interval of time.
For example, while u
= 1.5 may be appropriate
over some extended period, it is clearly inappropriate if a period represents
say, five seconds.
The adjustment to r,
the 1 plus risk-free interest rate, is the simplest. We want r raised to
the power of t to be the value of one
dollar at time t, so we define
r is
the 1 plus interest rate over a time interval of length n.
What about the adjustments to u and d?
These can be motivated as follows. Let ST
be the (random) stock price in period t. The expected, continuously compounded return from the stock
between time t and t+1
is E{log(St+1/St).
A common assumption is that this return is independently and
identically distributed, with mean
m
and variance s2. This
means that over n periods, the
expected return is nm
and the variance is ns2.
Suppose that in n
steps, we have j upticks and (n
- j) downticks. Then, we would
have:
so that by dividing both sides by S and taking the log for the continuously compounded return:
This can be rewritten as
Taking expectations of both sides with respect to the
binomial random variable j (the number
of realized upticks), we get:
If q is the true
probability of an uptick (which may be different from the risk-neutral
probability p),
then E(j) = nq. The true
probability determines the true expected return from the stock. The variance of the return is
If, in the limit, the
binomial model is to yield that the expected return over t
periods is mt
and the variance of the return is st,
we must choose u, d, and q so that
These conditions are met if
and
Previously, the value of a call option was determined to be:
where
(risk-neutral terminal value-weighted probability), and
(risk-neutral probability).
To determine the value of the call when n approaches infinity, we need to examine the limiting behavior of
both
.
The only other term that involves n is
r-n,
which we know converges to r -t.
We will sketch an argument that shows that by substituting
the values u and d, and taking the limit as n approaches infinity, the value of the
call option converges to the Black-Scholes formula. This formula is:
where
and
and N( ) is the
cumulative normal distribution function.
There are two essential parts to the argument that
characterizes the limiting behavior.
1) Use the
central limit theorem to derive the limiting distribution for the stock price.
2) Parameterize
the F's
in terms of the stock price distribution to obtain the limits of the
probabilities.
The Distribution of St
Let
and
where mt
and st are the
mean and variance, respectively, of the continuously compounded stock return
over the time interval [0,t].
The central limit theorem then implies that
where N(z) is the
cumulative standard normal distribution. This
requires a technical regularity condition, which is satisfied.
Thus, in the limit, log(St/S) is
normally distributed with mean mt
and variance s2t.
Note that we already knew that the mean and variance of this
return were mt and
s2t; what the
theorem gives us is that the return is normally distributed.
It implies that St is lognormally
distributed with mean (m
+ s2/2)t
and variance s2t.
The Binomial
Probabilities
In this section, we will go through a similar construction
using the risk-neutral probability, p, instead of the true probability, q.
Let
and
If
we have j upticks in n moves then
so
Also
so
F(m,n;p)
is the probability that a binomial random variable, which takes on the value 1
with probability p
and 0 with probability (1-p), leads to at
least m draws of 1 in n
attempts. Therefore, 1 - F(m,n;p) is
the probability of less than m draws
of 1 in n attempts.
If j is the number of draws of
1, or equivalently the sum of the n
draws, then j has a binomial distribution with mean np and variance np(1
- p).
Thus,
From
the equation immediately preceding this, we then get
Recall
that m is the smallest integer such that Sumdn-m >
X, so that we can write
for
some e
between 0 and 1.
Substituting
for m, and using the relations
and
it
is easy to verify that
Therefore,
The
limiting behavior then depends on the behavior of
and
It
is easy to see that
goes
to zero, since
The
final step in the argument is to show that
and
This can be done by examining the limit of
p as
n gets large.
Recalling the definition of p, both the numerator and denominator go to zero.
The limiting behavior is then determined by application of L'Hopital's
rule. We do not go through these
steps here.
The
result, then, is that
so
that
where
Symmetry
of the normal distribution implies 1 - N(z)
= N(-z), so we can write
Finally,
we obtain the value of the call in the limit:
This
is the Black-Scholes formula.
In the
next chapter, The Black-Scholes Option Pricing
Model, the continuous time approach is presented to derive the option
pricing model.