Now are going to evaluate Brownian motion as a model for asset prices or recall that the main reason
why we are looking at Brownian motion right now.
So we're hoping we can use it as a model for asset prices that will fix some of the problems there were
with the random walk.
So in this lecture we're going to take a look at just what Brownian motion has done for us in this regard.
First of all market efficiency.
Remember that one of the most compelling reasons for considering the random walk as a model for asset
prices was that it provided an implementation of the efficient market hypothesis by way of the independence
of the jumps.
Now it takes the place of the independence of Jump's for Brownian motion is the independence of the
increments in Brownian motion and just like for the random walk independence of the increments of Brownian
motion implies that the history of the Brownian motion process up until some time t.
So in other words the particular path that Brownian motion ends up following up until some time t the
information contained in that path is completely useless for making any predictions about what Brownian
motion is going to do next.
And that is the essential property of market efficiency.
That was that was contained in the random walk and that is retained by brownian motion.
In fact that is actually even stronger as we've noted before that is stronger than what's absolutely
necessary for market efficiency but it's certainly good enough.
And so like the random walk Brownian motion will be an implementation of the efficient market hypothesis.
And so we could put a put we can put a check mark in the box for market efficiency.
Brownian motion retains market efficiency and so that's one desirable feature from the random walk.
Brownian motion preserves Now some of the problems that there were that we've noted with the random
walk.
One of them was the discreteness of the random walk.
The fact that random walk is a discrete time stochastic process.
And although we did note earlier that that is not necessarily a serious limitation for the random walk
since quite often the underlying phenomenon that we're trying to model is itself discrete.
Like for instance the daily closing price some asset it's still desirable to have a continuous time
model available for certain applications like for instance market microstructure meaning the evolution
of the price of some asset over the course of a trading day.
So we still want to have a continuous time model if possible more serious problem with the random walk
was that it only allowed Jump's of some fixed pre-determined size at each time step.
So in particular the particular version of the random walk model that we have been considering at every
time step only one of two things can happen either the price can jump up by $1 or it can jump down by
$1.
That's a very crude model and that is a serious problem with around and walk the Brownian motion model
remedies these shortcomings Brownian motion is a continuous time model.
So as far as the discreetness issue in so far as it's an issue Brownian motion fixes that in fact Brownian
motion can be thought of as a continuous time version of a random walk and Brownian motion allows the
increment or the jump from any time to take any real value at all.
So Brownian motion also fixes the problem of the fix jump size.
So from time to time T-2 the jump in Brownian motion to minus W.T. one can take any real value and all
large jumps are less likely than spall small ones because that random variable W2 minus w T-1 the jump
and Brownian motion.
It has a normal distribution but the greater the time increment T-2 minus T-1 the more likely are large
jumps because the variance of that normal distribution will be exactly T-2 minus one is just a basic
property of Brownian motion.
So when we're looking at a jump from time to one to time t to a Brownian motion Brownian motion can
make any any value but the jump can take any value would call it larger jumps are less likely than shorter
jumps but the greater the time period between the over which the jump takes place the more likely are
larger jumps.
So here's a here's the depiction of what the distribution of the jump is from time to you and the time
T-2 the distribution of W.T. to minus W.T. one.
It's a normal distribution with mean 0 and variance T-2 minus T-1.
So the jump from time T1 to t2 w t2 minus w T-1 it can take any real value at all.
And the possible values it can take are going to be distributed as per the normal distribution as depicted
here.
So smaller jumps are more likely than larger jumps but larger jumps are still possible and the dispersion
of that normal distribution grows as the square root of T2 minus T-1.
So the greater the time period over which the jump takes place the more the the more dispersed is this
distribution which means the more likely are larger jumps.
And this is a perfectly reasonable properties for the model to have so in going from time to you and
to teach to the Brownian motion can jump by any real amount.
Tall but large jumps are less likely than shorter ones.
This is what you'd expect if this is a short time period you would expect that very large jumps should
be less likely but they're still possible.
So this is a much better model than the random walk model where all that can happen in each time step
is that the price can jump either up by $1 or down by $1.
So we remedied two of the major shortcomings of the random walk model by passing to the Brownian motion
model.
Now there are some other problems with the random walk model for which the Brownian motion has not fixed
the problem so well.
One problem was that random walks can with positive probability take negative values which we've noted
before is undesirable for an asset pricing model.
Also we had the serious problem that the jumps for and the jumps in the random walk do not scale with
the overall level of the random walk.
Now unfortunately Brownian motion continues to have these problems even with a drift term Brownian motion
still has a positive probability of taking negative values.
And we've observed that in various plots of Brownian sambil past that we've looked at also the volatility
of the price jumps is still insensitive to the absolute price level because regardless of the actual
value that the price is at that time T-1.
So regardless of the value of W of T-1 the variance of the jump in the process from time to you on to
T-2 is going to be equal to T-2 minus T-1.
And this is completely insensitive to what the actual value of the of the process or what the price
was at 91.
So what that in particular means is that over time increment from time over the time period from time
to time T-2 a jump of $10 is just as likely whether or not the stock is trading at $20 or $1000.
And that is as we have noted before essentially a nonsensical result.
So we still have some problems with this model with Brownian motion that we're going to have to sort
out somehow.
We're not done yet.
Our model still needs further refinement.