Financial Engineering is a multidisciplinary field involving finance and economics, mathematics, statistics, engineering and computational methods. The emphasis of FE & RM Part II will be on the use of simple stochastic models to (i) solve portfolio optimization problems (ii) price derivative securities in various asset classes including equities and credit and (iii) consider some advanced applications of financial engineering including algorithmic trading and the pricing of real options. We will also consider the role that financial engineering played during the financial crisis. We hope that students who complete the course and the prerequisite course (FE & RM Part I) will have a good understanding of the "rocket science" behind financial engineering. But perhaps more importantly, we hope they will also understand the limitations of this theory in practice and why financial models should always be treated with a healthy degree of skepticism.
Geometric Brownian Motion
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Из курса от партнера Columbia University
Financial Engineering and Risk Management Part II
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Martin HaughCo-Director, Center for Financial Engineering
Industrial Engineering & Operations Research
Garud IyengarProfessor
Industrial Engineering and Operations Research Department
>> In this module we're going to discuss Geometric Brownian Motion.
Geometric Brownian motion is a very important Stochastic process, a random
process that's used everywhere in finance. We have the following definition, we say
that a random process, Xt, is a Geometric Brownian Motion if for all t, Xt is equal
to e to the mu minus sigma squared over 2 times t plus sigma Wt, where Wt is the
standard Brownian motion. So we've discussed Brownian Motion, in a
separate module, so you can look at that module, if you'd like, to remind yourself
what a Brownian Motion is. But one thing to keep in mind with the
Brownian Motion, is that Wt, has got a normal distribution, with mean 0, and
variance t, This is one of the properties of a Brownian Motion.
Recall mu the drift, sigma the volatility, and write Xt till GBM mu sigma.
An interesting observation to make is the following, let's take a look at this
expression, but let's replace t with t plus s, if we do that, we'll see that Xt
plus s equals X0, and in fact I should have had an X0 here.
So Xt plus s equals X0, e to the mu, minus sigma squared over 2 times t plus s, plus
sigma plus Wt plus s. And now what we can do, is we can rewrite
this expression up here in the exponential.
We can subtract a minus Wt, and add a Wt here, and we can break this summation up
into t times this plus s times this. If we do that, we get this term here in
the right hand side, but what's interesting is that this quantity here, is
actually equal to Xt. So we can write Xt plus s equals Xt times
the exponential of mu minus sigma squared over 2 times s plus sigma times Wt plus s,
and this representation is very useful, it's in fact very useful for simulating
security prices, when those security prices follow a Geometric Brownian Motion.
This quantity here, Wt plus s minus Wt, well that's just a normal random variable
with mean 0, and variance s. Moreover, it is actually independent of Xt
and this follows from the independent increment property of Brownian motion that
we discussed in that other module on Brownian motion.
So that means for example, suppose that we wanted to generate values of a Geometric
Brownian Motion at time 0 and at time t, bit also may be at these intermediate
times may be delta, 2 delta, 3 delta, and so on.
Well, what we can do, so we want to generate x delta, x2 delta, x3 delta, and
so on. Well, what we can do is we can actually
simulate the Geometric Brownian Motion at these time periods by just simulating, and
zero delta random variables, that's very easy to do in standard software, you can
even do it easily in Excel. So you could generate a sample path of
your Geometric Brownian Motion or a sample path of your stock.
You start at time zero with x zero which you know, and then you get x delta, using
this formula here, with t equal to 0 and Wt plus s minus Wt, well that's just equal
to a normal mean 0 variance delta random variable, so that would give you x delta.
You could then get x2 delta by taking t equal to delta and s equal to delta, so
you will get x delta plus delta is x2 delta, that's equal to x delta times this
quantity again here. And again, to generate this term you could
just generate a standard normal random variable, with mean zero and variance
delta. So it's actually very useful simulating a
Geometric Brownian Motion and we may return to this again, later in the course.
Here's a question, suppose Xt is a Geometric Brownian Motion with parameters
mu and sigma, what is the expected value of Xt plus s given little t?
Well from equation 10 of the previous slide, we know that Xt plus s is equal to
Xt times the exponential of this term here.
Well, at time t, all of this is known to us, so we can take this outside the
expectation, and we're left with this times the expected value of each of the
sigma Wt plus s minus Wt. Well this term here, as I've already said,
is normal with mean 0 and variance s, so all you're trying to do when you compute
this expectation, is actually compute the moment generating function of a normal
rounding variable. How many have seen that before?
Suppose Zed is normal with mean a and variance b squared.
Then it implies that the expected value of e to the s times Zed is equal to e to the
a s plus a half, b squared times s squared, so this the moment generating
function of a normal rounding variable. And we can just use the standard result up
here, to recognize that this must be equal to e to the sigma squared over 2 times s.
This term, cancels with this term, and we get this expectation equals e to the mu s
times Xt, so that the expected growth rate of Xt, is in fact, mu.
Here are some sample paths of Geometric Brownian Motion.
The important thing to notice with these paths, is that they are continuous, they
are very jagged. If I was to zoom in, I would still see
that they are very jagged and they are continuous as I said, so they do not jump.
I can draw any one of these paths, by keeping my pen on the page.
The following properties of Geometric Brownian Motion, follow immediately from
the definition of Brownian Motion. Recall that we saw the following, so we
know that Xt plus s, is equal to Xt, e to the mu minus sigma squared, over 2 times
s, plus sigma times Wt plus s minus Wt. Okay so what are these three properties?
Well the first property states, that these ratios xt2 over xt1, xt3 over xt2 and so
on, they're mutually independent. And that follows, because if I divide
across here by Xt, I can see I've got the only random variable here is this
increment, and the independent property, independent increments property of
Brownian Motion will actually imply this first property here.
The second property is, the property I mentioned on the previous slide that is
that the paths of Xt our continuous as a function of t, they do not jump.
The third property states, that the log of Xt plus s over Xt has got a normal
distribution as follows, and that also follows from equation 10, which I've
rewritten here. So I can easily see that the log of Xt
plus s divided by Xt is equal to, well it's just this term up here in the
exponent, it's equal to mu, minus sigma squared over 2 times s, plus sigma times
Wt plus s, minus Wt. This guy is normal with mean 0 and
variance s, and so this quantity is normal with mean mu minus sigma squared over 2s,
and variance sigma squared s, which is exactly what we have here.
A couple of observations about Geometric Brownian Motion.
It is clear #1, that if Xt is greater than 0, than Xt plus s is always positive for
any value of s greater than 0. And again, let's write out equation 10
here just to see this more clearly. It's, so this is our equation 10 from an
earlier slide. We can see that if Xt is greater than 0,
then of course the exponential of this would be greater than 0, and so then Xt
plus s would be greater than 0. So if we are using a Geometric Brownian
Motion to model stock prices, then we can see that the limited liability of a stock
price, i.e., the fact that the stock price cannot go negative, is not violated.
Another observation, is that the distribution of Xt plus s divided by Xt,
only depends on s and not on Xt. In fact, this was clear, from the previous
slide where we had this result here. The log of Xt plus s is a normal
distribution, and this normal distribution does not depend on Xt, it only depends on
s and the parameters mu and sigma. And this is nice, because we wouldn't
expect returns to depend on Xt, so we can view this as being the return on a stock
the return between times t and t plus s and we don't expect in general that this
return should depend n the current value of the stock.
So again, this is another nice property that Geometric Brownian Motion has, that
is generally reflected in stock prices as well.
So these two properties suggest that Geometric Brownian Motion might be a
reasonable model for stock prices. And indeed, Geometric Brownian Motion is
the underlying model for the famous Black-Scholes option formula that we will
also see in this course.
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