**Society Investigating Mathematical
Mind-Expanding Recreations**

Pricing Financial Derivatives

Jeremy Questel

University of Toronto

This is a summary of what was presented and discussed
at the May 27th **SIMMER** meeting, along with some problems and
questions to think about.

But what are all these fancy new financial objects, and why are the banks and investment companies hiring hordes of fresh PhD's in math and physics (the so called `rocket scientists')? What are all these analysts doing?

To try to get some idea let's look at a couple of very simple examples. Simple as they are, these models are actually pretty close to the real models that are being used on Bay Street and Wall Street.

You also have the option to put your money in the bank and earn interest.
This involves no risk at all and $1 invested becomes $(1+*r*) at the
end of the year.
It is natural to assume that *d*<1+*r*<*u* because if 1+*r*>*u* there is no
point in buying
the stock, because you can do better with the bank for sure, and if 1+*r*<
*d* then there
is no point putting your money in the bank, because you can do better with
the stock
for sure.

Now a guy in a sharkskin suit makes you an offer. Instead of the
stock, he will sell you a call option on the stock with strike price *K*.
That
means that if after 1 year the stock is worth *S*_{1}>*K*, you can cash it
in for *S*_{1}-*K*. Otherwise it is not worth anything. In other words in
one year
the call option will be worth

(Note thatV_{1}=max(S_{1}-K, 0).

Of course a dollar a year from now is really worth 1/(1+p V_{1}(u) + (1-p)V_{1}(d) .

He offers to sell you the option for one half of that.

It looks like a great deal. Should you take it?

Suppose your initial wealth is $*V*_{0} and you are wondering whether to go
for it.
You actually have a completely different choice. You could buy
shares of
the stock and put the rest, in the bank. A year
later you would have

if the stock went up, and

if the stock went down.

On the other hand if you put the *V*_{0} into call options it is worth
*V*_{1}(*u*)
if the stock goes up, and *V*_{1}(*d*) if the stock goes down.

Let's try something. We have been given *S*_{0}, *u*, *d* and *r* and we
can choose
and *V*_{0}. So let's match things
after one year.

This is two equations in two unknowns, so we can solve it and the answer is

and

where *q *= (1+*r*-*d*)/(*u*-*d*).

If the price of the call option was more than *V*_{0}, why would you bother?
You would
do better for sure buying shares and putting the difference in
the bank.

On the other hand if the price of the call option was less than *V*_{0} then
who would
be so stupid as to sell them when they could do better for sure borrowing
money and investing
in the stock.

So the fair price for the option has to be *V*_{0}. You say it is the
*arbitrage price*
because if the price were any different many people in sharkskin suits
(known as
arbitrageurs) would have a field day making money on the difference.

But now we have two different answers for the price of the option, the
expectation price
and the arbitrage price. And the arbitrage price doesn't even depend on
the probabilities
*p* and 1-*p*, but uses some new `effective' probabilities *q* and 1-*q*
which only depend
on the interest rate and possible price changes.
These are called the *risk neutral probabilities*.

The market forces the arbitrage price to be the true price, so the strange
conclusion is
that if *p* is large enough, then the guys offer of half the expectation
price is a bad
deal!

**Question 2.** A stock price follows the following tree.

It starts at $20 and after three months either goes up to $22 or down to $18. If it went to $22 then three months later it either goes up to $24.2 or down to $19.8. If it went down to $18 then three months later it either goes up to $19.8 or down to $16.2. The interest rate is still 12% per year. What is the value of an option to buy the stock for $21 in six months?

**Question 3.** A stock price is currently $20 and it is known that at the
end of three months it will be either $22 or $20 or $18.
The interest rate is 12% per annum.
Can you determine
the value today of an option to buy the stock after three months for $21?

The simplest model which is reasonable is called geometric or exponential Brownian motion. Let's try to describe it. The main idea here is that the market is made up of a lot of very little investors, so price movements that we observe are sums of a great number of random small movements. From probability we know that a random quantity which is the sum of a lot of small random quantities always has a Gaussian (or Normal) disribution. But also the stock price grows at some rate . All prices also have to be relative to the total price. So in summary we expect

where *B*(*t*+*h*)-*B*(*t*) are mean 0 normal random variables. Since the
variance of a sum of
independent random variables is the sum of the variances, we have to have that
the variance of *B*(*t*+*h*)-*B*(*t*) is linear in *h*. plays the role
of the constant
and it is called the *volatility*. So *B*(*t*) itself is a random
function, called
Brownian motion and it is such that the increments *B*(*t*+*h*)-*B*(*t*) are
normal with
mean zero and variance one, and any two such increments which are not
overlapping are
independent. If you take the limit of the equation you get

called a *stochastic differential equation*.

Not bad, eh?

Stochastic differential equations are different from ordinary differential
equations
because the function *B*(*t*) turns out to be non-differentiable at every
single point
with probability one. So the equation is really strange and it took until
the 1950's
before people even started to make sense of it. In fact these functions
are so strange
that the ordinary rules of calculus, like the chain rule and the product
rule and
integration by parts are just not true. But there are analogues. For
example the
chain rule for Brownian motion turns out to be

In normal calculus the last term wouldn't be there. This is called Ito's formula.df(B(t)) =f'(B(t))dB(t) + (1/2)f"(B(t))dt.

You can price a call option with strike price *K* at time *T*
by analogy with the discrete time case. There is
risk neutral probability distribution under which the arbitrage price of
the option is

Using Ito's formula you can check that this can be determined by solving a certain partial differential equation. In this lucky case, one can even write down the solution to this equation. Black, Scholes and Merton did this in the 1970's, and the fact that they finally managed to compute the fair price for an option meant that the whole business could finally get rolling. It's been growing like crazy as more and more fancier types of derivatives are invented, and Merton and Scholes went on to win the Nobel prize in Economics for their work.

**Question 4.** How would you go about constructing a continuous function
which
was nowhere differentiable? (Hint: Think fractals.)

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