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>> In this module we're going to briefly discuss the Black-Scholes formula.

The Black-Scholes formula is of great significance.

Its used awful lot in industry, and indeed we can view the binomial model as an

approximation to the Black-Scholes formula.

Black and Scholes assumed a continuously compounded interest rate of R, so that $1

invested in the cash account at time 0 would be worth E to the R-T dollars at

time T. They also assumed Geometric Brownian

motion dynamics for the stock price, so that the stock price at time little t, is

equal to the initial stock price times the exponential at this quantity here where wt

is a standard Brownian motion. And for those of you who are interested we

have recorded separate modules on both Brownian motion and Geometric Brownian

motion; These modules are available on the course website.

They also assumed that the stock pays a dividend yield of c.

They assumed continuous trading with no transactions cost.

And they also assumed that short selling was allowed.

So here are some sample paths of Geometric Brownian motion.

They look quite similar to sample paths of Brownian Motion actually.

You can see that the stock price never jumps here.

So at no point you see the stock price, say jumping from here down to here.

So that's a property of Brownian Motion and Geometric Brownian Motion.

The sample paths are actually continues. Given the assumptions that Black and

Scholes made and that we listed two slides ago, they succeeded in deriving the price

for European coal option, with strike k and maturity t.

It is given to us by this quantity here. It's a somewhat complicated looking

formula, but it's easy to code up, and it is used everywhere in industry today.

The interesting thing to note about this formula is that, mu, which was the drift

of the Geometric Brownian motion two slides ago.

So recall, we assumed a drift of mu here. Well, if you notice, over in the

Black-Scholes formula, mu does not appear anywhere.

And this is similar to the fact that p, did not appear in the option pricing

formulas we derived in the context of the binomial model.

We'll return to that in a moment. So European put option prices, p zero, can

then be calculated from put call parity. So once we have the call option price up

here, we can price put options. In other words, we can derive P0 using put

call parity here. And this is the version of put call parity

that applies when we have a dividend yield.

Black and Scholes obtained their formula using a similar replicating strategy

argument to the one that we used for the binomial model.

In fact you can show that under the Black-Scholes Geometric Brownian motion

model, that we can write the price of the option C0, as being the expected

discounted pay off of the option using risk-neutral probabilities Q.

This is exactly the same formula we have for the binomial model.

In this case however, in the continuous time model of Geometric Brownian motion,

under Q we assume that the stock price is given to us by this here and the only

difference between this expression for St and the expression we have for St at

couple of slides ago. Which was mu minus sigma squared over 2

times T, plus sigma Wt is that we now have a factor r minus c appearing here, which

we don't have down here. And the true drift of the Geometric

Brownian motion mu, no longer appears in the option pricing formula.

This is exactly analogous to the fact that we use the risk mutual probabilities Q and

not the true probabilities P when we are pricing options in the binomial model.

So in fact, if you evaluate this rate this expectation, assuming St is equal to this,

you'll get the Black-Scholes formula. And for those of you who are interested,

it's actually not very difficult to do this, it involves an integral of a log

normal distribution. St here will have a log normal

distribution, so one can actually evaluate this, do some integration and actually get

the Black-Scholes formula that we showed on the previous slide.

The Black-Scholes formula is used a great deal in industry, in fact it is the way in

which option prices are actually quoted by industry practitioners.

The binomial model is often used as an approximation to the Black-Scholes model,

in which case one needs to translate the Black-Scholes parameters R sigma and so

on, into R familiar binomial model parameters.

This is often referred to as the calibration of a Binomial Model, so

suppose we are given some Black Scholes parameters we have R and we have Sigma,

and if you notice over here that's all we need, we have R we have Sigma, we also

have c of course. I'm going to see how to calibrate these

and rewrite these parameters in our binomial model.

So what we will do is, we will write rn and now we're going to have subscript n

here to denote the fact that these parameters in the binomial model, will

depend on the number of periods that we're using in the binomial model.

So recall, if t is the maturity, of the option, then t is equal to n times delta-t

where delta-t is the length of a period, In the binomial model.

Okay, so we're going to assume that rn is equal to e to the rt over n, Where n is

the number of periods, as we said. We'll assume that rn minus cn.

So this is rn minus c in the binomial model will be equal to e to the r minus c

times t over n. And a simple first order taylor expansion

will tell you that this is approximately equal to 1 plus r t over n minus c t over

n. So this of course, Is like our R factor,

and this is our C factor, in the binomial model.

We'll set U-n equal to this, and D n equal to 1 over u-n.

And now we can price European and American options, and futures, and so on, as

before, in the binomial model using these parameters.

The risk mutual para-, probabilities, will be calculated as, q subscript n.

Again, recognize the dependence of our paremeters on n, the number of periods.

So q subscript n will be equal to this here.

And using this approximation, we can see that this is approximately equal to our rn

minus cn minus dn, divided by un minus dn. So this the representation of the risk

mutual probabilities that we saw before in the binomial model.

We're actually going to use this in our binomial model now, when we're deriving

our parameters from a Geometric Brownian motion and the Black-Scholes Formula.

So our spreadsheet, will actually calculate binomial parameters, in this

way. I mentioned at the beginning of the

module, that the binomial model can be viewed as an approximation to geometric

Brownian motion, This is true, as delta t, the length of a period Goes to zero.

I'm just going to spend a couple of moments describing how you might go about

showing this. We certainly won't go through all the

calculations, but I'll do the first couple of steps of these calculations.

Recall that we can calculate European option prices with strike k, according to

this expression here. So this is the expression we had in our

binomial model. C0 equals the discount factor times the

expected pay-off of the option, using the risk neutral probabilities Q.

Well, if you recall, we also saw that we can write this expression, as we have

written here, so these are the binomial probabilities.

Okay, so this here, is equal to the probability inner binomial model of j

upmoves, and n minus j down-moves. And this then is St, the terminal stock

price, after the j up-moves and n minus j down-moves.

So therefore this here, is the expected value of the pay-off of the option.

So what you can do is, we can actually replace the summation which runs from j

equals zero, to run from j equals let's call it etta, say.

Where etta is the minimum j, such that the stock price again this is St, after j up

moves and n minus j down-moves. So 8 is the minimum number of up-moves

required to ensure that the stock price, is greater than or equal to k.

In that case, this maximum will always occur in the first argument here.

So we can remove the max, remove the 0 And just substitute this expression in here,

inside the summation, and then we can split the summation up into two components

as we've done here. So, what you have is, we can rewrite, in

the binomial model, we can write c0, the initial value of the option, as being

equal to s0 times some quantity minus k times some quantity.

And if you recall, that's exactly what you had in our expression for the black

Black-Scholes we, we had s0 times some quantity, minus K times some quantity.

What you can do, and we won't do it, but you can show that if n goes to infinity,

or equivalently, if delta-t goes to 0, remember capital T is fixed, so if in goes

to infinity, delta-t goes to 0. You can show that if n goes to infinite,

Then c0, as we have here, will actually converge to the Black-Sholes formula.

Very briefly there is some great history associated with the modeling of Brownian

motion and Geometric Brownian motion, Stochastic calculus and finance in the

pricing of options. There are many famous names, both

mathematicians and economists, who are associated with this history, you might

want to take a look at some of these names in your spare time.

I'll just throw out a couple of very interesting people.

Bachelier, back in 1900, was perhaps the first to model Brownian motion, he was

attempting to do so, with a view to pricing options on the Paris stock

exchange. Another fascinating person, is Doeblin,

another French mathematician, whose work was only recently Discovered.

He was actually very involved in the development of Stochastic calculus, but

his work wasn't discovered until very recently, as he died in World War 2 before

his work could be read. Another very influential person or

fascinating person is Ed Thorp! Ed Thorp is famous for card counting and

showing that you can actually card count and beat the system in Las Vegas.

After he was thrown out of Vegas, apparently he started trading options and

may well have been the first person to actually discover and use the

Black-Scholes formula. He didn't actually prove it was true, he

didn't have a model which derived the Black-Scholes formula, but it seems like

he somehow intuited that, that might have been the correct formula for pricing

options. And there's a host of other interesting

names here, that are well worth exploring.