0:02

In this module, we're going to review and discuss and Black -Scholes model in

geometric boundary and motion. Black and Scholes used this model way

back in their paper in the early 70s to derive European coal and production

prices. We're going to review them here.

Because we're going to be using the Black-Scholes model in later modules,

when we discuss the Greeks. The Greeks are the partial derivatives of

the option price with respect to the model parameters.

Such as the underlying security, time to maturity, the implied volatility, and so

on. So, it's very important that we know what

the Black-Scholes model is and that we know the assumptions behind the

Black-Scholes model as well. Recall that the Black-Scholes model

assumed a continuously compounded interest rate of, or they assumed

geometric Brownian motion for the dynamics of the stock price.

So that the stock price at time t, as little t say, is equal to the initial

stock price times e to the mu minus sigma squared over 2 times 2, plus sigma wt.

Where Wt is a standard Brownian motion. The stock price is assumed to pay a

dividend yield of c, and it also assumed that continuous trading is possible with

no transactions costs. And that short-selling is allowed.

So, this is a geometric Brownian motion model.

1:15

Here are some sample paths of geometric bounding in motion.

So, these are simulated paths of the geometric bounding in motion between

times t equals 0 and t equals two years. All three paths assume an initial stock

price s is zero of $100. Wanting to keep in mind here is that the

paths of the Brownian motion, while they're very jagged, they never jump.

So in other words, you can't have a path with Brownian motion going like this, and

then jumping down to another point here. So, the Brownian motion, and therefore,

the geometric Brownian motion moves continuously in time.

In comparison, here's an example of a binomial model with n equals 26 periods.

2:01

And here, I have shown you three simulated paths of the stock price here.

So there's a red, a blue, and a green path.

Now, it might not look very similar to Brownian motion or geometric Brownian

motion at this point. But imagine that instead of having 26

periods, that I have 260 periods. It's, or 2600 periods.

Well then, in that case, these simulated paths are going to look much more jagged.

And in fact, they will begin to look like these paths of geometric bounding motion.

And indeed, that is one of the properties that we mentioned before about the

binomial model. It can be viewed as an approximation to

geometric bounding motion. And indeed, if I let the number of

periods go to infinity, keeping the time horizon, T fixed.

Then, the binomial model will converge in an appropriate sense to geometric

bounding motion. We know in the binomial model that the

the call option price is given to us by this expression here.

It is equal to i equals, the sum from i equals 0 to n, n choose i times qu to the

power of i times qd to the n minus i times the maximum of 0, u to the power of

i and d to the n minus i, S0 minus k. And so, in our binomial model, this is

actually the fair value. I'm ignoring the discount factor here,

this should be an e to the minus or t in here.

But I will omit it because there's not room in the page this, but assuming it's

here. Then, this expression here is equal to

the price of the call option in the binomial model.

3:55

Now, we also mentioned before that we let the number of periods and we go to

infinity, then we're going to actually get the Black-Scholes formula.

In other words, this expression here will converge to the Black-Scholes formula

here. And this Black-Scholes formula is

arguably the most famous formula, the most important formula in all of

economics and finance. I say arguably becasue I'm sure some

people might disagree with that statement.

But nonetheless, it's certainly a very important formula with widespread

applications in practice. Now, a couple of things to keep in mind.

Note that mu does not appear in the Black-Scholes formula.

This is just analogous to the fact p, the true probability of an up move in the

binomial model. Does not appear in the risk-neutral

probabilities we calculated for the binomial model.

Now, this is certainly surprising, at least initially.

In fact, before we ever studied options pricing, if I was to ask you what

parameters the call option price depends on, well, you might have said the

following. You would have probably have said that

the call price depends on the following. S0 the initial stock price, the strike K,

the time to mature, T. Maybe there the risk-free interest rate

for discounting, the volatility sigma, the dividend yield c, and maybe I'm

guessing you would of said mu as well. And that's fair enough.

The vast majority of us would also agree with you, and I've assumed that the call

option price would also depend on the drift mu of the geometric Brownian

motion. But in fact, it's not true.

The call option price in the Black-Scholes model, actually depends is,

only on the first six parameters here. So in fact, it depends on S0, K, T or

sigma and c. So, mu does not appear in here.

That said, imagine for a second that some really positive news came through to the

markets about the stock price. So that mu became very large, maybe mu

became very, very large so that the market was anticipating that the stock

price will increase a lot. Well, what would happen in that situation

is that many people would buy the stock immediately in anticipation of this good

news. And therefore, the stock price would

increase. So, the way I like to think about this is

the following. The option price does not depend directly

on mu, but I think it is fair to say that S0, the stock price, now does depend on

people's views about the prospects of the stock.

And so, I like to write this as S0 of mu. So, I do believe that mu does enter

implicitly into the value of the call option.

It enters implicitly in the sense that the stock price depends on mu.

And so that, for me, is how to resolve this apparent contradiction that mu does

not enter in the Black-Scholes formula. Black and Scholes obtain their formula

using a similar replicating strategy to the strategy we used in the binomial

model. However, they did not use the binomial

model. The binomial model only came about a few

years after Black and Scholes wrote their original paper.

So, Black and Scholes actually did their replicating argument in the context of a

geometric Brownian motion model. If you want to prize European put option,

then you can simply use put-call parity, put call parity is given to us here.

We've seen it a few times now. So, if we know the call price, then we

can just bring this term over the right side to get the put price.

7:35

As I mentioned on the previous slide, the Black-Scholes formula is arguably the

most important and famous formula in all finance and economics.

It is used extensively in the financial industry.

It has also led to an enormous amount of acadmic work since it's publication.

What we're going to do is we're going to see how this is used in practice.

But we will emphasize now that the geometric Brownian motion model is not a

good approximation of security prices. And indeed, everybody in the marketplace

knows there are many problems with geometric Brownian motion and the

Black-Scholes model. Nonetheless, it is used extensively and

it is very important that people understand the limitations of

Black-Scholes, and how it is used in practice.