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.