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. 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. 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. 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. 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.
