>> 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.
