Here, we reached the level at which we can actually see how Black and Scholes derived their Black and Scholes formula for the call option in the Merton- Black- Scholes model, and haven't done that much in terms of sophisticated into calculus. We spent a couple of lectures on that, but still we have enough. What we will need is Ito's rule and we will need the idea of replication. That's all. Let's see how it works. The Black-Scholes model, actually, it was not originally due to black and Scholes or Merton. It goes back before they were you using it. But they were the ones that popularized it by successfully using it to price options. There will be a bank account which is the usual continuously compounded constant interest rate. B (t ) is e^rt, which is usually written in the differential form in terms of the dynamics of the bank account being dB is equal rBdt. We start the bank account with one. There is one stock. The stock satisfies a linear stochastic differential equation. The change in S is a constant mu times Sdt plus a constant Sigma times S, the change in Brownian motion. This is a linear stochastic differential equation there's a linear term here, linear term here which means that it's actually soluble like linear ordinary differential equations. In fact, the solution is here. I gave it for any initial time t and any future time u. I didn't want to just specify here from zero because sometimes we will need this expression. The stock price at the future time, as u is the stock price at today's time, times exponential or mu minus 1.5 Sigma squared u minus t plus sigma w of u minus w of t. How do we know that these two expressions are equivalent? Well, we can use it as rule on this one here. In fact, we already used it as a rule on exponential Brownian motion. This is exactly the same type of a process. It just different notation for the constants, but the same if you go back to Ito's rule and look at the example of the exponential over Brownian motion. Then you apply Ito's rule like there. You will see that you get this. In fact, most of the time people, when they write down the model, they write it in this form in the stochastic differential equation for one and for the bank and for the stock. But in fact, in the Black-Scholes model, we actually know that this is really this and the stock is really can be expressed explicitly as this. You can't always express explicitly solutions to stochastic differential equations. But in the Black-Scholes model, we can do that and it's here. It's useful to remember this formula. Sometimes it's easier to solve problems, for example, in your assignment. Sometimes it's easier to solve them using this expression rather than from this. That's the model we discussed it before. When I was using z instead of w in terms of the log-normal distribution which we can see that the distribution is log-normal here. We already gave some interpretation for mu and sigma. We basically have seen the model but not in a mathematically careful way by having the Brownian motion here. We've seen this version, not this. Fine, the model is there. I will show you two ways to derive the Black-Scholes formula. The first one is more or less, the way Black-Scholes did it, which is a partial differential equation approach. Then in the next set of slides, we will do the Martingale risk-neutral pricing approach. Let's do historically how it happened first in 1973, the paper was published and it was more or less like this, not quite, I will tell you what the difference is later on. We want to find the price of a European path independent claim, more precisely the claim pays at maturity, some function g, given function g of the final stock price. This includes call and put options, but it may be something else. I'm going to make some guesses, some conjectures, and then we will check that those gases are correct. The first thing we will guess is that the price will be a function, so the price at time t is going to be the function of time. In fact, maybe better to say time left to maturity, but it's a function of time and the function of the current stock price. Basically the guess is saying, I don't need to know the history of the stock price, I only need to know where the stock price is today to say what the price of the option should be today. At this moment, this is just a guess. It's going to turn, of course, to be correct. But at this point is a guess. I'm also going to guess that this function is going to be smooth enough to be able to apply Ito's rule. Let's apply Ito's rule. Applying Ito's rule, I get some dt terms and some dW terms. If you go back and look up Ito's rule, the dt terms are derivative respect to time plus 0.5, second derivative respect to S. Here I have Sigma squared, S squared rather than just Sigma squared. But the reason for that is because if you look back in the lectures, Merton model, previously I had dx is mu dt plus Sigma dW. But now my Sigma is really Sigma time S, and my mu is really mu times S. Instead of Sigma squared, I will have Sigma squared S squared, instead of mu, I will have mu times S. Whatever is next to dW, which is sigma times S, gets squared 0.5 second derivative plus mu times S replaces mu, mu times S times the first derivative dt. Then there is also first derivative times Sigma, which is now Sigma time S dW. This is just Ito's rule applied to function C. What do I want to do now? I'm guessing the value of my option is going to move randomly as time moves by and as the stock changes in this way. I'm basically guessing this is what's going to happen. Now I'm going to guess even more. I'm going to hope that in this model, I can actually replicate the value of this option by trading in the underlying stock price and in the bank account. If I can do that, then I'm going to use the law of one price and say, if I can replicate something, then the price is the cost of replication. I want to replicate the dynamics of this function C applied to t and S of t. For this, I have to see how a portfolio strategy in the bank account and the stock price, how the value of their strategy looks like in terms of its dynamics. Then I will compare to this, to dC. I need to do the following, which is going to be useful for the rest of the course and in other models too, not just in the Black-Scholes model. Let me denote by Pi of t, the amount invested in stock at time t. Here, instead of number of shares which I used before for a strategy Delta, I'm going to use amount. Not the number of shares invested but the amount invested in stock at time t. I'm going to call it Pi of t, Pi for portfolio. Then I claim that the changes in my wealth process, X is again the portfolio value process or the wealth process, is going to have this form. Let's look at this. It says Pi over S dS. What is this? Let's just consider the first term first. Pi over S is the amount over the value of the stock at that time. This is actually what we called Delta before. This is the number of shares. I'll write it here. I have number of shares times the change in S plus, and what I will have here is if you want the number of shares in the second asset, which is my bank account, times the change in the second asset. Why is this the number of shares in the bank account? Well, look at it, it's X minus Pi. X is my total wealth, Pi is the money in the stock. X minus Pi is the money in the bank. X minus Pi, that's exactly the money I have in the bank. Over the current value of the bank account, it's the number of shares, if you want, the number of units, in the bank account. Hopefully, this is intuitive. It says change in X is number of shares in the stock times the change in stock plus number of shares in the bank times the change in bank. I'm going to take this as a definition of self-financing in continuous time. In discrete time, there was a different definition which cannot be applied here. But as a consequence of that definition, we actually did have this version in discrete time in terms of discrete sums of profits and losses. When you add up gains over discrete time intervals, it was exactly adding these type of terms, adding number of shares in a stock times the change in the stock plus number of shares In other assets times the change in that other asset. This was then correct in discrete time and we're simply going to take it as a definition in continuous time. No proof, it's just definition. Hopefully intuitive enough. This expression, this second to last line here, it's true in any model. It's a definition in any model. It's not just true for the Black-Scholes. But for the Black-Scholes, it can be written more precisely. I know what dS over S is, I know what dS is, I know what dB is, so I'm going to plug it in. I go back just to remind myself what dS is. So dS and then if I divide by S, this S will cancel, this S will cancel. I will have Mu dt plus Sigma dW dB over B. This B will cancel, I will just have r dt. I'm just substituting dB from here and the S from here. In the next slide, substitute here, and if you do the algebra, which is simple enough, you will get this. The change in X is there is an interest rate component. I also have to subtract because not all the money is in the bank, some of it is in the stock. I had to subtract that, then there is a return rate of the stock depending on how much I invest in the stock, and then there is risk depending on how much I am investing in the stock. This is then the formula in the Black-Scholes model for the dynamics of my wealth process if I'm keeping Pi of t at every moment t in the stock. I'm not writing t's here, but you know what I mean.
