There is one more thing I want to mention here with partial differential equation, which is the partial differential equation as a mathematical object, is a deterministic object. There is nothing random here. This partial differential equation has to hold for every t and every s. Okay, so this should hold for all t and s. As I said before, while t less or equal to t and that s bigger than 0. And this is why we don't need randomness. Because whatever as turns out to be, whatever the stock price in the future happens to be, this will always hold for all a salty and therefore it will it will the replication will work okay. And another thing I'm not completely careful here with the notation. C is the C of t, s in these equations is really t of s. Also the partial derivatives of C respecter t depends on t and s. This one second derivative depends on t and s. The first derivative depends on t and s. So there is t and s arguments everywhere here. Just to make sure we understand that. All right, let's look at the Black-Scholes and the Black-Scholes formula. So I told you they came up with this special differential equation. And then for the call option, the function is S- K+ path or the maximum of S -K and 0. That's what the function is. And the partial differential equation has a solution. Now to find a solution, you have to be an expert in partial differential equations. Or go to textbook or partial differential equations and see how to construct a solution which is probably what Black and Scholes did. Or I can just give you the solution. And then if you want as an exercise you can do calculus, you can find partial derivatives of that solution. Check that. It really is the solutions. Okay, so I'm going to give you hear function. And in principle you can take derivatives of the function that are needed in the black shorts partial differential equation. And you would you could check that the left hand side is equal to 0. When you compute all those derivatives and add them up. Okay, so I'm not going to teach you how to solve partial differential equation. It's just I'm going to give you the solution they found. And if you read books on partial differential equations, maybe you could solve it on your own without me giving you the solution. And the solution looks like this. The function of t and S, t is composed of two parts, the stock price times, some N of the one which I have to tell you what it is. It's a probability minus discount the strike price times and of the two which is also some kind of probability which will describe in a second. It's not very easy to give intuition why the solution looks like this at this point. When we do the second approach, which is not what the black and shows did the second approach is going to be using martingale pricing using expected value. These probabilities will come up more naturally, but right now it's not so easy to give intuition for them. The only thing I can say, well, if you look at the payoff here, it's a difference of S and K. All right, you basically have at least when the option is in the money that's minus K. So yeah, here you have S adjusted by some probability minus discounted K, adjusted by some probability. So somehow there are these two terms difference of S and K. But they are adjusted by some probabilities. How exactly this probability has come up is going to be more clear when we do the other risk neutral pricing approach. In terms of the exact definition of these Ns, N is defined as the probability that the standard normal random variables. I'm using capital C for the standard normal and the variable. It's probability that standard normal is less or equal to X. So it's what is called the standard normal cumulative distribution function cdf which can be written precisely mathematically as this integral. It's an integral under the Goshen curve. So if you want a picture, try to do that. So normal density which is dysfunctional to the a- Y squared over two times a constant looks, it's a bell curve looks like this. So if your X is here and let me just that acts more carefully. So if your X here and you look at the probability the probability is this area here under the density. So it's this integral area. So that area is an X, right. That's your probability. Maybe if you're used to more mathematical notation to more typical mathematical notation in probability books, this would more often be denoted by capital F of X. It's the same thing in the option, an option theory usually people use capital and for normal distribution. But in mathematics books, you may see capital F more often. That's what N is and N is the cumulative standard normal distribution function. And at which point its evaluated the points, d1 and d2,d1 and d2 have formulas which are written down here and I don't want to go too much into this. Except that everything is determined if you know, volatility, sigma, time to maturity. Capital T-t Is the time left, maturity everywhere. It's time is in fact time to maturity. Strike price, current stock price, right. I'm looking at the current stock price and then interest rate and again volatility. So if you know those things but you don't need new right? You don't new. But if you know those other parameters of the option and of the model, you can compute the Black- Scholes formula and you can compute the value of the option. All right, so that's the famous black and scholes formula. They got the partial differential equation. They were able to solve it. And this is the solution. Let me just take a look and see how that looks in a graph. So for a call option, the payoff at the very end Is this black line here 0 before below the strike price and the s- K. Above the strike price, that's the perfect maturity. The value or the value of maturity, the value before maturity some small ts fixed here and I'm wearing here the stock price and the value before maturity is going to typically be looking like this blue line. So this is computed using the Black Scholes formula for some particular parameters, including some particular time before maturity. So as you get closer to maturity as capital T and a small t. Goes to zero. This blue line would get closer and closer to this. Pay off the call option. Piecewise linear, pay off of the call option at maturity. It's going to be a smooth function, which will get closer and closer to this non smooth piecewise linear functions. All right, so this blue line gives you the values of the option price at that particular time, depending on where the stock price is currently in. Different models would give you a different blue line. This is the Black Scholes model, gives you this blue line is computed from the formula. But other models will give you different values. And now for the put option you can actually easily get the formula for the put option because we have put call parity which is through theoretically in every model, in particular the model of the black shorts type. So we can get, we can compute put prices from the put call parity and for the put for the put it looks like this. So the black line again is the payoff, it maturity before maturity, it's going to be something like this. Okay. That's the protection. Fine, that's the famous Black -Sholes formula from 1973. Just to recap what we did or what they did. They suggested the model with this geometric brand emotion, exponential brand in motion for the stock price. They derived partial differential equation for the price from the logic that if you can replicate one asset by the other two assets, then you know what the price of what the value that asset has to be. In this lecture, we replicated the put option price, call option price by trading in the bank and the stock you get a differential equation. And then for some options like corruption, you can actually solve that differential equation. And that's the price of your options. All right. So that's it for this set of slides.