We are continuing on options pricing via transform techniques lecture two. In the previous lecture, if you remember, I asked about the- we went through definition of options, we looked at some scenarios, the buyers and sellers. Now, we are going to get more into valuation and the very, very first question is; how to value an option that's $5 or $8 that we had in the previous lecture. I'm wondering where that number actually is coming from. Now, how to value an option. Before we even discuss this, let's go through the so-called terms of the option contract. We mentioned these things, but now I am formulating it in a way. Today's price, which is known, we typically call that S naught and people call this one spot, or the spot price. The strike price, which is pre-specified, what we call typically as K, some books call it X, but I'm using K. The maturity, which is again pre-specified. When I say pre-specified that means they're known, or expiration, whatever you want to call it. Some people call it expiration. Be sure that's capital T. And the premium that or the fee that we are paying today for that strike and that maturity, we call it V naught K, T. The V naught means it's today's for this strike and at that maturity. In a way, I'm kind of will be changing this terminology because, typically, when we see a K instead of V, we call it C naught K, T. C for call. For the put, I will call it P naught K, T. But as I said, in general when I'm calling it V, V means the value of the option. There's a V for the value, but to distinguish between call and put some people call it CP as opposed to just V. But as I said, these are dummies, you can call it whatever you like. At the end of the day you just need to be careful, you need to know exactly what that terminology is for. An S sub t of course S naught was today's, in the future is what I call S sub T, capital T which is priced at maturity time, which is T, which this is absolutely unknown. Of course, if you know that wouldn't be an option then you would know where the market is setting, your not going to buy it. That's what's uncertain. Then this is today's price that you would agree to sell or buy at and the capital ST would be the price at maturity. Now, let's see when it comes to terminology, we pay something today which we call it today's premium. And as you see, immediately I call it C naught for the call. That's today's premium that I want to actually value. That's exactly what we're trying to do. But then the payoff at maturity should be cleared. The payoff at maturity is what we call as ST minus K the positive side. This simply means the maximum of ST minus K and zero. That means if you're above a strike you would exercise, if you are below it you wouldn't. That means you always look at these positives because it would never be zero. The payoff is never negative, it's either zero or positive. That's what we mean by the positive side. Now, pictorially, if you want to show it, the payoff- This red one is what we call ST minus K positive side. In this picture, I'm assuming a strike is 120 and this hockey-stick is what the payoff would be. As you see this is a C, I'm looking at various different scenarios. The strike either goes to zero or, and I'm assuming today is, say, 100, whatever that might be, to something like 400. That's the hockey stick. That means if you're here, you don't get any payoff, but then here these R becomes the so-called the payoff. The payoff, for example here, the payoff would be roughly about 175, something like that. The blue one is what we are trying to determine. It's called the so-called today's premium. You may wonder how come I go below the payoff here. This has to do with time value of the money. I'm not going to discuss this now. Then the engines or some of the modules I will be providing you, you should be able to actually do this yourself because I'm literally grabbing this from the Python code that I would be posting it for you. You can play with it yourself. Then pictorially, that's how this would look like. Let's look at this one for the put. For the put option against today premium is what I call as P naught KT, as I stated earlier. That's what you're willing to pay today. That's what we are trying to determine how to calculate. The payoff should be very clear, again is K minus ST the positive side. This is as opposed to the call, we shall call it ST minus K. Remember, there is a big difference between the two. For the put, you making when money when you go below the strike price which is K. That is why it is K minus C positive side. One more time, K minus ST. The positive part means the max of K minus ST and zero. That means if you go below K, you don't exercise, you simply would get a zero. That's what that means. Pictorially, again, that's how it looks like. Again, for a strike of 120, this red one is the hockey stick is the payoff for various scenarios from stock going literally to zero to a stock being at something close to 380, whatever that one is. The blue one is what is today's premium. That means the fee you are willing to pay today. Again, as you see there is a part of it goes below the payoff which has to do with the time value of the money. Now, what do we need for option pricing? Now, for option pricing definitely the very first thing you need is the payoff function. Now, in case of a call, you have the payoff, in the case of the put you have the payoff. Now, the other thing you would need is the so-called PDF, probability distribution function of a stock price at time T, which means distribution for a S cap T given today's spot. This is what we write it as F ST given S naught. That's the notation we are using. Because we've seen conditional on today's price, that's what I call conditional probability density function of the stock price. Now, that's the notation we are using for it. That means you having an assumption of if the today's price is, say, S naught, what would be the distribution for the future price? Of course, there is a range for it, but then you will see exactly how that will shape. Then what you would do is having the payoff, having this one, now you should be able to price it. And the way you would pricing it is you integrate that payoff against the distribution. What you would do is this, for example, for a call case you are saying ST minus K the positive side F of ST given S naught, over its entire distribution for this going from zero to infinity. That's what it means. That's your payoff, that's your conditional probability distribution. You're integrating again from zero and that would give you the price of the option. Having said this, I immediately ask, is something missing? The answer is yes. Because that gives you the value of the option at expiration. Because what you are going to pay today, the fee you paying is today, that means you have to discount it back. That's what am saying. We need to discount to get its today value. That means simply you're discounting it by just simply doing this. But let me make sure that it's clear what we are doing here. Assuming you have this rate. Which is risk-free rate, assuming you have the rate, you have the maturity, that's your discount factor. That means what you are paying in future you're discounting it back. Assuming you have this distribution, you simply do this integration. Now, one more time I want to show one actually pictorially how this works. I do this. That's the way we do it. On this side, I have two different scales. On this side, I am having F of ST given S naught, on that side I have the payoff. This should be actually- my apologies, it should be payoff. Then what you would do is this is your payoff, you're assuming this is your distribution. This is just one distribution, distribution could look very different ways. Then you are integrating and knowing that this is zero. That means you are just grabbing this piece of your PDF against that part. You multiplying, and you integrating, and you discounting that's what it means. Let me write it here that way. Then for the European call option, the way you would expressing it is you're saying today's value is discounted, payoff, expectation of the payoff. That's what we mean. By expectation means, I want to make sure that we note the notion of expectation because anytime you doing the expectation of this, that means zero. I'm sorry, let's change this into h, whatever that might be, because I don't want you to get confused with F. Then you doing h of x, f of x, d of x. That's the function which is your payoff like this, and that's your distribution which is over here. Of course, discounting I'm assuming is inside this that you took it out. Now, as you see, zero to infinity I change it to K to infinity just to drop this positive side because anything more than K this would become positive. I don't need to call it the positive side. Now, I do exact same thing for the put. Again, this side I'm having the payoff, put's payoff. On that side I'm having the scale for distribution. This is distribution, this is the payoff, this spot is zero. That means doesn't come into integration because anything multiplied by zero becomes zero. Then I'm just focusing on this part from the PDF. That payoff I'm integrating against each other, and what I would do is, again, discount the payoff for the put exact same PDF. PDF doesn't change no matter you use, you have a put or a call. The F of ST, S naught is the same. The only thing that's changing is the payoff. And one more time again, if I just do zero to K, you don't have to go to infinity because you just coming up to this point, don't forget. That is why exactly I'm saying zero to K naught to drop this positive part. I dropped it and that becomes your so-called your integration. Now, what I will be focusing on in the next lecture would be how to now value this integral. How easy to evaluate this integral and looking at maybe one example of the shape or the function that we're going to have for this conditional PDF. Thank you.