Well now, let's say a few words about options. Well like I said, a lot of people heard of this word, and now we are trying to add some understanding to this. So in this episode, we will talk about some examples. The definition, And we'll also show some payoff charts, And we'll study the parameters of options, And the value drivers. So we will prepare ourselves for the evaluation process that we will study later, but for now we have to just say a few of these words. Well first of all, an option by definition is a contract that gives its owner the right, but not the obligation to do something in the future. A classic stock call option is the contract that gives its owner the right to buy a share of stock. At a pre-specified price, that is fixed at the moment of the purchase of this option. And then that maybe at some future point in time, or over a period of time until some point in the future. So that is an option by definition, the put option on the other hand is the right to sell this share of stock at the pre-specified price at or up to a certain moment in the future. Now, you can see that clearly now, we better understand why options are so linked to our ability to make decisions in the future. Because if you decide to buy this share of stock in the case of a call option or sell in the case of a put option, that's called that you exercise an option. And we will see in the future that for stock options or listed options they rarely do that for some reason. So it's always more beneficial to just sell the option at some moment in the future than to exercise that. However, there are certain important exceptions and options do get exercised. And for that reason, we will see that unfortunately that act significantly influences the option availability. Now let's go a little bit further, And talk about payoffs. Now I will put some charts here, well this is the chart of a standard call option. So here, will be share of stock for which you have this option. And here, I will show what happens with the option. Well this is a special point, this is K, which is the exercised price or the strike price. This is the price at which I have the right to buy the share of stock in the future. Now, if we ignore the value of the option at the time of a purchase, then the payoff will look like this. Until the stock price reaches K, then there's nothing, and then it goes like this. So basically, if the share of stock now is $50, and if you bought the right to buy it at 30, then clearly, this option is immediately worth at least $20. This is called in the money. However, oftentimes people buy options that are at the money, that is 50 now and the price is $50, or out of the money. Let's say the price now is $50, but I buy the right to buy this share of stock at $70, that at the first glance, seems to be strange. So I'm sort of betting that the price of the stock will go up. But if we incorporate the price that we pay for that, let's say this is the price. So this is what you pay for this option, because clearly this right is valuable, then this angle changes to this one. So to get in the black, you have to wait until the point when the stock price not only goes over this strike price K, but also K plus the amount that you paid for the option, and this is at the time of expiration. However, if we talk about American options that are much more interesting and important. Then you can say that with time going by, you observe the picture like this. This is the value of the option and when the time goes by, this red dotted line sort of goes down, and finally, sits on this blue angle. Now if we looked at the pay output option, it's another story. Again, this is S, this is the price output, and then we will have everything a little bit different. But I will use the same K, which is just there's nothing special about that. And then it will be like this, without the price, then it will be like this, if we take something out. And for an American output, again, we will have something like that. Now, well basically you can see for an American option, this difference between this angle and the red dotted line that shows to us how we can use this value if we sell the option later. Now let's now proceed, I will come up with another important thing here that is called put call parity. Now I will put another chart. So this is K, and again for simplicity, I ignore for now the price of an option, so this is payoff. Sort of the call option looks like this, And this is long call, so we buy it. What if we sell the put, because we know that if that would be the long put, so it would be like this. But if I sell that, then short put will be like that. This is short put. And you can see that this line is basically, this is the stop. This line is basically like the stock, but the stock would be from zero. And here you can see this K, so we can put the following equation that the call plus PV of K is the same as the stock plus the put option. And that gives us the formula that is known as a put call parity, okay? I'll put that in red here that put is the same as call minus S + PV(k). This is a very well known thing that allows us, in many cases to value, let's say, a call option. And then to come up with a certain variation, we put option with the same strike pass. So all these things, from here, if we studied, let's say strategies in trading options, we could go much further in the analysis of these options and some combinations of that. But because our primary emphasis is on real investment projects we will not go further. Instead what we will do, we will now analyze what leads to any real value in the options what influences it, and that will be option, Value drivers. Well, these drivers, so we will use option parameters. And we'll see how they actually influence the price of an option, that will be for the call and that will be for the put. Now we first start with the stock price S, that is clearly an important value driver. And if the stock price goes up, then for the call option, this is a good thing, and for the put option, this is a bad thing. Because if the stock goes up significantly, then the put option at a certain point just looses its value. Now with the strike price K, this is a completely different story. This is minus for the call and then plus for food, because the higher this strike price, then you have to live up and over this price to profit from the call. And conversely when you do not reach this price, you are profiting from the put. Now come to other parameters that I will specifically put in red, because they are sort of the influence, the value of options regardless or whether its a call or a put. This is first of all sigma, this is the volatility of the underlying stock. And then this is time to expiration when you're dealing with an American option. Because clearly the idea of time for European option that can be at the point of expiration, does not make any sense. Now for both of this, I put them in red, this is positive. Well, why is that so? Why is this contribution important for volatility? Because if the behavior of the underlying is really volatile, then the probability that you end up at a certain point that is far from where we are right now is actually much higher. And therefore you can see that, although let's say you buy an out of the money option. So you bought a call option to buy a share of stock at 70, while the prices right now 50. If this is a very low volatility stock, then the probability that you will ever reach 70 is very low, and therefore the value of this option is low too. However, if the volatility is high then the chance that at some point in time the price will jump over 70 or even come close to 70. That is much higher, and therefore that contributes to the option value. By the same token if you talk about the American option that is long, then over this period of time, the probability of some, let's say previously unlikely outcomes goes up. Now there's another one very important thing which is the risk free rate. And the risk free rate is positive, for the call option a negative for the put option in the following way. Because a call option basically is equivalent to, so this is right, so you wait for some time and if you do decide to exercise that then you will pay the price for this share of stock at the time of exercising. So basically, you have some time to wait, or you can save this amount over a certain period of time, and earn a risk free rate on that. So you save this amount at in the case of a call option, in the put option it's quite the opposite, so basically that is why these signs. Now there are some other important things here, for example, dividends for the stock, because if you own an option, then you are not entitled to a dividend. So in order to be able to receive this dividend, you first have to exercise the option, and to become the title owner of the share of stock, that also contributes to the value. But for now I will say one thing that unfortunately is important in option evaluation and then we will come close to the way we'll proceed. We know from a previous weeks that all evaluation procedures, they employ the NPV approach. However, for options, NPV approach fails, why? Well first of all you can see that, well with cash flows it is let's say, difficult but doable. But when it comes to the discounting, that becomes absolutely unrealistic because the risk free rate for the option is stochastic. So you're dealing with stochastic discounting, and that unfortunately precludes you from using any static formula with the use of NPV, and that leads to significant aggravations. And therefore to come up with a sensible evaluation, there are two approaches that are used. One, this is a replication, so we, Create a synthetic portfolio for an option using stock and bonds. And then say that if this portfolio behaves exactly like the option then they have to be equivalent. And this approach, I will give some examples in the next episode. But this approach leads to the famous black and shells formula and another approach here, that is called the risk neutrality. That basically says that investors are, in general, risk neutral to the extend. If it is risk neutral, then if you create a portfolio that behaves like a riskless asset, and they expect to receive a risk free rate of return. And that in a more general way, that requires some outcomes, and that leads to the so called binomial approach. Well, not exactly leads, but it can be generalized this way. So in what follows, we will both these require some mathematical advancement. Although, I would say that here the formula looks more offensive, and here, we do not use that many formulas. But this approach is more universal, that is good even for stocks with dividends. And with some let's say, some specific limitations even for some bond options, which is as we will see later in this course that creates a very specific challenge. So here, I will wrap up the overall discussion or description of options. And then starting from the next episode, I will show on a simple example, how these approaches of option evaluation actually work.
