In this module, we are going to introduce you to options, and also introduce you to the idea of how simple no-arbitrage conditions can be used to provide some bounds on option prices, and using these bounds we'll show you that it's never optimal to execute an American-type call option before expiration. In this module, we're going to be talking about derivative securities called Options. They come in two varieties, a European option and an American option. If an Option is written to buy the underlying asset, it's called a call option. If the option is written to sell the underlying asset, it's called a put option. A European call option gives the buyer the right, but not the obligation to purchase one unit of the underlying asset at a specified price K, called the strike price at a specified time, D called expiration. An American type call option gives the right, but not the obligation to purchase one unit of the underlying at a specified price K, called the strike price, at any time until a specified time T, called expiration. The difference between a European Option and an American Option is that a European Option is written at time t equal to 0, let's say, and it can only be exercised at time T. So here is where you can exercise a European Option. An American Option can be exercised at any time here [NOISE]. And any time between time 0 to time T. A European put option gives the buyer the right but not the obligation to sell one unit of the underlined at a specified price K called the strike price, at a specified time T, called expiration. An American put option gives the buyer the right, but not the obligation to sell one unit of the underlying at a specified price K, called the strike price, at any time until a specified time T, called the expiration. Again, the difference between the European Put and American Put is the same as the difference between a European Call and an American Call. A Call gives the buyer the right at, but not the obligation to buy. A Put gives the buyer the right, but not the obligation to sell. So let's work through the payoff and the intrinsic value of a call option. The payoff of a European call option at expiration T depends on the spot price of the underlying at time T. If the spot price, S capital T, is less than K, you don't exercise the Option, because it's just cheaper for you to buy in the spot market. And the payoff from the Option is going to be 0, you're not obligated to buy, if you do not buy. If the price at time capital DST is strictly greater than K then you exercise the Option, and the option allows you to buy the underlying asset at price K which is cheap, and you immediately sell that in the spot market to get S T. So the the payoff is going to be S T minus K. Putting both of these cases together, the payoff from a European call option, at expiration T is going to be the maximum of the spite, spot price minus K and 0. It's important to note, but the payoff is non-linear in the spot price ST. This is the first time we have seen an instrument whose payoff is non-linear in the underlying asset and this is going to be important when we talk about hedging. The intrinsic value of a call option at some time little t, less than expiration D, is simply defined as the maximum of the spot price St minus k and zero. We say that the call option is in the money, if the spot price is greater than the exercise price K, it's at the money if the spot price is equal to K, and it's out of the money if the spot price is less than K. All of this works the other way for put options. So for a put option, you exercise when the price ST is less than K, because it allows you to sell at a higher price. And the payoff that you get is the difference between the exercise price K and ST. If the price and the spot market is greater than K, then you don't exercise the Option. It's better for you to just sell in the spot market and the payoff that you get is 0. The payoff from the put option therefore is the maximum of K minus ST and 0. Again, it's non-linear, and than the price ST. But it's a different non-linear function than that associated with the call option. The intrinsic value of a put option at some point t less than or equal to the expiration time is defined as maximum of K minus St, the exercise price minus the spot price and 0. It's in the money if the stock price is lower than the exercised price. It's at the money if the spot price is equal to the exercised price and its out of the money if the spot price is greater than the exercise price. So now, we want to see what can we do about pricing options. The payoff from the Options are nonlinear, and therefore we cannot price it without using some model for the underlying asset, and this model is going to come later on in this course. The model that we're going to first introduce is going to be the binomial model, which then we'll take it as a limit and go to a geometric Brownian motion model. In this particular module, we just want to see how much mileage can we get out of simple no arbitrage arguments. So let's put in some notation. A European put option with strike price K and expiration T, the price of such an option we'll call as pE, t K T. So, pE stands for a European put. Time is time. This is the strike price [NOISE] and that is the exercise time. [NOISE] cE t, K, T would stand for the price of a European call option, with strike K and expiration T. Now, when it becomes American style, we put a subscript A. So the price of an American put option with strike K and expiration T will be called p sub A. The price of an American call with strike K and expiration T would be called c sub A. Using the no-arbitrage argument, we can construct something called a put call parity. And here is the expression. It says that the price, of a, put option and the price of a European call option on a non-dividend paying stock are related using this expression. PE plus the stock price, the spot price of the asset, must equal cE plus K times the discount from time t to T. This is an expression I don't want you to pay too much attention to it. I will show you how this expression is constructed. But in a couple of slides, we are going to use this put-call parity, put-call parity stands for the fact that the put price and the call price for a European option have to be related to each other. We're going to use this put-call parity to show one important result, which has to do with American-style put options. Okay. So construct the following trading strategy. At time t, buy a European call with strike K and expiration T. Sell a European put with strike K and expiration T, short sell one unit of the underline, and buy it back at time T. Lend K times d t, T dollars up to time T. What happens to the cash flow at time T? Because I have bought a European call, I get its cash flow, which is going to be the maximum of ST minus K and 0. Because I have sold a European put, I have to pay out its cash flow, which is going to be maximum K minus ST and 0, but the negative sign in front says that I am responsible for paying, because I have sold it. I have to buy back the asset at time T, therefore minus ST is a cash flow associated with that. Whatever I had lent comes back to me. So ed, end up getting K dollars. And if you work out these different cases you will see that the cash flow at the time T is exactly equal to 0. The no-arbitrage argument tells me then the price that I paid for this particular deterministic cash flow should be exactly 0 at time little T. The cash that I get, the price that I paid for this particular portfolio at time t is going to be minus cE t, plus pE. The, this is the cash flow associated with buying the European call. This is the cash flow associated with selling the European put. This is the cash flow associated with short selling the asset. And this is the cash flow associated with lending money up to time T. The negative of this cash flow is the price. That should be equal to 0, which is equal and is saying that the cash flow at time t must also be equal to 0. If you rearrange these terms, you end up getting the put-call parity. So what we are going to do in this particular slide is connect, prices of European and American style options using put-call parity. The first thing we know is, the price of an American option has to be greater than equal to the price of a European option. Why does that happen? The American option gives me more choices on when to exercise, and therefore, I should be paying more for this freedom. So cA is going to be greater than or equal to cE. PA is going to be greater than or equal to pE. We can construct some lower bounds on European options using the put-call parity. So cE is going to be greater than equal to 0. Why? Because, you have the option of doing nothing. And therefore, the cost of such an option must be greater than or equal to 0. Using put-call parity, we end up getting that cE is equal ST plus pE minus Key a times D t, capital T. So, putting that together. Taking the maximum of that quantity and 0 gives me that cE must be equal to this quantity. Now, pE is a quantity greater than equal to 0. So if I drop that out, I end up getting that this is greater than equal to the maximum of ST minus K times the discount from t to T and 0. For the European put you end up getting the same story. You get one term from the put call parity, you get another term from the fact that the price of the European put is greater than equal to 0. Take the maximum [UNKNOWN] quantities and pE is equal to that. CE is greater than equal to 0. So if I drop that, I end up getting that pE which is the price of a European put has to be greater than equal to the maximum, of K times the discount minus ST and 0. You can get upper bounds of the European options using the stock price, since the maximum of ST minus K and 0 is less than or equal to ST, it means that the call price has to be less than the spot price of the underline. Since the payoff from the European put is less than or equal to K, it immediately implies that the price of the European put has to be less than or equal to K discounted backwards up to time t. If you have dividends and the put-call parity changes a little bit, it becomes pE plus St minus D, which is the present value of all dividends up to maturity, must equal cE plus K times the discount from t to T. These bounds by itself are not going to be very interesting. the only idea that I want you to take away from this slide is that using very simple ideas of freedom associated with American options versus European options, the put-call parity and very simple bounds on the fact that the cost of a call option or a put option has to be great, has to be, greater than equal to 0, the fact that, if I want to buy a stock, I could either get the option or buy the stock alright, outright and, therefore the price of the option must be less than equal to ST, and so on, gives me some nice bounds. In the next slide, I'm going to show you that these bounds can be used to compute an optimal strategy of execution for a American type call option, which gives you a very interesting result. So let's construct the bound on the price of an American type option, using the strab, stock price ST. Now I know that the price of an American call is greater than equal to the price of a European call, which using the bounds in the previous slide we know that its greater than equal to ST minus K, D t, T the discount N 0. Now, this discount d t, T is less than equal to 1. Therefore, if I replace that by 1, I get something smaller. So you end up getting that this quantity, is actually strictly greater than that quantity. But the last quantity there is the intrinsic value of an American call. This is equal to the intrinsic value [NOISE]. So this sequence of inequalities what it does mean, is that the price of an American call is strictly greater than its intrinsic value. It's strictly greater than whatever value that you could get, but exercising the American call immediately. And what that tells me is that it's never optimal to exercise the American call on a non-dividend paying stock early, which means that the price of an American call is exactly equal to the price of a European call. In order to construct this argument we did not need any model for the underline stock price process. Just by using put-call parity which came out of constructing a no other arbitrage argument, we were about to show you that an American call option is the same as a European call option. The prices are exactly the same. What is important though, is that the underlying asset is non-dividend paying. Let's try to do the same thing for the price of an American put. PA is greater than equal to pE, which is greater than equal to K times the discount from t to T minus S T and] 0. But the exercise value of an American put option is the maximum of K minus ST. These 2 are not related in the right direction, therefore these bonds, bounds don't tell us much about what is going to happen to an America put option. Using these bounds we cannot tell whether its optimal to exercise the put early or not. It turns out that if you use some model for the underlying stock price, there are situations in which it's optimal for you to exercise a put option earlier. In this last slide, I'm showing you what happens to the price of the put option as a function of the intrinsic value and the underline. So on the x axis is the price of the underlying. On the y axis I'm putting the price of the put option. The blue line here is the intrinsic value [NOISE]. So what this graph shows you, is at a price let's say a 120, the intrinsic value is 0, but the price of the put option is higher, which means it's clearly not optimal for you to exercise. Even at price 80, the intrinsic value is below the price or the value of the put option, which means again, it's not optimal for you to exercise. At the price 40, the intrinsic value is equal to the price or the value of the option which means that it's optimal for you to exercise. So the optimal exercise boundary is somewhere here. For all prices below here, optimal to exercise [NOISE]. For all prices larger than that, it's optimal to hold. [BLANK_AUDIO]
