So this shows an option on the day of expiration. The value of the option, is called the intrinsic value, this is a call option, and against the stock price. So if it's a call option and now it's the last day, you either exercise now or forget it. If the stock is worth less than $20, the option is worthless. You tear it up and throw it away. You would not pay $20 today, if the stock price were 15 you would not pay $20 to get something worth $15. But if it's in the money on the last day, on the exercise day, the option is worth the difference between the stock price and the option price. So if the stock price is $25, then you would definitely exercise because you pay $20 to exercise it and you can sell the stock immediately. So it's $5, like money in the bank money, lying on the street. Almost everybody does that. The only people who don't exercise in the money options on the exercise date are people who are not paying attention. And I think that's actually rare. Not many people make that mistake. So the value of the option on the last day has to follow this curve. If the stock price is less than the exercise price, here 20, the option is worthless. But if it's above it, it's equal to the stock price minus the exercise price. For put options, it's different. This is the right to sell. So you only exercise it if the option price is below the stock price. Now, there's something called the Put-Call Parity Relation which is a relationship enforced by arbitrage between a put price and a stock price that have the same underlying, the same strike price and the same exercise date. And that says that these two things are equivalent. The first two lines are equivalent. I just put the items in a different order. So the price of the stock has to equal the call price plus the present discounted value of the strike. Now, this is, at any day, and this is technically for European options, but it applies generally to both European and American options. The price of the stock equals the call price plus the present discounted value of the strike price, plus the present guided value of dividends coming in between now and the exercise date, minus the put price. It does hold up pretty well. Intel Corp, I showed you the example from the CBOE. And so, where was it back here. I'm using the first line here, strike price of 27. Now, this was the last price for the option. But, currently the market maker has a bid-ask spread between 6.05 and 6.20. I'll take the midpoint of the bid-ask spread as an indicator of the current market price. And similarly, the same strike price is available for the same date. It's expiring on January 19th of 2018. They're both expiring on the same day. And so, I'll take the midpoint of bid and ask for the put. And then, I'll go back to that slide here. Okay, so this is the midpoint of the call prices. The sum of the two values divided by two, plus the strike price. I'm assuming a zero interest rate to do this quickly in our head. So I'm not taking present values, while interest rates are pretty low now. So I'm being rough when I say that. Now, I have to figure out how many dividends are between now and January 19th 2018. And I didn't carefully figure out. I thought there's about eight of them. So 26 cents times eight is $2.08. And then, this is the mid point of the bid and ask spread for the puts. And I add them all up and I get $32.54. That's pretty close to the stock price of $31.63. Why isn't that exactly the same? Well first of all, most notably I didn't even do the interest rate calculations. So the interest rates are not exactly zero. So that would have bring down the present value of the strike price and the present value of the dividends. But also, there's just some non-synchrony here. I'm looking at the last price comparing that with the dealers, they didn't ask. There's a little timing, looseness here. So it doesn't work out exactly. But generally, it has to work out, that the so-called put-call parity relation has to hold because it's the same thing. You're pricing apples and oranges but it's really apples and apples. Think of it this way, the yellow line is the intrinsic value of a call. The pink line is the intrinsic value of a put. If you add the present value of the strike price to this sum of the puts and calls line, you get the stock price again. And dividends, as well, have to be brought in. So put-call parity is a fundamental relation that actually holds quite well if you do it exactly right in the options market. And what it really means is that, in fact, you don't even need both puts and calls. It's just for convenience. Because they're related to each other through the put-call parity relation. Now, what is the price of an option on a day before the last day. On the last day, it's all simple. The price of the option is the intrinsic value because there is no more risk. It's now. But it's only a negligible risk and over a matter of minutes that it would take you to sell. So this is the intrinsic value which is the value on the last day. On an earlier day, now we're talking about months or years before the exercise date, the stock option is, or whatever option, it's got to be worth more than the intrinsic value because it has option value. So consider here, suppose we're looking at a call option now with a strike price of 20. It says that there is value when the stock price is 15. There is value to the option. Why would it be worth anything if it's out of the money? Well, this is obvious. Because it might go up. So I'm willing to pay something for the option. Suppose the stock price goes up to 25, then my option price is going to be worth a lot on the exercise date. So the option has to be worth something even though it's out of the money. They're never worthless. It might be very minuscule, but there's always a chance that the stock will go up above the exercise price so it has to be worth something. But then, you also, why is it worth more than the underlying value when it's above the exercise price? Well, it's for the same reason that if the stock price will fall below the exercise price, you'd lose the full amount if you own the stock. But when you own the option, you still got something. You got the option value. The option isn't worthless if it has some time to expire, even though its intrinsic value is worthless. You understand what no arbitrage mean. No arbitrage mean no sure profits. Any profit that you make has to entail risk. There's no $10, you see any $10 bills lying on the floor? No, you don't. Why not? Somebody would have picked it up. Somebody at some point must have lost a $10 bill in this room. But it's not there anymore. Those things are rare that you'll ever find one because the first person to see it picks it up. So similarly, we don't expect to see the put-call parity relation violated. If that were violated, I can tell you. Here's a good job for you, drop out of college and invest in disparities between put-call parity and you make money for sure. So you might as well just push it to the limit and borrow millions of dollars and just do it on a big scale. So it's so simple and obvious, once you look at it. You can be sure that there are guys out there right now, arbitrageur making profits from the tiny discrepancies and put-call parity. But they eliminate the discrepancy. And when they do