# Put-call parity

## Video transcript

If we want to get the upside of owning a stock while still mitigating the downside, in case the stock price goes down, we saw that we could buy a stock and an appropriate put option. So that when the stock goes below some price, the put option starts to have value, and so it mitigates our downside. And just as a review, these payoff diagrams are the values of-- or at least the one on the left, is the value of our holdings at some future date. And we're defining that date to be the maturity date of the options under question. Now, and this one over here is the profit at that maturity date, and that's why we're subtracting the actual costs to enter the position on this one on the right. Now the question I want to answer in this video is how can we get the same payoff diagram without buying either stocks or puts? And as a bit of a clue, think about what happens if we were to just to buy a call option. Actually let me do it in that same color. So if you were to just have a call option, the payoff diagram would look like this. You would never exercise the call option at expiration, unless-- and we're assuming this is at expiration or at maturity. But if the stock price goes above $50, you would then exercise your option to buy it at$50. So then it starts to have value as the stock price goes above $50. If the stock price goes to$60, you would exercise your option to buy at $50, and then you could sell at$60 and you would make $10. So you start to get some of the upside. So how can we shift this graph up to get exactly the same payoff diagram? Well, we could have a call option, and we could own something that would essentially shift this entire graph up by$50. So we could have, essentially, a $50 bond, or a bond to that is worth-- let me write it this way. A bond that is worth$50 at option expiration. So if there's some interest we're getting, we might be able to buy it for a little bit less. If there's zero interest, then it's pretty much like cash, we would pay $50 for it. But the payoff diagram for a bond that will be worth$50 at this date, at maturity, or at expiration, the payoff diagram for just the bond would look like this. It would just be a straight line. It's guaranteed to pay you $50. So if you own the bond and the call option, below$50, the call option is worthless, so you're just going to have the bond over here. And then above $50, you still have the bond, but now the call option is worth something. So you have the value of the bond plus the call option. So at$60, the call option's worth $10, the bonds worth$50, the combination is worth \$60. And so the combination of the call option plus the bond, you'll see it here on the left, it's actually going to have the same payoff diagram as the stock plus the put. So you have the situation here that a stock plus an appropriately priced put or a put with a appropriate strike price is going to be the same thing when it comes to payoff, at a future date, at expiration, as a bond plus a call option. And this right here is called put call parity. And it shows the relationship between all of these different securities. And if any of the prices start to kind of not make this thing hold true, there might be an arbitrage opportunity. But we'll cover that in future videos.