# Put-callÂ 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.