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Studying for a test? Prepare with these 7 lessons on Options, swaps, futures, MBSs, CDOs, and other derivatives.

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# Put-call parity arbitrage I

Video transcript

Say stock XYZ is trading at $31. We have a call option on stock
XYZ with the $35 strike price. It's trading at $8. We have a put option on stock
XYZ with a $35 strike price. They have the same strike price. Trading at $12. And they both have the
same expiration over here. And then finally,
there's a bond. And this bond is
unrelated to stock XYZ. It's going to be
a risk free bond. So it could be some
type of a treasury bill. Worth $35 at option expiration. And you can buy it
right now for $30. And the reason why you can buy
it for less is you pay $30, you're going to get $35 in the
future at option expiration. So you're essentially getting
interest on that bond. So with these numbers, is there
a way to make risk free money? And to think about that,
let's think about the put call parity. We learned that a stock plus
a put at a given strike price, and the put is a put on
that stock, is equal to. It's going to have the
same value at expiration as a call with the
same strike price. A call with the same
underlying stock. Plus a bond, a risk free bond,
that's going to be worth that strike price at the expiration
of these two options. So since this is going to have
the same value, the same payoff in any circumstance,
as this at expiration, they really should be
worth the same thing. But when you look at
the numbers over here. Let's see if that works out. The stock is trading at $31. The put option is
trading at $12. So that's plus 12. So this on the left hand side
right now if you had to buy it, it's trading at $43. Is On the right hand side,
you have the call option is trading $8. And then the bond
is trading at $30. So this combination
is trading at $38. So even though they have
the exact same payoff at option expiration,
the call plus the bond is cheaper than the
stock plus the put. So you have an
arbitrage opportunity. You have an opportunity
to make profit from a discrepancy in
price from two things that are essentially equal. And what you always
want to do is you always want to
buy the cheaper thing. And you want to sell the more
expensive thing, especially when they are the same
thing, when they're going to have the exact
same payoff in the future. So you want to sell this. So buying is pretty
straightforward. What does it mean to
sell this over here? Well, you could short the stock. That's essentially,
you're selling the stock. And then you would
you essentially are shorting a put option. Or another way to think of it,
you could write a put option. So you short the stock
plus write a put. And so what would
happened there? Shorting the stock,
you're borrowing the stock and you are selling it. So you're going to get $31
from shorting the stock. And writing the put means
you literally are essentially creating a put option and
selling it to someone else. And so you're going
to get $12 for that. So you're going to get your $43. And then you're going to
buy the call and the bond. So you're going to spend $8
on the call, $30 on the bonds. So you are going to spend $38. And you're going to
make a profit of $5. And what we're going to
see in the next video is you make this profit upfront. And no matter what happens to
the stock price going forward, you're able to rearrange things
so that everything else just cancels out. And you can just keep your $5.