Arbitrage and Put-Call Parity
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Arbitrage and Put-Call Parity

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The word arbitrage sounds very fancy, but it's actually a very simple idea.
It's really just taking advantage of differences in price on essentially the same thing to make risk-free profits.
Let's think about it a little bit.
Let's say in one part of town, there's some type of a market.
Let's say it's a market for apples.
And let's say that in that market apples sell for $1 per apple.
And let's say in another part of town you have another market.
And in that other part of town, the apples sell for $1.50 per apple.
So how could you take advantage...?
And we're going to assume that these apples are completely identical apples.
How could you take advantage in this price difference on these identical things to make a risk-free profit?
Well ideally, you want to sell the apples in the more expensive market where you can get $1.50/apple.
And you want to buy apple in the less expensive market where you can get them for $1/apple.
And that's exactly what you would do.
You would go to this market over here....
You would buy apples; let's say you buy 10 apples for $10.
And then you would go maybe ride your bicycles to the other market over there
and you would sell your 10 apples.
So this is buy 10 apples for $10, and then you would sell those 10 apples for $15.
And so you would make an immediate risk-free profit of $5.
You are buying for $10 and selling for 15.
And you can just keep doing that over and over again
and on every trip as many apples as your bicycle can carry
You will continue to make money.
So this is what arbitrage is.
And just imagine the side effect
If someone did this enough,
then this would increase the supply of apples here
so supply would increase in this market
and in this market the demand will increase
because there's someone who keeps buying in this market and selling into that market
So what's eventually going to happen
with the demand increases the price will go up in this market
and when the supply increases in this market, the price will go down.
So in theory the more that you do this the more that you are going to make these prices come closer to each other.
And eventually you won't be able to make any profit at all.
Say stock XYZ is trading at 31 dollars.
We have a call option on XYZ with a $35 strike price.
It's trading at $8.
We have a put option on 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 finally there's a bond, unrelated to XYZ.
It's going to be a risk free bond, like a treasury bill
worth $35 at option expiration
and you can buy it right now at $30.
The reason you can get it for less is 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?
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
the put is a put on that stock
is equal to, is going to have the same value
at expiration as a call with the same strike price
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.
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.
On the left-hand side, right now,
it's trading at $43.
On the right-hand side, the call option
is trading at $8. And 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 aribtrage opportunity
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 buy the cheaper thing and you want to sell
the more expensive thing. Especially when they
are the same thing, when they are 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, essentially you are
shorting a put option. Another way to
think of it: you could write a put option.
So you would short the stock plus write a put.
So what would happen there?
Shorting the stock, you're borrowing the stock
and selling it. You're going to get $31
from shorting the stock.
And writing the put, literally means you are 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 you're going to buy the call and the bond.
You're going to spend $8 on the call
and $30 on the bond, you're going to spend $38.
And you're going to make a profit of $5.
We're going to see in the next video: you make
this profit up front, and no matter what happens
to the stock price going forward,
you're able to arrange things so everything else
cancels out and you can just keep your $5.


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