Friday, September 19, 2008

Time to review the Put-Call Parity Theorem

With the SEC's recent move to ban short-selling of politically-important securities, it's time to review the beautiful Put-Call Parity Theorem to understand the futility of doing so. Here's my phrasing and elegant explanation of it:

B(t,$X) = S + P(t,$X) - C(t,$X)

B is the value of a bond maturing at time t for $X.
S is the value of some asset, it doesn't matter which.
P is the value of right to sell the above asset at time t for $X. (In financial terminology, a put option dated at t with a strike price of $X.)
C is the value of the right to buy the above asset at time for $X. (In financial terminology, a call option dated at t with a strike price of $X.)

In this sign convention, negative means the counterparty to the security, so for example, if the bond term were negative, it would refer to the value to the borrower on that loan, while the negative call option refers to the person having the obligation to sell at $X to the call owner.

So, the equation means that, for some time t and some money amount $X, a bond maturing at t for $X is equal in value to some asset, plus the right to sell the asset at time t for $X, plus the obligation to sell it at time t for $X.

Proof: the left-hand side of the equation is worth $X at time t. The right-hand side is also worth $X at time t because if S were worth less than $X, the holder of the put could sell it for $X, while if it were worth more, the holder of the call could buy it for less. Q.E.D.

Note that if you find a case where the two sides are not equal, you profit through arbitrage buy buying the cheaper side and selling the more expensive side. In a discussion a few years ago, Gene Callahan claimed this was how he made money. You also might be interested to know that this theorem -- though of course it wasn't referred to in such terms -- was historically used to circumvent financial regulations such as bans on usury, since through clever rearrangement of the equation you can recreate any financial security. Here is a neat paper on that history.

Anyway, the point to remember is, let's say I want to take a short position in a stock. That would be represented by "-S" in the above equation. But let's say you found out that was banned! No problem. Just rearrange the equation! With the function arguments suppressed:

-S = -B + P - C

So, borrow money, buy a put, and write (sell) a call. Problem solved! (Except for the cost of fending off the SEC guy giving you an intimidating stare, of course.)


Bob Murphy said...

Can you talk us through what happens when a bunch of people buy puts, rather than shorting? How does that end up pushing the share price down? Does it set up pure profit opportunities for the current owners of the shares (when put-call parity is briefly violated by the massive buying of puts)?

Bob Murphy said...

Another point: If the ban on shorting really is pointless (except maybe for raising transaction costs), then why is Wall Street apparently so happy? Is Wall St. reacting to other news and not the specific restrictions on shorting?

Vilhelm S said...

I think the ban works out anyway. For you to be able to buy the put, someone else (usually a big investment bank) has to write it. If they do, they now stand to lose from the underlying stock going down, so they will need to delta-hedge by shorting the stock. Except that they can't because that's banned. Ergo, they will refuse to write the put in the first place (or at least price it much higher than it would be in a black-scholes world where delta-hedging is possible).

Silas Barta said...

Alright, ya'll win. This is above my pay grade. But I'll do my best.

-Puts lower the stock price the same way they always do.

-No, it doesn't set up pure profit opportunities, unless there's an asymmetric ability to see the implications for stock prices.

-The rally was probably because it signaled the government's willingness to help Wall Street. (Remember, I don't claim that it has no impact, just that it can't fundamentally eliminate the ability of non-holders to bet against a stock unless they do a lot more regulation.)

-If everyone delta-hedges everything, does no one ever actually take the risk of any price movements? Surely there have to be people that write puts without completely canceling the risk. It's not like there's an infinite regress going on.

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