Pricing far out of the money derivatives in the P measure January 24, 2014 at 12:03 pm
OK, not the most attractive title in the world I know but bear with me.
Most derivatives are priced in the Q, or risk neutral measure. This is the right thing to do when you are willing and able to hedge. Essentially in the Q measure the cost of a derivative is identified with the price, according to your theory, of hedging it.
You can’t do that sometimes, often because the necessary hedge instruments are not available. This is one practical distinction between derivatives and insurance: derivatives are hedgeable, insurance isn’t. Therefore you shouldn’t price insurance in the Q measure: instead you care about the real world (as opposed to the risk neutral) distribution of outcomes.
This is important when we think about things like Warren Buffett’s basketball trade. Buffett has written insurance (not a derivative*) to protect Quicken Loans against the risk that someone will correctly predict the winner of every game in the National Collegiate Athletic Association’s men’s basketball tournament – something Quicken has offered a billion dollars for.
This is actually an interesting thing to price. If you view the tournament as IID coin tosses, then of course the insurance is worth nothing. But of course it isn’t, partly because some teams are better than others, and partly because there will be entrants to the competition with private information. It isn’t much of an edge to know that a star player is off form, but excluding a favourite team skews the distribution for very unlikely outcomes, especially if you let insiders collude by assuming, say, that someone knows the outcome of every game involving, say, ten specific teams. Moreover, you don’t just need enough premium for this insurance to pay for the expected loss; you also need enough to pay for your cost of capital supporting unexpected loss, which will be a lot as the distribution is very fat tailed. Add in the cost of actually doing all this analysis, and pretty quickly you get to a multi-million dollar premium. Indeed, something I call (after a long-retired trader) Stavros’ law probably applies: never sell any put for less than a cent of premium, no matter what the model says it is worth.
*Don’t get me started, though, on the hypocrisy of Buffett’s public statements about derivatives vs. what Berkshire actually does.