Ingenious objections are always interesting, particularly when they are wrong. An actuary has objected to my post here, where I considered a put written at 90 on a price series which goes 95,96,95 etc. The point of the post was to show that the standard formula works perfectly well for such a distribution. The objector objects that if the series really does oscillate between 95 and 96, the price will clearly never reach 90, so the standard pricing formula must be wrong. The true price of the put must be zero.
The mistake is similar to the Buffett mistake I considered here, namely of applying statistical ideas to option pricing. We construct a ‘distribution’ of possible outcomes, each with an assigned probability, then use that to get average expected future cost of a defined set of outcomes, either by computer simulation, or a closed form solution if available. If the distribution consists of 95 or 96, each with a probability of 50%, then the value of the price being lower than 90 must be zero, according to this reasoning.
But the reasoning is faulty. In finance, a distribution is not a physical process, but reflects our uncertainty about a possible future outcome. If we had perfect knowledge of the future, we would certainly assign a price of zero to the put option. Indeed, if we had perfect knowledge of the future, we would probably not need prices at all. As Hayek neatly pointed out, prices are simply a kind of information about supply and demand.
Assume that somewhere in the world a new opportunity for the use of some raw material, say, tin, has arisen, or that one of the sources of supply of tin has been eliminated. It does not matter for our purpose—and it is very significant that it does not matter—which of these two causes has made tin more scarce. All that the users of tin need to know is that some of the tin they used to consume is now more profitably employed elsewhere and that, in consequence, they must economize tin. There is no need for the great majority of them even to know where the more urgent need has arisen, or in favor of what other needs they ought to husband the supply. 1
In such circumstances, tin prices are going to follow a distribution, but that merely reflects our lack of knowledge about what makes tin scarce, or plentiful. It is our lack of knowledge of the distribution that creates the distribution in the first place.
So it is with the price series I considered in the last post. We can observe the prices going 96, 96, 95 etc, once it has happened, but we don’t know what is going to happen. If we did, we wouldn’t need the option, whose purpose is a safeguard against the price ending below 90. The example shows how we can hedge against that, for a price, that the price of the hedge is the Black-Scholes price, and that the distribution is irrelevant.
Confusing a physical process with human uncertainty is an intellectual error. But it is an error that has real world consequences for investors in ERM firms, and (more importantly) for their policyholders, a vulnerable group of society that could suffer real harm as a result.