on options & epidemics

The pricing of call and put options seems to have nothing in common with attempts to control the spread of sexually transmitted diseases. But it turns out that, in both cases, identifying and influencing the variance of a probability distribution can be more important than identifying and influencing the mean.

It is easy to see that additional volatility in the underlying asset of a call option leads to greater option value. An option holder can cash in on the added gains from an upward fluctuation but loses no more if the price fluctuates wildly downward. For example, if you own an option to buy a share of Google stock at $200, you want Google’s stock price to fluctuate. In fact, option holders should be willing to trade a lower mean for a higher volatility. A person holding the Google call should prefer a world where there is an equal (1/3) chance that the price of Google stock will end up at $100, $200, or $300 than a world where there is an equal (1/3) chance that the price of Google stock will end up at $200, $210, or $220. Even though the latter distribution has a higher mean ($210 versus $200), the higher volatility of the former distribution has a bigger impact on the option value. Under the first distribution, the call option will be worth $33 (.33 x $100). The second distribution, even though it has a higher expected stock value, produces a lower call value of $10 (.33 x $10 + .33 x $20).

A folk theorem of finance theory is that whenever you identify an implicit option, there is almost always an interesting volatility story to tell. And there are implicit, or ‘real,’ options in all kinds of real-world settings. For example, consider an extremely stylized nuisance dispute. Imagine that Scholes and Samuelson are neighbors and that Scholes wants to stop Samuelson from singing in the morning. How should a court allocate the singing entitlement? One traditional answer (which is even codified into the Restatement (Second) of Torts § 826(a)) is that courts should give the entitlement to the litigant that the court believes to have the higher valuation. If Scholes values silence more than Samuelson values singing, then the court should give Scholes the entitlement to control whether his neighbor sings in the morning. This simple rule seems to make eminent economic sense.

But in resolving nuisance disputes, courts often go beyond merely deciding whether to enjoin singing (or pollution). Sometimes courts give the underlying entitlement to one party, but simultaneously give the other litigant a call option to buy the entitlement for a specified price. For example, in the famous case of Boomer v. Atlantic Cement Co., 257 N.E. 2d 870 (NY 1970), a court enjoined a factory’s pollution but simultaneously gave the factory the option to continue. If the factory paid their plaintiff-neighbors the court’s best assessment of the monetary value of the neighbors’ damages, the factory could resume polluting. In other cases, courts give the underlying nuisance entitlement to the defendant but simultaneously give the plaintiff the option of purchasing an injunction by paying the defendant a specified amount of damages. In these cases, the courts are allocating two entitlements: they are giving a call option to one side and the underlying nuisance entitlement (subject to the option) to the other.

In allocating this implicit option, courts would do well to consider the implicit volatility of litigants’ valuations. From the courts’ perspective, the litigant with the more speculative valuation has the higher volatility and therefore is likely to be the more efficient option holder. To see the importance of valuation volatility in a simple example, imagine that a court believes that a Resident’s harm from pollution is somewhere between $5 and $105, uniformly distributed, but that the Polluter’s costs of stopping pollution are somewhere between $40 and $60, uniformly distributed. Our first intuition might again be to give the initial entitlement to the Resident–because she has a higher expected value ($55 versus $50) –and the call option to the Polluter– to make up for the fact that parties may have trouble reaching agreement when the Polluter turns out to have the higher value.

But in this example, the Resident’s valuation has both a higher mean and a higher variance. Because options are worth more when the underlying entitlement is more variable, it turns out that giving the Polluter the entitlement and then giving the Resident a call option produces much higher allocative efficiency. Even though from the court’s perspective the Polluter has a lower expected valuation, giving it an entitlement subject to the Resident’s call is more efficient because the Resident with an option to enjoin pollution for $50 will do so whenever she has a particularly high valuation. If we give the Polluter the call option instead, we can end up with a truly inefficient outcome of pollution that creates $105 of damage. When we give the Resident the call option, this never happens. This simple and admittedly stylized example shows that valuation variance can be more important than the mean in deciding legal cases. When options are at stake, we need to attend to both.

The need to attend to volatility is important whenever options come into play. A number of years ago when I was teaching at Stanford, the university had a home mortgage program. The university would lend you half the purchase price of your house, if you give the university half the appreciation at the time of the sale. The program gave the university something akin to a call option on half your house. The university didn’t have to bear any cost of home depreciation, but got half the upside if the housing value increased. I had a choice of buying a house in an unincorporated (and unzoned) new section of Mountain View or a relatively staid and seasoned development just south of the campus called College Terrace. Attending to volatility, you should be able to tell which house was more subsidized.

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