The St. Petersburg Paradox and the Low Hanging Fruit

The most recent trading strategy I've been researching has been based on so-called volatility arbitrage, a horrible misnomer for the strategy of trading volatility like an asset by taking a view and getting exposure (generally) through options. But since the modern options market has all but adopted a notion of equivalence between volatility and price, trading volatility really amounts to determining the true fair value of options written on a particular underlying. It's not really a forecasting problem so much as a modeling problem, although elementary forecasting does play a role in a good model.

I shared a bit of this information with a reader of this blog last night, and to my surprise, he commented that vol arb had been "picked clean" -- a term applied to strategies whose popularity has eclipsed their profitability. This term gets thrown around a lot with reference to "statistical arbitrage," a similarly misnamed strategy by which traders exploit gaps between correlated securities and bet that the gap will disappear over time.

The reason statistical arbitrage (as it is now known) is possible is because of two Nobel Prize winners named Robert Engle and Clive Granger. They won the prize for the development of a number of related time-series analysis techniques, one of which was the measure of "cointegration," which aptly embodies the "gap-ungap" property of a good stat arb pair.

The study of cointegration was borne of Engle and Granger's realization that correlation was really a terrible metric in most cases of statistical analysis. Thus, the statistical arbitrage opportunity was borne of a simple realization that one of the core beliefs of the markets -- that correlation is the correct measure of similarity -- was wrong. A paradigm shift takes place, and an opportunity arises until information diffusion has the opportunity to destroy it.

Once enough people knew how to implement a cointegration test, they could implement stat arb. And stat arb got picked flat. And it will remain flat until another paradigm shift comes along, and we are able to discover similar time series with great precision or quality, and then stat arb is back in full force.

The point is: opportunity is created by widespread misunderstanding. And to demonstrate just how much work there is left to do in quantitative finance, and how many opportunities still exist to exploit widespread ignorance, let me guide you through some of the theory of options valuation, and present some startling contradictions that you may find quite shocking.

But first, let me bore you with some mathematical history. The St. Petersburg Paradox is a problem worked on jointly by the Bernoulli brothers. It describes a lottery in which one buys a ticket, and the "dealer" of sorts flips a coin in succession until a tails comes up. If a tails comes up on the first flip, the gambler gets $1. If it comes up on the second, he gets $2, and the third, $4, and so on ad infinitum.

The problem: determine the fair price of a ticket. Most people would do this by taking expectations over the probability distribution defined by the payoffs, but get this -- the expectation is infinite! That implies that a human being would pay any price for a ticket!

Clearly, a human being wouldn't pay any price for a ticket. The Bernoulli brothers solve the problem by expressing the payoffs in terms of a diminishing marginal utility function (logarithmic) and show the fair value to be $2, which is a lot more sensible. The point of the paradox is how nonsensical an answer you can get by naively taking expectations.

Now let's turn our attention to the original work of Black and Scholes, something I intend to discuss plenty on this blog. Black and Scholes begin by dreaming up a portfolio containing a delta-hedged option position, and by no-arbitrage arguments reason that the portfolio must yield the risk-free rate, since it is itself risk-free.

This is the celebrated Black-Scholes differential equation, the form of which is the basis for the great majority of all derivative pricing methods. Since the equation has no risk-dependent parameters in it, like the stock's drift rate, they reason that the derivative's value is not dependent on the level of risk, or even the expected return, on a stock.

This is where it starts to get weird. They reason that since the derivative's value is independent of risk, that the value of an option in our world is the same as that of a derivative in a hypothetical universe called the "risk neutral world."

The risk-neutral world is a bit like Eden. All investors are 100% tolerant of risk. All securities earn the risk-free rate. Securities are all equal to their discounted expected value under the correct probability measure.

They reason that since the value of the derivative in our world is the same as in theirs, we may as well make life simple and value the derivative in their world. So they do -- they combine the differential equation obtained above with the solution to an Ito integral representing the stock price's lognormal diffusion (with drift equal to the risk free rate). That yields the Black-Scholes solution.

I hope the St. Petersburg paradox is contrasting nicely with this Nobel Prize-winning reasoning of Black and Scholes. We're going to make the comparison even more explicit by extending the discussion to the pricing of American options, which have the capability to be exercised at any point during their lifetime, not just at expiry.

The Black Scholes-style way to value an American option is using a binomial tree representation, popularized initially by Cox, Rubinstein and Ross. A tree is constructed in order to represent the lognormal process described above; each leaf node is annotated with its value; the other nodes are annotated with the discounted expected value of their children under the tree's induced probability measure.

Now, if the stock doesn't pay a dividend, we have a bizarre situation. Under the induced probability measure of the binomial tree, the discounted price process (by the Black-Scholes theory) is a martingale. Thus, we know there is no optimal stopping time. So, the value of the early exercise never exceeds the discounted expected value, and therefore is never taken. Therefore, the value of an American option on a non-dividend paying stock is equal to the value of the same European option on that stock.

You might need a minute to digest that. Yes, it's true. Under the Black-Scholes assumptions, an American option has the same value as a European one so long as there is no dividend. And these same assumptions that yield such a terrifying paradox also underlie the rest of the models used for pricing and risk management of derivative contracts.

The source of the contradiction is similar to the case of the St. Petersburg Paradox; the reckless taking of expectations, that reduction of a probability distribution to a scalar, is such a dangerous approach. And much more importantly, the idealistic assumptions of Black-Scholes don't just manifest themselves in an unrealistic price process with skinny tails, as people frequently point out, but in the very modeling of human behavior that underlies the model.

See, with European options, there's no behavior to model, since nobody really has much of a choice. The only choice happens at expiry, and you can bet your bottom dollar that the option will be exercised if and only if the option is in the money. With American options, on the other hand, the option holder is forced to make the choice of whether to exercise every second that the option is in the money. As in the case of the St. Petersburg Paradox, the assumption that humans are rational, expectations-taking automata is highly inappropriate in finance. In fact, options investors will tell you it's common knowledge to never hold an option to expiry; how ironic, then, that the behavior modeling of Black and Scholes leads to a scenario in which this is not just possible but unavoidable.

The real point of this article isn't about American options or the St. Petersburg Paradox, but the "standard of proof" in our nascent field. Will we accept gross generalizations, and ignore their shortcomings to the point we award the Nobel Prize to the mathematicians who created the above model? Or will we enforce a standard of rigor, look at problems on their face, and imagine creative solutions in real-life conditions?

The question isn't so much about proof as assumptions. It is one of the assumptions of Black and Scholes that leads to the notion of risk-neutral valuation, and hence this odd model of human behavior.

When I ask people what they think the worst assumption in Black-Scholes is, you usually get some swagger about fat tails and normal distributions, or maybe complete markets from those in the know. But here's the bigger one, the much, much bigger one:

Black and Scholes ignore transaction costs.

Is this a fair assumption? In most of economics, it is. Let's examine the case of stocks and bonds. Is it valid to ignore transaction costs when discussing a deal in stocks or bonds? I think so. It's possible that eventually transaction costs for simple assets like this will be driven to zero if humans are more able to find the appropriate counterparty due to greater access to information. And in the meantime, to be honest, it's really a fair assumption. Transaction costs here are low relative to the cost of carry and cost of buying the security, and they don't make much of a difference.

But options? Options aren't like stocks, in my mind. Remember, the basis of the Black-Scholes thinking was the delta-hedged portfolio yielding a risk-free rate. But in order to actually create that kind of portfolio, you need to hire a trader to sit there and manually hedge and re-hedge. In fact, the Black-Scholes differential equation represents a continuously re-hedged portfolio, implying virtually infinite transaction costs. And, since the derivative didn't exist in the first place, the transaction cost also includes writing the option and selling it. For options, transaction costs aren't a negligible factor -- they're everything.

It's clear that quantitative finance professionals have a long hill to climb in order to engender the kind of progress that would legitimize our work to the rest of the scientific community. But in this lack there lies a great amount of opportunity, not just for monetary gain, but for intellectual exploration and the fun of pushing the envelope. Clearly, there's something amiss about the theory underlying the most common of derivative pricing formulas. But it's unclear whether an easy solution exists. It seems there are still many questions to answer, not just mathematical, but behavioral and psychological. There is no doubt that quantitative finance has developed into a scientific discipline, but we may be surprised to find the aftertaste of other fields in papers to come.