Holes in Black-Scholes

Betting on Getting Flattened

It’s a rite of passage these days to diss the Black-Scholes option pricing model. Maybe this is because it’s the best known of the quantitative models that underlie the problems in global finance, although the fact it’s known to fail in unusual market conditions probably doesn’t help. On the flip side, if we know that Black-Scholes doesn’t work at market extremes you’d have thought there’d be ways of making money out of its failures.

Unsurprisingly it turns out that there are people looking at this. It also turns out that only a few have the mental discipline to achieve success because you spend a lot of time losing. You’re throwing pennies under the steamroller and watching others gather them while betting big money on the penny-gathers getting flattened eventually. It’s a discipline requiring courage, intelligence, patience and knowledge of history. So that rules most traders out, then.

Black, Scholes and Merton

Back in the 1970’s Fischer Black and Myron Scholes developed a method of pricing options which, with the typical inventiveness of economists, they called the "Black-Scholes method." Following adaptations by Robert Merton it’s become the standard method of pricing options and earned all three a Nobel Prize. It’s also become the target of many other economists who either wish they’d thought of it first or reckon it shouldn’t ever have been thought of at all.

Options allow us to take a bet on the future price of some asset like a share without actually going to the trouble of buying it. They give the buyer the right, but not the obligation, to buy or sell the underlying asset at a pre-agreed price by a pre-agreed date. They’re a type of derivative – their value is derived from the underlying asset. Derivatives have been given a bad name over the years, being associated with many of the truly horrible things that the nastier investment gnomes are apt to get up to. However, used correctly, they can be hugely beneficial as they allow the risk associated with any asset to be transferred from where it can’t be dealt with to where it can be.

Essentially an option is a trade in uncertainty. If you can’t, for instance, afford to take the risk that your share portfolio will fall below a certain value then you can use options to protect yourself against this. If the portfolio falls below the trigger point you sell the option and pocket the cash. If it doesn’t then the option will expire and the money you spent on it disappears. Used like this options are a form of insurance. Of course, some financiers have found other more ‘creative’ ways of using them.

Volatility, Liquidity and LTCM

Black-Scholes allows traders to plug in values for asset prices, dividends, interest rates, time and volatility to produce a valuation. Volatility – the amount by which an asset’s price may vary over any given period – is often the critical factor. An asset with high volatility is more attractive to buyers of options because there’s a greater probability that, at some point, it’ll be in the money. Conversely assets with low volatility are more likely to be favoured by option sellers.

Black-Scholes came in for criticism due to its involvement in the collapse of Long Term Capital Management (LTCM) in 1998. LTCM imploded when Russia defaulted on its bonds because the model didn’t cope with the “impossible” liquidity crisis that followed: it didn’t handle the extreme conditions that resulted in no one being willing to accept the other side of LTCM’s deals.

Peer under the covers of Black-Scholes and you find our old friend, the Gaussian distribution, assuming that extreme events are impossible instead of just rather unlikely. The unlikely happens all the time in markets, usually because of human behavioural biases which kick in at extreme moments and lead to sustained overshoots in valuations and liquidity.

Taleb on Black-Scholes

Nicolas Taleb is a vehement critic of Black-Scholes as can be seen in "Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula":

"Option traders call the formula they use the “Black-Scholes-Merton” formula without being aware that by some irony, of all the possible options formulas that have been produced in the past century, what is called the Black-Scholes-Merton “formula” ... is the one the furthest away from what they are using. In fact of the formulas written down in a long history it is the only formula that is fragile to jumps and tail events."

Taleb profits from the deficiencies of the model by betting that the markets will go mad, eventually. He may lose small amounts of money most of the time but when things go really wrong he makes a huge killing. However, he’s not the only famed investor to look at the inadequacies of the model.

Buffett on Black-Scholes

Just this year Warren Buffett has shown that Black-Scholes can lead to irrational pricing, even during non-extreme conditions, over very long periods. By looking at the effect including volatility has on option valuations he concludes that this leads to stupid outcomes:

"The ridiculous premium that Black-Scholes dictates in my extreme example is caused by the inclusion of volatility in the formula and by the fact that volatility is determined by how much stocks have moved around in some past period of days, months or years. This metric is simply irrelevant in estimating the probability weighted range of values of American business 100 years from now. (Imagine, if you will, getting a quote every day on a farm from a manic-depressive neighbor and then using the volatility calculated from these changing quotes as an important ingredient in an equation that predicts a probability-weighted range of values for the farm a century from now.)"

Buffett’s suggesting a value investing approach to making money from the indiscriminate application of such models – invest over long enough periods to make short-term human irrationality, aka volatility, an irrelevance. Unfortunately most of us don’t have the lifespans to wait the decades he envisages, but maybe there’s yet another way of profiting from irrational option pricing.

Longshot-Favourite Bias versus Black-Scholes

Back in 1947 Richard Griffith showed that there was an irrational behavioural bias being applied to horseracing odds. The favourite-longshot bias states that favourites are consistently underpriced and longshots consistently overpriced. So a 2-1 favourite is more likely to win than its odds suggest and the 100-1 outsider is less likely to win. This finding has been replicated time and again. In fact you wouldn’t need to add much in the way of skill and judgement to make betting on the favourite consistently a winning strategy. Some other researchers have wondered whether the bias also applies to people investing in options.

Options are either “out of the money” – i.e. trading at a level where they won’t make any money if exercised at the moment – or “in the money” – trading at a level where they are currently worth exercising. Buying an option that’s a long way out of the money is the equivalent of betting on a longshot, buying a deep in the money option, a favourite.

Black-Scholes predicts that calls which are further out of the money will provide greater returns, the opposite of what favourite-longshot bias suggests. The limited evidence so far suggests that it’s the bias which drives the price rather than the model. This would be consistent with irrational investors doing irrational things – exactly the conditions under which Black-Scholes might be expected to fail.

It isn’t yet clear that this research amounts to a way of actually making money but it does, at least, add weight to two conclusions. Firstly, before you put your trust in any automated model make sure you understand it. And secondly, always assume that irrational people will find a way of breaking it, no matter how careful you are.


Related articles: Black Swans, Tsunamis and Cardiac Arrests, Alpha and Beta - Beware Gift Bearing Greeks, Risky Bankers Need Swiss Cheese Not VaR