## The quantum joke

The volatility smile can seem illogical from the perspective of classical finance, but applying quantum logic allows us to see that it accurately describes a genuine aspect of the market

If there is a single idea which sums up the field of mainstream economics, it is that prices are drawn by the forces of supply and demand – aka the invisible hand – to a stable equilibrium, at which the price accurately reflects intrinsic value. A logical consequence is that price changes must be due to extrinsic effects, such as news which affects the value of a stock. Because such events are random, prices should perform what the statistician Karl Pearson called a random walk.

Pearson illustrated the problem in a 1905 paper with an example of a drunken man, who takes a step in one direction, then another step in a different direction, and so on. The expected distance travelled is seen to grow with the square root of time – but “the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point!”

The same idea had already been used by the French mathematician Louis Bachelier in his 1900 dissertation Theorié de la Spéculation, to argue that an investor’s expected profit or loss was zero. Prices move randomly up and down, but the best forecast for an asset’s future price is its current price. The typical size of the step was described by a parameter which he called the market’s “nervousness” and is now known as the volatility.

In the 1950s Bachelier’s thesis was dusted off and improved on by economists who were trying to model stock prices, and in particular compute the correct price of financial options (those instruments which give one the right to buy or sell a security in the future at a set price, known as the strike). The goal was realised in 1973 by the Black-Scholes option pricing model. Today, versions of this model are used to price options and other derivatives in global financial markets – and volatility has become the magic number used to evaluate financial risk.

**From ear to ear**

A key assumption of the Black-Scholes model, when it was derived 50 years ago, was that the volatility could be treated as constant. In practice though, traders assigned prices to options, which did not perfectly agree with the theoretical prices, but departed from them in a predictable way. Options which only paid off in the case of extreme price changes attracted a higher price. Since the option price depended on volatility, another way to look at this was that markets were assigning higher volatility numbers (known as implied volatility) to these options.

Volatility has become the magic number used to evaluate financial risk

The result of all this was that, if you plotted implied volatility against strike price (the price at which the option can be exercised), you got a curve that resembled a smile. Economists have long debated the reasons for the volatility smile, since it seems illogical from the perspective of classical finance. However, it makes more sense if you use a different kind of logic. The quantum sort.

In quantum economics (which is based on quantum probability, not quantum physics), the price of something like a stock is fundamentally indeterminate, so there is a base level of uncertainty (the financial version of the uncertainty principle).

Price perturbations occur due to transactions, which measure the price but also affect it. And in the quantum model, the uncertainty measured by volatility is not a constant, as in the traditional random walk model, but varies depending on the state of the market.

When markets are out of balance, as when there are more buyers than sellers, then the price is affected – in this case it goes up – but the volatility increases as well. And if you plot volatility over a period such as a month, against price change over the same period, then you get exactly the same volatility smile (though somewhat steeper). In other words, the smile is not some kind of illogical anomaly, or artefact caused by trader behaviour. It is a reflection of a real phenomenon.

**Mispricing risk**

The consequences of this for financial markets are no laughing matter. For example, the main index used to measure market volatility is the VIX index, which weights the implied volatilities of a range of options on the S&P 500 stock market index to arrive at a single number for the implied volatility. However, in reality there is no single number, because implied volatility is described by a curve.

The formula used to derive the VIX makes sense in the random walk world of economists’ imagination, but the actual world seems more in tune with quantum logic. The result is that risk is consistently mispriced by conventional models.

Economists have long known that the market is smiling, but because of their rigid obsession with classical ideas such as equilibrium, they didn’t know why. Quantum models, it seems, are in on the joke.