QEF14 – Is the volatility smile real or imaginary?

The answer to this question, according to quantum finance, is both.

The volatility smile refers to the phenomenon in options trading where the implied volatility has a smile-like shape as a function of strike price (see Figure 1 below for an example). The volatility is lowest for at-the-money options where the strike price is the same (after discounting) as the current price, and is higher for out-of-the-money options.

The volatility smile is usually viewed as something of a conundrum, since it seems to violate the idea - which forms the basis of quantitative finance - that prices undergo a random walk with a volatility that can be described by a single number. The Black-Scholes model for option pricing, for example, assumes that prices follow a lognormal distribution with constant volatility.

This assumption is so widespread that in quantitative finance, volatility is usually expressed in terms of inverse square-root of time, because this happens to hold for a random walk where variance (i.e. volatility squared) increases linearly with time.

The quantum model differs from this classical random walk in a number of respects. In the classical model, a normal distribution is used to describe the possible range of prices at a particular time. This is an essentially static picture, which does not reflect the fact that price is the result of investors buying and selling, and assumes that markets are balanced. The quantum oscillator model replaces the normal distribution with a complex wave function, which rotates around the real axis. Since this is a dynamic model - the quantum version of a spring - it can be used to model this turnover process (it literally turns over in the imaginary plane), and capture what happens when markets are out of balance.

For example, a large order will perturb the price by an amount which depends on the square-root of the relative order size - which is just the well-known square-root law of market impact. However another effect is that the wave function also distorts in shape, leading to higher volatility.

In other words, market imbalance between buyers and sellers affects both price and volatility - so price and volatility are correlated, in a manner which happens to match the volatility smile seen in options trading.

Now, the volatility smile usually refers to the volatility that is implied by the price paid for options, so you could argue that it is based on traders' subjective projections about the future, and is just a figment of their imagination.

But an easy way to test the hypothesis is to plot volatility versus price change for different time periods. For a lognormal distribution there is no correlation (and it is actually a bit tricky to produce an artificial data set which gives the right properties, but you can do it with the quantum model) but historical market data follows the predicted volatility equation.

Figure 1. Right panel shows volatility as measured by the VIX index versus monthly price change for the S&P 500 January 1992 to August 2022. Left panel is same for an artificial lognormal distribution. Red line shows average volatility, black line is the theoretical smile curve. The lognormal distribution shows no smile, while the market data smile is well approximated by the quantum formula.

Of course what we really care about here is the effect on option pricing. As mentioned, implied volatility usually refers to the volatility implied by the cost of options. But if the model is correct, then the cost should equal the expected payout. So another test is to ask, what volatility is implied - not by traders' projections - but by after-the-fact market outcomes? In other words, what is the correct volatility to use in the Black-Scholes model so that the option cost calculated from the formula equals the expected payout, as determined from historical data?

The Black-Scholes model calculates option prices by assuming that the price distribution is lognormal. If the theory is correct, then the market implied volatility" should be just the usual volatility, which does not depend on price change. The average option cost using this volatility should therefore equal the average payout (with of course an allowance for noise).

However the experiments show that while the Black-Scholes model does work for a lognormal data set - i.e. the model option cost equals the average payout - it produces systematic errors when historical data are used. It therefore fails a basic calibration test for a predictive model. Results can be improved by making volatility dependent on price change according to the smile equation.

Figure 2. Left panel shows Black-Scholes model cost (blue line) for 1-month straddle options, versus average payout for a lognormal distribution. Right panel shows the same for S&P 500 historical data. The red line shows the quantum model where volatility follows the smile curve. The Black-Scholes model is well calibrated for the artificial lognormal data, as expected, but the quantum model is a better fit for the market data.

So to summarize, the volatility smile is definitely real - even if the oscillations which produce it in the model take place in the imaginary plane.

For details, see the SSRN discussion papers:

A quantum oscillator model of stock markets

Keep on smiling: Market imbalance, option pricing, and the volatility smile

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