The Future of Everything

March 2, 2021

QEF10 – A Quantum Option Pricing Model

We’ve talked about a quantum walk model of asset price changes and a quantum model of supply and demand so let’s put them together into a quantum model of financial option pricing.

Traditional option-pricing models such as Black-Scholes assume that market prices obey a stochastic differential equation with a drift $\mu$ and volatility $\sigma$. Solving the discrete version of this equation yields the classical binomial model, which was first developed by Cox, Ross and Rubinstein in 1979, and exists in different versions.

In the usual model, the steps up and down in log price are given by $u = \left( r - \frac{\sigma^2}{2} \right) \Delta_t+\sigma \sqrt{\Delta_t}$ and $d = \left( r - \frac{\sigma^2}{2} \right) \Delta_t-\sigma \sqrt{\Delta_t}$

This can be interpreted as a random walk with $u = \sigma \sqrt{\Delta_t}$ and $d=-\sigma \sqrt{\Delta_t}$, in combination with a drift term $r \Delta_t$ due to the risk-free interest rate, and an adjustment term $-\frac{\sigma^2}{2} \Delta_t$. We can follow a similar procedure for the quantum walk, to get a price probability distribution, as in the screenshot below from the online app. Instead of having a normal distribution for our stock prices we’re going to end up with a quantum walk with the distinct peaks on either side. In the quantum walk the distribution depends on the initial condition so if you start with a balanced initial condition then you end up with a balanced distribution. We can also skew it a little bit which has been done here to introduce a small degree of bias in order to reflect optimism on the part of the buyer. So this buyer is looking at this asset price into the future and they’re thinking there are two main possibilities. One is it’s going to grow at a certain linear rate like several percent a year into the future, but that’s also balanced with the idea that it can also go down by by a similar amount. On the other hand there’s a more objective view, which is closer to how asset prices actually do tend to behave over time, which assumes they will follow a roughly normal distribution (red line in the figure).

Now, consider a European-style call option with strike price $K$. After $n$ time steps, the expected value of the payoff is $\left[ exp(S_n) - K \right]^+$ (the exponential is required because the stock price $S_n$ is logarithmic). The option price $V_n$ is the expected payoff discounted to time zero, or $V_n = \left< \frac{1}{(1+r)^n} \left[ exp(S_n) - K \right]^+ \right>$.

Since the quantum walk model reflects subjective factors, while the classical model (i.e. the model with decoherence) is a better match for objective reality, we will use the quantum model to represent the buyer (assumed to be more affected by subjective factors) , and the classical model to represent the seller (assumed to be more objective). The reason for this is that sellers are generally going to have a more objective view based on market data, which you can think of as a kind of decoherence effect which will collapse the model down to the classical model. But people buying options are usually doing so because they have some position that it’s going to go up or down by a certain amount.

We therefore have these two different pictures of how asset prices are going to evolve in the future. For either of them we can calculate what would be the fair option price, and we find that the most noticeable difference is for normalized strike prices which are close to one. For these prices the quantum model is saying the price should be higher and therefore the consensus price is going to appear to be a better deal.

You would therefore expect more interest in these options. One way to picture this is through implied volatility, but another way is through volume. A good thing about the quantum model is that, because it’s based on propensity, we can use it to estimate the volume of transactions at different prices and expiration dates. The result is a plot which looks like the one in the screenshot from the video below. It is highest as we would expect for strike prices close to one. This result agrees quite well with actual empirical data for option prices.

The quantum model incorporates a subjective model of human cognition which is useful because it can accommodate various quantum effects such as interference. But perhaps most importantly from the modeling point of view, it gives us a sense of the probability of transactions and therefore the volume. Finally from a purely technical standpoint a big advantage of the quantum walk model is that it is native to a quantum device so it can be run very fast on a quantum computer or anything that can perform a quantum walk.

A quantum walk down Wall Street, The Economist (Nov, 6, 2021).

Orrell D (2021) A quantum walk model of financial options. Wilmott 2021(112): 62-69.

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