A Quantum Introduction to Quantitative Finance

A physics degree is expensive, both financially and emotionally. In return, graduates leave with a brain packed full of abstract math, sharp logical thinking, years of sleep debt, and, if lucky, only a few grey hairs. But outside research, where are those skills actually worth something? One answer is quantitative finance: using mathematical models, statistics, and computer programming to analyze markets and price assets (Cesa 2017). This work often relies on probability, differential equations, and stochastic calculus: the same mathematical toolkit physicists use to model physical phenomena.

In the 1990s, financial markets began being described by physical models which flourished into a research field called econophysics (Mantegna and Stanley 1999). While these often came from thermodynamics and statistical physics, quantum physics slowly found a spot too. One simple example is Zhang and Huang’s (2010) model, which treats a stock’s price in the Chinese market like a quantum particle trapped in a box.

In quantum mechanics, a particle doesn’t have one definite position until it’s measured. Instead, it’s described by a wave function (ψ), which spreads across all the places the particle could be. The Born rule tells us that taking its modulus squared and integrating over some regions of space gives the probability of finding the particle within that region. Upon particle, the spread-out wavefunction “collapses” to one definite value. Before a trade, a stock’s price, and hence rate of return (ROR), is a spread of possible values, and the trade itself is the measurement that collapses it to one value. Zhang and Huang treat a stock as a particle and its position as its ROR as shown in Equation 1.

The time-independent Schrodinger equation determines the shape of the wavefunction for a given potential energy function. Zhang and Huang adapt it as shown in Equation 2, where m is the “stock inertia a measure of how readily the price of the stock moves. Solving this equation gives the form of the wavefunction for that specific potential. 

The Chinese stock market imposes a price limit rule, capping daily ROR at ±10%. To model this, the potential is set to be zero between two walls; set to be ±10% away from the origin and infinity outside. The expression for the potential is shown in Equation 3.

Within these walls, Equation 2 reads as shown below.

Solving Equation 4 yields the wavefunction in the form:

With the equation for the wavefunction, Equation 1 can be used to find the probability of getting a specific ROR within the confines of the “price well”.

It is important to note that ℏ is fine-tuned for quantum systems, not financial markets, so it would have to be derived from market data. Similarly, the stock inertia is not well defined and would also have to come from market data. Additionally, the time-dependent part was omitted here due to its length and complexity, but the authors use it to incorporate periodic fluctuations in ROR due to the news.

Using the three-dimensional version of the Schrodinger equation and changing the form of the potential are just a few of the many ways to improve this kind of model and adapt it to all kinds of stocks and markets. While this is far from a get-rich-quick model, it serves as a clear example of how quantum models can capture aspects of financial market behaviour. 

References

Cesa, Mauro. 2017. “A Brief History of Quantitative Finance.” Probability, Uncertainty and Quantitative Risk 2 (1): 6. https://doi.org/10.1186/s41546-017-0018-3.

Mantegna, Rosario N., and H. Eugene Stanley. 1999. Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge University Press. https://doi.org/10.1017/CBO9780511755767.

Zhang, Chao, and Lu Huang. 2010. “A Quantum Model for the Stock Market.” Physica A: Statistical Mechanics and Its Applications 389 (24): 5769–75. https://doi.org/10.1016/j.physa.2010.09.008.