Quantum Random Walk – Could this be a model of Financial Assets?
December 27, 2008
It is more than three decades now that we have been using the model of geometric Brownian motion for stochastic modelling asset prices and pricing financial derivatives and about a century since Louis Bachelier first used the model of arithmetic Brownian motion to model stock prices. This is the classical Random Walk hypothesis that has characterized the financial derivatives and credit derivatives modelling and valuation in the banks and financial institutions.
On 15th September 2008 as Lehman Brothers, the 150 years old venerable Wall Street investment bank, was filing for bankruptcy I sat before a group of market risk quants at WestLB bank in Tokyo running a Monte Carlo simulation code to value some exotic equity structured notes. By mistake, one of the trainees had forgotten to multiply the daily volatility by square root of 252 days to convert it into annualized volatility. He had simply multiplied it by 252 days (forgotten the square root part) and gotten an absurdly high annualized volatility of around 100%.
It was then that Tanaka san had mentioned, jokingly of course, about the “Lehman Walk”. He had watched enough Lehman bankers at the Mori Tower lobby to conceptualize a walk which could be called a “Lehman Walk”. We have written an article about this topic on this site: "Random Walk" or the "Lehman Walk". And while the Monte Carlo code kept running to complete a million simulations, we took a wild excursion in the abstract space and discussed how would such a walk look like?
One of the ideas that we bounced around was: what if the volatility of an asset did not scale with the square root of time but with time itself? What if the random walk did not converge to any limiting distributions? What, if via some tunneling mechanism an asset reaches a state, say, an absorbing state, at an exponential speed?
Unwittingly, and by accident, we had entered into the space of what is known as a Quantum Random Walk. A quantum random walk is the equivalent of a classical random walk (Brownian motion without a drift) in the quantum mechanical space of sub-atomic particles. Now this is a lot of physics talk but that is what modern finance theory is all about.
In 1965 Richard Feynman had introduced a random walk process that was predicated on Dirac equation in the one dimensional case. However, to this day, this remains a new theory, even within the sphere of quantum mechanics and quantum computing. But a quantum random walk displays certain very interesting characteristics and maybe, just maybe, it is better suited to describe the properties of asset price diffusion process in the financial markets.
Asset prices, such as stocks, currencies, commodities, interest rates, etc. are modeled using a geometric or an arithmetic Brownian motion – which are "classical random walks" – very similar to the ones used in statistical mechanics whereby the Laplacian operator is the classical diffusion operator (diffusion equation) which produces a characteristic Brownian motion signature where the distance, d is proportional to the square root of time:.
When the classical random walk hypothesis is applied to financial markets and asset prices, this translates into a theorem that states that volatility of an asset price scales with the square root of time. If one day volatility of a stock is 1% then one year (252 days) volatility will be . So the classical diffusion process – classical random walk – produces the following signature: .
In a relativistic quantum mechanics the Dirac equation uses a different diffusion operator that produces a signature of distance, d, proportional to time:. Due to this property, where the distance traveled by the random variable in a quantum random walk is proportional to the time, instead of the square root of time as in a classical random walk, for particular absorbing points quantum walks could become significantly faster than their classical counterpart.
Applied to financial assets this could mean that volatility scales with time, instead of square root of time as in a classical walk. Thus . In this scheme of things a 1% daily volatility would translate into a 252% annualized volatility ().
A key property of a quantum random walk is that the standard deviation of the random variable is whereas in a classical random walk, the current model being applied to financial assets, the standard deviation is of order . Therefore, a quantum random walk propagates quadratically faster. Also, the position probability distribution of a quantum random walk depends on the initial quantum state as opposed to a classical random walk.
In a quantum random walk, a walker – say, an asset such as a stock or a currency pair – may be in a “superposition” of positions and because of quantum interference the walk may spread significantly faster, or even slower at times, than its classical equivalents. Further, quantum random walks are unitary and reversible and in a single dimension the hitting time is quadratically faster; on an n-dimensional hypercube the speed up in the hitting time is exponential..
Could this then be the appropriate model of financial assets? Should the entire stochastic framework of asset pricing for financial derivatives then be predicated on Quantum Random Walk rather than a Classical Random Walk? Perhaps, then it would be easier to explain a Lehman bankruptcy, a Madoff blowup or any of those seemingly Black Swan events?
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