Derivative Pricing using Quantum Computers
by Juan Adame and Fabio Sanches
If you just want to download the notebook, click here.
2019 saw the birth of one of the all-time greats in the history of financial derivatives: Keith Patrick Gill. You might recognize him by one of his pseudonyms, either Roaring Kitty or DFV. In mid 2019, Gill began buying a derivative called an American call option — a contract that gives the owner the right to buy an asset at a fixed price, called the strike price, before a set date (the expiration date). More specifically, Gill began buying call options on GameStop (GME) shares. By trading in these contracts, Gill was able to turn an initial investment of about fifty thousand dollars into nearly fifty milliondollars in just over a year! This remarkable anecdote naturally raises the questions: what is the price of these contracts, and how does one determine these prices?
The related but more general question of how much any general derivative (not necessarily an American call option) should be worth is a very challenging question to answer. The first satisfactory answer to this problem didn’t come until 1973 thanks to the work of Fisher Black, Myron Scholes, and Robert Merton. Scholes and Merton would go on to win the 1997 Nobel Memorial Prize in Economic Sciences for this work (unfortunately, Black passed away a few years before this prize was awarded). Although the above anecdote shows how derivatives can be used for speculation, in fact derivatives also play a critical role for institutions’ risk management. If we use size as a partial indicator of their significance, according to the Bank of International Settlements, the gross market value of all over-the-counter (OTC) derivatives in the second half of 2021 was about 12 trillion USD.
As it turns out, the price of these derivatives is often boiled down to computing an expected value. Often these expected values cannot be evaluated analytically, so one must resort to Monte Carlo simulations to estimate them. These simulations can quickly become computationally costly even for the world’s most powerful classicalcomputers.
At this point quantumcomputers enter the discussion, and in 2015, Ashley Montanaro published a quantum algorithm to accelerate Monte Carlo methods. This speedup, however, requires quantum computers to be sufficiently mature, with very low error rates. It also requires a large number of qubits, since this advantage only exists when solving complex problems that cause classical methods to struggle.
However, even with more recent improvements in both the hardware and the algorithmic requirements, the largest hardware experiments are still only able to showcase toy models using 3 or 4 qubits. This not only emphasizes the importance of improving qubit quality, but also suggests that a more targeted approach can yield significant improvements.
To highlight the steps involved in applying the quantum version of Monte Carlo to derivative pricing, and the hardware requirements needed for running such an algorithm, we’ve written a demonstration that walks through the steps needed to implement the quantum algorithm for Monte Carlo methods in the context of a simple European call option. You can download the notebook here.
The circuits for the various steps outlined in the demonstration are built using generic gates and techniques. As we’ve written previously, Bleximo is focusing on building application-specific quantum hardware, along with the appropriate software layers needed to successfully run these high-value applications. An algorithm such as this one is our starting point. We then focus on every step — the circuit encoding the algorithm, the compilation techniques, the gate set and processor layout, and the control systems — to build a high-performance integrated solution for these applications.
If you’re interested in learning more, reach out to us!
Originally published at https://medium.com on March 17, 2022.
