Brief Introduction to Quantum Option Pricing
In March 2021, I was selected to be a part of the third cohort of the Quantum Open Source Foundation by my mentor Dr. Maria Kieferova. I have completed my bachelors degree in Economics and am currently doing my masters in Computer Science, so Quantum Finance was both an interesting field to explore and the most natural segue into the world of Quantum Computing. Our original intention was to extend quantum algorithm for pricing European options to American options. Unfortunately, our research has went over the time limit. However, we collected a substantial amount of information on the topic and decided that we can make a brief pedagogical introduction to Quantum Option Pricing. First, we will talk about the field of quantitative finance and introduce the financial instruments called options. We are going to outline how the American options are priced classically and follow by introducing the prevailing quantum approach in pricing European Options.
Quantitative Finance is a field of applied mathematics that aims to create mathematical models of financial markets. Quantitative finance emerged as a field in the 1970s building on the seminal work of Fischer Black, Myron Scholes, and Robert Merton on option pricing theory. There are two distinct branches of the field: the Q World, which concerns itself with derivative pricing and the P World, which focuses on the risk and portfolio management. Quantitative analysts have variety of tools at their disposal, many of which are taken directly out of physics such as Wiener processes.
An option is a type of financial instrument that gives its holder the right but not the obligation to buy (call option) or sell (put option) the underlying asset at a prespecified price on a specified date. For example, Imagine that there is an arrangement such as person A will sell person B 10 oranges for 10 dollars a month from now. In that time market price of oranges might change and 10 oranges will cost 12 dollars, this means that person A will make a profit of 2 dollars from this arrangement. These financial instruments can become very difficult to value, despite having a relatively simple function. One must take into account multiple factors, such as the value of the underlying asset, time until the contract expiration, market volatility, and many other factors when attempting to value an option. Given such a high degree of flexibility in the structuring of an option contract, it is not surprising to see that options have many different types.
We are going to focus our attention on the most common type of options: European and American options. Both of those options are called ‘vanilla’ and are the most fundamental types of an option contract. Majority of the exotic options present some alteration to them. Time evolution of a financial instrument can be modeled using a Wiener process called Geometric Brownian motion. We must make explain the distinction between Regular Brownian motion and Geometric Brownian motion. Perhaps, the quickest way to understand the practicality of such choice is by presenting code and providing a visual example.
We are going to use a penny stock of a Canadian aerospace company called Bombardier(OTCMKTS: BDRBF). When we run our simulations with 1000 paths we can see a crucial difference between Brownian motions. In the case of a Regular Brownian motion by day 50 we observe that several stock paths trend towards negative values. It is impossible for a stock to cost less than $0.00, therefore we must conclude that Regular Brownian motion is unsuitable for financial prognosis. However, when we add a logarithm of the randomly varying quantity with drift, we get a Geometric Brownian motion, which not only makes values less than zero a mathematical impossibility, but also resembles a stock behavior more than a Regular Brownian motion. One can notice how paths in the RBM are uniformly spread, whereas in GBM there are several paths with much more volatile behavior.
To properly understand the mathematical intuition behind the behavior of Brownian motions, one should familiarize themself with Itô calculus. This branch of mathematics applies methods of calculus to stochastic processes, therefore it is uniquely suited for solving the stochastic differential equations. A stochastic process St is said to follow GBM when it satisfies:
In this equation μ represents the percentage drift and σ represents the percentage volatility. Both μ and σ are constants. Using Itô interpretation we achieve an analytical solution. It is also worth noting that one can enhance the realism of this model by dropping the assumption that volatility is constant, this will transform model into a local volatility model.
Once we establish stock price simulation for an American option we can further our study into the methods of classical pricing of these financial instruments. Perhaps, the most well known way of pricing American Options is described in a seminal paper called “Valuing American Options by Simulation: A Simple Least-Squares Approach” by Longstaff and Schwartz of UCLA. In this paper Longstaff and Schwartz framed the problem into an optimal stopping problem. Their approach relies on finding a contingent expected value of not exercising the option early and continuously evaluating a tradeoff between exercising and not exercising the option. This is done by using a cross-sectional information in the simulated paths to find the conditional expectation function. The Least Squares Monte Carlo (LSM) algorithm is an iterative two step process that first uses a backward induction to recursively assign value at every timestep using least squares regression against the market price of the option at that timestep. The value of the option is a sum of the value of the exercise possibilities and the timestep value of exercising the option at that timestep. Once all states ate valued at every timestep, the value of the option is calculated by moving through the timesteps and making a decision on optimal exercise at every step. We can find an implementation of the Longstaff-Schwartz algorithm in an eponymously named Python package by luphord
One can see certain mathematical ingenuity in the LSM approach to solving the optimal stopping problem in the setting of financial markets. On that note, we can conclude our brief introduction to the classical methods in American Option pricing and we are now ready to evaluate the current state of Quantum Computing for finance. For further studies into the topic I recommend a book called “Stochastic Calculus and Financial Applications” by Dr. Steele.
Quantum Finance is still in nascent stages of development and researchers from variety of agencies are looking into ways of how quantum computers can provide a competitive edge. Currently, there exists research into the quantum methods for pricing European options. Notable examples of this research are:
- “Quantum computational finance: Monte Carlo pricing of financial derivatives” by Rebentrost, Gupt and Bromley,
- “Quantum Risk Analysis” by Woerner and Egger,
- “Option Pricing using Quantum Computers” by Dr. Stamatopoulos, Egger, Sun, Zoufal, Iten, Shen and Woerner.
Dr. Rebentrost in his paper points out that quantum speedup is achieved through connection of the desired expectation value to an eigenfrequency of an oscillating quantum system and then use of another quantum degree of freedom (such as another register of qubits) as a probe to extract the eigenfrequency. Crucially, Amplitude Estimation algorithm is utilized to provide a quadratic speed up for the number of repetitions. We must observe that in the case of the algorithm proposed in the Rebentrost paper there exists a quadratic dependence in ε, which is analogous to the classical Monte Carlo methods.
Unsurprisingly, Quantum Finance has also found home in the large financial institutions such as investment banks. Their work is primarily geared towards near term applications of quantum computing. Currently, the state of the art is Noisy Intermediate-Scale Quantum (NISQ) devices, which have many limitations. Nevertheless, NISQ devices, while imperfect, are still useful and banks are looking into the ways how to apply them to variety of financial problems. In 2019 at Q2B Dr. Stamatopoulos gave a talk about how quantum computers are used for option pricing
For the purposes of this article we are going to direct our attention towards the Qiskit Finance application module by IBM, more specifically we are going to look at the quantum computing approach for pricing European Call options. In the Qiskit tutorial we can look at the gradual construction of a circuit specific for European Call.
Above code produces the following quantum circuit:
In this diagram, we see implementation of the Log-normal distribution on the first layer and Linear Amplitude Estimation function on the second layer. In both European call and European put implementations a central algorithm is Iterative Amplitude Estimation. Crucially, this implementation does not rely on Quantum Phase Estimation, instead it bases only on select Grover iterations. Essentially, we are interested in the probability of measuring ∣1⟩ in the last qubit.
We might ask why there is no quantum method for pricing American Options. This question was central to my research project and we are still searching for the right answer. We were able to establish a vision for future research into the topic. Our approach will be a modification of the Rebentrost algorithm. Crucially, algorithm will apply LSM for each path to calculate the payoffs using a quantum computer. Afterwards, expected payoffs for American options can be calculated by applying quantum Monte Carlo on the registers encoding paths.
While this research project has run beyond the time limit imposed by QOSF, I have collected substantial information on the topic and reached a conclusion that these ideas can be tested in the real world. In June, my bid was accepted to the Creative Destruction Lab, where I intend to take my research project to the next level.
