Derivative Valuation

Derivatives pricing is one of the most crucial aspects of quantitative finance. The gross market value of derivatives contracts stood at $12.4 trillion (OTC derivatives statistics at end-December 2021 – bis.org), making it one of the most lucrative and challenging problems to simulate and optimize. Although many aspects of derivative pricing can be computed using classical computing, gate-based quantum computers can be an efficient resource when multiple computations are required, due to their ability to parallelize and handle multiple assets. This chapter looks into the theoretical aspects of derivatives pricing and examines its real-time applications through classical computing and gate-based quantum computers.

This chapter addresses the need to explain option pricing from a layman’s point of view. Various aspects of option pricing will be described in an easy-to-understand manner. Different case studies and anecdotes will be incorporated to...

Derivatives pricing – the theoretical aspects

Learning the factors that go into setting derivatives pricing is crucial. An in-depth knowledge of financial product pricing is essential to make sound investing decisions, whether you’re on the buying or selling side of the market. After all, knowing how a product’s attributes interact to produce value is necessary to decide what to offer or bid for a financial product. It is crucial to comprehend the price of financial assets. The capital asset pricing model and its variants, based on discounted cash flow, help establish values for financial assets. However, unlike traditional assets such as stocks and bonds, derivatives have their challenges, but they also have several unexpectedly straightforward properties. In the next section, we will discuss the important concept of money’s time value, which forms the basis for securities pricing.

The time value of money

The value of money fluctuates over time. What...

Machine learning

Machine learning in derivative pricing employs complex algorithms to predict future derivative prices, drawing from a vast dataset of historical trading data. By modeling market dynamics and identifying patterns, it provides more accurate price forecasts than traditional models. This not only reduces financial risk but also optimizes trading strategies. Furthermore, it provides insights into market behavior, assisting in the development of more resilient financial systems.

Geometric Brownian motion

We must model the underlying equities before estimating the price of derivative instruments based on their value. The geometric Brownian motion (GBM), also called the Wiener process, is the method often uses to model the stochastic process of a Brownian motion, driving the future values of an asset. It helps create trajectories that the asset price of the underlying stock may take in the future.

A stochastic or random process, here defined as the time-dependent...

Quantum computing

As we saw in the previous section, in order to estimate the future price of an equity, many iterations over potential future prices need to be run. However, what if we could establish a way to load those potential distributions into quantum states and evaluate them using quantum devices? The following subsection will dig into different ways to load those future price distributions into quantum states using existing solutions for direct loading, such as Qiskit functionalities and adversarial training using PennyLane, which might be better suited for ML tasks due to its differentiable programming approach (similar to TensorFlow or PyTorch in the classical ML domain).

Implementation in Qiskit

As discussed in Chapter 2, Qiskit is one of the most mature quantum computing frameworks available and counts with higher-level modules so that specific applications can be easily translated to the quantum regime. This is the case with Qiskit Finance, which we will explore...

Summary

In this chapter, we explored a complex use case to forecast the future, the future of how markets will evolve, how stock prices will evolve, and so on a non-trivial task indeed. We showed how classical statistics extensively used in the field can be used in a quantum regime. This brings some benefits but also, given its current status and the limitations of the devices themselves, poses other impediments we would need to work around.

Encoding a probability distribution on a gate-based quantum device entails some resolution loss. Probability distributions may need to be truncated, and value ranges will be rendered as the same discrete bin that the future price of an option may be placed. This limitation will probably change when bigger devices are made available so that larger distributions can be encoded. However, it is indeed a limitation we should have present in our minds when adopting these techniques.

Of course, there are benefits, as quantum devices are capable...

Further reading

A quintessential part of what we have discussed in this chapter relates to some of the foundational algorithms in quantum computing. Grover’s algorithm (Jozsa 1999) and QAE (Rao et al. 2020) are not only key contenders for financial use cases but also for numerous applications pertaining to quantum algorithms.

More and more, QML is gaining relevance, as it allows the exploitation of existing data to create those embeddings or dynamics that quantum algorithms often require. Chapter 6 will examine in more detail these techniques. However, for those already knowledgeable about classical generative models such as GANs, variational autoencoders, and neural networks in general, there is plenty of literature that can be found to help their adaptation to the Quantum regime (Lloyd and Weedbrook, 2018). New ways that QNNs can be exploited for financial applications (Tapia et al. 2022) or different perspectives on how a price projection can be tackled constantly appear...

References

• Glasserman, P. (2004). Monte Carlo methods in financial engineering (Vol. 53, pp. xiv+-596). New York: Springer.
• Jozsa, R. (1999). Searching in Grover’s algorithm. arXiv preprint quant-ph/9901021.
• Kingma, D. P., & Ba, J. (2014). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980.
• Lloyd, S., & Weedbrook, C. (2018). Quantum generative adversarial learning. Physical review letters, 121(4), 040502.
• MacBeth, J. D., & Merville, L. J. (1979). An empirical examination of the Black-Scholes call option pricing model. The journal of finance, 34(5), 1173-1186.
• Moguel, E., Berrocal, J., García-Alonso, J., & Murillo, J. M. (2020, October). A Roadmap for Quantum Software Engineering: Applying the Lessons Learned from the Classics. In Q-SET@ QCE (pp. 5-13).
• Singh, P. (2021). FQTSFM: A fuzzy-quantum time series forecasting model. Information Sciences, 566, 57-79.
• Stojkoski, V., Sandev, T., Basnarkov, L....