Financial Payoff Functions and Potentials

Patrick Rebentrost, Brajesh Gupt, Thomas R. Bromley

This work presents a quantum algorithm for the Monte Carlo pricing of financial derivatives. We show how the relevant probability distributions can be prepared in quantum superposition, the payoff functions can be implemented via quantum circuits, and the price of financial derivatives can be extracted via quantum measurements. We show how the amplitude estimation algorithm can be applied to achieve a quadratic quantum speedup in the number of steps required to obtain an estimate for the price with high confidence. This work provides a starting point for further research at the interface of quantum computing and finance.

Osama M. Raisuddin, Suvranu De

K Soveet Kumar Prusty, Kartikeyan Premkumar, Shweta S. Aladakatti +1 more

Quantum Amplitude Estimation (QAE) is an attractive financial derivative pricing algorithm with the potential for a quadratic speedup over classical Monte Carlo approaches. Yet, quantum advantage will only be found if not just fault-tolerant hardware but also an accurate representation of the financial problem on a quantum circuit is achieved. This paper offers a reproducible, diagnostic examination of the default Qiskit Finance application to European option pricing. We present a critical diagnostic marker: the classical discretized price, which is the classically computed expected payoff on the quantum model's actual grid. Our convergence analysis of low-qubit models (4, 5, and 6 uncertainty qubits) shows that for this particular configuration, representation error, which arises due to imprecise trigonometric encoding of the option payoff function, is the prevalent source of error, much larger compared to the discretization error. In a 6-qubit model, the representation error was 0.17699 USD, which is an order of magnitude larger than the discretization error of -0.01200 USD. We illustrate how a different Piecewise Linear (PWL) encoding can successfully remove this representation error, but at the expense of more circuit resources. Our results, for the vanilla, low-qubit Qiskit Finance circuits we used, indicate that the creation of high-fidelity, resource-conscious problem encodings is a pressing and essential problem on the road to applicable quantum advantage in finance.

Martin Pieper

Nicholas Bornman

The financial sector is anticipated to be one of the first industries to benefit from the increased computational power of quantum computers, in areas such as portfolio optimisation and risk management to financial derivative pricing. Financial mathematics, and derivative pricing in particular, are not areas quantum physicists are traditionally trained in despite the fact that they often have the raw technical skills needed to understand such topics. On the other hand, most quantum algorithms have largely focused on qubits, which are comprised of two discrete states, as the information carriers. However, discrete higher-dimensional qudits, in addition to possibly possessing increased noise robustness and allowing for novel error correction protocols in certain hardware implementations, also have logarithmically greater information storage and processing capacity. In the current NISQ era of quantum computing, a wide array of hardware paradigms are still being studied and any potential advantage a platform offers is worth exploring. Here we introduce the basic concepts behind financial derivatives for the unfamiliar enthusiast as well as outline in great detail the quantum algorithm routines needed to price a European option, the simplest derivative. This is done within the context of a quantum computer comprised of qudits and employing the natural higher-dimensional analogue of a qubit-based pricing algorithm with its various subroutines. From these pieces, one should relatively easily be able to tailor the scheme to more complex, realistic financial derivatives. Finally, the entire stack is numerically simulated with the results demonstrating how the qudit-based scheme's payoff quickly approaches that of both a similarly-resourced classical computer as well as the true payoff, within error, for a modest increase in qudit dimension.