# Towards Pricing Financial Derivatives with an IBM Quantum Computer

## TL;DR

This paper demonstrates how quantum principal component analysis on IBM quantum computers can reduce complexity in simulating interest rate derivatives, paving the way for practical quantum finance applications.

## Contribution

It introduces a quantum approach to efficiently simulate the Heath-Jarrow-Morton model using principal component analysis on a 5-qubit quantum computer.

## Key findings

- Quantum PCA reduces noisy factors in interest rate models.
- Experimental estimation of correlation matrices on IBMQX2.
- First step towards quantum algorithms for derivative pricing.

## Abstract

Pricing interest-rate financial derivatives is a major problem in finance, in which it is crucial to accurately reproduce the time-evolution of interest rates. Several stochastic dynamics have been proposed in the literature to model either the instantaneous interest rate or the instantaneous forward rate. A successful approach to model the latter is the celebrated Heath-Jarrow-Morton framework, in which its dynamics is entirely specified by volatility factors. On its multifactor version, this model considers several noisy components to capture at best the dynamics of several time-maturing forward rates. However, as no general analytical solution is available, there is a trade-off between the number of noisy factors considered and the computational time to perform a numerical simulation. Here, we employ the quantum principal component analysis to reduce the number of noisy factors required to accurately simulate the time evolution of several time-maturing forward rates. The principal components are experimentally estimated with the $5$-qubit IBMQX2 quantum computer for $2\times 2$ and $3\times 3$ cross-correlation matrices, which are based on historical data for two and three time-maturing forward rates. This manuscript is a first step towards the design of a general quantum algorithm to fully simulate on quantum computers the Heath-Jarrow-Morton model for pricing interest-rate financial derivatives. It shows indeed that practical applications of quantum computers in finance will be achievable in the near future.