Quantum advantage for multi-option portfolio pricing and valuation adjustments

A critical problem in the financial world deals with the management of risk, from regulatory risk to portfolio risk. Many such problems involve the analysis of securities modelled by complex dynamics that cannot be captured analytically, and hence rely on numerical techniques that simulate the stochastic nature of the underlying variables. These techniques may be computationally difficult or demanding. Hence, improving these methods offers a variety of opportunities for quantum algorithms. In this work, we study the problem of Credit Valuation Adjustments (CVAs), which has significant importance in the valuation of derivative portfolios. As a variant, we also consider the problem of pricing a portfolio of many different financial options. We employ quantum algorithms that accelerate statistical sampling processes to approximate the price of the multi-option portfolio and the CVA under different measures of dispersion. Our resulting algorithms are based on enhancing the quantum Monte Carlo (QMC) algorithms by Montanaro with an unbiased version of quantum amplitude estimation. We analyse the conditions for a quantum advantage and demonstrate the application of QMC techniques on the CVA estimation when particular bounds for the variance of the CVA are known. As a concrete example, we derive the variance bound for CVA in the Cox-Ross-Rubinstein model. Moreover, we perform numerical experiments to demonstrate the performance of our quantum algorithms and discuss the potential for practical quantum advantage. Although our quantum algorithms are designed for finite event spaces, they serve as good approximations of the continuous case when the number of qubits is sufficiently large.