Quantum computational finance for martingale asset pricing in incomplete markets

Abstract

A derivative is a financial asset whose future payoff is a function of underlying assets. Pricing a financial derivative involves setting up a market model, finding a martingale ("fair game") probability measure for the model from the existing asset prices, and using that probability measure to price the derivative. When the number of underlying assets and/or the number of market outcomes in the model is large, pricing can be computationally demanding. In this work, we first formulate the pricing problem in a linear algebra setting, including the realistic setting of incomplete markets where derivatives do not have a unique price. We show that the problem can be solved with a variety of quantum techniques such as quantum linear programming and the quantum linear systems algorithm. While in previous works the martingale measure is assumed to be given, here it is extracted from market variables akin to bootstrapping, a common practice among financial institutions. We discuss the quantum zero-sum game algorithm and the quantum simplex algorithm as viable subroutines. For quantum linear systems solvers, we formalize a new market assumption milder than market completeness, which allows the potential for large speedups. Towards prototype use cases, we conduct numerical experiments motivated by the Black-Scholes-Merton model.

Figure 1

Range of admissible prices of the derivative while changing different parameters. The parameters are the drift $\mu$ in the top-left panel, the volatility $\sigma$ in the top-right panel, the strike price Z in the bottom-left panel and the stock price $\mathrm{\Pi }$ in the bottom-right panel.

Figure 2

Changing the regularization parameter. (Left panel) The Radon–Nikodym derivative obtained analytically is compared to the solution vector obtained from solving the linear program. (Right panel) The regularization parameter constrains the solution space and hence limits the admissible prices of the derivative.