A Quantum Mechanics for Interest Rate Derivatives Markets

Belal Ehsan Baaquie, Pan Tang

Tang, Hao, Wu, Wenxun, Xian‐Min Jin

The LIBOR Market Model (LMM) is a widely used model for pricing interest rate derivatives. While the Black-Scholes model is well-known for pricing stock derivatives such as stock options, a larger portion of derivatives are based on interest rates instead of stocks. Pricing interest rate derivatives used to be challenging, as their previous models employed either the instantaneous interest or forward rate that could not be directly observed in the market. This has been much improved since LMM was raised, as it uses directly observable interbank offered rates and is expected to be more precise. Recently, quantum computing has been used to speed up option pricing tasks, but rarely on structured interest rate derivatives. Given the size of the interest rate derivatives market and the widespread use of LMM, we employ quantum computing to price an interest rate derivative, caps, based on the LMM. As caps pricing relates to path-dependent Monte Carlo iterations for different tenors, which is common for many complex structured derivatives, we developed our hybrid classical-quantum approach that applies the quantum amplitude estimation algorithm to estimate the expectation for the last tenor. We show that our hybrid approach still shows better convergence than pure classical Monte Carlo methods, providing a useful case study for quantum computing with a greater diversity of derivatives.

Belal Ehsan Baaquie, Yang Cao

Paul McCloud

Belal Ehsan Baaquie

Interest rate instruments form a major component of the capital markets. The Libor market model (LMM) is the finance industry standard interest rate model for both Libor and Euribor, which are the most important interest rates. The quantum finance formulation of the Libor market model is given in this paper and leads to a key generalization: all the Libors, for different future times, are imperfectly correlated. A key difference between a forward interest rate model and the LMM lies in the fact that the LMM is calibrated directly from the observed market interest rates. The short distance Wilson expansion [Phys. Rev. 179, 1499 (1969)] of a Gaussian quantum field is shown to provide the generalization of Ito calculus; in particular, the Wilson expansion of the Gaussian quantum field A(t,x) driving the Libors yields a derivation of the Libor drift term that incorporates imperfect correlations of the different Libors. The logarithm of Libor phi(t,x) is defined and provides an efficient and compact representation of the quantum field theory of the Libor market model. The Lagrangian and Feynman path integrals of the Libor market model of interest rates are obtained, as well as a derivation given by its Hamiltonian. The Hamiltonian formulation of the martingale condition provides an exact solution for the nonlinear drift of the Libor market model. The quantum finance formulation of the LMM is shown to reduce to the industry standard Bruce-Gatarek-Musiela-Jamshidian model when the forward interest rates are taken to be exactly correlated.

Emmanuel E. Haven