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.
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.
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.
The prices of Libor options are obtained for the quantum finance Libor Market Model. The option prices show new features of the Libor Market Model arising from the fact that, in the quantum finance formulation, all the different Libor payments are coupled and (imperfectly) correlated. Black's caplet formula for quantum finance is given an exact derivation. The coupon and zero coupon bond options as well as the Libor European and Asian swaptions are derived for the quantum finance Libor Market Model. The approximate Libor option prices are derived using the volatility expansion developed in Section 3.14. The BGM–Jamshidian expression for the Libor interest rate caplet and swaption prices is obtained as the limiting case when all the Libors are exactly correlated. Introduction The Libor option prices are obtained from the Libor zero coupon bonds B L (t, T) – obtained from the Libor ZCYC curve Z L (t, T) discussed in Section 7.9 – and the benchmark three-month Libor L ( t, T ). For notational convenience, Libor zero coupon bonds B L (t, T) will be denoted by B ( t, T ). All the options are defined to mature at future calendar time T 0 , with present time given by t 0 = T − k ; the notation of present being denoted by t 0 is used to simplify the notation. It is natural for these options to choose B ( t, T 0 ) as the forward bond numeraire. In other words, the forward bond numeraire is B ( t, T I +1 ) with I = −1 and Libor drift is calculated for this numeraire. Libor calendar and future time are shown in Figure 6.1 and Libor times t 0 = T − k , T 0 , and T n are shown in Figure 8.1.
Options are financial instruments whose payoffs are determined by the value of an underlying asset. These underlying assets are typically stocks, interest rates, or exchange rates, but can also include commodities such as energy or agricultural products. Options serve diverse purposes, including hedging against price volatility and speculating on market movements. Additionally, they play a critical role in assessing certain positions on the balance sheets of insurance companies, particularly in the context of solvency capital requirement calculations. As an option’s payoff lies in the future and is stochastic in nature, finding a fair price today is a non-trivial task. The focus of this work is finding quantum algorithms for calculating these prices and compare them to algorithms designed for classical computers. This work follows two different directions on how to improve on classical methods for option pricing. The first direction is improving on Monte-Carlo (MC) methods and the second one is adapting Fourier based pricing algorithms. The Amplitude Estimation (AE) algorithm promises a quadratic speedup compared to classical MC. This fact makes this algorithm an interesting candidate for improvements by quantum algorithms. It was shown in the literature, that simple options with very crude discretization of the underlying stochastic models, can be implemented on a quantum device and evaluated with AE. One big challenge is the efficient implementation of the option’s payoff profile. To solve this problem colleagues and I have introduced a multi-objective genetic algorithm to automatically find well performing and efficient circuits to implement the payoff of non-path-dependent European options on one or more underlyings. Another strategy for pricing options is based on the fact, that in many models the characteristic function of the underlying is numerically more tractable than its distribution function. There are methods that make use of this fact and the performance of the Fast Fourier Transform (FFT) to efficiently calculate prices for European call options for many strikes in one go. The quantum version of the FFT, the Quantum Fourier Transform (QFT), is in some sense, even faster than its classical counterpart. We have adapted the existing classical method to make use of QFT, to benefit from its performance. Finally, the different strategies are benchmarked and compared regarding their accuracy, their potential for scaling and their adaptability to different styles of derivatives and models. The result of this comparison are requirements on the quantum hardware for the methods to be viable as well as use cases where each method has its benefits over the others.
Since the beginning of the 21st century, the scale and complexity of financial derivatives have increased significantly. Meanwhile, the pricing models of derivatives have become increasingly complex, and have involved larger-scale data, posing unprecedented challenges to algorithms and computing speed. Although classical computing can still solve the pricing problem of some complex derivatives by adopting the big data simulation algorithms, its speed has become more and more difficult to meet the needs of real-time computing. In recent years, the emerging quantum computing makes full use of the advantages of quantum superposition to improve the computing speed and accuracy. It has been proven to be an efficient computing mode in big data simulation. This paper takes the pricing of European options and defaultable bonds as examples to explore the application of Quantum Monte Carlo Simulation of complicated derivative pricing in big data. The innovative comprehensive method of pricing derivatives by Quantum Monte Carlo Simulation provides new development for the application of quantum computing in the pricing of complicated financial derivatives (taking defaultable bond as example) with big data simulation. In the case of defaultable bond pricing, the Quantum Amplitude Estimation has been employed to speed up the Monte Carlo Simulation and the price of it has been successfully obtained with at least the same accuracy, which demonstrates that the Quantum Monte Carlo Simulation is a prospective approach to price the complicated derivatives based on the big data since the quantum bits has dramatically increased.
Pricing financial derivatives, in particular European-style options at different time-maturities and strikes, is a relevant financial problem. The dynamics describing the price of vanilla options when constant volatilities and interest rates are assumed, is governed by the Black-Scholes model, a linear parabolic partial differential equation with terminal value given by the pay-off of the option contract and no additional boundary conditions. Here, we present a digital quantum algorithm to solve Black-Scholes equation on a quantum computer for a wide range of relevant financial parameters by mapping it to the Schrodinger equation. The non-Hermitian nature of the resulting Hamiltonian is solved by embedding the dynamics into an enlarged Hilbert space, which makes use of only one additional ancillary qubit. Moreover, we employ a second ancillary qubit to transform initial condition into periodic boundary conditions, which substantially improves the stability and performance of the protocol. This algorithm shows a feasible approach for pricing financial derivatives on a digital quantum computer based on Hamiltonian simulation, technique which differs from those based on Monte Carlo simulations to solve the stochastic counterpart of the Black Scholes equation. Our algorithm remarkably provides an exponential speedup since the terms in the Hamiltonian can be truncated by a polynomial number of interactions while keeping the error bounded. We report expected accuracy levels comparable to classical numerical algorithms by using 10 qubits and 94 entangling gates on a fault-tolerant quantum computer, and an expected success probability of the post-selection procedure due to the embedding protocol above 60\%.
In a global derivatives market with notional values in the hundreds of trillions of dollars, the accuracy and efficiency of pricing models are of fundamental importance, with direct implications for risk management, capital allocation, and regulatory compliance. In this work, we employ the Black-Scholes-Merton (BSM) framework not as an end in itself, but as a controlled benchmark environment in which to rigorously assess the capabilities of quantum machine learning methods.
We propose a fully quantum approach to option pricing based on Quantum Neural Networks (QNNs), and, to the best of our knowledge, present one of the first implementations of such a methodology on currently available quantum hardware. Specifically, we investigate whether QNNs, by exploiting the geometric structure of Hilbert space, can effectively approximate option pricing functions.
Our implementation utilizes a compact 2-qubit QNN architecture evaluated across multiple state-of-the-art quantum processors, including IBM Fez, IQM Garnet, IonQ Forte, and Rigetti Ankaa-3. This cross-platform study reveals distinct hardware-dependent performance characteristics while demonstrating that accurate pricing approximations can be achieved consistently across different devices despite the constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware.
The results provide empirical evidence that QNN-based approaches constitute a viable framework for derivative pricing. While the analysis is conducted within the BSM setting, the broader significance lies in the potential extension of these methods to more realistic and computationally demanding models, including local volatility, stochastic volatility, and interest rate frameworks commonly used in practice.