Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

Miyamoto, Koichi

doi:10.1140/epjqt/s40507-022-00124-3

Cited by 10 publications

(7 citation statements)

References 60 publications

(141 reference statements)

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“…We also have the CI of a, which contains the true value of a with high probability. Next, for some k ∈ N, we repeat generating G k | and the measurement on it, and from the measurement outcomes we get an estimate of a k := sin 2 ((2k + 1)θ a ) (7) and then that of a. Here, due to the periodicity of a k as the function of a, the 2k + 1 different values in [0, 1] can be the maximum likelihood estimates of a.…”

Section: Iterative Quantum Amplitude Estimationmentioning

confidence: 99%

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On the bias in iterative quantum amplitude estimation

Miyamoto

2024

EPJ Quantum Technol.

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Quantum amplitude estimation (QAE) is a pivotal quantum algorithm to estimate the squared amplitude a of the target basis state in a quantum state $|{\Phi}\rangle $ | Φ 〉 . Various improvements on the original quantum phase estimation-based QAE have been proposed for resource reduction. One of such improved versions is iterative quantum amplitude estimation (IQAE), which outputs an estimate â of a through the iterated rounds of the measurements on the quantum states like $G^{k}|{\Phi}\rangle $ G k | Φ 〉 , with the number k of operations of the Grover operator G (the Grover number) and the shot number determined adaptively. This paper investigates the bias in IQAE. Through the numerical experiments to simulate IQAE, we reveal that the estimate by IQAE is biased and the bias is enhanced for some specific values of a. We see that the termination criterion in IQAE that the estimated accuracy of â falls below the threshold is a source of the bias. Besides, we observe that $k_{\mathrm{fin}}$ k fin , the Grover number in the final round, and $f_{\mathrm{fin}}$ f fin , a quantity affecting the probability distribution of measurement outcomes in the final round, are the key factors to determine the bias, and the bias enhancement for specific values of a is due to the skewed distribution of $(k_{\mathrm{fin}},f_{\mathrm{fin}})$ ( k fin , f fin ) . We also present a bias mitigation method: just re-executing the final round with the Grover number and the shot number fixed.

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Section: Iterative Quantum Amplitude Estimationmentioning

confidence: 99%

“…For example, it is used in the quantum algorithm for Monte Carlo integration (QMCI) [3], which estimates the expectation of a random variable quadratically faster than the classical counterpart. Furthermore, QMCI has many applications in industry, e.g., derivative pricing [4][5][6][7][8][9] in finance.…”

Section: Introductionmentioning

confidence: 99%

On the bias in iterative quantum amplitude estimation

Miyamoto

2024

EPJ Quantum Technol.

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“…• Pricing of standard ('vanilla'), path-dependent (e.g. barrier and Asian) and multiasset options in a Black-Scholes [9] and local volatility [28] framework [4,18,33,38,51,54,64,72,73,75,76,80] • Pricing of options under a stochastic volatility [16] and jump-diffusion process [95] • Pricing of American-style options [27,62] • Pricing of interest-rate derivatives with a multi-factor model [58,84] • Pricing of collateralised debt obligations (CDOs) [83] Risk measurement. In a financial context, 'risk' usually refers to an adverse event associated with a (financial) loss.…”

Section: Research Landscape In Quantitative Financementioning

confidence: 99%

Quantum computing for financial risk measurement

Wilkens¹,

Moorhouse

2023

Quantum Inf Process

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Quantum computing allows a significant speed-up over traditional CPU- and GPU-based algorithms when applied to particular mathematical challenges such as optimisation and simulation. Despite promising advances and extensive research in hard- and software developments, currently available quantum systems are still largely limited in their capability. In line with this, practical applications in quantitative finance are still in their infancy. This paper analyses requirements and concrete approaches for the application to risk management in a financial institution. On the examples of Value-at-Risk for market risk and Potential Future Exposure for counterparty credit risk, the main contribution lies in going beyond textbook illustrations and instead exploring must-have model features and their quantum implementations. While conceptual solutions and small-scale circuits are feasible at this stage, the leap needed for real-life applications is still significant. In order to build a usable risk measurement system, the hardware capacity—measured in number of qubits—would need to increase by several magnitudes from their current value of about $$10^2$$ 10 2 . Quantum noise poses an additional challenge, and research into its control and mitigation would need to advance in order to render risk measurement applications deployable in practice. Overall, given the maturity of established classical simulation-based approaches that allow risk computations in reasonable time and with sufficient accuracy, the business case for a move to quantum solutions is not very strong at this point.

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“…In recent years, applications of quantum computers have been discussed in financial engineering. Specifically, the applications include portfolio optimization [1-3], risk measurement [4][5][6][7][8], and derivative pricing [9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Comprehensive reviews of these topics are presented in Refs.…”

Section: Introductionmentioning

confidence: 99%

Pricing multi-asset derivatives by variational quantum algorithms

Kubo¹,

Miyamoto²,

Mitarai³

et al. 2022

Preprint

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Pricing a multi-asset derivative is an important problem in financial engineering, both theoretically and practically. Although it is suitable to numerically solve partial differential equations to calculate the prices of certain types of derivatives, the computational complexity increases exponentially as the number of underlying assets increases in some classical methods, such as the finite difference method. Therefore, there are efforts to reduce the computational complexity by using quantum computation. However, when solving with naive quantum algorithms, the target derivative price is embedded in the amplitude of one basis of the quantum state, and so an exponential complexity is required to obtain the solution. To avoid the bottleneck, the previous study [Miyamoto and Kubo, IEEE Transactions on Quantum Engineering, 3, 1-25 ( 2022)] utilizes the fact that the present price of a derivative can be obtained by its discounted expected value at any future point in time and shows that the quantum algorithm can reduce the complexity. In this paper, to make the algorithm feasible to run on a small quantum computer, we use variational quantum simulation to solve the Black-Scholes equation and compute the derivative price from the inner product between the solution and a probability distribution. This avoids the measurement bottleneck of the naive approach and would provide quantum speedup even in noisy quantum computers. We also conduct numerical experiments to validate our method. Our method will be an important breakthrough in derivative pricing using small-scale quantum computers.

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Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

Cited by 10 publications

References 60 publications

On the bias in iterative quantum amplitude estimation

On the bias in iterative quantum amplitude estimation

Quantum computing for financial risk measurement

Pricing multi-asset derivatives by variational quantum algorithms

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