Option Pricing using Quantum Computers

Nikitas Stamatopoulos1, Daniel J. Egger2, Yue Sun1, Christa Zoufal2,3, Raban Iten2,3, Ning Shen1, and Stefan Woerner2

1Quantitative Research, JPMorgan Chase & Co., New York, NY, 10017
2IBM Quantum, IBM Research – Zurich
3ETH Zurich

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We present a methodology to price options and portfolios of options on a gate-based quantum computer using amplitude estimation, an algorithm which provides a quadratic speedup compared to classical Monte Carlo methods. The options that we cover include vanilla options, multi-asset options and path-dependent options such as barrier options. We put an emphasis on the implementation of the quantum circuits required to build the input states and operators needed by amplitude estimation to price the different option types. Additionally, we show simulation results to highlight how the circuits that we implement price the different option contracts. Finally, we examine the performance of option pricing circuits on quantum hardware using the IBM Q Tokyo quantum device. We employ a simple, yet effective, error mitigation scheme that allows us to significantly reduce the errors arising from noisy two-qubit gates.

► BibTeX data

► References

[1] John C. Hull, Options, futures, and other derivatives, 6th ed. (Pearson Prentice Hall, Upper Saddle River, NJ [u.a.], 2006).
https:/​/​doi.org/​10.1007/​978-1-4419-9230-7_2

[2] Fischer Black and Myron Scholes, ``The pricing of options and corporate liabilities,'' Journal of Political Economy 81, 637–654 (1973).
https:/​/​doi.org/​10.1086/​260062

[3] Bruno Dupire, ``Pricing with a smile,'' Risk Magazine , 18–20 (1994).

[4] Phelim P. Boyle, ``Options: A Monte Carlo approach,'' Journal of Financial Economics 4, 323–338 (1977).
https:/​/​doi.org/​10.1016/​0304-405X(77)90005-8

[5] Paul Glasserman, Monte Carlo Methods in Financial Engineering (Springer-Verlag New York, 2003) p. 596.
https:/​/​doi.org/​10.1007/​978-0-387-21617-1

[6] Michael A. Nielsen and Isaac L. Chuang, Cambridge University Press (2010) p. 702.
https:/​/​doi.org/​10.1017/​CBO9780511976667

[7] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta, ``Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets,'' Nature 549, 242 (2017).
https:/​/​doi.org/​10.1038/​nature23879

[8] Abhinav Kandala, Kristan Temme, Antonio D. Corcoles, Antonio Mezzacapo, Jerry M. Chow, and Jay M. Gambetta, ``Error mitigation extends the computational reach of a noisy quantum processor,'' Nature 567, 491–495 (2019).
https:/​/​doi.org/​10.1038/​s41586-019-1040-7

[9] N. Moll, P. Barkoutsos, L. S. Bishop, J. M. Chow, A. Cross, D. J. Egger, S. Filipp, A. Fuhrer, J. M. Gambetta, M. Ganzhorn, A. Kandala, A. Mezzacapo, P. Müller, W. Riess, G. Salis, J. Smolin, I. Tavernelli, and K. Temme, ``Quantum optimization using variational algorithms on near-term quantum devices,'' Quantum Science and Technology 3, 030503 (2018).
https:/​/​doi.org/​10.1088/​2058-9565/​aab822

[10] M. Ganzhorn, D.J. Egger, P. Barkoutsos, P. Ollitrault, G. Salis, N. Moll, M. Roth, A. Fuhrer, P. Mueller, S. Woerner, I. Tavernelli, and S. Filipp, ``Gate-efficient simulation of molecular eigenstates on a quantum computer,'' Phys. Rev. Applied 11, 044092 (2019).
https:/​/​doi.org/​10.1103/​PhysRevApplied.11.044092

[11] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, ``Quantum algorithm for linear systems of equations,'' Phys. Rev. Lett. 103, 150502 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.103.150502

[12] Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost, ``Quantum principal component analysis,'' Nature Physics 10, 631–633 (2014).
https:/​/​doi.org/​10.1038/​nphys3029

[13] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd, ``Quantum machine learning,'' Nature 549, 195–202 (2017).
https:/​/​doi.org/​10.1038/​nature23474

[14] Vojtech Havlicek, Antonio D. Corcoles, Kristan Temme, Aram W. Harrow, Abhinav Kandala, Jerry M. Chow, and Jay M. Gambetta, ``Supervised learning with quantum-enhanced feature spaces,'' Nature 567, 209 – 212 (2019).
https:/​/​doi.org/​10.1038/​s41586-019-0980-2

[15] Roman Orus, Samuel Mugel, and Enrique Lizaso, ``Quantum computing for finance: Overview and prospects,'' Reviews in Physics 4, 100028 (2019).
https:/​/​doi.org/​10.1016/​j.revip.2019.100028

[16] Patrick Rebentrost and Seth Lloyd, ``Quantum computational finance: quantum algorithm for portfolio optimization,'' (2018), arXiv:1811.03975.
arXiv:1811.03975

[17] Stefan Woerner and Daniel J. Egger, ``Quantum risk analysis,'' npj Quantum Information 5, 15 (2019).
https:/​/​doi.org/​10.1038/​s41534-019-0130-6

[18] Patrick Rebentrost, Brajesh Gupt, and Thomas R. Bromley, ``Quantum computational finance: Monte carlo pricing of financial derivatives,'' Phys. Rev. A 98, 022321 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.022321

[19] Christa Zoufal, Aurélien Lucchi, and Stefan Woerner, ``Quantum generative adversarial networks for learning and loading random distributions,'' npj Quantum Information 5, 1–9 (2019).
https:/​/​doi.org/​10.1038/​s41534-019-0223-2

[20] Ana Martin, Bruno Candelas, Angel Rodriguez-Rozas, Jose D. Martin-Guerrero, Xi Chen, Lucas Lamata, Roman Orus, Enrique Solano, and Mikel Sanz, ``Towards pricing financial derivatives with an ibm quantum computer,'' (2019), arXiv:1904.05803.
arXiv:1904.05803

[21] Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp, ``Quantum Amplitude Amplification and Estimation,'' Contemporary Mathematics 305 (2002), 10.1090/​conm/​305/​05215.
https:/​/​doi.org/​10.1090/​conm/​305/​05215

[22] Yohichi Suzuki, Shumpei Uno, Rudy Raymond, Tomoki Tanaka, Tamiya Onodera, and Naoki Yamamoto, ``Amplitude estimation without phase estimation,'' Quantum Information Processing 19, 75 (2020).
https:/​/​doi.org/​10.1007/​s11128-019-2565-2

[23] Reuven Y. Rubinstein, Simulation and the Monte Carlo Method, Wiley Series in Probability and Statistics (Wiley, 1981).
https:/​/​doi.org/​10.1002/​9780470316511

[24] Daniel S Abrams and Colin P Williams, ``Fast quantum algorithms for numerical integrals and stochastic processes,'' (1999), arxiv:quant-ph/​9908083.
arXiv:quant-ph/9908083

[25] Ashley Montanaro, ``Quantum speedup of monte carlo methods,'' Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 471 (2015), 10.1098/​rspa.2015.0301.
https:/​/​doi.org/​10.1098/​rspa.2015.0301

[26] A. Yu. Kitaev, ``Quantum measurements and the Abelian Stabilizer Problem,'' (1995), arXiv:quant-ph/​9511026.
arXiv:arXiv:quant-ph/9511026

[27] R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, ``Quantum algorithms revisited,'' Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 339–354 (1998).
https:/​/​doi.org/​10.1098/​rspa.1998.0164

[28] Paul Glasserman, Philip Heidelberger, and Perwez Shahabuddin, ``Efficient Monte Carlo Methods for Value-at-Risk,'' in Mastering Risk, Vol. 2 (2000) pp. 5–18.

[29] Robert C. Merton, ``Theory of rational option pricing,'' The Bell Journal of Economics and Management Science 4, 141–183 (1973).
https:/​/​doi.org/​10.2307/​3003143

[30] Lov Grover and Terry Rudolph, ``Creating superpositions that correspond to efficiently integrable probability distributions,'' (2002), arXiv:quant-ph/​0208112.
arXiv:quant-ph/0208112

[31] Adriano Koshiyama, Nick Firoozye, and Philip Treleaven, ``Generative adversarial networks for financial trading strategies fine-tuning and combination,'' (2019), arXiv:1901.01751.
arXiv:arXiv:1901.01751

[32] Blanka Horvath, Aitor Muguruza, and Mehdi Tomas, ``Deep learning volatility,'' SSRN Electronic Journal (2019), 10.2139/​ssrn.3322085.
https:/​/​doi.org/​10.2139/​ssrn.3322085

[33] Martin Plesch and Časlav Brukner, ``Quantum-state preparation with universal gate decompositions,'' Phys. Rev. A 83, 032302 (2011).
https:/​/​doi.org/​10.1103/​PhysRevA.83.032302

[34] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio, ``Generative adversarial nets,'' in Advances in Neural Information Processing Systems 27, edited by Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger (Curran Associates, Inc., 2014) pp. 2672–2680.
arXiv:1406.2661

[35] Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. Divincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter, ``Elementary gates for quantum computation,'' Phys. Rev. A 52, 3457–3467 (1995).
https:/​/​doi.org/​10.1103/​PhysRevA.52.3457

[36] Steven A Cuccaro, Thomas G Draper, Samuel A Kutin, and David Petrie Moulton, ``A new quantum ripple-carry addition circuit,'' (2004), arxiv:quant-ph/​0410184.
arXiv:quant-ph/0410184

[37] Gadi Aleksandrowicz, Thomas Alexander, Panagiotis Barkoutsos, Luciano Bello, Yael Ben-Haim, David Bucher, Francisco Jose Cabrera-Hernández, Jorge Carballo-Franquis, Adrian Chen, Chun-Fu Chen, Jerry M. Chow, Antonio D. Córcoles-Gonzales, Abigail J. Cross, Andrew Cross, Juan Cruz-Benito, Chris Culver, Salvador De La Puente González, Enrique De La Torre, Delton Ding, Eugene Dumitrescu, Ivan Duran, Pieter Eendebak, Mark Everitt, Ismael Faro Sertage, Albert Frisch, Andreas Fuhrer, Jay Gambetta, Borja Godoy Gago, Juan Gomez-Mosquera, Donny Greenberg, Ikko Hamamura, Vojtech Havlicek, Joe Hellmers, Łukasz Herok, Hiroshi Horii, Shaohan Hu, Takashi Imamichi, Toshinari Itoko, Ali Javadi-Abhari, Naoki Kanazawa, Anton Karazeev, Kevin Krsulich, Peng Liu, Yang Luh, Yunho Maeng, Manoel Marques, Francisco Jose Martín-Fernández, Douglas T. McClure, David McKay, Srujan Meesala, Antonio Mezzacapo, Nikolaj Moll, Diego Moreda Rodríguez, Giacomo Nannicini, Paul Nation, Pauline Ollitrault, Lee James O'Riordan, Hanhee Paik, Jesús Pérez, Anna Phan, Marco Pistoia, Viktor Prutyanov, Max Reuter, Julia Rice, Abdón Rodríguez Davila, Raymond Harry Putra Rudy, Mingi Ryu, Ninad Sathaye, Chris Schnabel, Eddie Schoute, Kanav Setia, Yunong Shi, Adenilton Silva, Yukio Siraichi, Seyon Sivarajah, John A. Smolin, Mathias Soeken, Hitomi Takahashi, Ivano Tavernelli, Charles Taylor, Pete Taylour, Kenso Trabing, Matthew Treinish, Wes Turner, Desiree Vogt-Lee, Christophe Vuillot, Jonathan A. Wildstrom, Jessica Wilson, Erick Winston, Christopher Wood, Stephen Wood, Stefan Wörner, Ismail Yunus Akhalwaya, and Christa Zoufal, ``Qiskit: An open-source framework for quantum computing,'' (2019).
https:/​/​doi.org/​10.5281/​zenodo.2562110

[38] Vlatko Vedral, Adriano Barenco, and Artur Ekert, ``Quantum networks for elementary arithmetic operations,'' Phys. Rev. A 54, 147–153 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.54.147

[39] Thomas G Draper, ``Addition on a Quantum Computer,'' (2000), arXiv:quant-ph/​0008033.
arXiv:quant-ph/0008033

[40] Thomas G Draper, Samuel A Kutin, Eric M Rains, and Krysta M Svore, ``A logarithmic-depth quantum carry-lookahead adder,'' Quantum Information and Computation 6, 351–369 (2006), arXiv:quant-ph/​0406142.
arXiv:quant-ph/0406142

[41] Ghasem Tarmast, ``Multivariate Log-Normal Distribution,'' International Statistical Institute Proceedings: 53rd Session, Seoul (2001).

[42] Emmanuel Gobet, ``Advanced monte carlo methods for barrier and related exotic options,'' Handbook of Numerical Analysis, 15, 497 – 528 (2009).
https:/​/​doi.org/​10.1016/​S1570-8659(08)00012-4

[43] Pavel V. Shevchenko and Pierre Del Moral, ``Valuation of barrier options using sequential monte carlo,'' Journal of Computational Finance 20, 107–135 (2014).
https:/​/​doi.org/​10.2139/​ssrn.2529539

[44] M. Žnidarič, O. Giraud, and B. Georgeot, ``Optimal number of controlled-not gates to generate a three-qubit state,'' Physical Review A 77, 032320 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.77.032320

[45] Vivek V Shende, Stephen S Bullock, and Igor L Markov, ``Synthesis of quantum-logic circuits,'' IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006).
https:/​/​doi.org/​10.1109/​TCAD.2005.855930

[46] Raban Iten, Oliver Reardon-Smith, Luca Mondada, Ethan Redmond, Ravjot Singh Kohli, and Roger Colbeck, ``Introduction to UniversalQCompiler,'' (2019), arXiv:1904.01072.
arXiv:1904.01072

[47] A. Dewes, F. R. Ong, V. Schmitt, R. Lauro, N. Boulant, P. Bertet, D. Vion, and D. Esteve, ``Characterization of a two-transmon processor with individual single-shot qubit readout,'' Phys. Rev. Lett. 108, 057002 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.108.057002

[48] Kristan Temme, Sergey Bravyi, and Jay M. Gambetta, ``Error mitigation for short-depth quantum circuits,'' Phys. Rev. Lett. 119, 180509 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.180509

[49] E. F. Dumitrescu, A. J. McCaskey, G. Hagen, G. R. Jansen, T. D. Morris, T. Papenbrock, R. C. Pooser, D. J. Dean, and P. Lougovski, ``Cloud quantum computing of an atomic nucleus,'' Phys. Rev. Lett. 120, 210501 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.210501

[50] Mark Broadie and Paul Glasserman, ``Estimating security price derivatives using simulation,'' Management Science 42 (1996), 10.1287/​mnsc.42.2.269.
https:/​/​doi.org/​10.1287/​mnsc.42.2.269

[51] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland, ``Surface codes: Towards practical large-scale quantum computation,'' Phys. Rev. A 86, 032324 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.86.032324

[52] Miroslav Dobšíček, Göran Johansson, Vitaly Shumeiko, and Göran Wendin, ``Arbitrary accuracy iterative quantum phase estimation algorithm using a single ancillary qubit: A two-qubit benchmark,'' Phys. Rev. A 76, 030306 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.76.030306

[53] C. J. O'Loan, ``Iterative phase estimation,'' Journal of Physics A: Mathematical and Theoretical 43 (2010), 10.1088/​1751-8113/​43/​1/​015301.
https:/​/​doi.org/​10.1088/​1751-8113/​43/​1/​015301

[54] Krysta M Svore, Matthew B Hastings, and Michael Freedman, ``Faster phase estimation,'' Quantum Information & Computation 14, 306–328 (2014), arXiv:1304.0741.
arXiv:1304.0741

[55] S.S. Bullock and I.L. Markov, ``Smaller circuits for arbitrary n-qubit diagonal computations,'' Quantum Information & Computation 4, 027–047 (2004), arXiv:quant-ph/​0303039.
arXiv:quant-ph/0303039

Cited by

[1] Brian S. Haney, "Quantum Machine Learning: A Patent Review", SSRN Electronic Journal (2020).

[2] Koichi Miyamoto and Kenji Shiohara, "Reduction of qubits in a quantum algorithm for Monte Carlo simulation by a pseudo-random-number generator", Physical Review A 102 2, 022424 (2020).

[3] Daniel J. Egger, Ricardo Gacía Gutiérrez, Jordi Cahué Mestre, and Stefan Woerner, "Credit Risk Analysis using Quantum Computers", arXiv:1907.03044.

[4] Dmitry Grinko, Julien Gacon, Christa Zoufal, and Stefan Woerner, "Iterative Quantum Amplitude Estimation", arXiv:1912.05559.

[5] Almudena Carrera Vazquez and Stefan Woerner, "Efficient State Preparation for Quantum Amplitude Estimation", arXiv:2005.07711.

[6] Julia E. Rice, Tanvi P. Gujarati, Tyler Y. Takeshita, Joe Latone, Mario Motta, Andreas Hintennach, and Jeannette M. Garcia, "Quantum Chemistry Simulations of Dominant Products in Lithium-Sulfur Batteries", arXiv:2001.01120.

[7] Adam Holmes and A. Y. Matsuura, "Efficient Quantum Circuits for Accurate State Preparation of Smooth, Differentiable Functions", arXiv:2005.04351.

[8] Lee Braine, Daniel J. Egger, Jennifer Glick, and Stefan Woerner, "Quantum Algorithms for Mixed Binary Optimization applied to Transaction Settlement", arXiv:1910.05788.

[9] Sergi Ramos-Calderer, Adrián Pérez-Salinas, Diego García-Martín, Carlos Bravo-Prieto, Jorge Cortada, Jordi Planagumà, and José I. Latorre, "Quantum unary approach to option pricing", arXiv:1912.01618.

[10] Juan José García-Ripoll, "Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations", arXiv:1909.06619.

[11] Christa Zoufal, Aurélien Lucchi, and Stefan Woerner, "Variational Quantum Boltzmann Machines", arXiv:2006.06004.

[12] Iordanis Kerenidis, Anupam Prakash, and Dániel Szilágyi, "Quantum Algorithms for Portfolio Optimization", arXiv:1908.08040.

[13] Austin Gilliam, Stefan Woerner, and Constantin Gonciulea, "Grover Adaptive Search for Constrained Polynomial Binary Optimization", arXiv:1912.04088.

[14] Pauline J. Ollitrault, Guglielmo Mazzola, and Ivano Tavernelli, "Non-adiabatic molecular quantum dynamics with quantum computers", arXiv:2006.09405.

[15] Filipe Fontanela, Antoine Jacquier, and Mugad Oumgari, "A Quantum algorithm for linear PDEs arising in Finance", arXiv:1912.02753.

[16] Ikko Hamamura and Takashi Imamichi, "Efficient evaluation of quantum observables using entangled measurements", npj Quantum Information 6, 56 (2020).

[17] Samuel Mugel, Carlos Kuchkovsky, Escolastico Sanchez, Samuel Fernandez-Lorenzo, Jorge Luis-Hita, Enrique Lizaso, and Roman Orus, "Dynamic Portfolio Optimization with Real Datasets Using Quantum Processors and Quantum-Inspired Tensor Networks", arXiv:2007.00017.

[18] Eric G. Brown, Oktay Goktas, and W. K. Tham, "Quantum Amplitude Estimation in the Presence of Noise", arXiv:2006.14145.

[19] Julien Gacon, Christa Zoufal, and Stefan Woerner, "Quantum-Enhanced Simulation-Based Optimization", arXiv:2005.10780.

[20] Kazuya Kaneko, Koichi Miyamoto, Naoyuki Takeda, and Kazuyoshi Yoshino, "Quantum Pricing with a Smile: Implementation of Local Volatility Model on Quantum Computer", arXiv:2007.01467.

[21] Tomoki Tanaka, Yohichi Suzuki, Shumpei Uno, Rudy Raymond, Tamiya Onodera, and Naoki Yamamoto, "Amplitude estimation via maximum likelihood on noisy quantum computer", arXiv:2006.16223.

[22] Xavier Vasques, "The data center of tomorrow is made up of heterogeneous accelerators", arXiv:2003.10950.

[23] Adam Holmes and A. Y. Matsuura, "Entanglement Properties of Quantum Superpositions of Smooth, Differentiable Functions", arXiv:2009.09096.

The above citations are from Crossref's cited-by service (last updated successfully 2020-10-20 20:10:18) and SAO/NASA ADS (last updated successfully 2020-10-20 20:10:19). The list may be incomplete as not all publishers provide suitable and complete citation data.