Quantum pricing with a smile: implementation of local volatility model on quantum computer
Tóm tắt
Quantum algorithms for the pricing of financial derivatives have been discussed in recent papers. However, the pricing model discussed in those papers is too simple for practical purposes. It motivates us to consider how to implement more complex models used in financial institutions. In this paper, we consider the local volatility (LV) model, in which the volatility of the underlying asset price depends on the price and time. As in previous studies, we use the quantum amplitude estimation (QAE) as the main source of quantum speedup and discuss the state preparation step of the QAE, or equivalently, the implementation of the asset price evolution. We compare two types of state preparation: One is the amplitude encoding (AE) type, where the probability distribution of the derivative’s payoff is encoded to the probabilistic amplitude. The other is the pseudo-random number (PRN) type, where sequences of PRNs are used to simulate the asset price evolution as in classical Monte Carlo simulation. We present detailed circuit diagrams for implementing these preparation methods in fault-tolerant quantum computation and roughly estimate required resources such as the number of qubits and T-count.
Tài liệu tham khảo
Orus R et al.. Quantum computing for finance: overview and prospects. Rev Phys. 2019;4:100028.
Hull JC. Options, futures, and other derivatives. New York: Prentice Hall; 2012.
Shreve S. Stochastic calculus for finance I: the binomial asset pricing model. Berlin: Springer; 2004.
Shreve S. Stochastic calculus for finance II: continuous-time models. Berlin: Springer; 2004.
Montanaro A. Quantum speedup of Monte Carlo methods. Proc R Soc Ser A. 2015;471:2181.
Miyamoto K, Shiohara K. Reduction of qubits in quantum algorithm for Monte Carlo simulation by pseudo-random number generator. Phys Rev A. 2020;102:022424.
Rebentrost P et al.. Quantum computational finance: Monte Carlo pricing of financial derivatives. Phys Rev A. 2018;98:022321.
Stamatopoulos N et al.. Option pricing using quantum computers. Quantum. 2020;4:291.
Ramos-Calderer S et al.. Quantum unary approach to option pricing. Phys Rev A. 2021;103:032414.
Chakrabarti S et al.. A threshold for quantum advantage in derivative pricing. Quantum. 2021;5:463.
Black F, Scholes M. The pricing of options and corporate liabilities. J Polit Econ. 1973;81:637.
Merton RC. Theory of rational option pricing. Bell J Econ Manag Sci. 1973;4:141.
Dupire B. Pricing with a smile. Risk. 1994;7:18–20.
Grover L, et al. Creating superpositions that correspond to efficiently integrable probability distributions. quant-ph/0208112.
Campbell ET et al.. Roads towards fault-tolerant universal quantum computation. Nature. 2017;549:172.
Egger DJ, et al. Credit Risk Analysis using Quantum Computers. 1907.03044.
Amy M et al.. A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits. IEEE Trans Comput-Aided Des Integr Circuits Syst. 2013;32(6):818–30.
Selinger P. Phys Rev A. 2013;87:042302.
Maruyama G. On the transition probability functions of the Markov process. Rend Circ Mat Palermo. 1955;4:48.
Bassard G et al.. Quantum amplitude amplification and estimation. Contemp Math. 2002;305:53.
Suzuki Y et al.. Amplitude estimation without phase estimation. Quantum Inf Process. 2020;19:75.
Nakaji K. Faster Amplitude Estimation. 2003.02417.
Giurgica-Tiron T, et al. Low depth algorithms for quantum amplitude estimation. 2012.03348.
Plekhanov K, et al. Variational quantum amplitude estimation. 2109.03687.
Herbert S. The problem with grover-rudolph state preparation for quantum Monte-Carlo. Phys Rev E. 2021;103:063302.
Vedral V et al.. Quantum networks for elementary arithmetic operations. Phys Rev A. 1996;54:147.
Beckman D et al.. Efficient networks for quantum factoring. Phys Rev A. 1996;54:1034.
Draper TG. Addition on a quantum computer. quant-ph/0008033.
Cuccaro SA et al.. A new quantum ripple-carry addition circuit. In: The eighth workshop on quantum information processing. 2004.
Takahashi Y et al.. A linear-size quantum circuit for addition with no ancillary qubits. Quantum Inf Comput. 2005;5(6):440–8.
Van Meter R et al.. Fast quantum modular exponentiation. Phys Rev A. 2005;71(5):052320.
Draper TG et al.. A logarithmic-depth quantum carry-lookahead adder. Quantum Inf Comput. 2006;6(4):351.
Takahashi Y et al.. Quantum addition circuits and unbounded fan-out. Quantum Inf Comput. 2010;10(9–10):0872.
Portugal R et al.. Reversible Karatsubas algorithm. J Univers Comput Sci. 2006;12(5):499.
Alvarez-Sanchezet JJ et al.. A quantum architecture for multiplying signed integers. J Phys Conf Ser. 2008;128(1):012013.
Takahashi Y et al.. A fast quantum circuit for addition with few qubits. Quantum Inf Comput. 2008;8(6):636.
Thapliyal H. Mapping of subtractor and adder-subtractor circuits on reversible quantum gates. Transactions on Computational Science XXVII. 2016;10.
Thapliyal H, Ranganathan N. Design of efficient reversible logic based binary and BCD adder circuits. ACM J Emerg Technol Comput Syst. 2013;9:17.
Lin C-C et al.. Qlib: quantum module library. ACM J Emerg Technol Comput Syst. 2014;11(1):7:1–7:20.
Babu HMH. Cost-efficient design of a quantum multiplier-accumulator unit. Quantum Inf Process. 2016;16(1):30.
Jayashree HV et al.. Ancilla-input and garbage-output optimized design of a reversible quantum integer multiplier. J Supercomput. 2016;72(4):1477.
Muñoz-Coreas E, Thapliyal H. Quantum circuit design of a T-count optimized integer multiplier. IEEE Trans Comput. 2019;68:5.
Khosropour A et al.. Quantum division circuit based on restoring division algorithm. In: Information technology: new generations (ITNG), 2011 eighth international conference on. Las Vegas: IEEE; 2011. p. 1037–40.
Jamal L, Babu HMH. Efficient approaches to design a reversible floating point divider. In: 2013 IEEE international symposium on circuits and systems (ISCAS2013). 2013. p. 3004–7.
Dibbo SV et al.. An efficient design technique of a quantum divider circuit. In: 2016 IEEE international symposium on circuits and systems (ISCAS). 2016. p. 2102–5.
Thapliyal H et al.. Quantum circuit designs of integer division optimizing T-count and T-depth. In: IEEE transactions on emerging topics in computing. 2019.
Amy M, Maslov D, Mosca M. IEEE Trans CAD. 2014;33(10):1476.
Maslov D. On the advantages of using relative phase Toffolis with an application to multiple control Toffoli optimization. Phys Rev A. 2016;93:022311.
O’Neill ME. PCG: A Family of Simple Fast Space-Efficient Statistically Good Algorithms for Random Number Generation. Harvey Mudd College Computer Science Department Tachnical Report. 2014. http://www.pcg-random.org/.
Hörmann W, Leydold J. Continuous random variate generation by fast numerical inversion. ACM Trans Model Comput Simul. 2003;13(4):347.
Haner T, et al. Optimizing Quantum Circuits for Arithmetic. 1805.12445.
Muñoz-Coreas E, Thapliyal H. T-count and qubit optimized quantum circuit design of the non-restoring square root algorithm. ACM J Emerg Technol Comput Syst. 2018;14:3.
Kliuchnikov V et al.. Practical approximation of single-qubit unitaries by single-qubit quantum Clifford and T circuits. IEEE Trans Comput. 2016;65(1):161.