Abstract
1. INTRODUCTION
The pricing of financial derivatives is a prominent problem that requires extensive computational resources, as the stochastic nature of the underlying assets requires precise modeling. One of the typical financial derivatives is the option, which is a contract that allows the holder to buy or sell assets at a pre-established price (strike) at or before a specified date (maturity date). The payoff of an option relies heavily on the stochastic evolution of asset price. The traditional option pricing model, Black–Scholes–Merton (BSM) [1], usually oversimplifies market dynamics, which limits its practical application to real-life scenarios. As such, numerical methods such as the Monte Carlo method are typically employed for handling more realistic stochastic fluctuations. However, the Monte Carlo method requires extensive computation resources and is slow to predict complicated options. Reducing the computational resources required for models and speeding up option pricing could have significant implications for the financial industry.
Recently, quantum algorithms have shown promise in facilitating computationally hard financial problems such as trading, portfolio optimization, and risk profiling [2,3], and specifically, quantum amplitude amplification can accelerate option pricing with quadratic speedups [4–10]. The unique advantages of quantum algorithms will make up for the shortcomings of classical algorithms to a certain extent, enabling massive high-speed data services in the financial industry. However, current experimental demonstrations using binary approaches and standard quantum circuit models on superconducting devices [11] require dense chip connections and high gate fidelity, making it difficult for practical applications in the near future without a universal quantum computer [12,13]. In addition, superconducting devices require bulky, energy-intensive, and expensive peripherals such as cooling systems, making industrial-scale applications have poor prospects.
For specialized application tasks such as option pricing, there is no need to use universal quantum computers. Photonic circuits can provide fundamental functions that can be combined to implement specific algorithms [14–20], which would be practical and efficient for user-cased application scenarios. Moreover, the reduced energy costs of photonic computing have been a driving force behind works on dedicated photonic chips for machine learning and algebra [16,21–24]. Therefore, we demonstrate a unary (versus binary) approach in a photonic chip for option pricing. Compared to the binary approach, the unary approach [9] has a remarkably simplified structure and reduced depth of quantum circuits and is especially suitable for linear optical circuit realizations in photonic chips. The unary scheme also allows a post-selection strategy for error mitigation. Additionally, we demonstrate generative adversarial learning to upload the probability distribution implicitly given by data samples into the photonic chip. Generative adversarial learning has previously been demonstrated only in superconducting and optoelectronics devices [25–30]. Compared with traditional Monte Carlo methods, our approach shows high accuracy and significant speedup. It provides a promising avenue for interdisciplinary research in quantum machine learning and financial problems, paving the way for the development of practical photonic processors for quantitative financial applications. It can greatly improve the efficiency and quality of financial services, which is of great significance to the rapid and steady development of the financial industry.
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2. CHIP DESIGN FOR UNARY OPTION PRICING
In this work, we focus on European option pricing, and the expected payoff of options is given by
Figure 1.Schematic of the unary approach to option pricing, compared to the classical Monte Carlo method. (a) Integrated photonic chip with the unary algorithm, consisting of a generator of the generative adversarial network (GAN), payoff calculation, and quantum amplitude estimation for acceleration. (b) Monte Carlo simulation on a classical computer, which first generates the future asset price paths based on random variables, and then calculates the payoff. The accuracy relies on extensive simulations of random walk asset paths. (c) Expected acceleration of the convergence of payoff errors, compared to classical Monte Carlo simulations. Shaded areas in the top inset indicate statistical uncertainty.
The unary approach to option pricing encodes an asset price distribution into the unary basis of a quantum register, as shown in Fig. 2. A binning scheme is applied such that Monte Carlo paths that would belong to the same interval of asset prices end up in the same bin. Each bin is then mapped to an element of the unary basis, whose coefficient is the ratio of the number of Monte Carlo paths in that bin to the total number. The accuracy of unary encoding is bounded by the number of bins that can be stored in a quantum state, i.e., the usable dimension of the high-dimensional unary state. Based on the unary basis, Fig. 3(a) depicts the algorithmic model for unary option pricing, which consists of three modules: a distribution loading module that loads the asset price distribution into a quantum state, a payoff calculation module that computes the expected return, and a quantum amplitude estimation module to gain quadratic speedup over classical sampling to reach a target accuracy.
Figure 2.Mapping of asset prices to unary basis. (a) Classical Monte Carlo paths partitioned into different unary bases. (b) Probability density function (PDF) according to the defined unary basis. (c) Payoff value calculated according to the PDF and asset prices.
Figure 3.Photonic chip design for the unary option pricing algorithm. (a) Algorithmic model of unary option pricing. The input state consists of an
Figure 3(b) depicts the optical circuit model, whereby each module of the unary algorithm is mapped to a linear optical operator. We represent the high-dimensional state by path encoding a single photon using optical waveguides. The superposition of a single photon traveling through different waveguides directly encodes the unary basis. This high-dimensional state can be written as , where represents the probability of observing a photon in the waveguide mode , and these probabilities conform to . The payoff calculation requires an ancilla qubit to store the expected return for each asset price, expanding the Hilbert space of the algorithm to . To avoid non-local controlled gates in the photonic chip implementation, we instead add an ancillary waveguide to each of the unary waveguide modes to represent the effect of the ancilla qubit. Each element of the unary basis is now represented by two waveguides. This way, the controlled operations of the original algorithm are converted to linear transformations on the optical circuit. The architecture of the photonic processor with the detailed chip design is shown in Fig. 3(c), which replaces each linear optical operator with the corresponding waveguide structure. The entire chip is reconfigurable via wire bonds and integrated thermo-optic phase shifters; see the experimental setup in Appendix B.
In the distribution loading module , a single photon is incident into the chip from a waveguide in the middle of the circuit, which encodes the ancilla in its state, e.g., for a three-asset case, the initial input state can be written as the tensor product of the middle unary qudit and the ancilla qubit as . The distribution of asset prices is then uploaded to the different waveguides using a linear depth circuit. This distribution loading circuit spreads the superposition to the neighboring basis using swp operators:
The payoff calculation module encodes the expected payoff as the probability of measuring the photon in the waveguides encoding the ancilla in state , using rotation operations between the two waveguides of each element of the unary basis. The rotations encode the expected return for each asset price in the distribution. This action, labeled , can be written as
A quantum amplitude estimation module is applied to achieve quantum speedups. Various amplitude estimation techniques have been presented that are friendly to NISQ devices [31–33]. Here, we implement an amplitude estimation algorithm without quantum phase estimation in the photonic circuit, following the technique used in Ref. [9]. Increasing steps of amplitude amplification are applied to estimate the relevant amplitudes with up to a square root advantage oversampling from the original distribution. This amplification module is performed by applying the following operators. First, identifies the amplitudes that encode the expected payoff and reverses their signs. Explicitly, for the three-asset example at hand, such an operation is , and is realized experimentally by applying a phase shift of on the second waveguide of each element of the unary basis. Then, the original operations are reversed, that is, the inverse of the payoff calculator and the distribution loading are applied. An operator follows, which reverses the sign of the initial state of the computation. Experimentally, it is applied by introducing a phase shift of to the waveguide where the photon was introduced. The last step is to repeat the distribution loading and the payoff calculator modules. The amplitude amplification operator is repeated a different number of times, and the results are processed to estimate the expected payoff. This technique provides up to quadratic speedups over ordinary sampling in the number of calls to the and operators to reach the same confidence level; see theoretical derivations in Appendix C.
3. GAN FOR DISTRIBUTION UPLOADING
A GAN is implemented in the distribution loading module with on-chip training for real-time noise perception. The goal of the GAN is to obtain an intelligent generator at the chip parameter level that captures the probability distribution behind the given market data without simulating enormous random paths, accumulating data statistics, and then fitting them into the chip architecture. With the GAN, we can efficiently load the classical data, i.e., the probability distribution underlying market data, into quantum states and obtain more precise payoff calculations with the presented unary option pricing methods.
GANs train a generator (G) to synthesize semantically meaningful data from standard signal distributions, as well as a discriminator (D) to distinguish real samples in the training dataset from fake ones produced by the generator [34], as depicted in Fig. 4(a). As its adversary, the generator aims at deceiving the discriminator by producing more realistic samples. Training GANs involve the search for a Nash equilibrium of a two-player game between a generative and a discriminative network, which can be formulated as
Figure 4.GAN on the photonic chip for precise asset distribution uploading. (a) Algorithm of GAN, composed of a generator and a discriminator. (b) Generator implemented by a variational photonic circuit, which is trained on-chip in real time. The probability distributions accumulated on the waveguide paths are used as fake samples. Real samples are the training targets taken from market data in real applications. (c) Classical discriminator consisting of sequential convolutional layers and trained by a gradient descent algorithm. The discriminator aims to distinguish the source of the input sample, from the generator or a real distribution. The cost function is calculated from the discriminator output and used to train the discriminator itself and the generator. (d) The generator is trained by an evolutionary optimization procedure where populations (e.g., different configurations of the generator ansatz) are generated, evaluated, and iterated. The evaluation is accomplished using the scores granted by the discriminator. New generations are produced via the operators of selection, crossover, and mutation of current populations.
We develop a hybrid GAN implementation that consists of a generator network in the photonic chip, a classical discriminator network, and a control system that communicates between the classical computer and photonic chip, all depicted in Figs. 4(b)–4(d). The generator is parameterized by the angles on the phase shifters that are reconfigurable through the thermo-optic effect, induced by applying tiny electrical power to the integrated heaters. Instead of a noise distribution as input, we utilize the uncertainty of photons appearing at different waveguide modes to achieve the equivalent randomness for the generator. The fake samples are the probability distribution of the photons at different waveguide modes. The real samples are drawn from the desired probability distribution, a log-normal or normal distribution for the examples presented in Fig. 5. Fake and real samples sequentially enter the classical discriminator to achieve the classification results. The discriminator is a classical neural network implemented with TensorFlow. We then explicitly discuss the training of the GAN with data samples drawn from the log-normal distribution and the normal distribution.
Figure 5.Experimental training performance of the GAN under Wasserstein distance. (a), (c) Comparison between the probability distributions obtained experimentally from the generator (solid line with data points) and the target distribution (histogram). (b), (d) Evolution of the
The training process of the GAN in a photonic chip introduces two challenges, the difficulty of obtaining gradients due to the stochastic nature of measurements, and the phenomenon that the discriminator easily overpowers the generator. To circumvent these problems, we propose a hybrid training strategy, where the generator is optimized under a gradient-free evolutionary algorithm, while the classical discriminator uses a gradient descent optimizer. Additionally, the Wasserstein distance [35,36] is used to train the GAN, which changes the dynamic between the generator and the discriminator. In this new GAN scheme, the discriminator acts as a critic, instead of classifying. It aims to give a high score to real instances over fake ones, effectively alleviating the problem of unstable GAN training.
4. RESULTS AND DISCUSSION
A. GAN Results
Our chip can accommodate the entire option pricing process of distribution loading, payoff calculation, and amplitude estimation for three option assets. However, training three bins is too trivial to demonstrate the ability to implement GANs on a photonic chip. Here, to demonstrate the GAN, we employ a chip that supports up to eight bins to demonstrate the generation of probability distribution. Figure 5(a) shows the probability distribution of the generator output compared to the real log-normal distribution. Figure 5(b) shows the convergence of the norm between the fake and real samples, of 100 training iterations. For generator output and real distribution , the norm is defined as
By training this generative model directly on the photonic chip, we bypass the need to solve the BSM equations while capturing the nuances that the simplified method overlooks. Concurrently, it incorporates environmental elements that are hard to model, such as cross talk and chip imperfections into the GAN training. Another feature of using GANs for the amplitude distribution step is that we can tailor the variational ansatz to construct short-depth circuits for a given degree of accuracy, even in a more general case with multiple photons.
B. Unary Option Pricing Results
As a proof of principle, the fabricated photonic chip supports an option pricing problem with three asset values, whose schematic diagram is shown in Fig. 6(a). The chip has six waveguide inputs: each pair represents one element of the unary basis and the ancilla qubit state. The chip is divided into distribution loading, payoff calculation, and then runs of amplitude amplification. To stay within the depth constraints of this proof of concept, the circuit’s unitary matrix is multiplied and uploaded into the photonic chip at a constant depth. The single-photon measurement is performed at the waveguide modes that represent ancilla state for asset prices larger than the strike value. The comparison between the theoretical payoff and estimation achieved experimentally is shown in Fig. 6(b), with increasing iterations of amplitude estimation. The performance of the amplitude estimation is shown in Figs. 6(c) and 6(d). In Fig. 6(c), the dotted line represents the theoretical payoff expectation, the solid line with data points represents the experimental results, and the shaded area represents the standard deviation (STD) of 50 measurements performed in each step of amplitude estimation. The progression of from 0 to 50 ( being a classical sampling of payoff calculation) demonstrates the convergence of the STD. Similarly, in Fig. 6(d), we visualize the convergence of the payoff error with more amplitude estimation runs. The amplitude estimation improves the accuracy of the expected payoff for a certain number of circuit runs.
Figure 6.Experimental results of option pricing with three asset values. (a) Illustration of the optical chip with payoff calculation and amplitude estimation module. Operator
The structure of the unary algorithm allows a simple but efficient design of photonic chips, especially when loading probability distributions into quantum registers since only local interactions between neighboring waveguides are required. This, however, is inaccessible for the binary alternative, where high connectivity is required to offset the exponential Hilbert space available; see the unary and binary comparison in Appendix D. The optical circuit that implements the unary approach requires a linear number of waveguides scaling with the required precision, which coincides with the remarkable scalability of photonic chips. Instead, avoiding the use of controlled operations via ancilla waveguides bypasses one of the main bottlenecks of photonic chips in quantum computing: the obstacle of realizing photon interactions.
Speedup is achieved in our work much akin to proposals for quantum search without entanglement [37,38], whereby polynomial speedup of unstructured search is achieved with a single photon at the cost of exponential resources. In particular, isomorphisms exist between a system of qubits and a qudit residing in a Hilbert space (systems presented in this thought experiment and our implementation) [39]; thereby the unary implementation on a photonic chip displays entanglement in the path encoding of the single photon. Coherent light can achieve a similar effect with a high sampling rate, which is advantageous in near-term use case scenarios, with some trade-offs for the random behavior of single photons in the generator part of the GAN.
The presented avenue to achieve speedup in option pricing is scalable in the photonic chip. It transforms the unary algorithm’s need for increasing qubits into a need for waveguide paths, which are highly scalable in photonic chips. For further scalability, in the presented experiment, the photon detectors placed in ancilla waveguides could be combined into a single one, as only the counts of photons in any ancilla qubit are needed, hence significantly reducing the resources needed to scale this approach to meaningful problems. The energy efficiency of photonic chips also promises a relevant advantage beyond a complexity separation between quantum and classical algorithms. Given an energy budget instead of a shot budget, the photonic implementation of the unary approach to option pricing can yield a significant advantage in the number of operations performed.
5. CONCLUSION
This work is the first demonstration of photonic chips for financial applications. As a proof of concept, we implement the unary option pricing algorithm in a photonic chip for European options, which includes the generation of amplitude distribution of the asset value, evaluation of expected return, and amplitude estimation. We prove the high accuracy in calculating the payoff function, as well as the effectiveness of amplitude estimation in reducing the number of evaluations to reach the same degree of accuracy when compared to classical sampling. The unary representation remarkably simplifies the structure and reduces the depth of quantum circuits in the linear optical circuit implementation. Such photonic devices could eventually be an eco-friendly alternative to electronic circuits. Furthermore, we demonstrate an on-chip training of a GAN that successfully captures important market dynamics in real-life scenarios, bypassing the simplified assumptions in the BSM model that limit its accuracy, as well as the computational burden in solving differential equations. The photonic chip could be potentially employed for other options pricing, paving the way for developing dedicated processors in finance applications.
APPENDIX A: EUROPEAN OPTION PRICING MODEL
The Black–Scholes model is a typical economic model used to calculate the evolution of asset prices in financial markets, known as the European option pricing problem. In this model, the evolution of option price at time is decided by two market properties: interest rate and volatility , which are expressed by the stochastic differential equation
APPENDIX B: EXPERIMENTAL SETUP AND SINGLE-PHOTON GENERATION
The entire packaged chip is shown in Fig.
Figure 7.Fabricated quantum photonic chip.
A degenerated photon pair is used in our experiment. The pump laser is generated from the Ultrafast Optical Clocks device (PriTel) with a repetition rate of 500 MHz, central wavelength of 1550.116 nm, and bandwidth of 1.9 nm. A dual pump scheme is employed to generate pairs of identical photons on chip with a degenerated spontaneous four-wave mixing (SFWM) process. On the chip, the desired state is generated out of the two-photon N00N state , by configuring the phase value when interfering the two photons.
APPENDIX C: THEORY OF UNARY OPTION PRICING
By solving the aforementioned BSM model, the probability density function of the option price can be described by a log-normal distribution. We map this continuous price distribution into discrete values, which are the amplitudes of orthogonal quantum state basis, by using a probability loading operator acting on an initial state as
Then, the expected payoff of the option price is related to the amplitude of the ancilla qubit in the form of
Therefore, the measurement of the ancilla qubit under basis after repeated amplitude estimation would yield the results of for us to infer the payoff of the option price with improved accuracy. The amplitude estimation scheme we use here is an iterative approach [
Figure 8.Simulation of the scaling of quantum AE and classical MC.
APPENDIX D: UNARY AND BINARY COMPARISON
The utilization of a unary approach [ Unary and Binary ComparisonAspect Unary Approach Binary Approach Representation Intuitive: single symbol repeated multiple times Compact: base-2 system with 0 and 1 symbols Chip architecture Simple: first-nearest neighbor connectivity Complex: full connectivity Amplitude estimation Without phase estimation: feasible in linear optical circuits Phase estimation required: not feasible in linear optical circuits Gate count Linear (advantageous for near-term devices with Logarithmic, requiring Toffoli gate Distribution loading error due to single-qubit error KL divergence of KL divergence of Payoff deviation due to single-qubit error
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