1 INTRODUCTION

Quantum computing for finance applications is an emerging field with quickly growing popularity. The finance industry involves various numerical and analytical tasks, for example, derivative pricing, credit rating, Forex algorithm trading, portfolio optimization, and so forth. They all demand heavy quantitative work, and the improved calculation speed and precision would bring significant social value. Quantum computing aims at these very targets.¹ Early studies focused on improving finance models with basic quantum mechanics.^2-4 Schrodinger equations and Feynman's path integral were suggested to solve stochastic differential equations for pricing interest rate derivatives,² and Heisenberg uncertainty principle was used to interpret the leptokurtic and fat-tailed distribution of stock price volatilities.⁴ Recent studies tend to utilize quantum advantages as a faster computing machine. Algorithms that can be implemented in quantum circuits, such as amplitude estimation,⁵ quantum principle component analysis (PCA),⁶ quantum generative adversarial network (QGAN),⁷ the quantum-classical hybrid variational quantum eigensolver (VQE),⁸ and quantum-approximate-optimization-algorithm (QAOA),⁹ spring up and begin to be applied to various financial quantitative tasks.^10-15

Within all sectors of quantitative finance, the Monte Carlo simulation always plays a significant role,^16-18 as only a few stochastic equations for derivative pricing have found analytical solutions,^{19, 20} while most can only be solved numerically by repeating random settings a great many times in an uncertainty distribution (e.g., normal or log-normal distribution), which therefore consumes much time. The quantum amplitude estimation (QAE) algorithm was raised⁵ in 2002. It is newly suggested as a promising alternative to the Monte Carlo method, as it shows a quadratic speedup comparing to the latter.¹⁰ So far, applications of QAE for option pricing¹¹ and credit risk analysis¹² have been demonstrated.

Considering the wide use of Monte Carlo simulation and the large variety of pricing models, the involvement of quantum techniques in finance is still at its infancy. Credit derivatives are frequently mentioned financial instruments because of the strong demand for tackling default risks in finance industry. Collateralized debt obligation (CDO) is a multi-name credit derivative backed on a pool of portfolios of defaultable assets (loans, bonds, credits, etc.). CDO then packages the portfolio into several tranches with different returns and priorities to suffer the default loss.¹⁸ CDO can effectively protect the senior tranche from the loss, but too many default events in the pool would still make the CDO collapsed, which was the case during the subprime financial crisis in 2008. Many voices were then made for improving the CDO pricing model and strengthening regulations in various aspects. Nonetheless, the CDO itself is a useful credit instrument that can work out and redistribute credit risks in a very quantitative way. So far, however, the implementation of complex credit instruments like CDO in quantum algorithms has never been reported.

In this work, we present the first quantum circuit implementation for CDO pricing using IBM Qiskit.²¹ To address the correlations among a large number of assets in the CDO pool, we use both the common Gaussian copula model²² and an improved model, the normal inverse Gaussian (NIG) copula model^{23, 24} that can interpret the skewness and kurtosis of the real markets which the Gaussian distribution cannot portray.^25-27 We follow a conditional independence approach to load the correlated distributions in the quantum circuits, and then use quantum comparators and QAE algorithm to calculate the losses in different tranches. We demonstrate the quantum computation results for a CDO that matches the classical Monte Carlo method, suggesting a promising approach for pricing various derivatives.

2 THE CDO STRUCTURE AND PRICING MODELS

2.1 The CDO tranche structures

The CDO pool is normally divided into three tranches: the equity, mezzanine, and senior tranche. As shown in Figure 1, when defaults occur, the equity tranche investors bear the loss first, then the mezzanine tranche investors if the loss is greater than the first attachment point. Only when the loss is greater than the second attachment point, will the senior tranche investors lose money. Therefore, senior tranche has the priority of receiving principle and interest payment, and the best protection from risk while having the lowest return.

Let and denote the lower and upper attachment point for tranche k, respectively. When defaults occur, the buyer of the tranche k will bear the loss in excess of , and up to .

Let L denotes the total loss for the portfolio and denotes the loss suffered by the holders of tranche k. There is: . As there are various default scenarios under some uncertainty distribution, we evaluate the expectation value of the tranche loss for each tranche k: . Then we can get the fair spread for this tranche denoted as :

(1)

where is the notional value of tranche k of the portfolio, which can be calculated by . To arrive at a fair price of a CDO, the return for investors of each tranche should be consistent with the expected loss the investors would bear. Therefore, such a fair spread is considered as the return for this tranche.

2.2 The conditional independence approach

Usually the pool in CDO is a portfolio of correlated assets. Their default events are not independent, which can be modeled using the single-factor Gaussian copula.

Meanwhile, through years' practice on the Gaussian model, it is found not to well portray the phenomena in real CDO markets, for example, the “correlation smile.”²⁵ In 2005, the NIG model was introduced to CDO pricing. In fact, price volatilities in derivative markets seldom show perfect Gaussian distribution. NIG can flexibly introduce a target skewness and kurtosis which the Gaussian model cannot achieve.^25-27 Explanation for NIG distribution and its probability density function (pdf) can be seen in Supplementary Appendix I.

For either the Gaussian copula or NIG copula model, both of them can use the conditional independence approach²⁸ originally developed by Vašíček^{29, 30} for the multivariate distribution problems. Consider a portfolio that comprises n assets, each with an independent default risk , and a correlation with the systematic risk Z. The latent variables can be used: , where s are correlation parameters that can be obtained by calibrating the market data. , , and Z generally follow the same type of uncertainty distribution, that is, the three all follow a Gaussian-type distribution in the Gaussian copula model.

Let be the original default probability for asset i that is uncorrelated to Z. Via detailed derivation,²⁸ the default probability under z, a detailed certain level of the systematic risk Z, follows:

(2)

Equation (2) is derived for very general scenarios.²⁸ F stands for the cumulative distribution function that Z follows, which can be any continuous and strictly increasing distribution function, and in this content, they are Gaussian for the Gaussian copula model or NIG for the NIG copula model. stands for the inverse of distribution F.

Using this conditional independence model, each asset i would default with a probability . The default event has a Bernoulli distribution, that is, means to default and for no default. Given , the loss that would incur for asset i when default happens, the total loss is certainly . Consider the probability for is , the expectation for the total loss would be:

(3)

where is the probability density function (PDF) of a systematic risk level z. For Gaussian distribution with a variance , integrating z from to would cover 99.73% of the distribution. After obtained the expected total loss from Equation (3), we can refer to Equation (1) to get the tranche loss and hence fulfill CDO pricing for each tranche. More derivations for the conditional independence approach and the Monte Carlo method for calculating the tranche loss are given in Supplementary Appendix II.

3 QUANTUM CIRCUIT CONSTRUCTION

The quantum circuit framework is demonstrated in Figure 2. To apply quantum computation for CDO pricing, the primary task is to load the correlated default risk for each asset of the portfolio into the quantum circuit. Either the Gaussian or NIG model can be loaded following a previous circuit approach¹² for the conditional independence model. This involves the operator , , and , and then sum up the total loss using operator .

3.1 Load uncorrelated default using operator

We first load the uncorrelated asset default event using linear Y-rotation gate. The default probability for each asset i can be obtained from its historical performance. The operator involves qubits to load the independent assets. For each of the qubits, operator inputs the initial state and outputs the state:

(4)

so that the probability for state encodes the default probability . See Supplementary Appendix III for the circuit for operator .

3.2 Load Z distribution using operator

We need to note that the Gaussian or NIG distribution for systematic risk Z has to be loaded using operator before operator . We use qubits to discretize the distribution to slots. The y axis for these slots is the probability of Z, that is, the PDF function . For Gaussian distribution function, the values can be loaded using the built-in codes of uncertainty model in Qiskit, and we contribute the similar codes for NIG distribution. Essentially, operator inputs the state for these qubits, and outputs , a superposition of entangled states:

(5)

where z is an integer in the binary form, for example, for , ranges from to , corresponding to 3 to 3 of the Gaussian distribution. Operator is constructed via a series of controlled-not gates and unitary rotations. See Supplementary Appendix IV for the circuit of operator and a brief derivation via matrix calculations.

3.3 Load correlated default using operator

Then we need to load the default correlation using operator . In short, operator inputs and from and , respectively, and outputs the further entangled state:

(6)

We use affine mapping¹² to encode the influence of a z value for the lower qubits. For instance, with qubits, for , Qubit 1 and Qubit 3 turn on their controlled gates, while Qubit 2 does not switch on its controlled gate, so that the value is considered for the qubits, and there is a probability of for z being 4. Meanwhile, there are also many linear Y-rotation gates working on the qubits, which change the probability for state from to . The expression for as a function of z and the correlation-free just follows Equation (2), which derives the slope and offset for the rotation gate for operator , that is, . See derivation of slope and offset in Supplementary Appendix V. The quantum circuit for operator is provided in Supplementary Appendix VI.

3.4 Load total loss using operator

Furthermore, we set an operator to sum up the loss due to all default events in this asset pool. The sum of loss equals to , where is the loss given default for asset i, and is 1 if asset i default and is 0 vice versa. The amplitude for at the quantum state is . That is, the probability for is , where is given by operator . The maximum loss would be when all assets default, so ensuring the maximum loss to be encoded needs qubits that . The operator uses qubits following the previous design of sum operator.¹² It inputs the state: , and outputs the state:

(7)

The first qubits are used to load the sum of loss . Given the probability for is , the expectation of total loss is . The next qubits are used as the carry qubits . See details on the circuit and a simple proof that the sum operator indeed loads the expectation value of the total loss in Supplementary Appendix VII.

In short, the output after operator S is consistent with the expression for total loss given in Equation (3). The next step is to compare the total loss with the attachment points for each tranche and work out the tranche loss.

3.5 Load tranche loss using operator

We use the comparator operator (, 2, and 3) to compare the sum of loss with the fixed lower attachment point for each tranche k. The comparator has been used to compare the underlying asset value with the striking price for option pricing in a recent work.¹¹ The operator would flip the comparator ancilla qubit from to if , the sum of loss under a systematic risk level z, is higher than , and would keep otherwise.

Meanwhile, a piecewise linear rotation operator will also rotate the state of an objective qubit under the control of the comparator ancilla qubit. The operator inputs the state and outputs the state :

(8)

where , and c is a scaling factor. , where can be implemented using controlled Y-rotations, and it is mapped to integer value . Note that there is an upperbound breakpoint as well, so we set another comparator operator that encodes , and finally reads as:

(9)

With such settings would be in the range , and by choosing a small scaling parameter c, which is generally set as 0.1 in this work, we can ensure in a monotonously increasing regime. See Supplementary Appendix VIII for the quantum circuit of operator .

Then the probability at state is expressed as , and it is found to have a relationship with the tranche loss:

(10)

where is the expectation of loss for a certain tranche k, for instance, the loss for equity tranche when setting and . See detailed derivation for Equation (10) in Supplementary Appendix IX.

4 RESULT ANALYSIS

We consider an example to show the pricing for CDO tranches. As listed in Table 1, the CDO pool has four assets, each showing a default probability , a sensitivity to the systematic risk and a loss given default .

TABLE 1. The relevant parameters for each asset

Asset i
1	2	0.3	0.05
2	2	0.1	0.15
3	1	0.2	0.1
4	2	0.1	0.05

The CDO is divided into three tranches: the equity, mezzanine, and senior tranches. Values for the lower attachment point and upper attachment point for three tranches are provided in Table 2.

TABLE 2. The attachment points for each tranche

Tranche name	Lower	Upper
Equity	0	1
Mezzanine	1	2
Senior	2	7

For this task, we need qubits to represent the four assets in operator , and qubits in operator to make slots for the uncertainty distribution of systematic risk Z. We implement Gaussian (Figure 3A) and NIG (Figure 3B) distribution for Z.

For NIG distribution, by setting the parameters given in Supplementary Appendix I, it shows a skewness of 1 and kurtosis of 6, which are consistent with a real CDO market.²⁵ Comparing with Gaussian distribution, this is narrower and centered to the left.

The step after loading distribution is to calculate the cumulative loss. The maximal loss is for this portfolio. Therefore, we can use qubits to encode the total loss in the weighted sum operator .

The pricing of the tranche loss is similar to the call option pricing, where there is a linear “payoff function” that goes up from zero after the option striking price or, for the CDO tranche, the attachment point. The tranche loss as a function of the total cumulative loss is given in Figure 3C–E for this specific example. We can use the built-in code named “PwlObjective” for the piecewise linear rotation function in operator . The built-in function uses the “breakpoints” array to record the attachment points, and uses the “slopes” and “offsets” arrays in which slope k and offset k correspond to these for the line segment between breakpoint and breakpoint k. Note the offset is the y-axis value for the starting point of the line segment, instead of the intercept by extending the line segment to the y axis. The breakpoints, slopes and offsets for the tranche loss function in this specific case study are shown in each figure in Figure 3C–E, which can be very straightforwardly calculated. These are used as the input parameters for the built-in piecewise linear rotation function.

We then use IQAE to estimate and convert it to the tranche loss according to Equation (4). We use the QASM cloud backend that is in the noisy intermediate-scale quantum (NISQ) environment. Figure 4 demonstrates the calculated tranche loss for an NIG distribution (Figure 3B) using IQAE (with a precision parameter and a confidence interval parameter ), the expected wavefunction results from the quantum circuit and the classical Monte Carlo method. The results for different approaches match well. The IQAE results, as indicated by a blue bar with different widths, give a confidence level of , while QAE estimates a few different values, each with a certain probability (see Figure A10 in the Supplementary Appendix).

When Z follows Gaussian distribution (Figure 3A), consistent results have also been obtained, as shown in Figure A11 in the Supplementary Appendix. Still, the NIG results slight differs from the Gaussian results with a relatively lower tranche loss, especially for the senior tranche loss, which is 0.2233 for NIG and 0.2301 for Gaussian distribution, both obtained via the matrix calculation result for related quantum circuits. This can be due to the skewed distribution for NIG, which makes more positive z values than the Gaussian one, so that expected total loss will be relatively lower considering a negative relationship given in Equation (2) and a positive relationship given in Equation (4). Therefore, if the real market follows an NIG distribution while we use Gaussian distribution to model it, we would overestimate the expected tranche loss.

With the calculated tranche loss, we can price the CDO tranche return according to Equation (1). The notional value N for the equity tranche, mezzanine tranche, and senior tranche is 1, 1, and 5, respectively, by calculating for each tranche. For equity, mezzanine, and senior tranche, the expected tranche loss via IQAE gives 52.0%, 41.7% and 23.8%. Then the tranche return for these tranches are 52.0%, 41.7% and 4.76%, respectively. The low return for the senior tranche is consistent with the practice in reality. Such a low value is first due to the last sequence to bear the loss, and second owing to its large notional value, which is normally above 80% of the sum for the three tranches.

It is worth noting that the returns for equity and mezzanine tranche in the case study of the main text are a bit too high, comparing to the custom returns that would be around 15%–25% and 5%–15% for the equity and mezzanine tranche, respectively.¹⁸ It is partially because that default probabilities s are a bit high. One more reason is that we ignore the recovery rate of the asset in order to focus on the essential structure. The recovery rate , which is generally set as 40%, means that when asset defaults, some values can be recovered by ways like selling real estates to get funds to compensate investors. Then the maximum loss would equal to the total notional value multiplies (). In this example, the loss given default to would become 1.2, 1.2, 0.6, and 1.2, while the tranche attachment points keep unchanged. This would bring down the tranche loss.

We further conduct a robustness analysis on the quantum computation method for CDO tranche pricing, with details given in Supplementary Appendix XI. We have noticed that the scaling factor c is introduced in the operator , and it has to be small enough to make the approximation satisfied. Therefore, a smaller value of c tends to be more accurate. The IQAE parameters, including the confidence interval parameter and the precision parameter , are investigated as well. We find that does not have a prominent influence on the range of the result, that is, the bar width, while has a severe impact, and the result is satisfactory only when goes down to 0.002 or below. For all these quantum computing parameters, the senior tranche is most sensitive to the changes. This can be due to its last sequence to bear loss. While little fluctuation would not change the result for other tranches, a slight decrease of total loss can possibly exempt the loss due for the senior tranche. We also introduce up to fluctuations of either the independent probability of default or the correlation to systematic risk, , and find that the fluctuation of has a stronger influence on the tranche loss comparing to . We show that each tranche loss shows an increasing and decreasing trend with and , respectively, which is consistent with the theoretical conditional independence approach in Eq.(2). In addition, an analysis is given in Supplementary Appendix XII to show how different operators scale with , , and . It suggests that and consume heavy circuit depth and it is worth investigation for further optimization for those operators.

5 DISCUSSION AND CONCLUSION

The CDO is a relatively advanced and complex structured finance product, and the credit market plays a significant role in the finance industry. Therefore, despite there were some disputes on CDOs during the 2008 financial crisis, CDOs are still widely studied in quantitative finance, and inspire the work of pricing more credit derivatives, which now include a wide range of products like credit default swaps (CDS) and credit valuation adjustment (CVA).

The CDOs are being improved with various financial models considering the inadequacy of early single-factor Gaussian copulas. In this work, we implement the NIG model as an alternative to the Gaussian model. In general, the heavy-tailed distribution of factors in a one-factor copula model may always help to solve the correlation smile problem of the Gaussian copula model. For instance, there is also the variance gamma model that was first applied to option pricing³⁵ and later found to be a good model for CDO pricing.³⁶ Such improved models can also be calculated via quantum computation.

We have stated that the Monte Carlo is useful as only a few stochastic equations for derivative pricing have analytical solutions. It is also worth to note that there are many numerical methods which are being investigated and may work faster than Monte Carlo, for example, finite difference and Fourier transform methods. They would set up a higher classical benchmark that calls for further improvement of corresponding quantum algorithms.

Note that the quantum adaption of generative adversarial network^{7, 37} has now been considered as an effective way to load any distribution in quantum circuits¹⁵ and can be applied to more finance models. Besides, the parameter shift rule^{38, 39} has been raised to solve the issue of encoding gradients in quantum circuits, which facilitates the mapping of machine learning techniques in quantum algorithms. Furthermore, the trendy variational quantum algorithms that are suitable for NISQ environment, and the alternative approach using quantum annealing,⁴⁰ may work on a large variety of optimization tasks in finance. In all, there's much room to explore for quantum computation in finance applications.

CONFLICT OF INTEREST

The authors declare no competing interests.

AUTHOR CONTRIBUTIONS

Hao Tang and Xian-Min Jin conceived and supervised the project. Hao Tang and Anurag Pal designed the scheme. Anurag Pal wrote the Qiskit code. Hao Tang did Monte Carlo simulation. Hao Tang, Anurag Pal, Lu-Feng Qiao, Tian-Yu Wang, Jun Gao, and Xian-Min Jin analyzed the data and presented the figures. Hao Tang, Anurag Pal, and Tian-Yu Wang enriched the quantum circuit analysis in the Supplementary Appendix. Hao Tang wrote the article, including the Supplementary Appendix, with input from all the other authors.