Quantum computational finance for martingale asset pricing in incomplete markets

Abstract

A derivative is a financial asset whose future payoff is a function of underlying assets. Pricing a financial derivative involves setting up a market model, finding a martingale (“fair game”) probability measure for the model from the existing asset prices, and using that probability measure to price the derivative. When the number of underlying assets and/or the number of market outcomes in the model is large, pricing can be computationally demanding. In this work, we first formulate the pricing problem in a linear algebra setting, including the realistic setting of incomplete markets where derivatives do not have a unique price. We show that the problem can be solved with a variety of quantum techniques such as quantum linear programming and the quantum linear systems algorithm. While in previous works the martingale measure is assumed to be given, here it is extracted from market variables akin to bootstrapping, a common practice among financial institutions. We discuss the quantum zero-sum game algorithm and the quantum simplex algorithm as viable subroutines. For quantum linear systems solvers, we formalize a new market assumption milder than market completeness, which allows the potential for large speedups. Towards prototype use cases, we conduct numerical experiments motivated by the Black–Scholes–Merton model.

Introduction

Financial markets often involve complex scenarios and challenging computational problems. A significant part of computational finance involves the pricing of financial securities such as derivatives^1,2,3 in stock, interest rate, and credit derivative markets. Some of these securities are tailor-made between two parties and are often called over-the-counter (OTC) derivatives. Such securities have complex payoffs that usually depend on underlying liquid, frequently-traded assets, which introduce their own dynamics due to market and economic fluctuations. These derivatives are exotic and illiquid, in the sense that they are usually not frequently traded to allow adequate price determination by the market itself⁴. Pricing is computationally expensive when the number of underlying liquid assets is large and/or the pricing model takes into account a large number of possible market outcomes, motivating the investigation of quantum algorithms^5,6,7,8,9.

Given a liquid financial market for certain assets, the task is to price fairly a new financial instrument that is not currently traded in the market. The risk-neutral pricing framework^2,3 allows a determination of arbitrage-free prices of financial derivatives within model assumptions. In a theoretical formulation called the Black–Scholes–Merton (BSM)^3,10,11 model, a market is composed of one or multiple “risky” stocks driven by random processes (e.g., Brownian motions) and one safe asset (“bond” or bank account). The BSM model is a starting point for modeling real-world markets and has the advantage that it allows for an analytic formula for the arbitrage-free price of certain financial derivatives such as put and call options on a single stock. Beyond Black–Scholes–Merton, a generalized situation for markets includes a large number of underlying assets, a large sample space, and complex price dynamics^2,12. The number of assets N is large when many stocks and in addition frequently-traded derivatives are included. The size K of the sample space is large when the probabilistic model includes many events. To price a derivative, risk-neutral probability measures are extracted from the market variables. Such measures turn the discounted price process of each of the assets into a martingale, and are used to compute the price of the derivative via an expectation value.

In this work, we present three quantum algorithms for risk-neutral pricing in a single-period framework. These algorithms price illiquid financial securities with the potential for significant speedups over classical algorithms. Our algorithms are particularly suited for incomplete markets: markets where not every derivative has a unique price. These algorithms also extract descriptions of martingale measures from the market data, i.e., perform bootstrapping or calibration of a model to market data, a common practice among financial institutions. The first two quantum algorithms rely on a linear programming (LP) formulation of finding the range of admissible prices of a derivative, and we show how quantum algorithms for solving linear programs can be employed. To illustrate the setting, we study numerically a use case motivated by the BSM model, where we compare to the analytically solvable BSM model with regularization techniques. Moreover, we investigate the use of quantum linear system solvers for the pricing problem. To this end, we propose a new market model which provides a sufficient condition for employing quantum algorithms for linear systems of equations. Our work opens up the possibility for large speedups and the use of near-term algorithms for derivative pricing.

Related works

Quantum speedups have been discussed for solving linear systems^13,14, general singular value transformations, and for many problems in optimization such as linear and semidefinite programming^15,16. Options pricing with Monte–Carlo estimation on quantum computers was discussed in⁵, given knowledge of the martingale measure. Similarly, an algorithm for risk management¹⁷ and further works on derivative pricing¹⁸ with amplitude estimation and principal component analysis¹⁹ were shown with proof-of-principle implementations on quantum hardware. Reviews are given in^6,7,20. Zero-sum games are a particular case of minmax games, specifically, they are the $\ell _1$–$\ell _1$ minmax games. More general quantum algorithms for solving minmax games can be found in^21,22.

Mathematical preliminaries

Define $[N] := \{ 1,\dots , N \}$. We use boldface for denoting a vector $\varvec{v}$, where the i-th entry of the vector is denoted by $v_i$. With $\varvec{v}_i$ we denote the i-th vector in a list of vectors. Denote by $\mathbbm {R}_+$ the non-negative real numbers. The Kronecker delta is $\delta _{\omega \omega '}$ for some integers $\omega , \omega '$. Define the simplex of non-negative, $\ell _1$-normalized K-dimensional vectors and its interior by $\Delta ^{K} := \left\{ \varvec{p} \in \mathbbm {R}^K : \sum _{k=1}^K p_k =1, p_k \ge 0\right\}$ and $\textrm{int}(\Delta ^{K}) := \left\{ \varvec{p} \in \mathbbm {R}^K : \sum _{k=1}^K p_k =1, p_k > 0\right\}$. For a matrix M, the spectral norm is $\Vert M\Vert$, the Frobenius norm is $\Vert M \Vert _F$, and the condition number is $\kappa (M):=\Vert M\Vert \Vert M^+\Vert$, where the pseudoinverse is denoted by $M^+$. Denote by $M_{i,\cdot }$ the i-th row of the matrix. We use the notation ${\widetilde{O}}\left( a \right)$ to hide factors that are polylogarithmic in a and the notation ${\widetilde{O}}_b(a)$ to hide factors that are polylogarithmic in a and b. We assume basic knowledge of quantum computing and convex optimization (in particular linear programming), and we use the same notation that can be found in²³ and²⁴. In Supplementary Material, we recall facts in linear algebra and linear programming and review useful definitions and subroutines in quantum linear algebra.

Regarding probability theory, we refer to the literature^25,26 for the definitions of $\sigma$-algebra, measurable spaces, measurable functions, and probability space. A probability space is given by a triple $(\Omega , \Sigma , \mathbb {P})$. Our first assumption is the existence of a sample space $\Omega$ which is finite (and hence countable). We take the $\sigma$-algebra $\Sigma$ to be the set of all subsets of $\Omega$. For most of the paper, we do not require the knowledge of the probability measure $\mathbb {P}$, a setting which is called Knightian uncertainty²⁷. We only require the assumption that $\mathbb {P}[\{\omega \}] > 0$ for all $\omega \in \Omega$: it is reasonable to exclude singleton events from $\Omega$ that have no probability of occurring. Equivalence of two probability measures $\mathbb {P}$ and $\mathbb {Q}$ in our finite context amounts to the statement that for $\omega \in \Omega$, $\mathbbm {P}[\{ \omega \}]=0$ if and only if $\mathbbm {Q}[\{ \omega \}] = 0$. Since we assume that $\mathbbm {P}[\{ \omega \}]>0$, equivalent measures $\mathbbm {Q}$ are all measures for which $\mathbbm {Q}[\{ \omega \}] > 0$. Random variables are measurable functions from $\Omega$ to some domain. We denote with $\mathbb {E}_\mathbb {P}[X]$ the expectation value of the random variable X with respect to the probability measure $\mathbb {P}$. A $\mathbbm {P}$-martingale in the single-period model is a random variable $X_1$ and a fixed $X_0$ such that under a probability measure $\mathbbm {P}$ it hold that ${\mathbb {E}}_{\mathbbm {P}}[|X_1|] < \infty$ and ${\mathbb {E}}_{\mathbbm {P}}[X_1] = X_0$.

Introduction to the financial setting

We consider a single-period model consisting of a present time and a future time. We consider a market with $N+1$ primary assets: N risky assets and one safe asset. The risky assets could be ${N}_{\textrm{stocks}}$ stocks and possibly other financial instruments on these stocks, such as simple call options, e.g., $\max (x-Z,0)$, or options involving several stocks, e.g., $\max (x+y-Z,0)$, where Z is the strike price. The number of possible options may scale with the number of subsets of the set of stocks $\sim 2^{N_{\textrm{stocks}}}$, and some of them could be liquid assets. The safe asset acts like a bank account and models the fact that money at different time instances has a different present value due to inflation and the potential for achieving some return on investment. Discount factors are used for comparing monetary values at different time instances. Without loss of generality, we assume that the discount factors are 1, which corresponds to a risk-free interest rate of 0². Hence, our bank account does not pay any interest. At the present time, the prices of all assets are known. The prices are denoted by $\Pi _i \in \mathbbm {R}_+$ for $i\in [N+1]$, where $\Pi _1=1$ for the safe asset. We assume that we can buy and short-sell an arbitrary number and fractions of them at no transaction cost. At the future time, the random variables of the future asset prices are defined on the sample space $\Omega$. We assume that $\Omega$ is finite with $K:=\vert \Omega \vert$ elements.The prices at the future time are given by the random variables $S_i: \Omega \rightarrow {\mathbb {R}}_+$, for $i \in [N+1]$, where $S_1: \Omega \rightarrow 1$, i.e., there is no randomness for the safe asset.

Definitions of market model, derivatives, and market completeness

Arrow securities^28,29 are imaginary securities that pay one unit of currency, say $\$1$, in case some event in $\Omega$ occurs and pay nothing otherwise. Arrow securities have historical and theoretical relevance in finance and asset pricing, and in some cases can model existing financial securities such as digit options that pay when a stock price is in a certain range or credit default swaps that pay when the underlying’s default occurs. They allow us to express the future stock prices as a sum of Arrow securities multiplied by respective scaling factors. This separation then naturally leads to a matrix description of the problem of pricing a derivative.

Definition 1

(Arrow security ²⁹) The Arrow security for $\omega \in \Omega$ is the random variable $A_{\omega }: \Omega \mapsto \{0,1\}$ such that in the event $\omega ' \in \Omega$ the payoff of the security is: $A_{\omega } (\omega ') = \delta _{\omega \omega '}.$

Arrow securities are often not traded in the market. Their prices, also called state prices³⁰, may be inferred from the traded securities. Pricing the Arrow securities is equivalent to finding a probability measure, which in turn can be used for pricing other assets. Arrow securities can be considered as a basis of vectors from which the other financial assets are expanded, as in the next definition. For this definition, we also require the concept of redundancy: here, a redundant asset is a primary asset $i \in [N+1]$ which can be expressed as a linear combination of the other N primary assets.

Definition 2

(Payoffs, payoff matrix, price system) Let there be given one safe asset with current price of $\Pi _1 = 1$ and payoff $S_1: \Omega \mapsto \{1\}$. Let there be given N non-redundant assets with current price $\Pi _i \in \mathbbm {R}_+$ and future payoff $S_i: \Omega \mapsto \mathbbm {R}_+$ for $i\in \{ 2,\cdots , N+1\}$. We denote by $\varvec{S}: \Omega \rightarrow \mathbbm {R}_+^{N+1}$ the associated vector of asset payoffs and by $\varvec{\Pi }= [1, \Pi _2, \cdots , \Pi _{N+1}]^T$ the price vector. Define the payoff matrix $S \in \mathbbm {R}_+^{(N+1)\times K}$ such that the assets are expanded as

$$\begin{aligned} S_{1 \omega } = 1\quad \text {for}\ \omega \in \Omega , \quad S_i = \sum _{\omega \in \Omega } S_{i \omega } A_{\omega }\quad \text {for}\ i\in \{ 2,\cdots , N+1\}. \end{aligned}$$

(1)

Finally, we call $(\varvec{\Pi }, S)$ a price system.

The matrix S specifies the asset price for each asset for each market outcome. An interpretation of a single row of the payoff matrix is as a portfolio that holds the amounts $S_{i \omega }$ in Arrow security $A_\omega$. Hence, the random variable $S_i$ is replicated with a portfolio of Arrow securities. However, because of the non-redundancy assumption, it cannot be replicated with the other primary assets. Non-redundancy implies that the matrix S has always rank $N+1$.

A financial derivative in the context of this work is a new asset whose payoff can be expanded in terms of Arrow securities and which does not have a current price.

Definition 3

(Financial derivative²) A derivative is a financial security given by a random variable $D : \Omega \rightarrow \mathbb {R}_+$. The derivative D can be written as an expansion of Arrow’s securities with expansion coefficients $D_\omega , \forall \omega \in \Omega$, as: $D=\sum _{\omega \in \Omega } D_{\omega } A_\omega$. We denote by $\varvec{D} \in \mathbb {R}_+^K$ the vector of $D_\omega$ arising from the random variable D.

This definition includes the usual case when the derivative has a payoff that is some function of one or multiple of the existing assets. As an example, let the function be $f: {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+$ and the derivative be $D_{[f]} := f(S_j)$, for some asset $j \in [N+1]$. Then, we have the expansion in Arrow securities $D_{[f]} =\sum _{\omega \in \Omega } f(S_{j \omega }) A_\omega$.

Market completeness means that every possible payoff is replicable with a portfolio of the existing assets. Hence it is an extremal case where every derivative is redundant.

Definition 4

(Complete market ², Definition 1.39) An arbitrage-free market model is called complete if every derivative D (Definition 3) is replicable, i.e., if for all $\varvec{D}\in \mathbbm {R}^K$ there exists a vector $\varvec{\xi }\in \mathbbm {R}^{N+1}$ such that $\varvec{D} = S^T \varvec{\xi }$ .

Problem statement

Risk-neutral pricing has two conceptual components. The first component is to find probability measures under which the existing asset prices are martingales. Such probability measures are derived from the market prices of the existing assets, and are denoted by $\mathbbm {Q}$ in the probability theory context, $\varvec{q}$ in the vector notation, and are elements of a set denoted by ${\mathcal {Q}}$. The second component is to use probability measures $\mathbb {Q}\in \mathcal {Q}$ to determine the range of fair prices of the non-traded financial security by computing expectation values. Note the "Methods" section for the martingale setting and the linear programming formulation. For incomplete markets, finding the range of possible prices for a derivative can be reduced to a maximization and a minimization problem, which can be solved via linear programming.

Definition 5

(Problem statement) With the definitions above, solve the linear programming formulations of the pricing problem $\max \limits _{\varvec{q} \in \mathbbm {R}^K} \varvec{D}^T \varvec{q}$ and $\min \limits _{\varvec{q} \in \mathbbm {R}^K} \varvec{D}^T \varvec{q}$ subject to $\varvec{q} \ge 0$ and $S\varvec{q} = \varvec{\Pi }$.

The solutions of the two optimization problems give the range of possible prices of the new asset, along with the martingale measures used to compute the prices.

Results

This section provides a summary of the results in this work. More details are found in the "Methods" section.

Derivative pricing with quantum algorithms for linear programming

We adapt to our settings two quantum algorithms for solving linear programs. The first is a quantum algorithm for zero-sum games³¹, which is asymptotically one of the best algorithms (in terms of query complexity) for approximating the solution of a linear program. We identify the conditions under which this quantum algorithm outperforms its classical counterpart. As an alternative to the zero-sum game algorithm, in Supplementary Material, we discuss the quantum version of the simplex method for this problem. Simplex methods are often used for solving linear programs in practice: they provide exponential worst-case complexity, but they can be very efficient on real-world problems. The performance of the two algorithms can be summarized informally as follows. The formal version for the quantum zero-sum game can be found in Theorems 2, and that for the quantum simplex algorithm can be found in the Theorem E.3 in the Supplementary Material.

Result

(Linear-programming approach, informal) Assume to have appropriate quantum access to a price vector $\varvec{\Pi }\in \mathbb {R}^{N+1}$ of assets, the payoff matrix $S \in \mathbb {R}^{(N+1) \times K}$ of a single-period market, and a derivative D. Then, with a absolute error $\varepsilon$ we can estimate the range of implied present values of the derivative

• With a quantum zero-sum games algorithm, with cost proportional to $\left( \sqrt{N} + \sqrt{K}\right) /\varepsilon ^3$,

• With a quantum simplex algorithm with cost proportional to $\kappa (S_B)\sqrt{K}\mu (S_B)+N$ per iteration, a number of iterations similar to the classical algorithm, and a final step with cost proportional to $\frac{\kappa (S)\mu (S)}{\varepsilon }$ operations.

Here $S_B$ is an invertible matrix obtained in the simplex method, and $\mu (S)$ ($\mu (S_B)$) is a function of the matrix S ($S_B$), upper bounded by the Frobenius norm of S and reported in Definition B.3 in the Supplementary Material. For the zero-sum gate algorithm we show a condition for a speedup based on the number of assets of the replicating portfolio of the derivative. This provides a characterization of the size parameters of linear programming in the context of derivative pricing.

Result

(Theorem 3, informal) If the portfolio that optimally replicates the derivative requires at most $\varepsilon (\sqrt{N+K})$ asset allocations, then the quantum zero-sum game algorithm performs fewer queries to the input oracles compared to the classical algorithm.

Martingale pricing for Black–Scholes–Merton assets

We show a case study of pricing a European call option in our framework. Noting that only the final time point determines the payoff of the option, we discretize the sample space of the Black–Scholes–Merton (BSM) stock price at the final time point. In this case, the payoff matrix can be efficiently computed, see Eq. (17). Hence, access to large amounts of additional data is not required. The consequence is that instead of having quantum access to S directly—for example from a QRAM of size O(KN)—we only require access to data of size O(N). We compare our approach to the analytical price derived from the BSM model. For better comparison, we develop two main ideas. First, we consider a formulation of the linear program that instead of finding the measure $\varvec{q}$ it finds a measure change. The measure change is here given by a vector defined by $q_i/p_i$, where $\varvec{p}$ is a given initial measure. Second, to find solutions closer to the BSM analytic price of the call option we develop a regularization technique. Based on an key property of the BSM measure change from Girsanov’s theorem, this regularization is performed by including a constraint for the slope of the measure change into the LP, which leads to a narrowing of the range of admissible prices. Results are presented in Fig. 1, where we show the derivative price as a function of the drift, volatility, strike, and stock price. We find that for a broad range of parameters, the analytic price of the derivative is contained in the interval of admissible prices given by the maximization and minimization linear programs. The regularization is successful in recovering the BSM price for sufficiently strong regularization parameters. In that sense, the linear programming framework contains the BSM model and allows modeling effects beyond the BSM model. While we considered the example of a single stock and the European call option, the framework can be extended to multiple stocks with correlated price dynamics and more complex options. Adding multiple non-redundant stocks generally achieves pricing that is consistent with the market and narrows the range of admissible prices by making the market more complete². We discuss additional details and steps towards a quantum implementation in the "Methods" section, and leave more complex scenarios for future work.

Figure 1

Range of admissible prices of the derivative while changing different parameters. The parameters are the drift $\mu$ in the top-left panel, the volatility $\sigma$ in the top-right panel, the strike price Z in the bottom-left panel and the stock price $\Pi$ in the bottom-right panel.

Derivative pricing in least-square markets

Taking insight from the fact that the sought-after measure $\varvec{q}$ is a vector satisfying the linear system $\varvec{\Pi }= S \varvec{q}$, we consider the quantum linear systems algorithm (originally proposed in¹³, and successive improvements^32,33) for solving the pricing problem. The right assumptions about the data input and output may allow for a large speedup (up to exponential for special cases) for the pricing problem. However, the quantum linear systems algorithm cannot directly be used to find positive solutions for an underdetermined linear equations system³⁴. The constraint $\varvec{q} \ge 0$ involves K constraints of the form $q_\omega \ge 0$, which all have to be enforced. For applying the quantum linear systems algorithm, we demand a stronger no-arbitrage condition, as will be discussed in the present section.

We introduce the definition of a least-squares market, a market model that defines the martingale measure as the solution to a least-square optimization problem obtained by the price system $(\varvec{\Pi }, S)$. We prove that market completeness is a sufficient condition for having a least-square market. As the converse is usually not true, a least-squares market is generally a weaker assumption than a complete market.

Result

(Definition 9 and Theorem 4, informal) An arbitrage-free market model is called a least-squares market if the pseudoinverse is a martingale probability measure. Least-squares markets include complete markets but not all incomplete markets.

We show that with appropriate assumptions on the data input, quantum linear systems solvers are able to provide the derivative price for least-squares markets with the potential of a large speedup over classical algorithms.

Result

(Theorem 5, informal) Assume to have appropriate quantum access to a price vector $\varvec{\Pi }$ of assets, the payoff matrix S of a single-period market, and a derivative D. If the market is a least-squares market (Definition 9) we can estimate the price of the derivative with a quantum matrix-inversion algorithm with cost proportional to $\frac{\kappa (S)\mu (S)}{\varepsilon }$.

Discussion

The study of quantum algorithms in finance is motivated by the large amount of computational resources deployed for solving problems in this domain. This work has explored the use of quantum computers for martingale asset pricing in one-period markets, with specific focus on incomplete markets. Contrary to other works in quantum finance, the problems considered here cannot be solved by a direct application of quantum speedups of Monte Carlo algorithms³⁵, albeit some of the employed algorithms use similar subroutines (e.g., amplitude amplification and estimation). One algorithm discussed here is based on the quantum zero-sum games algorithm (Theorem 2), one is based on the pseudoinverse algorithm (Theorem 5), and one is based on the quantum simplex method (Theorem E.3 in the Supplementary Material). The zero-sum game and the simplex algorithm return a value that maximizes (or minimizes) the price over the convex set of martingale measures, the linear system algorithm returns a single price consistent with the least-squares market assumption. We studied the conditions under which the quantum zero-sum game is better than the classical analogue and under which the simplex method is expected to beat the zero-sum game method.

All three quantum algorithms in this work apply to the single-period setting. Important problems in finance, such as dynamic hedging, naturally involve multi-period or continuous settings. Often continuous problems are discretized³⁶ and multi-period problems are solved by using subroutines for single periods. An example is American option pricing, which typically is performed in the discretized “Bermudan” setting³⁷ and reduces to estimating single-period expectation values and dynamic programming^37,38. Existing quantum algorithms for dynamic problems either rely on quantum algorithms for Monte Carlo^36,38, achieving at most quadratic quantum speedups, or use quantum neural networks³⁹ without provable guarantees.

We discuss additional technical points. In the simplex method and the linear systems algorithms, we have a factor of $\mu (S)$, which arises from the matrix inversion subroutine. This factor depends on the way the block-encoding of the matrices is created, referring to the matrix S for the least-square markets, and the matrix $S_B$ for the simplex algorithm. Possible bounds for $\mu (S)$ consist of the Frobenius norm or other more complex matrix functions, including tighter bounds for sparse matrices^13,14. We note that the payoff matrices of complete markets must be full-column rank, but they can be sparse. This leaves the possibility of market models admitting large speedups over the best classical algorithms. We remark again that all algorithms will only approximately return an equivalent martingale measure. Moreover, the quantum zero-sum game algorithm returns an $\varepsilon$-feasible measure, which is to be interpreted as a solution where every constraint is satisfied up to a tolerance $\varepsilon$. This will be ${\widetilde{O}}\left( \frac{1}{\varepsilon ^2} \right)$ sparse, which follows directly from the total number of iterations. However, this output is sufficient to provide an approximation of the derivative price with the desired precision. The quantum algorithms based on the simplex and the pseudo-inverse algorithm are able to create quantum states that are $\varepsilon$-close to an equivalent martingale measure.

We have provided a characterization of a class of market models, for which the quantum linear systems solvers^13,14 can be applied. We stress that a least-square market (Definition 9) is a weaker condition than market completeness and may allow for the large speedups made possible in principle by the quantum linear systems algorithms. It remains to be seen if the market condition can be weakened further for a broader application of the quantum linear systems algorithms. It is also interesting to estimate the quantum resources for the algorithms proposed in this manuscript in real-world settings, i.e., in a realistic market model for which the parameters such as volatilities have been estimated from historical data and in view of more realistic assumptions on the quantum hardware. We also leave as future work the study of other quantum algorithms such as interior-point methods^40,41.

Methods

Martingale measure

The martingale property of an asset means that its current price is fair, that is, it reflects the discounted expected future payoffs. Martingale probability measures are often denoted by $\mathbbm {Q}$ in the financial literature. We want to find martingale measures which are equivalent to the original probability measure $\mathbbm {P}$. In our setting, equivalency holds if for the new probability measure all probabilities are strictly positive. The probability measure is hence $\in \textrm{int}(\Delta ^K)$ and denoted by $\varvec{q} \in \textrm{int}(\Delta ^K)$. The martingale property here reads as: $\varvec{\Pi }= \mathbbm {E}_{\varvec{q}}[\varvec{S}].$ The corresponding probability measure $\varvec{q}$ is called equivalent martingale measure or risk-neutral measure, whose existence is discussed below. Finding a risk-neutral measure is equivalent to finding the fair prices of the Arrow securities, which are also called state prices³⁰. The state prices are in general not $\ell _1$-normalized, while the martingale measure is $\ell _1$-normalized. Under the risk-neutral measure, the expectation value of an Arrow security is the probability of the respective event,

$$\begin{aligned} \mathbbm {E}_{\varvec{q}}[A_{\omega }]= q_{\omega }. \end{aligned}$$

(2)

Putting together the martingale property, the expansion into Arrow securities of Eq. (1), and the property of the Arrow securities of Eq. (2) we obtain for the prices of the traded securities

$$\begin{aligned} \Pi _i= \mathbbm {E}_{\varvec{q}}[S_i] = \sum _\omega S_{i\omega } q_\omega . \end{aligned}$$

(3)

In matrix form this equation reads as $\varvec{\Pi }= S\varvec{q}$. The set of solutions that are contained in the simplex is given by

$$\begin{aligned} {{\mathcal {Q}}}:= \{ \varvec{q} \in \textrm{int}(\Delta ^K): S \varvec{q} = \varvec{\Pi }\}. \end{aligned}$$

(4)

Depending on the problem this set is either empty, contains single or multiple elements, or an infinite number of elements. The existence of a solution is discussed below via a no-arbitrage argument. We note that linear programming formulations do not admit optimization over open sets, i.e., cannot work with strict inequality constraints. Thus, we work with the closed convex set

$$\begin{aligned} \overline{{\mathcal {Q}}}:= \{ \varvec{q} \in \Delta ^K: S \varvec{q} = \varvec{\Pi }\}. \end{aligned}$$

(5)

As all algorithms in this work are approximation algorithms, the sets ${\mathcal {Q}}$ and $\overline{\mathcal {Q}}$ will be considered the same in practice.

Existence of martingale measure

Now we discuss the existence of a martingale measure, a property that is closely related to the non-existence of arbitrage in the market. Arbitrage is the existence of portfolios that allow for gains without the corresponding risk. A standard assumption in finance is that the market model under consideration is arbitrage-free.

Definition 6

(Arbitrage portfolios and no-arbitrage assumption) An arbitrage portfolio is defined as a portfolio with non-positive present value $\varvec{v}^T \varvec{\Pi }\le 0$ and non-negative future value for all market outcomes $S^T \varvec{v}\ge 0$, and at least one $j\in [K]$ such that $(S^T\varvec{v})_j > 0$. The no-arbitrage assumption, denoted by (NA), states that in a market model such portfolios do not exist.

No arbitrage implies the existence of a non-negative solution $\varvec{q}\ge 0$ via Farkas’ lemma.

Lemma 1

(Farkas’ lemma²⁴) For a matrix $S \in \mathbbm {R}^{(N+1)\times K}$ and a vector $\varvec{\Pi }\in \mathbbm {R}^{N+1}$ exactly one of the two statements hold. Either there exists a $\varvec{q} \in \mathbbm {R}^{K}$ such that $S\varvec{q} = \varvec{\Pi }$ and $\varvec{q} \ge \varvec{0}$, or there exists a $\varvec{v} \in \mathbbm {R}^{N+1}$ such that $\varvec{v}^T \varvec{\Pi }< 0$ and $S^T \varvec{v} \ge \varvec{0}$.

The condition $\varvec{q}\ge 0$ is sufficient in our case, given the discussion for Eq. (5). Farkas’ lemma can be extended to the case $\varvec{q}>0$^42,43 (see for example,²⁴, Example 2.20 & Exercise 2.23). Expressing Lemma 1 in words, in the first case the vector $\varvec{\Pi }$ is spanned by a positive combination of the columns of S. In the second case, there exists a separating hyperplane defined by a normal vector $\varvec{v}$ that separates $\varvec{\Pi }$ from the convex cone spanned by the columns of S. This lemma is immediately applicable to the arbitrage discussion. By Definition 6, existence of arbitrage is

$$\begin{aligned} \exists \, \varvec{v} \in \mathbbm {R}^{N+1} \text { such that: } \varvec{v}^T \varvec{\Pi }\le 0, \, S^T \varvec{v} \ge \varvec{0} \text { and } \exists j\in [K] \text { such that } (S^T\varvec{v})_j > 0. \end{aligned}$$

(6)

It is easy to see that the following condition implies Eq. (6),

$$\begin{aligned} \exists \, \tilde{\varvec{v}} \in \mathbbm {R}^{N+1} \text { such that: } \tilde{\varvec{v}}^T \varvec{\Pi }< 0 \text { and } S^T \tilde{\varvec{v}} \ge \varvec{0}, \end{aligned}$$

(7)

since from $\tilde{\varvec{v}}$, we can construct the $\varvec{v}$ in Eq. (6) by adding the amount $\vert \tilde{\varvec{v}}^T \varvec{\Pi }\vert$ to the first component². Then, by negation, if $(\textbf{NA})$ holds, Farkas’ lemma implies that case (i) holds, as case (ii) is ruled out. Since there exists a $\varvec{q} \ge 0$ such that $S\varvec{q} = \varvec{\Pi }$, the set $\overline{{\mathcal {Q}}}$ is by its definition not empty. In conclusion, assuming the absence of arbitrage implies that $|\overline{{\mathcal {Q}}}| \not = 0$. Together with the reverse direction, one obtains the “first fundamental theorem of asset pricing”².

Linear programming formulation of asset pricing

We reduce the problem of derivative pricing to two linear programs, corresponding to the finding of the maximum and the minimum price of the derivative. For pricing, we use any risk-neutral probability vector $\varvec{q} \in {\mathcal {Q}}$, if the set is not empty (i.e., under the no-arbitrage assumption). Lacking any other information or criteria, any element in ${\mathcal {Q}}$ is a valid pricing measure. The derivative is priced by computing the expectation value of the future payoff under the risk-neutral measure $\varvec{q}$, (discount factors are taken to be 1)

$$\begin{aligned} \Pi _{D,\varvec{q}} := \mathbbm {E}_{\varvec{q}}[{D}] = \sum _{\omega \in \Omega } D_\omega q_\omega . \end{aligned}$$

(8)

The price $\Pi _{D,\varvec{q}}$ is in our setting an inner product between the pricing probability measure $\varvec{q}$ and the vector describing the random variable of the financial derivative. Since ${\mathcal {Q}}$ is convex², it follows that the set of possible prices of the derivative is a convex set, i.e., an interval. We can formulate as a convex optimization problem the problem of obtaining the maximum and minimum price $\Pi _{{D},\min }$ and $\Pi _{{D},\max }$, respectively, which are possible over all the martingale measures. We obtain two optimization programs:

$$\begin{aligned} \Pi _{{D},\max }:= & {} \max _{\varvec{q} \in {\mathcal {Q}}} \mathbbm {E}_{\varvec{q}}[{D}], \end{aligned}$$

(9)

$$\begin{aligned} \Pi _{{D},\min }:= & {} \min _{\varvec{q} \in {\mathcal {Q}}} \mathbbm {E}_{\varvec{q}}[{D}] . \end{aligned}$$

(10)

when the solution of the optimization problems is a large range of prices, additional constraints can be added to narrow the range of admissible prices. We discuss a simple case study in the main part (with further details in Methods), where for a highly incomplete market we can narrow the range of admissible prices of a derivative to its analytic price with a regularization technique. In general, additional non-redundant assets reduce the number of valid risk-neutral measures by making the market more complete, and thus reduce the range of admissible prices.

In the following, we discuss the maximizing program Eq. (9), as the minimizing program is treated equivalently. We make explicit the LP formulation of the problem in Eq. (9). The first step is to replace the open set ${\mathcal {Q}}$ with the closed set $\overline{{\mathcal {Q}}}$. In that sense, we do not find probability measures that are strictly equivalent to $\mathbbm {P}$. However, the probability measures in $\overline{{\mathcal {Q}}}$ are arbitrarily close to the equivalent probability measures in ${{\mathcal {Q}}}$. We use vector notations and the price system to write the linear program

$$\begin{aligned} \max \limits _{\varvec{q} \in \mathbbm {R}^K} \varvec{D}^T \varvec{q} \quad \text {s.t.} \quad \varvec{q} \ge 0,\ S\varvec{q} = \varvec{\Pi }. \end{aligned}$$

(11)

From Supplementary Material, we see that the dual of the maximizing program is given by a linear program for the vector $\xi \in \mathbbm {R}^{N+1}$ as

$$\begin{aligned} \min \limits _{\xi \in \mathbbm {R}^{N+1}} \varvec{\Pi }^T \varvec{\xi }\quad \text {s.t.} \quad \varvec{D} \le S^T \varvec{\xi }. \end{aligned}$$

(12)

The financial interpretation is the following. The dual program finds a portfolio of the traded assets $\varvec{\xi }$ which minimizes today’s price $\varvec{\Pi }^T \varvec{\xi }$. The portfolio attempts to replicate (or “hedge”) the derivative future payoff. Ideal replication would mean $\varvec{D} = S^T \varvec{\xi }$, i.e, the payoff of the portfolio of assets is exactly the derivative outcome for all market outcomes. The linear program attempts to find a portfolio $\xi$ that “superhedges” the derivative in the sense that $\varvec{D} \le S^T \varvec{\xi }$.

In the Supplementary Material, we discuss in Lemma 4 how to cast the LP optimization problem in standard form, as per Definition A.5 in the Supplementary Material. To map the problem into a zero-sum game we require the parameters defining the LP to assume values in the interval $[-1, 1]$. For this, we need to assume knowledge of the following quantities.

Assumption 1

(Assumptions for conversion to standard form LP) Assume the following:

• Knowledge of $D_{\max }:= \max \limits _\omega D_\omega$, the maximum payoff of the derivative under any event.

• Knowledge of $S_{\max }:= \max \limits _{ij} S_{ij}$, the maximum entry of the payoff matrix S.

Quantum input model

We discuss the input model of our quantum algorithm and a way of obtaining quantum sampling access to probability distributions. We assume access to the price system and the derivative in the standard quantum query model⁴⁴ as follows.

Definition 7

(Quantum access to price system and derivative) Consider a price system as in Definition 2 and a derivative as in Definition 3. We say that we have quantum access to the price system and derivative if we have access to unitaries that perform $\vert jk\rangle \vert 0\rangle \rightarrow \vert jk\rangle \vert S_{jk}\rangle$, $\vert j\rangle \vert 0\rangle \rightarrow \vert j\rangle \vert \Pi _{j}\rangle$, and $\vert k\rangle \vert 0\rangle \rightarrow \vert k\rangle \vert D_{k}\rangle$ at unit cost, where $j\in [N_1]$, $k\in [N_2]$, and all numbers are encoded to fixed-point decimal precision.

In many practical cases such as conventional European options, the query access to the derivative $\varvec{D}$ can be obtained from a quantum circuit computing the function used to compute the derivative and query access to the payoff matrix S. We also define quantum vector access as the ability to construct a unitary that provides access to quantum states proportional to any vector in a subspace.

Definition 8

(Quantum vector access) We say that we have quantum vector access to d-dimensional vectors if, for any $\varvec{x} \in \mathbb {R}^d$ with $\Vert \varvec{x}\Vert _0 = s$ and $\Vert \varvec{x} \Vert _2 \le \beta$, we can construct a unitary in time $O\left( s\ \textrm{poly} \log d \right)$ that performs the mapping

$$\begin{aligned} \vert 0\rangle \mapsto \frac{1}{\sqrt{\beta }}\sum _{i=1}^d x_i \vert i\rangle \vert 0\rangle + \vert G\rangle \vert 1\rangle , \end{aligned}$$

in $O\left( \textrm{poly} \log d \right)$ time, where $\vert G\rangle$ is an sub-normalized arbitrary quantum state.

Further discussion on the quantum input model can be found in the Supplementary Material.

Quantum zero-sum game

The quantum algorithm for zero-sum games uses amplitude amplification and quantum Gibbs sampling to achieve a speedup compared to the classical algorithm. At the end of this section, we discuss the run time of the classical algorithm for solving LP based on the reduction to a zero-sum game. The quantum computer is used to sample from certain probability distributions created at each iteration. We state here the main result from³¹. An $\varepsilon$-feasible solution to our LP problem means that we can find a solution such that the constraint is relaxed to $A\varvec{q} \le \varvec{c} + \varvec{1} \varepsilon$.

Theorem 1

(Dense LP solver³¹, Theorem 13) Assume to have suitable quantum query access to a normalized LP in standard form as in Lemma 4, with r, R (defined in Lemma 5) known, along with quantum vector access as in Definition 8 to two vectors $\varvec{x} \in \mathbb {R}^{N_1}, \varvec{y} \in \mathbb {R}^{N_2}$. For $\varepsilon \in (0,1)$, there exists a quantum algorithm that finds an $\varepsilon$-optimal and $\varepsilon$-feasible $\varvec{y}$ with probability $1-\delta$ using ${\widetilde{O}}\left( (\sqrt{K}+\sqrt{N}) \left( \frac{R(r+1)}{\varepsilon } \right) ^3 \right)$ quantum queries to the oracles, and the same number of gates.

Theorem 1, can be directly adapted to the martingale pricing problem to obtain Algorithm 1 which we detail in the Supplementary Material. We formalized the embedding of a LP into a zero-sum game in Lemma D.1 in the Supplementary Material. The following statement follows from Theorem 1.

Theorem 2

(Quantum zero-sum games algorithm for martingale pricing) Let $(\varvec{\Pi }, S)$ be a price system with payoff matrix $S \in \mathbb {R}^{(N+1) \times K}$ and price vector $\varvec{\Pi }\in \mathbb {R}^{N+1}$, and let $\varvec{D}\in \mathbb {R}^{K}$ be a derivative. Assume to have quantum access to the matrix S and the vectors $\varvec{D}$ and $\varvec{\Pi }$ through Oracle 7, and two vectors $\varvec{x} \in \mathbb {R}^{N_1}$, and $\varvec{y} \in \mathbb {R}^{N_2}$ through quantum vector access as in Definition 8. Under Assumption 1, for $\varepsilon \in (0,1/2)$, Algorithm 1 in the Supplementary Material estimates $\Pi _{D, \max }$ with absolute error $\varepsilon$ and with high probability using

$$\begin{aligned} {\widetilde{O}}\left( (\sqrt{K}+\sqrt{N}) \left( \frac{(r+1)}{\varepsilon }D_{\max }\right) ^3 \right) \end{aligned}$$

quantum queries to the oracles, and the same number of gates. The algorithm also returns quantum access to an $\varepsilon$-feasible solution $\widetilde{\varvec{q}}$ as in Definition 8.

Along with the price, this algorithm also returns a $\varepsilon$-feasible solution $\widetilde{\varvec{q}}$. This solution could be used to price other derivatives $\varvec{D}$. Note that the original algorithm of Theorem 1 results in an absolute error $\varepsilon$ for $\alpha$, for the normalized problem, which then results in an absolute error of $\varepsilon D_{\textrm{max}}$ for the value of the derivative. Finding a relative error results from combining the algorithm described in Theorem 2 with standard techniques in randomized algorithms. We refer to⁴⁵ for more details. The procedure involves executing the algorithm with different values for the absolute error, in an exponential search for a lower bound for the quantity $\Pi _{D,\max }$. The expected runtime to obtain a relative error is

$$\begin{aligned} {\widetilde{O}}\left( (\sqrt{K}+\sqrt{N}) \left( \frac{(r+1)}{\varepsilon }\rho \right) ^3\log (\rho )\log (\log (\rho )) \right) , \end{aligned}$$

where $\rho = \frac{D_{\max }}{\Pi _{D, \max }}$. Recently, a new quantum algorithm for solving zero-sum games has been proposed⁴⁶. We discuss the implication of this algorithm in the Supplementary Material.

Classical zero-sum games algorithm and condition for quantum advantage

As the quantum algorithm, the classical algorithm requires a number of iterations $T=O\left( \left( \frac{(r+1)}{\varepsilon } D_{\max } \right) ^2 \log (\frac{NK}{\delta }) \right)$ to achieve the same error $\varepsilon$ for $\Pi _D$. The cost of the computation for a single iteration is dominated by the construction of a sampling data structure for two probability vectors of size $O\left( N \right)$ and $O\left( K \right)$, respectively. Hence, the classical run time of the algorithm is $O\left( T (N+K) \right) = O\left( \frac{(N+K)(r+1)^2 }{\varepsilon ^2}D_{\max }^2\log {\frac{NK}{\delta }} \right)$. Now we focus on a condition that allows for the quantum algorithm to be faster than the classical counterpart. The run time of the algorithms depends on the parameter r, which we can relate to the number of assets required to hedge a derivative.

Remark 1

Consider the context of Lemma A.6 in the Supplementary Material and its dual formulation. For any vector $\varvec{\xi }\in \mathbbm {R}^{N+1}$, let $r(\varvec{\xi })=\sum _{j=1}^{N+1} \xi _j$ and note that $\sum _{j=1}^{N+1} \xi _j \le \Vert \varvec{\xi }\Vert _1$. In addition, if $\vert \xi _j\vert \le 1$, we have $\Vert \varvec{\xi }\Vert _1 \le \Vert \varvec{\xi }\Vert _0$. In our setting, where $\varvec{\xi }^*$ is the optimal portfolio and $\Vert \varvec{\xi }^*\Vert _0$ is the number of assets in the optimal portfolio, we hence can use $r \le \Vert \varvec{\xi }^*\Vert _0$.

To obtain an advantage in the run time of the quantum algorithm we have to presuppose the following for $\varvec{\xi }$.

Theorem 3

If the derivative requires $\Vert \varvec{\xi }^*\Vert _0 \le \frac{\varepsilon (\sqrt{N+K})}{D_{\max }}$ asset allocations to be replicated, then Algorithm 1 performs fewer queries to the input oracles compared to the classical algorithm.

Proof

To find the upper bound for r which up to logarithmic factors still allows for a speedup, the value r needs to be such that $(\sqrt{N}+\sqrt{K})\left( \frac{(r+1)}{\varepsilon }D_{\max }\right) ^3 < \frac{(N+K) (r+1)^2 }{\varepsilon ^2}D_{\max }^2$. It is easy to see that $r+1 \le \frac{\varepsilon (\sqrt{N+K})}{D_{\max }}$. By Remark 1, we have an advantage in query complexity if the number of assets in the optimal portfolio $\Vert \varvec{\xi }^*\Vert _0$ is bounded by $\frac{ \varepsilon \sqrt{N+K}}{D_{\max }}$. $\square$

Linear programming martingale pricing with the Black–Scholes–Merton model

The Black–Scholes–Merton (BSM) model obtains the price for certain financial derivatives by analytically evaluating an expectation value of the future payoff under a risk neutral measure $\mathbbm {Q}$. We show a use case in the context of our linear programming formulation starting from a discretization of the standard BSM framework. In our example the payoff matrix can be efficiently computed, and hence access to a large QRAM may be avoided. We show how additional constraints in the LP regularize the model, so to narrow the range of admissible prices closer to the BSM price. The range of prices can be made comparable to the analytic value of the derivative. In the BSM model, we know an initial probability measure ${\mathbb {P}}$, which may not be a martigale measure. In the linear programming framework, it is hence more appropriate to consider a measure change from this initial probability measure.

Pricing with a measure change

An importation notion in pricing is the change of measure from the original measure to the martingale measure. The change of measure is described by the Radon-Nikodym derivative, which is denoted by the random variable $X:=\frac{d\mathbbm {Q}}{d\mathbbm {P}}$. We extend the main part of the paper by formulating the problem of pricing a derivative when we are given an initial probability measure $\mathbbm {P}$ via a vector $\varvec{p} \in \Delta ^K>0$. We consider the measure change to a new probability measure $\mathbbm {Q}$ via the vector $\varvec{q} \in \Delta ^K$. The Radon-Nikodym derivative in this context is an $\varvec{x} \in \mathbbm {R}^K_+$ for which $x_\omega := \frac{q_\omega }{p_\omega }, \forall \omega \in \Omega$. The martingale property for the assets under $\mathbbm {Q}$ and the pricing of a derivative are formulated as

$$\begin{aligned} \Pi _j = \mathbb {E}_{\mathbbm {Q}}[S_j]= & {} \mathbb {E}_{\mathbbm {P}}[X S_j] = \sum _{\omega \in \Omega } p_\omega x_\omega S_{j\omega }, \end{aligned}$$

(13)

$$\begin{aligned} \mathbb {E}_{\mathbbm {Q}}[D]= & {} \mathbb {E}_{\mathbbm {P}}[X D] = \sum _{\omega \in \Omega } p_\omega X_\omega D_\omega . \end{aligned}$$

(14)

Hence, using $\circ$ for the Hadamard product (element-wise product) between vectors, $\varvec{D}[\varvec{p}] := \varvec{p} \circ \varvec{D}$ and $(S[\varvec{p}])_{j} := \varvec{p}\circ S_{j}$, we obtain the linear program for the maximization problem of Eq. (9) of the Radon-Nikodym derivative

$$\begin{aligned} \max \limits _{\varvec{x} \in \mathbbm {R}^K} \varvec{D}[\varvec{p}]^T \varvec{x} \quad \text {s.t.} \quad \varvec{x} \ge 0,\ S[\varvec{p}]\varvec{x} = \varvec{\Pi }. \end{aligned}$$

(15)

The minimization problem is obtained similarly.

Discretized Black–Scholes–Merton model for a single time step

In the BSM model, the market is assumed to be driven by underlying stochastic processes (factors), which are multidimensional Brownian motions. Multi-factor models have a long history in economics and finance⁴⁷. The dimension and other parameters of the model, like drift and volatility, can be estimated from past market data^48,49,50. We perform our first drastic simplification and set the number of risky assets to be $N=1$. This makes the market highly incomplete but provides the analogue to the simplest BSM case. For the randomness, while the Brownian motion stochastic process has a specific technical definition, in the single-period model a Brownian motion becomes a Gaussian random variable. Hence, we assume here that we have a single Gaussian variable driving the stock. Under the measure $\mathbbm {P}$, let B be a Gaussian random variable, also denoted by $B\sim {\mathcal {N}}(0,1)$. The future stock price $S_2$ is modeled as a function of B. In BSM-type models, asset prices are given by a log-normal dependency under $\mathbbm {P}$,

$$\begin{aligned} S_2 = \Pi \ e^{\sigma B + \mu -\frac{1}{2}\sigma ^2}, \end{aligned}$$

(16)

using today’s price $\Pi \in \mathbbm {R}_+$, the volatility $\sigma \in \mathbbm {R}_+$, and the drift $\mu \in \mathbbm {R}_+$. The stock price is not a martingale under $\mathbbm {P}$, which can be checked by evaluating $\mathbbm {E}_{\mathbbm {P}}[S_2] = \Pi e^\mu \ne \Pi$. However, it can be shown that under a change of probability measure, the Gaussian random variable can be shifted such that the resulting stock price is a martingale. The following lemma is a simple version of Girsanov’s theorem for measure changes for Gaussian random variables.

Lemma 2

(Measure change for Gaussian random variables⁵¹) Let $B \sim {\mathcal {N}}(0,1)$ under a probability measure $\mathbbm {P}$. For $\theta \in \mathbbm {R}$ define $B':= B + \theta$. Then, $B' \sim {\mathcal {N}}(0,1)$ under a probability measure $\mathbbm {Q}$, where this measure is defined by $\frac{d \mathbbm {Q}}{d \mathbbm {P}} = \exp \left( -\theta B - \frac{1}{2} \theta ^2\right)$.

In standard Black–Scholes theory, we use $\theta = \mu /\sigma$ to obtain $B' \sim {\mathcal {N}}(0,1)$ under $\mathbbm {Q}$ where $\frac{d \mathbbm {Q}}{d \mathbbm {P}} = \exp \left( -\frac{\mu }{\sigma } B - \frac{1}{2} \left( \frac{\mu }{\sigma }\right) ^2\right)$. Note that $S_2$ satisfies the key property of a martingale as $\mathbbm {E}_{\mathbbm {Q}}[S_2] = \mathbbm {E}_{\mathbbm {P}}\left[ \frac{d \mathbbm {Q}}{d\mathbbm {P}} S_2\right] = \Pi \ \mathbbm {E}_{\mathbbm {P}}[e^{\beta B -\frac{1}{2} \beta ^2}] = \Pi$, with $\beta := \sigma - \mu /\sigma$.

We use the analytically solvable BSM model as a benchmark for the linear programming formulation. In order to fit this model into our framework, we discretize the sample space. We take a truncated and discretized normal random variable centered at 0 on the interval $[-6, 6]$. We take the sample space to be $\Omega := \{-K_0,\cdots ,0,\cdots , K_0\}$ and $K_0 \in \mathbbm {Z}_+$. The size of the space is $K = \vert \Omega \vert = 2K_0+1$. The probability measure for $\omega \in \Omega$, is $p_\omega = e^{-\frac{36\omega ^2}{2 K_0^2}}/ \Sigma$, where $\Sigma = \sum _{\omega \in \Omega } e^{-\frac{36\omega ^2}{2 K_0^2}}$. In analogy to the BSM model, we define the payoff matrix of future stock prices as

$$\begin{aligned} S_{1\omega }:= & {} 1, \nonumber \\ S_{2\omega }:= & {} \Pi \ e^{ {\sigma } B(\omega ) + \mu - \frac{1}{2}\sigma ^2}, \end{aligned}$$

(17)

where $B(\omega ) = \frac{6\omega }{K_0}$. An important property of this example is that each $S_{2\omega }$, can be computed in $O\left( \log K \right)$ time and space via Eq. (17). For settings with more than a single asset $N>1$, if we have more than one Gaussian random variables driving the assets, $d>1$, we expect this cost to be $O\left( d\log (NK) \right)$.

Regularization

Our optimization framework allows for a much larger set of Radon–Nikodym derivatives (all entry-wise positive K-dimensional vectors) than the set implied from the log-normal BSM framework. We observe that this freedom allows for a large range of prices corresponding to idiosyncratic solutions for the measure change. One method to overcome this issue is to include a larger number of non-redundant risky assets into the model. The assets make the market less incomplete and generally narrow the range of prices. As this strategy will also increase the complexity of the model, we pursue an alternative regularization strategy here.

To find solutions closer to the Black–Scholes measure change, we develop a regularization technique. This regularization is performed by including a constraint for the slope of $\varvec{x}$ into the LP. The regularization leads to a narrowing of the range of admissible prices. The linear program in Eq. (15) optimizes over the set of measure changes

$$\begin{aligned} {\mathcal {M}}:= \left\{ \varvec{x} \in \mathbbm {R}^K \right\} , \end{aligned}$$

(18)

with the additional constraints given by the linear program. On the other hand, the Black–Scholes formalism essentially uses a particular measure from the one-parameter set of Gaussian measure changes (see Lemma 2)

$$\begin{aligned} {\mathcal {M}}_{\textrm{BS}}:= \left\{ \varvec{x}(\theta ) \in \mathbbm {R}^K\ \big |\ x_\omega (\theta ) = \exp \left( -\theta B(\omega ) - \frac{1}{2} \theta ^2\right) , \theta \in \mathbbm {R}, \omega \in \Omega \right\} . \end{aligned}$$

(19)

Optimizing over ${\mathcal {M}}$ finds a much larger range of prices, as the set ${\mathcal {M}}_{\textrm{BS}}$ is of course much smaller than ${\mathcal {M}}$. Via a regularization technique, we can find measures that are closer to the set ${\mathcal {M}}_{\textrm{BS}}$. Specifically, we add a first derivative constraint. Viewing $\omega$ continuously, for the Girsanov measure change it holds that

$$\begin{aligned} \frac{\partial x_{\omega }}{\partial \omega } = \frac{\partial x_{\omega }}{\partial B} \frac{\text {d}B}{\text {d}\omega } = -\theta x_\omega \Delta , \end{aligned}$$

(20)

where $\Delta := 6/K_0$. Since $\omega$ is discrete, we obtain $x_{\omega +1} - x_{\omega } \approx -\theta _0 x_{\omega } \Delta$ using $\theta _0:= \mu /\sigma$. Employing this relationship as a constraint and using $1/\eta >0$ as a closeness parameter, we optimize over the set

$$\begin{aligned} {\mathcal {M}}_{\textrm{reg}}(\eta ):= \left\{ \varvec{x}(\theta ) \in \mathbbm {R}^K\ \big |\ \left| (1 - \theta _0 \Delta ) x_\omega - x_{\omega +1}\right| \le \frac{1}{\eta }, \forall \omega \in \Omega \setminus \{K_0\} \right\} . \end{aligned}$$

(21)

Instead of using this particular regularization, there could be other approaches to narrow the range between admissible prices. As mentioned, an approach founded in theory² is to complete the market with assets for which we know the price analytically/from the market. When there are more benchmark assets, we expect the range of prices to narrow, and in the limit of a complete market, every derivative has a unique price. We leave this approach for future work.

Call option

We define the common case of an European call option. The European call option on the risky asset gives the holder the right but not the obligation to buy the asset for a predefined price, the strike price. Hence, the future payoff is defined as $D:= \max \left\{ 0, S_2 - Z \right\} ,$ where $Z \in \mathbbm {R}_+$ is the strike price. Using the stock prices from Eq. (17), for a specific event $\omega \in \Omega$ the payoff under $\mathbbm {P}$ is

$$\begin{aligned} D(\omega ) = \max \left\{ 0, \Pi _i e^{ \sigma B(\omega ) + \mu -\frac{1}{2}\sigma ^2} - Z \right\} . \end{aligned}$$

(22)

Numerical tests

We test this approach for the call option, using the linear program in Eq. (15), its minimization form, and the payoff matrix Eq. (17). We compare the approach to the analytical price derived from the Black–Scholes–Merton model. We find that for a broad enough range of parameters, the analytic price of the derivative is contained in the interval of admissible prices given by the maximization and minimization linear programs of Eq. (15). The regularization is successful in recovering the BSM price for strong regularization parameter. In that sense, this linear programming framework contains the BSM model and allows modeling effects beyond the BSM model.

In the experiments as assume a payoff matrix of size $S\in \mathbb {R}^{2 \times 100}$, i.e. $N=1$ and $K=100$, with the first row representing risk-less asset with payoff 1 in all the events. In our plots, we start with fixing the parameters $\Pi =10$, drift $\mu =0.5$ (and $\mu =0.1$ for Fig. 1), volatility $\sigma =1$ and strike price for the derivative of $Z=10$. In most of the plots, the analytic value of the derivative is always within the range of the admissible prices of the minimization and maximization problem. The analytical dependencies are the standard BSM results⁴⁸. Changing the drift does not influence the analytical price of the derivative, because the change of measure makes the price of the call option independent of $\mu$. For benchmarking, in Fig. 2 we test different value for the regularization parameter. In picture (a) we find that the regularization parameter in the range of $\eta \in \{1, 2, 5\}$ allows to get close to the analytic solution of the Radon-Nikodym derivative. In panel (b) we scan the value of the regularization parameter for $\eta \in [0.1, 100]$. We observe that in our example, the solutions from the two LPs get closer to the analytic price as $\eta$ goes larger.

Figure 2

Changing the regularization parameter. (Left panel) The Radon–Nikodym derivative obtained analytically is compared to the solution vector obtained from solving the linear program. (Right panel) The regularization parameter constrains the solution space and hence limits the admissible prices of the derivative.

Towards a quantum implementation

The requirement for an efficient quantum algorithm is to have quantum query access for the vectors (as in Definition 7) . For the payoff matrix S we can build efficient quantum circuits, without requiring large input data. To create the mapping $\vert i,j\rangle \mapsto \vert i,j,S_{ij}\rangle$, we require classical access to input quantities such as drift and volatility, quantum access to the prices, and apply the arithmetic operations in Eq. (17). Following the same reasoning, since the vector $\varvec{D}$ is efficiently computable from the matrix S, we can build efficient query access to its entries. We remark that such efficient computation can also hold for more complicated models, i.e., models that are more complicated than log-normal such as Poisson jump processes or Levy heavy-tailed distributions.

Quantum risk-neutral pricing in least-squares markets

With quantum linear systems algorithms, we can prepare a quantum state that is proportional to $S^+ {\varvec{\Pi }}$, where $(\varvec{\Pi }, S)$ is a price system and $S^+$ is the pseudo-inverse of the matrix S. The pseudo-inverse of S finds the minimum $\ell _2$ norm solution to a linear system of equations; see also Theorem A.2 in the Supplementary Material. Unfortunately, this solution can lie outside the positive cone, and thus outside $\textrm{int} (\Delta ^K)$, so we need a stronger no-arbitrage condition to guarantee to find a valid probability measure. Existence of an element in ${\mathcal {Q}}$ follows from the no-arbitrage assumption as discussed after Definition 6. We have to assume that one of the probability measures in ${\mathcal {Q}}$ is indeed also the minimum $\ell _2$-norm solution. A market satisfying this condition is what we call a least-square market. We note that the literature on incomplete markets has considered the pseudo-inverse of the payoff matrix before³⁰. In general, positive solutions are obtained by adding suitable vectors from the nullspace to the pseudo-inverse solution⁵².

Definition 9

(Least-squares market) An arbitrage-free market model is called a least-squares market if $S^+\varvec{\Pi }\in \mathcal {Q}$.

Under this assumption, the pseudo-inverse is guaranteed to find a positive solution to $S \varvec{q}= \varvec{\Pi }$. This assumption is obviously stronger than the no-arbitrage assumption, which only implies that $|\mathcal {Q}|\ne 0$. However, it is not as strong as assuming market completeness, which means that every possible payoff is replicable with a portfolio of assets. From the second fundamental theorem of asset pricing (see Theorem A.3 in the Supplementary Material), there exists a unique martingale measure if and only if every financial derivative is redundant, i.e., replicable via a unique portfolio of traded assets. If the market is not complete, a derivative is in general not replicated. The next theorem states that market completeness is a stronger assumption than the assumption of a least-square market.

Theorem 4

Let $(\varvec{\Pi }, S)$ be a price system. If the market model $S \in \mathbb {R}^{(N+1) \times K}$ is complete, it is also a least-squares market.

Proof

Market completeness, via Theorem A.3 in the Supplementary Material, implies that $|\mathcal {Q}|=1$, so there exists a unique measure $\textbf{q}^*\in \mathcal {Q}$ such that $\varvec{q}^*\in \textrm{int}(\Delta ^K)$ and $S \varvec{q}^*= \varvec{\Pi }$. Furthermore, market completeness (via Lemma A.4 in the Supplementary Material), implies that the matrix S is square and has full rank. Define the pseudo-inverse solution as $\varvec{q}^+:= S^{+} \varvec{\Pi }$. Since $N+1=K$, and the assets are non-redundant, $\varvec{q}^+$ satisfies $S \varvec{q}^+= \varvec{\Pi }$ and is the unique solution to the equation system. Thus, $\varvec{q}^+$ must also have the property of being in $\textrm{int}(\Delta ^K)$ and $\varvec{q}^+ = \varvec{q}^*$. $\square$

Market completeness is usually a rather strong hypothesis, while the least-square market includes a broader set of markets. Both the market completeness and the weaker least-squares market conditions enable the use of quantum matrix inversion for finding a martingale measure. Even weaker conditions may also suffice for the use of quantum matrix inversion. For instance, markets where some $\varvec{q} \in {\mathcal {Q}}$ is $\varepsilon$-close to $S^+\varvec{\Pi }$ may also be good candidates. Another case could arise in the situation where there is a vector $\varvec{v}$ in the null space of S such that $(\varvec{q}^+ + \varvec{v}) \in \mathcal {Q}$. We leave the exploration of weaker assumptions for future work. In the following theorem, we assume block-encoding access to the matrix S, as per Definition B.2 in the Supplementary Material.

Theorem 5

(Quantum pseudo-inverse pricing in incomplete least-squares markets) Let $(\varvec{\Pi }, S)$ be the price system of a least-squares market with the extended payoff matrix $S\in \mathbb {R}^{(K+N+1)\times (K+N+1)}$ satisfying $\Vert {S}\Vert _2 \le 1$ and $\kappa (S) \ge 2$. Let $U_{S}$ be a $(\mu (S), 0)$-block-encoding of matrix S, and assume to have quantum access as Definition B.1 in the Supplementary Material to the extended price vector $\varvec{\Pi }\in \mathbb {R}^{(K+N+1)}$ and the extended derivative vector $\varvec{D} \in \mathbb {R}^{(K+N+1)}$. Let $\gamma \in [0,1]$ and $\sqrt{\gamma } \le \Vert P_{\mathrm{{col}}}(S)\vert \varvec{\Pi }\rangle \Vert _2$, where $P_{\mathrm{{col}}}(S)$ is the projector into the column space of S. For $\varepsilon > 0$ and $\varvec{q}^+ := S^+ \varvec{\Pi }$, there is a quantum algorithm that estimates $\Pi _{D,\varvec{q}^+}:= \varvec{D}^T \varvec{q}^+$ with absolute error $\Vert {\varvec{\Pi }}\Vert _2\Vert {\varvec{D}}\Vert _2 \varepsilon$ and high probability using ${\widetilde{O}}\left( \frac{\kappa (S)\mu (S)}{\varepsilon \sqrt{\gamma }} \right)$ queries to the oracles.

We leave the proof in the Supplementary Material. Interestingly, when the market is complete, the factor $\sqrt{\gamma }$ is 1, so we can think of this parameter as a quantifier of market incompleteness, which is reflected in the run time of our algorithm (and can be estimated with amplitude estimation). In addition, we can recover an classical approximation of the risk-neutral measure $\varvec{q}^+$. Obtaining an estimate of $\varvec{q}^+$ allows us to price derivatives classically. We can use tomography (Theorem B.6 in the Supplementary Material) which in general incurs a cost of K, or we can use weaker models of tomography such as $\ell _\infty$ tomography or classical shadows to extract useful information.