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arxiv: 2606.31076 · v1 · pith:RM6U6RJ5new · submitted 2026-06-30 · 🪐 quant-ph · cs.NA· math.NA

Quantum Derivative Pricing for SPDEs via BDSDE Representation

Xinmiao Li , Yanqiao Wang , Rundi Lu , Zhengwei Liu , Jin-Peng Liu This is my paper

Pith reviewed 2026-07-01 05:53 UTC · model grok-4.3

classification 🪐 quant-ph cs.NAmath.NA
keywords quantum derivative pricingBDSDE representationQA-MLMCstochastic partial differential equationsGreeks estimationmultilevel Monte Carlostochastic volatility modelsquantum algorithms
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The pith

Quantum-accelerated multilevel Monte Carlo methods reduce sampling complexity for SPDE derivative pricing and Greeks from quadratic to linear in the error tolerance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes quantum speedups for pricing derivatives whose underlying dynamics follow stochastic partial differential equations by rewriting the problems as backward doubly stochastic differential equations. It introduces conditional and nested quantum-accelerated multilevel Monte Carlo estimators that cut the number of samples needed from order one over epsilon squared to order one over epsilon, for additive error epsilon. The same framework supplies quantum estimators for first- and second-order Greeks via likelihood-ratio and Malliavin-weight methods, and it is illustrated on Heston-type stochastic-volatility models. A family of Forward-Backward Taylor schemes is constructed to discretize the stochastic integrals so that the overall estimators retain order-one strong convergence, which is required for the quadratic quantum advantage to appear. If these constructions succeed, high-precision pricing calculations that are currently prohibitive on classical computers become feasible on quantum hardware.

Core claim

By representing SPDE derivative pricing problems through their BDSDE equivalents and applying conditional and nested QA-MLMC to the resulting expectations, together with Forward-Backward Taylor discretization that achieves global strong-error order one, the sampling complexity of classical Monte Carlo improves from ilde O(\epsilon^{-2}) to ilde O(\epsilon^{-1}) for both prices and Greeks within additive error \epsilon.

What carries the argument

Conditional and nested quantum-accelerated multilevel Monte Carlo (QA-MLMC) estimators applied to BDSDE representations, supported by Forward-Backward Taylor discretization schemes that deliver global strong-error order-one convergence for pricing and Greek estimators.

If this is right

  • Quantum estimators exist for first-order and second-order Greeks using both likelihood-ratio and Malliavin-weight representations.
  • The same complexity improvement applies to Heston-type stochastic-volatility models.
  • Numerical experiments confirm that the discretization schemes attain the convergence orders needed for the full quadratic quantum speedup.
  • The framework supplies both pricing and sensitivity analysis under a single set of quantum-accelerated estimators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same BDSDE-plus-QA-MLMC structure could be tested on other high-dimensional path-dependent claims whose classical Monte Carlo cost is dominated by nested expectations.
  • If the oracle construction cost scales favorably, the linear sampling regime might combine with quantum linear-system solvers to produce end-to-end quantum pipelines for entire pricing workflows.
  • The order-one discretization property may extend the approach to Greeks of even higher order without losing the quadratic quantum advantage.

Load-bearing premise

Efficient quantum circuits or oracles can be built for the conditional and nested expectations that appear in the BDSDE representations, and the Forward-Backward Taylor schemes really attain the stated order-one strong convergence.

What would settle it

A concrete counter-example or numerical test in which the Forward-Backward Taylor discretization fails to achieve global strong-error order one for the price or Greek estimators, or in which no quantum circuit of the assumed size exists for the required conditional expectations.

Figures

Figures reproduced from arXiv: 2606.31076 by Jin-Peng Liu, Rundi Lu, Xinmiao Li, Yanqiao Wang, Zhengwei Liu.

Figure 1
Figure 1. Figure 1: FIG. 1. Multilevel bias, variance, and cost for the pricing estimator (left), first-order Greek estimator (middle), and second [PITH_FULL_IMAGE:figures/full_fig_p035_1.png] view at source ↗
read the original abstract

We study quantum speedups of derivative pricing for stochastic partial differential equation (SPDE) models through their backward doubly stochastic differential equation (BDSDE) representations. We develop conditional and nested quantum-accelerated multilevel Monte Carlo (QA-MLMC) methods for estimating the resulting conditional and nested expectations, improving the sampling complexity of classical Monte Carlo methods from $\widetilde{O}(\epsilon^{-2})$ to $\widetilde{O}(\epsilon^{-1})$ within additive error $\epsilon$. We apply the framework to derivative pricing and sensitivity analysis, providing quantum-accelerated estimators for prices as well as first-order and second-order Greeks, likelihood-ratio and Malliavin-weight representations for Greeks, and Heston-type stochastic-volatility models. To enable efficient multilevel coupling, we construct a family of Forward--Backward Taylor discretization schemes for the stochastic integrals arising in the BDSDE representations and establish global strong-error order one convergence for pricing and Greek estimators. Numerical experiments showcase our schemes for first-order and second-order Greeks can reach the required orders for the full quadratic quantum speedups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops BDSDE representations for derivative pricing and Greek computation in SPDE models, then introduces conditional and nested QA-MLMC estimators that reduce classical Monte Carlo sampling complexity from ilde{O}(\epsilon^{-2}) to ilde{O}(\epsilon^{-1}). It constructs Forward-Backward Taylor discretization schemes claimed to achieve global strong-error order one for both pricing and Greek estimators, applies the framework to Heston-type models and likelihood-ratio/Malliavin representations, and reports numerical experiments confirming the required convergence orders for quadratic quantum speedups.

Significance. If the end-to-end complexity claims hold, the work would extend quantum Monte Carlo advantages to SPDE-based pricing and sensitivity analysis, with the order-one convergence of the Taylor schemes and numerical validation for first- and second-order Greeks constituting concrete strengths. The explicit treatment of nested expectations and stochastic-volatility extensions adds technical value.

major comments (2)
  1. [Complexity analysis] § on complexity analysis (near the QA-MLMC definition): the sampling-complexity reduction from ilde{O}(\epsilon^{-2}) to ilde{O}(\epsilon^{-1}) is derived under the assumption that each quantum oracle for the conditional/nested BDSDE expectations has gate cost polylog(1/\epsilon) independent of spatial dimension. No explicit gate-complexity bound is supplied for the oracles arising from spatial truncation of the SPDE (finite-element or spectral) to a high-dimensional SDE; this assumption is load-bearing for whether the quadratic advantage survives once oracle construction cost is counted.
  2. [Discretization and convergence] § on Forward-Backward Taylor schemes and Theorem establishing order-one convergence: while global strong error O(h) is proved for the discretization, the multilevel coupling argument for the nested QA-MLMC estimators is not shown to preserve the same order when the inner conditional expectations are replaced by quantum oracles; this step is required to justify that the full estimator attains the stated ilde{O}(\epsilon^{-1}) complexity.
minor comments (2)
  1. [Preliminaries] Notation for the BDSDE coefficients and the precise definition of the nested expectation operators could be collected in a single preliminary section to improve readability.
  2. [Numerical experiments] Numerical experiments report convergence orders for Greeks but do not tabulate the effective dimension of the spatial discretization or the observed gate counts; adding these would strengthen the validation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address each major comment below with honest responses and indicate planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Complexity analysis] § on complexity analysis (near the QA-MLMC definition): the sampling-complexity reduction from ilde{O}(\epsilon^{-2}) to ilde{O}(\epsilon^{-1}) is derived under the assumption that each quantum oracle for the conditional/nested BDSDE expectations has gate cost polylog(1/\epsilon) independent of spatial dimension. No explicit gate-complexity bound is supplied for the oracles arising from spatial truncation of the SPDE (finite-element or spectral) to a high-dimensional SDE; this assumption is load-bearing for whether the quadratic advantage survives once oracle construction cost is counted.

    Authors: We acknowledge that the end-to-end complexity claim relies on the polylog gate cost assumption for the quantum oracles, and the manuscript does not supply an explicit bound accounting for the cost of spatial truncation to a high-dimensional SDE. The analysis in the paper focuses on sampling complexity under standard quantum oracle assumptions common in the QA-MLMC literature. For a fixed spatial discretization (constant number of modes or elements), the effective dimension is independent of \epsilon and the oracle costs remain polylog(1/\epsilon) by existing quantum SDE simulation results under Lipschitz conditions. We agree this point merits clarification and will add a remark in the complexity section of the revised manuscript noting the fixed-dimension assumption and that full space-time discretization costs are beyond the current scope. This constitutes a partial revision. revision: partial

  2. Referee: [Discretization and convergence] § on Forward-Backward Taylor schemes and Theorem establishing order-one convergence: while global strong error O(h) is proved for the discretization, the multilevel coupling argument for the nested QA-MLMC estimators is not shown to preserve the same order when the inner conditional expectations are replaced by quantum oracles; this step is required to justify that the full estimator attains the stated ilde{O}(\epsilon^{-1}) complexity.

    Authors: The referee is correct that the order-one strong convergence is established for the classical Forward-Backward Taylor schemes, and the manuscript does not explicitly verify that the multilevel coupling for the nested QA-MLMC preserves this order once inner expectations are approximated by quantum oracles. The construction ensures oracle accuracy is set to not exceed the discretization error at each level, allowing the variance decay and coupling to carry over. However, we recognize an explicit lemma bridging the classical coupling to the quantum-oracle case is missing. We will add a short argument in the revised manuscript (near the QA-MLMC definition and the convergence theorem) showing that the oracle error bounds suffice to maintain the required multilevel properties and thus the ilde{O}(\epsilon^{-1}) complexity. This will be incorporated as a revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims rest on external quantum-oracle assumptions

full rationale

The paper's core contribution is the application of QA-MLMC to BDSDE representations of SPDE pricing problems, together with Forward-Backward Taylor schemes whose strong order-1 convergence is proved. No equation in the provided abstract or description reduces a claimed prediction to a fitted input by construction, nor does any load-bearing step collapse to a self-citation whose content is itself unverified. The stated complexity improvement from ilde{O}(ε^{-2}) to ilde{O}(ε^{-1}) is conditional on the separate assumption that oracles for the conditional/nested expectations can be realized with polylog cost; that assumption is flagged explicitly as the weakest link and is not derived inside the paper. Hence the derivation chain remains non-circular at the level of the enumerated patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated or can be extracted.

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    M. Kac, On distributions of certain wiener functionals, Transactions of the American Mathematical Society65, 1 (1949). 40 Supplementary Materials Appendix A: Terminal-value SPDE and Its BDSDE Representation It is well known that the Feynman-Kac formula provides a classical probabilistic representation for solutions of linear parabolic PDEs [76]. Given a t...

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    Proof of Proposition 3 Proposition(Strong-error order for the direct pricing payoff).Fix(t, x)∈[0, T]×R d and a uniform grid{t k}N k=0 withh= (T−t)/N

    Direct Pricing Estimator a. Proof of Proposition 3 Proposition(Strong-error order for the direct pricing payoff).Fix(t, x)∈[0, T]×R d and a uniform grid{t k}N k=0 withh= (T−t)/N. Let Ptk = Φ(t, tk)G X t,x tk +Y tk , whereY tk is the exact accumulated payoff Ytk = Z tk t Φ(t, r) F(r, X t,x r ) +d(r)H(r, X t,x r ) dr+ Z tk t Φ(t, r)H(r, X t,x r ) d← −B r. L...

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    Moreover,S int satisfies the accumulated stability estimate defined in Definition 4

    The strong-error orders ofS X,S Φ andS int arep X , pΦ, pint respectively. Moreover,S int satisfies the accumulated stability estimate defined in Definition 4

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    There existsM >0, independent ofh, such thatΦ (h) k andG(X t,x tk )are uniformly bounded inL 4 W,B byM. 3.Gis globally Lipschitz, i.e., there existsL G >0such that|G(x)−G(y)| ≤L G|x−y|. Then the payoff approximation satisfies the joint strong-error bound sup 0≤k≤N Ptk −P (h) k L2 =O(h p), p= min{p X , pΦ, pint}. Proof.First, for each 0≤k≤N, we have Ptk −P...

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    First-order Greek Estimators a. Proof of Proposition 6 Proposition(Strong-error order for the first-order Greek payoff).Fix(t, x)∈[0, T]×R d and1≤i≤dand define P (i) tk =Φ(t, tk)∇G(X t,x tk )⊤J t,x tk ei +Y (i) tk , where Y (i) tk = Z tk t Φ(t, s)∇ x F(s, X t,x s ) +d(s)H(s, X t,x s ) ⊤ J t,x s ei ds+ Z tk t Φ(t, s)∇ xH(s, X t,x s )⊤J t,x s ei d← −B s. Le...

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