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Path-dependent option pricing with two-dimensional PDE using MPDATA

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper shows that arithmetic-average Asian options can be priced by casting the two-dimensional Black-Merton-Scholes PDE as an advection-only transport problem and solving it with the MPDATA finite-difference scheme, yielding…

desk verdict Clean, honest 2D MPDATA proof-of-concept for Asian options with reusable test case; loose validation but no load-bearing flaw. read the letter →

arxiv 2505.24435 v1 pith:GLQOILSN submitted 2025-05-30 q-fin.CP

Pawe{l} Magnuszewski , Sylwester Arabas This is my paper
classification q-fin.CP MSC 91G2091G6065M06
keywords AsianoptionsarithmeticaverageMPDATAadvection-diffusionpseudo-velocityfinitedifferencepath-dependentBlack-Merton-ScholesPDE
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a uniform numerical treatment for Asian-style options, whose payoff depends on the average price of the underlying over time. The authors transform the two-dimensional Black-Merton-Scholes PDE for fixed-strike arithmetic-average options into a homogeneous advection equation, then solve it with MPDATA, a conservative, sign-preserving, non-oscillatory explicit finite-difference scheme. In all reported test cases the MPDATA prices agree with published reference values within a 20% relative tolerance, comparable to the paper's Monte Carlo estimates, while the underlying upwind step alone deviates at the first significant digit. The contribution matters because it shows that a single advection solver can handle a path-dependent pricing problem without dimensionally split or mixed numerical methods, with an intuitive stability condition and open-source implementation.

What carries the argument

The carrying mechanism is the pseudo-velocity transformation together with MPDATA's iterative antidiffusive corrections. The diffusion term $-\nu\,\partial^2 \Psi/\partial x^2$ is rewritten as an advective flux by adding a velocity component proportional to $(\partial_x \Psi)/\Psi$, yielding the transport equation $\partial_t \Psi + \nabla\cdot(\vec u\,\Psi)=0$. This is discretised with an upwind (donor-cell) scheme, and MPDATA's corrective passes add an antidiffusive Courant number derived from modified equation analysis, with the non-oscillatory variant applying flux-corrected-transport limiting. The scheme's multidimensional formulation means a two-dimensional step is not merely two one-dimensional passes, which the authors identify as a particular advantage for the 2D Asian pricing PDE.

What would settle it

A concrete check: for one of the Table 3 cases, run MPDATA with half the grid spacing and twice the corrective iterations while keeping $\Delta t$ at the stability limit, and compare against a high-precision Monte Carlo estimate with one million paths; if the MPDATA price moves outside the 20% band, loses positivity, or develops oscillations, the claimed robustness and second-order accuracy for the 2D problem would be contradicted.

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Extended reading notes

Core claim

The central claim is that the two-dimensional Asian option pricing problem can be recast as an advection-only transport problem and integrated forward in time with MPDATA, giving valuations that match established references. The transformation substitutes $x=\ln S$, $y=A$, $\Psi=e^{-rt}f$, and absorbs the diffusion term through the pseudo-velocity $u - \nu\,(\partial_x \Psi)/\Psi$, so that every term of the pricing equation is represented by a single advection operator. MPDATA's corrective iterations then remove the numerical diffusion of the upwind step, and the non-oscillatory variant enforces monotonicity. On the test cases tabulated against [BP96], MPDATA (two corrective iterations) reproduces the reference values within a 20% relative tolerance, with accuracy comparable to Monte Carlo estimates, and the paper presents the scheme as conservative, positive-definite, and inherently multidimensional.

Load-bearing premise

The load-bearing premise is that MPDATA's second-order accuracy, positivity, and non-oscillatory behavior, established for one-dimensional Black-Scholes integration in prior work, carry over to the two-dimensional Asian pricing PDE with the pseudo-velocity term at the chosen grid spacing and time step, though no 2D convergence study is shown.

Editorial extensions

If this is right

  • Asian options can be priced with one explicit advection solver, replacing mixed PDE/Monte Carlo frameworks for the tested instrument type.
  • UPWIND without corrective iterations is insufficient: the corrections are what move prices from first-significant-digit errors to agreement with references.
  • The same transport formulation extends to American-style Asian options through the free-boundary MPDATA treatment documented in prior work.
  • A third-order-accurate MPDATA variant can be dropped in for the second-order version, offering higher accuracy within the same framework.
  • The 2D Asian case provides a useful benchmark for advection schemes, with analytic geometric-average bounds and Monte Carlo solutions available for comparison.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not report a convergence study for the two-dimensional case; a natural extension is to measure the observed order of accuracy against a high-resolution Monte Carlo reference on refined grids.
  • The pseudo-velocity construction could be applied to other path-dependent instruments whose auxiliary state variable satisfies a transport equation, such as lookback or barrier options, although the paper does not explore this.
  • Because MPDATA is fully multidimensional, adding a third factor (for example stochastic volatility) would not require dimensional splitting, and the same solver machinery could be reused.
  • A sharper validation would compare against a high-accuracy arithmetic-average reference rather than the geometric-average analytic bound, isolating the discretisation error of the advection scheme.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes solving the two-dimensional PDE for fixed-strike arithmetic-average Asian options by casting the Black-Merton-Scholes equation as a homogeneous advection-diffusion problem and then, using the pseudo-velocity technique, as an advection-only transport equation. The authors apply the MPDATA finite-difference scheme, with an upwind base step and corrective iterations, and validate against the Barraquand-Pudet reference table, Monte Carlo simulations, and the geometric-average analytic formula. The central claim is that, in all presented cases, MPDATA valuations match the BP96 reference values within a relative tolerance of 20% and are comparable in accuracy to Monte Carlo, while improving substantially on first-order UPWIND.

Significance. If substantiated, the paper would provide a useful explicit, conservative, positive-definite and non-oscillatory solver for a genuinely two-dimensional pricing PDE, with no fitted parameters and with an openly archived PyMPDATA implementation. The presentation of the transformation and the discussion of boundary conditions are clear and helpful. However, the validation currently does not support the stronger claims: the reported tolerance is loose enough to admit first-order upwind errors, no 2D convergence study is provided, and Monte Carlo standard errors are absent. The contribution is therefore at the proof-of-concept stage, and the central quantitative claims need additional support.

major comments (3)
  1. [§4.3, Table 1] The 20% relative tolerance is too loose to support the claims that MPDATA 'matches' the [BP96] references and is 'comparable in accuracy' to Monte Carlo. For example, in the σ=0.2, T=6, K=105 call, MPDATA gives 2.65 versus [BP96] 2.24, an 18.3% error, while MC with N=100000 gives 2.18, a 2.7% error; the MPDATA error is several times the MC error. Conversely, several UPWIND entries also pass the 20% test, e.g., the σ=0.4, T=12, K=100 call has UPWIND 12.5 versus [BP96] 11.2, an 11.6% difference. The claimed discrimination between MPDATA and first-order upwinding and the claimed comparability to Monte Carlo are therefore not established by the reported numbers.
  2. [§2.4, §4.2] The paper asserts second-order accuracy for the 2D MPDATA solution, but no grid-convergence study is provided for the two-dimensional Asian pricing problem. The second-order property is inherited from [AF20] for the one-dimensional Black-Scholes problem and from MPDATA theory for constant-coefficient advection. Here the pseudo-velocity field in eq. (9) depends on Ψ through the term -ν ∂xΨ/Ψ, discretized via eq. (14), so the error behaviour cannot be assumed identical. In addition, the grid resolution used for Table 1 is not stated anywhere; only the time step Δt=1/1760 is given, while Fig. 1 reports a 21×31 grid. Without a convergence study in both Δx, Δy and Δt, the central accuracy claims are unsupported.
  3. [§3.2, §4.3] The Monte Carlo reference values are reported without standard errors or confidence intervals. The statement that convergence is confirmed by 'lack of significant variation across N=10000 and N=100000 paths' is not a substitute, because the absence of a visible change between two sample sizes does not quantify the statistical error of either estimate. Since the 'comparable to Monte Carlo' claim is a central part of the validation, the authors need to provide standard errors or confidence intervals for the MC estimates, and preferably a longer run for a subset of cases.
minor comments (5)
  1. [Throughout] The text repeatedly refers to 'Table 3' (e.g., in Sections 4.1, 4.2 and 4.3), but the table in the manuscript is labelled 'Table 1'. The numbering should be made consistent.
  2. [Figure 2 caption] The caption of Figure 2 says 'as in panel (d) in Fig. 2', but the referenced panel is in Figure 1. This should be corrected.
  3. [§4.2] The sentence 'The impact of the number of MPDATA iterations by comparing 2 and 4 iterations in Fig.1 and presenting UPWIND results in all cases (i.e. 1 iteration, no corrections)' is grammatically incomplete and should be rephrased.
  4. [§4.3] The 'relative tolerance of 20%' is not defined precisely; the authors should state explicitly that the error is measured relative to the [BP96] reference value.
  5. [§2.5] The stability criterion (2νΔt)/Δx² ≤ 1/2 is stated without derivation; since the pseudo-velocity depends on Ψ and its spatial derivative, it would strengthen the paper to explain how this criterion is obtained and whether it is sufficient for the variable-coefficient case.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the MPDATA Asian-option results are compared with independent Monte Carlo and Barraquand–Pudet references; self-citations are background and tooling, not inputs to the derivation.

full rationale

The paper's derivation chain is self-contained in the sense required by the circularity analysis. The governing PDE (3) is the standard augmented Black-Merton-Scholes equation; the transformation (6) and the pseudo-velocity flux form (8)-(9) are algebraic reformulations of that PDE, with the advective velocity defined so that expanding (8) reproduces (7) term by term. No unknown parameter is fitted to the validation data: the timestep dt=1/1760 is stated to be chosen arbitrarily within the stability limit (Section 4.2), and the spatial grid is fixed by the reported domain and figures (Sections 2.4 and 4.2). The target claim — that MPDATA valuations agree with reference values from Barraquand and Pudet [BP96] and with Monte Carlo estimates (Table 3, Section 4.3) — is tested against external benchmarks: [BP96] is an independent published method, and the Monte Carlo simulations (Section 3.2) are produced by a standard path-simulation algorithm with no calibration to the PDE solution. The self-citations to [AF20] and to the PyMPDATA library are used for background, for the origin of the transport formulation, and for previously documented convergence of the same scheme in the one-dimensional Black-Scholes case; they do not assume the two-dimensional Asian-option price that is being reported. The paper itself flags the proof-of-concept character and the arbitrary discretisation parameters, and the 20% relative tolerance and absence of a two-dimensional convergence study are legitimate concerns about the strength of the validation, but under the stated rules those are correctness and evidence-strength issues, not circularity. Accordingly, no reduction of a prediction to a fit, no self-referential uniqueness argument, and no equation that is equal to its own input were found.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard financial model assumptions and on numerical choices that are not fitted to target prices. No new physical entities are introduced. The main unverified load is transferring MPDATA guarantees from 1D and standard advection problems to this 2D pseudo-velocity formulation.

free parameters (3)
  • time step = 1/1760 for Table 3, 1/500 for figures
    Chosen arbitrarily within the stability limit (Section 4.2). Affects accuracy and is not determined by the model.
  • spatial domain and grid resolution = S from 50 to 200, A from 0 to 200; 21x31 grid for figures
    Hand-selected domain in Section 4.1. No sensitivity analysis is provided, so truncation error at S0=100 is unquantified.
  • number of MPDATA corrective iterations = 2 corrective iterations (4 in Figure 1 comparison)
    Method control setting. More iterations reduce error, but no 2D convergence study supports the chosen value.
assumptions (5)
  • domain assumption Black-Merton-Scholes frictionless geometric-Brownian-motion market and risk-neutral valuation framework
    The governing PDE (3) and the pricing problem inherit these standard financial assumptions from the cited model.
  • domain assumption Arithmetic average modeled as a normalized running sum A=(1/T)∫₀ᵗ S dτ, implying v=S/T
    Section 2.1, equations (4) and (5). This prices continuous-average Asian options, whereas market contracts use discrete observations; the Monte Carlo validation uses 1000 time steps as an approximation.
  • standard math Exactness of the substitution x=ln S, Ψ=e^{-rt}f and the pseudo-velocity advection-only transformation, requiring Ψ>0 in the denominator
    Sections 2.2 and 2.3. Where Ψ is zero, the scheme sets the offending term to zero (Section 2.4), an ad hoc regularization.
  • domain assumption Inflow boundary conditions as spatial extrapolation of Ψ and constant extension of the Courant field
    Section 2.6. The paper does not test sensitivity of the y=0 transect to this approximation, even though the option value is read at A=0.
  • domain assumption MPDATA stability and convergence results from the literature, including [AF20] and [SC86], extend to the 2D Asian PDE
    Sections 2.4 and 2.5. Convergence is cited from a 1D Black-Scholes study; no 2D convergence or grid-refinement study is presented here.

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Pith. "Pith review of Path-dependent option pricing with two-dimensional PDE using MPDATA." pith.science (2026). https://pith.science/paper/GLQOILSN

@misc{pith2026250524435,
  author       = {Pith},
  title        = {Pith review of: Path-dependent option pricing with two-dimensional PDE using MPDATA},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GLQOILSN}},
  note         = {Machine review of arXiv:2505.24435}
}
read the original abstract

In this paper, we discuss a simple yet robust PDE method for evaluating path-dependent Asian-style options using the non-oscillatory forward-in-time second-order MPDATA finite-difference scheme. The valuation methodology involves casting the Black-Merton-Scholes equation as a transport problem by first transforming it into a homogeneous advection-diffusion PDE via variable substitution, and then expressing the diffusion term as an advective flux using the pseudo-velocity technique. As a result, all terms of the Black-Merton-Sholes equation are consistently represented using a single high-order numerical scheme for the advection operator. We detail the additional steps required to solve the two-dimensional valuation problem compared to MPDATA valuations of vanilla instruments documented in a prior study. Using test cases employing fixed-strike instruments, we validate the solutions against Monte Carlo valuations, as well as against an approximate analytical solution in which geometric instead of arithmetic averaging is used. The analysis highlights the critical importance of the MPDATA corrective steps that improve the solution over the underlying first-order "upwind" step. The introduced valuation scheme is robust: conservative, non-oscillatory, and positive-definite; yet lucid: explicit in time, engendering intuitive stability-condition interpretation and inflow/outflow boundary-condition heuristics. MPDATA is particularly well suited for two-dimensional problems as it is not a dimensionally split scheme. The documented valuation workflow also constitutes a useful two-dimensional case for testing advection schemes featuring both Monte Carlo solutions and analytic bounds. An implementation of the introduced valuation workflow, based on the PyMPDATA package and the Numba Just-In-Time compiler for Python, is provided as free and open source software.

Figures

Figures reproduced from arXiv: 2505.24435 by the authors.

Figure 1
Figure 1. Overview of advector and advectee fields for a sam [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Depiction of values at t=0 along y=0 transect of the sample valuation domain as in panel (d) in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Works this paper leans on

2 extracted references · 2 canonical work pages

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    doi: 10.1166/jcsmd.2015.1089. [CZ12] M.J. Capiński and T. Zastawniak. Numerical Methods in Finance with C++. Cambridge University Press, 2012. doi: 10.1017/CBO9781139017404. [De 08] F. De Weert. Exotic Options T rading. Wiley Finance,

    work page arXiv 2015

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    A critique of the Crank-Nicolson scheme, strengths and weaknesses for financial instrument pricing

    isbn: 978-0470517901.url: https://wiley.com/ Exotic+Options+Trading-p-9781119995180. [Duf04] D. Duffy. “A critique of the Crank-Nicolson scheme, strengths and weaknesses for financial instrument pricing”. Willmott Magazine 4 (2004). (the paper is available as a part of the following book/collection (ISBN 978-0-470- 03145-2): https://www.wiley.com/en-us/Th...

    work page arXiv 2004

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