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arxiv: 1906.10888 · v1 · pith:WHJ5KZQXnew · submitted 2019-06-26 · 💱 q-fin.PR · math.PR

European Option Pricing of electricity under exponential functional of L\'evy processes with Price-Cap principle

Martin Kegnenlezom , Patrice Takam Soh , Antoine-Marie Bogso , Yves Emvudu Wono This is my paper

Pith reviewed 2026-05-25 15:20 UTC · model grok-4.3

classification 💱 q-fin.PR math.PR
keywords electricity pricingLévy processEuropean optionviscosity solutionPIDEfinite differencesprice cap principle
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The pith

European electricity options are priced as the unique viscosity solution to a PIDE under an exponential Lévy model.

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

This paper models electricity asset prices using an exponential functional of a jump Lévy process that incorporates the price-cap principle. The model is designed to reflect both mean reversion and price jumps seen in electricity markets. The value of a European option on this asset is established as the unique viscosity solution of a partial integro-differential equation. A finite difference method is used to approximate the solution, with proofs of consistency, stability, and convergence provided. Numerical examples are given for illustration.

Core claim

The authors define the asset price as an exponential functional of a jump Lévy process under the price-cap principle. They prove that the European option value is the unique viscosity solution of the PIDE associated with this price process. The finite difference scheme for the PIDE is consistent, stable, and convergent, allowing numerical computation of option prices.

What carries the argument

Exponential functional of a jump Lévy process, which determines the asset price dynamics and the form of the PIDE governing option prices.

If this is right

  • The option price can be numerically computed via the finite difference scheme.
  • The scheme converges to the unique viscosity solution.
  • Mean reversion and jumps are both accounted for in the pricing.
  • The price-cap principle is integrated into the Lévy-based dynamics.

Where Pith is reading between the lines

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

  • This model may allow better risk management in electricity trading by accounting for jumps.
  • The PIDE approach could be compared to other pricing methods like Monte Carlo simulation.
  • Regulatory price caps might be analyzed for their impact on option values in this framework.
  • The method might apply to valuing other derivatives in energy markets with similar dynamics.

Load-bearing premise

Electricity prices follow an exponential functional of a jump Lévy process.

What would settle it

Empirical test: fit the Lévy process parameters to electricity price data and check if market-observed European option prices equal the viscosity solution of the PIDE.

Figures

Figures reproduced from arXiv: 1906.10888 by Antoine-Marie Bogso, Martin Kegnenlezom, Patrice Takam Soh, Yves Emvudu Wono.

Figure 1
Figure 1. Figure 1: Call price for four diffrents values of spot price a [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of call values for four diffrents value [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Call price for four different remaining time to mat [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of call price for four different versus [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Call price for three diffrents values of strike ver [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Call price versus strike price K, the other paramet [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Call price versus strike price K, the other paramet [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Call price versus remaining time and spot price [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Call price versus remaining time and spot price [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
read the original abstract

We propose a new model for electricity pricing based on the price cap principle. The particularity of the model is that the asset price is an exponential functional of a jump L\'evy process. This model can capture both mean reversion and jumps which are observed in electricity market. It is shown that the value of an European option of this asset is the unique viscosity solution of a partial integro-differential equation (PIDE). A numerical approximation of this solution by the finite differences method is provided. The consistency, stability and convergence results of the scheme are given. Numerical simulations are performed under a smooth initial condition.

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 proposes a price-cap model in which the electricity spot price is defined as an exponential functional of a jump Lévy process. It asserts that the value of a European call option on this asset is the unique viscosity solution of the associated PIDE obtained from the infinitesimal generator of the process. A finite-difference scheme is constructed for the PIDE; consistency, stability, and convergence of the scheme are stated, and numerical illustrations are given for smooth initial data.

Significance. If the uniqueness result and the convergence of the scheme hold under the capped dynamics, the work supplies a modeling framework that simultaneously incorporates mean reversion, jumps, and an explicit price cap—features that are practically relevant for electricity markets. The combination of a viscosity-solution characterization with a provably convergent numerical method would constitute a useful contribution to quantitative pricing in energy derivatives.

major comments (2)
  1. [PIDE derivation and uniqueness statement] Abstract and the section deriving the PIDE: the claim that the option value is the unique viscosity solution requires an explicit verification of the comparison principle for the nonlocal operator under the state-dependent price cap. Standard uniqueness theorems for PIDEs impose integrability conditions on the Lévy measure and modulus-of-continuity requirements that may be altered by the truncation; the manuscript does not appear to re-establish these conditions for the capped exponential functional, which is load-bearing for the central theoretical claim.
  2. [Finite-difference scheme and convergence analysis] Numerical approximation section: the consistency, stability, and convergence statements for the finite-difference scheme are asserted, yet the error analysis does not detail how the nonlocal integral term is discretized near the cap boundary or whether the truncation affects the uniform ellipticity or monotonicity properties used in the convergence proof. This verification is necessary to support the numerical results that accompany the viscosity-solution claim.
minor comments (2)
  1. [Model setup] The precise functional form of the price cap (whether it is applied before or after the exponential) should be written explicitly in the model definition to remove any ambiguity in the generator.
  2. [Numerical simulations] The numerical examples would benefit from stating the exact Lévy measure and parameter values used, together with a quantitative convergence table (e.g., L^∞ or L^2 errors versus mesh size) rather than qualitative plots alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help strengthen the theoretical and numerical foundations of the paper. We address each major point below and will incorporate the necessary clarifications in a revised version.

read point-by-point responses

  1. Referee: [PIDE derivation and uniqueness statement] Abstract and the section deriving the PIDE: the claim that the option value is the unique viscosity solution requires an explicit verification of the comparison principle for the nonlocal operator under the state-dependent price cap. Standard uniqueness theorems for PIDEs impose integrability conditions on the Lévy measure and modulus-of-continuity requirements that may be altered by the truncation; the manuscript does not appear to re-establish these conditions for the capped exponential functional, which is load-bearing for the central theoretical claim.

    Authors: The referee correctly identifies that the uniqueness statement would benefit from an explicit check. The price cap is realized as a bounded, Lipschitz truncation of the exponential functional; this leaves the Lévy measure unchanged and preserves the required integrability and modulus-of-continuity conditions on the nonlocal operator. Consequently the standard comparison principle for PIDEs with bounded coefficients continues to apply. We will insert a short dedicated paragraph immediately after the derivation of the PIDE that records this verification. revision: yes

  2. Referee: [Finite-difference scheme and convergence analysis] Numerical approximation section: the consistency, stability, and convergence statements for the finite-difference scheme are asserted, yet the error analysis does not detail how the nonlocal integral term is discretized near the cap boundary or whether the truncation affects the uniform ellipticity or monotonicity properties used in the convergence proof. This verification is necessary to support the numerical results that accompany the viscosity-solution claim.

    Authors: We agree that the discretization near the cap boundary and its effect on the monotonicity/ellipticity properties should be spelled out. The quadrature for the nonlocal integral is truncated at the cap level and the resulting scheme remains monotone because the cap modification is Lipschitz continuous. Uniform ellipticity is unaffected since the diffusion coefficient is state-independent. We will expand the consistency and convergence subsections with these explicit arguments and a brief remark confirming that the convergence rate is preserved. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation of PIDE and viscosity uniqueness is self-contained from model generator

full rationale

The paper defines the asset price as an exponential functional of a jump Lévy process with price cap, derives the associated PIDE via the infinitesimal generator, and states that the European option value is its unique viscosity solution. This follows standard Feynman-Kac-type arguments for Lévy-driven processes and does not reduce any quantity to a fitted parameter, self-citation chain, or definitional tautology. The numerical finite-difference scheme and its consistency/stability/convergence analysis are presented separately. No load-bearing step collapses by construction to the inputs; the uniqueness claim is a mathematical assertion about the PIDE, not a renaming or imported ansatz. External verification of comparison principles is a correctness issue, not circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Lévy process model assumption and the mathematical theory of viscosity solutions for PIDEs, which are standard but the specific application is new. No free parameters or invented entities are explicitly detailed in the abstract.

free parameters (1)
  • Lévy process parameters
    Parameters of the jump Lévy process are likely chosen or fitted but not specified in abstract.
axioms (1)
  • domain assumption Electricity prices follow an exponential functional of a jump Lévy process under price-cap principle
    Core modeling assumption stated in the abstract.

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Reference graph

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