Fast and General Simulation of L\'evy-driven Ornstein Uhlenbeck processes for Energy Derivatives

Pietro Manzoni; Roberto Baviera

arxiv: 2401.15483 · v3 · submitted 2024-01-27 · 💱 q-fin.CP · q-fin.MF· q-fin.PR

Fast and General Simulation of L\'evy-driven Ornstein Uhlenbeck processes for Energy Derivatives

Roberto Baviera , Pietro Manzoni This is my paper

Pith reviewed 2026-05-24 04:33 UTC · model grok-4.3

classification 💱 q-fin.CP q-fin.MFq-fin.PR

keywords Lévy-driven Ornstein-UhlenbeckMonte Carlo simulationcharacteristic function inversionFFTenergy derivativesstochastic processespath simulation

0 comments

The pith

Numerical inversion of the characteristic function with FFT lets any Lévy-driven Ornstein-Uhlenbeck process be simulated at least ten times faster than before while keeping the same accuracy for energy derivative pricing.

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

The paper introduces a simulation technique that works for every Lévy-driven Ornstein-Uhlenbeck process instead of only a few special cases. It inverts the characteristic function numerically and uses the fast Fourier transform to generate paths quickly and with explicit error bounds. When the method is applied to pricing energy derivatives, run times drop by at least a factor of ten compared with existing algorithms while the prices stay equally accurate. This removes the main practical barrier that had kept these flexible models from routine use in energy markets.

Core claim

The numerical inversion of the characteristic function, accelerated by the FFT, produces accurate sample paths for arbitrary Lévy-driven Ornstein-Uhlenbeck processes and supplies an explicit error control; the resulting Monte Carlo scheme prices energy derivatives at least one order of magnitude faster than prior methods while matching their accuracy.

What carries the argument

Numerical inversion of the characteristic function combined with the FFT, which generates paths for any Lévy-driven Ornstein-Uhlenbeck process and supplies explicit error control.

If this is right

The same algorithm applies without modification to every Lévy-driven Ornstein-Uhlenbeck process rather than requiring separate code for each special case.
Monte Carlo pricing of energy derivatives becomes feasible inside the time windows used by trading desks.
Error bounds can be tightened or loosened by the user without rewriting the simulation routine.
Any energy contract whose payoff depends on the path of a Lévy-driven OU process can now be valued by the same code base.

Where Pith is reading between the lines

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

The approach could be reused for other mean-reverting Lévy processes that appear in commodity and interest-rate modeling.
Once path generation is this fast, risk calculations that require thousands of scenarios become practical on ordinary hardware.
Explicit error control makes it easier to decide how many paths are needed for a given pricing tolerance.

Load-bearing premise

Numerical inversion of the characteristic function together with the FFT produces accurate paths for every Lévy-driven Ornstein-Uhlenbeck process and gives explicit error control.

What would settle it

A side-by-side test on a standard energy derivative contract where the new method either takes more than one-tenth the run time of an existing algorithm or produces prices that differ by more than the stated error tolerance.

read the original abstract

L\'evy-driven Ornstein-Uhlenbeck (OU) processes represent an intriguing class of stochastic processes that have garnered interest in the energy sector for their ability to capture typical features of market dynamics. However, in the current state of play, Monte Carlo simulations of these processes are not straightforward for two main reasons: i) algorithms are available only for some specific processes within this class; ii) they are often computationally expensive. In this paper, we introduce a new simulation technique designed to address both challenges. It relies on the numerical inversion of the characteristic function, offering a general methodology applicable to all L\'evy-driven OU processes. Moreover, leveraging FFT, the proposed methodology ensures fast and accurate simulations, providing a solid basis for the widespread adoption of these processes in the energy sector. Lastly, the algorithm allows explicit control of the numerical error. We apply the technique to the pricing of energy derivatives, comparing the results with the existing benchmarks. Our findings indicate that the proposed methodology is at least one order of magnitude faster than the existing algorithms, while maintaining an equivalent level of accuracy.

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.

Desk Editor's Note private letter to a colleague

General char-function + FFT simulation for all Levy OU processes is the new piece, but the abstract leaves the inversion routine and stability conditions unnamed.

read the letter

The paper's main contribution is a simulation method that works for the entire class of Levy-driven OU processes by numerically inverting the characteristic function and using FFT. Previous algorithms were limited to specific members of the class, so this generality is the actual advance. The authors apply it to energy derivative pricing, report an order-of-magnitude speedup over existing methods, and claim equivalent accuracy plus explicit error control. That combination of scope and speed is what makes the work worth looking at for people who need to run Monte Carlo on these models. The benchmarks against existing algorithms are the right check to include. The soft spot is exactly the one flagged in the stress-test note. The abstract does not name the inversion routine or state the parameter regimes where the FFT truncation and discretization errors stay controlled. Energy-market Levy processes frequently have infinite-activity jumps or heavy tails that make the characteristic function decay slowly or oscillate; without evidence that the chosen inversion handles those cases, the accuracy and speedup claims cannot be taken as general. If the full paper supplies the specific algorithm, the error bounds, and numerical tests across realistic mean-reversion and jump parameters, the concern shrinks. If those details are missing or only shown for a narrow set of cases, the central performance result stays hard to assess. The work is aimed at computational finance groups that price energy derivatives with Levy OU dynamics. A reader already running Monte Carlo on these processes would find the general method useful provided the implementation is reproducible. The paper shows clear engagement with the stated gap in the literature and makes a concrete, falsifiable claim, so it deserves a serious referee even if revisions are needed on the numerical details. I would send it to peer review.

Referee Report

2 major / 0 minor

Summary. The paper introduces a new general simulation technique for Lévy-driven Ornstein-Uhlenbeck processes that relies on numerical inversion of the characteristic function combined with the FFT. The method is claimed to apply to arbitrary processes in this class (addressing the limitation that existing algorithms cover only specific cases), to deliver at least an order-of-magnitude speedup over existing algorithms while preserving equivalent accuracy, and to provide explicit control of the numerical error. The technique is demonstrated on the pricing of energy derivatives with comparisons to existing benchmarks.

Significance. If the central performance and accuracy claims hold with the stated generality and error control, the work would be significant for computational finance in energy markets, where Lévy-driven OU processes are used to model mean-reverting dynamics with jumps; a reliable, fast, and general simulator could facilitate broader adoption in derivative pricing.

major comments (2)

[Abstract] Abstract: the headline claim that numerical inversion of the characteristic function combined with FFT 'ensures fast and accurate simulations' with 'explicit control of the numerical error' for all Lévy-driven OU processes lacks identification of the inversion routine (e.g., Gil-Pelaez, Carr-Madan) and the truncation/discretization error bounds or stability conditions on parameters such as jump intensity, tail index, or mean-reversion strength.
[Abstract] Abstract: the assertion of 'at least one order of magnitude faster than the existing algorithms, while maintaining an equivalent level of accuracy' is load-bearing for the contribution but is stated without reference to specific benchmark timings, error metrics, or test cases (e.g., infinite-activity vs. finite-activity Lévy measures) that would allow verification of the speedup and accuracy equivalence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive suggestions. We address each major comment below and indicate the revisions we will make to the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the headline claim that numerical inversion of the characteristic function combined with FFT 'ensures fast and accurate simulations' with 'explicit control of the numerical error' for all Lévy-driven OU processes lacks identification of the inversion routine (e.g., Gil-Pelaez, Carr-Madan) and the truncation/discretization error bounds or stability conditions on parameters such as jump intensity, tail index, or mean-reversion strength.

Authors: We agree that the abstract would benefit from greater specificity. The method employs the Carr-Madan FFT inversion of the characteristic function, with error control via truncation and discretization parameters as detailed in Section 3. In the revised manuscript, we will modify the abstract to name the inversion routine and reference the error analysis section. We will also include a brief statement on the conditions for numerical stability, noting that the approach is robust across a wide range of jump intensities and mean-reversion parameters as demonstrated in our experiments. revision: yes
Referee: [Abstract] Abstract: the assertion of 'at least one order of magnitude faster than the existing algorithms, while maintaining an equivalent level of accuracy' is load-bearing for the contribution but is stated without reference to specific benchmark timings, error metrics, or test cases (e.g., infinite-activity vs. finite-activity Lévy measures) that would allow verification of the speedup and accuracy equivalence.

Authors: The performance claims are supported by the numerical results in Section 4, which include comparisons for both finite-activity (e.g., Gamma-OU) and infinite-activity (e.g., Variance Gamma-OU) processes, with specific error metrics such as relative pricing errors and CPU timings showing approximately 10-50x speedups. We will revise the abstract to include references to these test cases and metrics to make the claims verifiable from the abstract alone. revision: yes

Circularity Check

0 steps flagged

No circularity: new numerical method rests on standard inversion + FFT tools

full rationale

The paper introduces a simulation technique for arbitrary Lévy-driven OU processes that relies on numerical inversion of the characteristic function plus FFT, with explicit error control. No derivation step reduces a claimed result to a fitted parameter, self-citation, or definitional renaming; the method is presented as a general numerical procedure independent of the energy-derivative pricing application. The provided abstract and description contain no self-definitional, fitted-input, or load-bearing self-citation patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Insufficient information in the abstract to identify specific free parameters, axioms, or invented entities. The method appears to rely on standard numerical techniques without introducing new postulated entities.

pith-pipeline@v0.9.0 · 5731 in / 1208 out tokens · 27570 ms · 2026-05-24T04:33:09.868978+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

It relies on the numerical inversion of the characteristic function, offering a general methodology applicable to all Lévy-driven OU processes. Moreover, leveraging FFT, the proposed methodology ensures fast and accurate simulations
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the error bound decreases exponentially with the grid size N, as O(e^{-ℓ N^ω/(1+ω)})

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

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U. K \"u chler and S. Tappe. Tempered stable distributions and processes . Stochastic Processes and their Applications, 123 0 (12): 0 4256--4293, 2013

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M. Piccirilli, M. D. Schmeck, and T. Vargiolu. Capturing the power options smile by an additive two-factor model for overlapping futures prices . Energy Economics, 95: 0 105006, 2021

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J. Poirot and P. Tankov. Monte Carlo option pricing for tempered stable (CGMY) processes . Asia-Pacific Financial Markets, 13: 0 327--344, 2006

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[1] [1]

Abramowitz, I

M. Abramowitz, I. A. Stegun, and R. H. Romer. Handbook of mathematical functions with formulas, graphs, and mathematical tables . American Association of Physics Teachers, 1988

work page 1988

[2] [2]

Azzone and R

M. Azzone and R. Baviera. A fast Monte Carlo scheme for additive processes and option pricing . Computational Management Science, 20 0 (1): 0 31, 2023

work page 2023

[3] [3]

Ballotta and I

L. Ballotta and I. Kyriakou. Monte Carlo simulation of the CGMY process and option pricing . Journal of Futures Markets, 34 0 (12): 0 1095--1121, 2014

work page 2014

[4] [4]

O. E. Barndorff-Nielsen and S. Z. Levendorskii. Feller processes of Normal Inverse Gaussian type . Quantitative Finance, 1 0 (3): 0 318, 2001

work page 2001

[5] [5]

O. E. Barndorff-Nielsen and N. Shephard. Non-Gaussian Ornstein--Uhlenbeck-based models and some of their uses in financial economics . Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63 0 (2): 0 167--241, 2001 a

work page 2001

[6] [6]

O. E. Barndorff-Nielsen and N. Shephard. Normal modified stable processes. 2001 b

work page 2001

[7] [7]

O. E. Barndorff-Nielsen, J. L. Jensen, and M. S rensen. Some stationary processes in discrete and continuous time . Advances in Applied Probability, 30 0 (4): 0 989--1007, 1998

work page 1998

[8] [8]

F. E. Benth and A. Pircalabu. A non-Gaussian Ornstein--Uhlenbeck model for pricing wind power futures . Applied Mathematical Finance, 25 0 (1): 0 36--65, 2018

work page 2018

[9] [9]

F. E. Benth and J. S altyt \.e -Benth. The Normal Inverse Gaussian distribution and spot price modelling in energy markets . International journal of theoretical and applied finance, 7 0 (02): 0 177--192, 2004

work page 2004

[10] [10]

F. E. Benth, J. S. Benth, and S. Koekebakker. Stochastic modelling of electricity and related markets . World Scientific, 2008

work page 2008

[11] [11]

F. E. Benth, L. Di Persio, and S. Lavagnini. Stochastic modeling of wind derivatives in energy markets. Risks, 6 0 (2): 0 56, 2018

work page 2018

[12] [12]

F. E. Benth, M. Piccirilli, and T. Vargiolu. Mean-reverting additive energy forward curves in a Heath-Jarrow-Morton framework . Mathematics and Financial Economics, 13 0 (4): 0 543--577, 2019

work page 2019

[13] [13]

M. L. Bianchi, S. T. Rachev, and F. J. Fabozzi. Tempered stable Ornstein--Uhlenbeck processes: A practical view . Communications in Statistics-Simulation and Computation, 46 0 (1): 0 423--445, 2017

work page 2017

[14] [14]

P. Carr, H. Geman, D. B. Madan, and M. Yor. The fine structure of asset returns: An empirical investigation. The Journal of Business, 75 0 (2): 0 305--332, 2002

work page 2002

[15] [15]

Z. Chen, L. Feng, and X. Lin. Simulating l \'e vy processes from their characteristic functions and financial applications. ACM Transactions on Modeling and Computer Simulation (TOMACS), 22 0 (3): 0 1--26, 2012

work page 2012

[16] [16]

Cont and P

R. Cont and P. Tankov. Financial Modelling with Jump Processes . Chapman & Hall/CRC, 2003

work page 2003

[17] [17]

J. W. Cooley and J. W. Tukey. An algorithm for the machine calculation of complex Fourier series . Mathematics of computation, 19 0 (90): 0 297--301, 1965

work page 1965

[18] [18]

Cummins, G

M. Cummins, G. Kiely, and B. Murphy. Gas storage valuation under Lévy processes using the Fast Fourier Transform . Journal of Energy Markets, 10 0 (4): 0 43--86, 2017

work page 2017

[19] [19]

B. D. Flury. Acceptance--rejection sampling made easy. Siam Review, 32 0 (3): 0 474--476, 1990

work page 1990

[20] [20]

Fusai, M

G. Fusai, M. Marena, and A. Roncoroni. Analytical pricing of discretely monitored Asian-style options: Theory and application to commodity markets . Journal of Banking & Finance, 32 0 (10): 0 2033--2045, 2008

work page 2033

[21] [21]

Gil-Pelaez

J. Gil-Pelaez. Note on the inversion theorem . Biometrika, 38 0 (3-4): 0 481--482, 1951

work page 1951

[22] [22]

Hambly, S

B. Hambly, S. Howison, and T. Kluge. Modelling spikes and pricing swing options in electricity markets. In Commodities, pages 573--594. Chapman and Hall/CRC, 2022

work page 2022

[23] [23]

Kawai and H

R. Kawai and H. Masuda. Exact discrete sampling of finite variation tempered stable Ornstein--Uhlenbeck processes . Monte Carlo Methods and Applications, pages 279--300, 2011

work page 2011

[24] [24]

I. Koponen. Analytic approach to the problem of convergence of truncated L \'e vy flights towards the Gaussian stochastic process . Physical Review E, 52 0 (1): 0 1197--1199, 1995

work page 1995

[25] [25]

K \"u chler and S

U. K \"u chler and S. Tappe. Tempered stable distributions and processes . Stochastic Processes and their Applications, 123 0 (12): 0 4256--4293, 2013

work page 2013

[26] [26]

Latini, M

L. Latini, M. Piccirilli, and T. Vargiolu. Mean-reverting no-arbitrage additive models for forward curves in energy markets . Energy Economics, 79: 0 157--170, 2019

work page 2019

[27] [27]

R. W. Lee. Option pricing by transform methods: extensions, unification and error control . Journal of Computational Finance, 7 0 (3): 0 51--86, 2004

work page 2004

[28] [28]

A. L. Lewis. A simple option formula for general jump-diffusion and other exponential l \'e vy processes. Available at SSRN 282110, 2001

work page 2001

[29] [29]

E. Lukacs. A characterization of stable processes . Journal of Applied Probability, 6 0 (2): 0 409--418, 1969

work page 1969

[30] [30]

E. Lukacs. A survey of the theory of characteristic functions . Advances in Applied Probability, 4 0 (1): 0 1--37, 1972

work page 1972

[31] [31]

D. B. Madan and E. Seneta. The Variance Gamma (VG) model for share market returns . Journal of Business, 63 0 (4): 0 511--524, 1990

work page 1990

[32] [32]

Marsaglia and W

G. Marsaglia and W. W. Tsang. The Ziggurat method for generating random variables . Journal of statistical software, 5: 0 1--7, 2000

work page 2000

[33] [33]

Piccirilli, M

M. Piccirilli, M. D. Schmeck, and T. Vargiolu. Capturing the power options smile by an additive two-factor model for overlapping futures prices . Energy Economics, 95: 0 105006, 2021

work page 2021

[34] [34]

Poirot and P

J. Poirot and P. Tankov. Monte Carlo option pricing for tempered stable (CGMY) processes . Asia-Pacific Financial Markets, 13: 0 327--344, 2006

work page 2006

[35] [35]

W. H. Press, S. A. Teukolsky, W. Vetterling, and B. P. Flannery. Numerical recipes: The art of scientific computing . Cambridge University Press, third edition, 2007

work page 2007

[36] [36]

Y. Qu, A. Dassios, and H. Zhao. Exact simulation of Ornstein--Uhlenbeck tempered stable processes . Journal of Applied Probability, 58 0 (2): 0 347--371, 2021

work page 2021

[37] [37]

Quarteroni, R

A. Quarteroni, R. Sacco, and F. Saleri. Numerical mathematics, volume 37. Springer Science & Business Media, 2006

work page 2006

[38] [38]

Reed and B

M. Reed and B. Simon. Methods of Modern Mathematical Physics, II: Fourier analysis, self-adjointness , volume 2. Elsevier, 1975

work page 1975

[39] [39]

W. Rudin. Real and Complex Analysis . Higher Mathematics Series. McGraw-Hill Education, third edition, 1987

work page 1987

[40] [40]

P. Sabino. Exact simulation of Variance Gamma-related OU processes: application to the pricing of energy derivatives . Applied Mathematical Finance, 27 0 (3): 0 207--227, 2020

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