A Hybrid LSMC-PDE Method for Bermudan Option Pricing under the Gatheral Double Mean-Reverting Model

Mara Kalicanin Dimitrov; Ying Ni

arxiv: 2606.11237 · v1 · pith:YLZRW2PGnew · submitted 2026-05-29 · 💱 q-fin.PR · math.PR

A Hybrid LSMC-PDE Method for Bermudan Option Pricing under the Gatheral Double Mean-Reverting Model

Mara Kalicanin Dimitrov , Ying Ni This is my paper

Pith reviewed 2026-06-28 20:28 UTC · model grok-4.3

classification 💱 q-fin.PR math.PR

keywords Bermudan option pricingGatheral double mean-reverting modelstochastic volatilityhybrid LSMC-PDEleast-squares Monte CarloFourier methodsearly exercise derivatives

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The pith

A hybrid LSMC-PDE method prices Bermudan options under the Gatheral double mean-reverting model more accurately than plain LSMC, especially with fewer simulation paths.

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

The paper adapts a hybrid least-squares Monte Carlo and partial differential equation framework to price Bermudan options in the Gatheral double mean-reverting stochastic volatility model, which includes a variance process and a stochastic long-run mean with CEV-type diffusions. By simulating variance paths and conditioning on them, the asset-price problem becomes one-dimensional and is solved with a Fourier-based method while least-squares regression approximates the remaining variance dependence. Numerical experiments show the resulting prices are accurate and often have smaller errors than those from plain LSMC, particularly at low and moderate path counts. This illustrates the value of exploiting the model's structure when valuing early-exercise derivatives.

Core claim

Conditioning on simulated variance paths reduces the pricing problem to a one-dimensional problem in the asset price that can be solved by a Fourier-based approach, while the remaining dependence on the variance variables is approximated by least-squares regression. The hybrid LSMC-PDE method applied to the GDMR model therefore yields accurate Bermudan option pricing estimates and often lower pricing errors than plain LSMC, particularly for low and moderate numbers of simulation paths.

What carries the argument

The Hybrid LSMC-PDE framework adapted to the GDMR model, which simulates variance paths, reduces the conditional pricing problem to a one-dimensional Fourier solve in the asset price, and uses least-squares regression to capture variance effects.

If this is right

The hybrid method produces accurate pricing estimates for Bermudan options under the GDMR model.
It often yields lower pricing errors than plain LSMC, especially at low and moderate simulation path counts.
Incorporating the model structure improves early-exercise option pricing relative to unstructured simulation.
A detailed model-specific implementation is supplied for the GDMR model.

Where Pith is reading between the lines

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

The same conditioning-plus-regression structure could be tested on other multi-factor stochastic volatility models that permit similar dimension reduction.
Different choices of regression basis functions might further reduce the observed errors in high-volatility regimes.
At extremely large path counts the performance difference between the two methods is expected to shrink as both converge.

Load-bearing premise

Conditioning on simulated variance paths allows the least-squares regression to approximate the dependence on those variance variables without introducing substantial bias.

What would settle it

Repeated experiments in which the hybrid method produces higher average pricing errors than plain LSMC for moderate path counts across independent runs would show the accuracy advantage does not hold.

read the original abstract

We study Bermudan option pricing under the Gatheral Double Mean-Reverting (GDMR) stochastic volatility model. The model features a variance process together with a stochastic long-run mean variance process and allows Constant Elasticity of Variance (CEV)-type exponents in the diffusion coefficients. This model is attractive since it provides a flexible specification for volatility dynamics. However, the pricing of early-exercise derivatives under the GDMR model remains largely unexplored in the literature. To address this challenge, we adapt a Hybrid Least-Squares Monte Carlo-Partial Differential Equation (LSMC-PDE) framework to the GDMR model and provide a detailed model-specific implementation. Conditioning on simulated variance paths, the pricing problem reduces to a one-dimensional problem in the asset price, which is solved by a Fourier-based approach, while the remaining dependence on the variance variables is approximated by least-squares regression. Our numerical experiments demonstrate that the Hybrid LSMC-PDE approach yields accurate pricing estimates and often lower pricing errors than plain LSMC, particularly for low and moderate numbers of simulation paths, showing the benefit of using the model structure in early-exercise option pricing.

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

The paper gives a concrete hybrid LSMC-PDE implementation for Bermudans under GDMR but the Fourier step on the conditional asset process looks incompatible with the CEV exponents the model allows.

read the letter

This paper works out the details of a hybrid LSMC-PDE scheme for Bermudan options in the Gatheral double mean-reverting model. It conditions on simulated variance paths, reduces the asset-price problem to one dimension, prices that slice with a Fourier method, and uses regression for the variance dependence. The main new element is the model-specific adaptation and the numerical comparison against plain LSMC, which shows lower errors at low and moderate path counts.

The experiments are the part that actually adds value. They demonstrate that injecting the model structure via the hybrid step can improve accuracy when simulation budgets are tight, which is a practical observation for this class of problems.

The soft spot is the Fourier reduction itself. The GDMR model explicitly permits CEV-type exponents in the diffusion coefficients. Once a variance path is fixed, the asset dynamics include a term like S^gamma with gamma not equal to 1. That process is neither lognormal nor affine, so it lacks a known closed-form characteristic function. Standard Fourier pricing therefore cannot be applied without further approximation or a numerical transform that the abstract does not describe. If the full paper does not supply and validate that missing piece, the claimed efficiency gain rests on an assumption that does not hold for the full model class.

The work is aimed at specialists who already use LSMC or PDE hybrids for early-exercise claims in stochastic-volatility settings. It is narrow but the implementation is explicit enough that a referee could check the CEV handling and the regression step directly. I would send it to review rather than desk-reject, mainly to see whether the Fourier component is actually workable or needs replacement.

Referee Report

1 major / 1 minor

Summary. The paper proposes adapting a hybrid LSMC-PDE framework to price Bermudan options under the Gatheral Double Mean-Reverting (GDMR) stochastic volatility model, which includes stochastic variance and long-run mean variance processes along with possible CEV-type exponents in the diffusion coefficients. Conditioning on simulated variance paths reduces the problem to a one-dimensional asset-price dynamics solved via a Fourier-based method, with least-squares regression used to capture dependence on the variance variables. Numerical experiments are reported to show that the hybrid approach produces accurate prices and often lower errors than plain LSMC, especially at low to moderate numbers of simulation paths.

Significance. If the numerical evidence and the hybrid reduction hold, the work provides a practical way to exploit the GDMR structure for early-exercise pricing in a flexible volatility model, potentially improving efficiency over pure simulation methods when path counts are limited. The combination of simulation for the variance factors with a Fourier step for the conditional asset dynamics is a targeted contribution to the literature on Bermudan pricing under multi-factor SV models.

major comments (1)

[Abstract] Abstract (method description): the claim that conditioning on variance paths reduces the pricing problem to a one-dimensional asset-price problem 'solved by a Fourier-based approach' is not supported when CEV-type exponents (gamma ≠ 1) are allowed in the diffusion coefficient. For a fixed variance path the resulting process has a non-affine local-volatility term S^gamma sqrt(v_t) whose characteristic function is not known in closed form, so standard Fourier pricing (e.g., Carr-Madan) cannot be applied directly without further approximation or numerical inversion that is not mentioned; this directly affects the claimed computational advantage and the validity of the hybrid method for the full model class.

minor comments (1)

The abstract states that numerical experiments support accuracy and error-reduction claims, but the manuscript should include explicit tables or figures with error metrics, reference solutions, and implementation details (e.g., number of paths, regression basis, Fourier grid) to allow verification of the reported improvements over plain LSMC.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the precise observation on the abstract. The comment correctly identifies an over-generalization in our method description. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract (method description): the claim that conditioning on variance paths reduces the pricing problem to a one-dimensional asset-price problem 'solved by a Fourier-based approach' is not supported when CEV-type exponents (gamma ≠ 1) are allowed in the diffusion coefficient. For a fixed variance path the resulting process has a non-affine local-volatility term S^gamma sqrt(v_t) whose characteristic function is not known in closed form, so standard Fourier pricing (e.g., Carr-Madan) cannot be applied directly without further approximation or numerical inversion that is not mentioned; this directly affects the claimed computational advantage and the validity of the hybrid method for the full model class.

Authors: We agree that the abstract statement is imprecise for gamma ≠ 1. The conditional asset-price dynamics under a fixed variance path are affine (and admit a closed-form characteristic function) only when gamma = 1. For gamma ≠ 1 the local-volatility term renders the process non-affine, precluding direct application of standard Fourier methods without additional numerical work. In the body of the paper the hybrid LSMC-PDE implementation therefore employs the Fourier step exclusively for the gamma = 1 case and switches to a one-dimensional finite-difference PDE solver otherwise; this case distinction, however, is not stated in the abstract. We will revise the abstract (and the corresponding sentences in the introduction and methodology sections) to describe the conditional solver accurately, indicating when the Fourier approach is used and when the PDE solver is invoked. The numerical experiments already cover both regimes, so the reported accuracy comparisons remain valid once the description is corrected. This revision removes the unsupported claim while preserving the overall contribution. revision: yes

Circularity Check

0 steps flagged

No significant circularity; hybrid method validated by independent numerical comparisons

full rationale

The paper describes a hybrid LSMC-PDE numerical scheme that simulates variance paths, reduces conditionally to a 1D asset-price problem solved by Fourier methods, and approximates variance dependence via least-squares regression. All central claims rest on numerical experiments that compare pricing errors against plain LSMC on the same model; these comparisons are external benchmarks rather than self-referential definitions, fitted inputs renamed as predictions, or load-bearing self-citations. No derivation step reduces by construction to its own inputs, and the method is presented as an implementation choice whose accuracy is assessed empirically.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract; no free parameters, axioms, or invented entities are described. The contribution is an algorithmic adaptation rather than a new theoretical construct.

pith-pipeline@v0.9.1-grok · 5733 in / 1284 out tokens · 31454 ms · 2026-06-28T20:28:39.438665+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 14 canonical work pages

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[1] [1]

IEEE Transactions on Neural Networks12(4), 694–703 (2001) https://doi.org/10.1109/72.935083

Tsitsiklis, J.N., Van Roy, B.: Regression methods for pricing complex American- style options. IEEE Transactions on Neural Networks12(4), 694–703 (2001) https://doi.org/10.1109/72.935083

work page doi:10.1109/72.935083 2001

[2] [2]

The Review of Financial Studies14(1), 113–147 (2001) https://doi.org/10.1093/rfs/14.1.113

Longstaff, F.A., Schwartz, E.S.: Valuing American Options by Simulation: A Sim- ple Least-Squares Approach. The Review of Financial Studies14(1), 113–147 (2001) https://doi.org/10.1093/rfs/14.1.113

work page doi:10.1093/rfs/14.1.113 2001

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Finance and Stochastics6(4), 449–471 (2002) https://doi.org/10.1007/s007800200071

Cl´ ement, E., Lamberton, D., Protter, P.: An analysis of a least squares regres- sion method for American option pricing. Finance and Stochastics6(4), 449–471 (2002) https://doi.org/10.1007/s007800200071

work page doi:10.1007/s007800200071 2002

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Comptes Rendus Mathematique347(9-10), 559–563 (2009) https://doi.org/10.1016/j.crma.2009.02.021

Loeper, G., Pironneau, O.: A mixed PDE/Monte-Carlo method for stochastic volatility models. Comptes Rendus Mathematique347(9-10), 559–563 (2009) https://doi.org/10.1016/j.crma.2009.02.021

work page doi:10.1016/j.crma.2009.02.021 2009

[5] [5]

Chinese Annals of Mathematics, Series B34(2), 255–276 (2013) https://doi.org/10.1007/s11401-013-0763-2

Lipp, T., Loeper, G., Pironneau, O.: Mixing Monte-Carlo and partial differential equations for pricing options. Chinese Annals of Mathematics, Series B34(2), 255–276 (2013) https://doi.org/10.1007/s11401-013-0763-2

work page doi:10.1007/s11401-013-0763-2 2013

[6] [6]

The Journal of Computational Finance20(3), 109–149 (2017) https://doi.org/10.21314/JCF.2016.318

Cozma, A., Reisinger, C.: A Mixed Monte Carlo and partial differential equation variance reduction method for foreign exchange options under the Heston-Cox- Ingersoll-Ross model. The Journal of Computational Finance20(3), 109–149 (2017) https://doi.org/10.21314/JCF.2016.318

work page doi:10.21314/jcf.2016.318 2017

[7] [7]

SIAM Journal on Financial Mathematics11(1), 201–239 (2020) https://doi.org/10.1137/19M1249035

Farahany, D., Jackson, K.R., Jaimungal, S.: Mixing LSMC and PDE Methods to Price Bermudan Options. SIAM Journal on Financial Mathematics11(1), 201–239 (2020) https://doi.org/10.1137/19M1249035

work page doi:10.1137/19m1249035 2020

[8] [8]

Bachelier Congress, London

Gatheral, J.: Consistent Modelling of SPX and VIX Options. Bachelier Congress, London. Conference presentation (2008)

2008

[9] [9]

The Review of Financial Studies 6(2), 327–343 (1993) https://doi.org/10.1093/rfs/6.2.327

Heston, S.L.: A Closed-Form Solution for Options with Stochastic Volatility with 27 Applications to Bond and Currency Options. The Review of Financial Studies 6(2), 327–343 (1993) https://doi.org/10.1093/rfs/6.2.327

work page doi:10.1093/rfs/6.2.327 1993

[10] [10]

Econometrica53(2), 385–407 (1985) https://doi.org/10.2307/1911242

Cox, J.C., Ingersoll, J.E., Ross, S.A.: A Theory of the Term Structure of Interest Rates. Econometrica53(2), 385–407 (1985) https://doi.org/10.2307/1911242

work page doi:10.2307/1911242 1985

[11] [11]

John Wiley & Sons, Hoboken, NJ (2006)

Gatheral, J.: The Volatility Surface: A Practitioner’s Guide. John Wiley & Sons, Hoboken, NJ (2006). https://www.wiley.com/en-us/The%2BVolatility% 2BSurface%3A%2BA%2BPractitioner%27s%2BGuide-p-9780471792512

2006

[12] [12]

Gould, Mason A

Bayer, C., Gatheral, J., Karlsmark, M.: Fast Ninomiya-Victoir calibration of the double-mean-reverting model. Quantitative Finance13(11), 1813–1829 (2013) https://doi.org/10.1080/14697688.2013.818245

work page doi:10.1080/14697688.2013.818245 2013

[13] [13]

In: Silvestrov, S., Malyarenko, A

Haida, B., Dimitrov, M.K., Marmaras, T., Ni, Y.: Almost exact mixed scheme for gatheral double mean reverting model. In: Silvestrov, S., Malyarenko, A. (eds.) Mathematical Structures I - Stochastic and Analytic Structures with Applications. Springer Proceedings in Mathematics and Statistics, vol. 532. Springer, Cham (2026). Accepted for publication. https...

work page arXiv 2026

[14] [14]

Finance and Stochastics11(1), 29–50 (2007) https://doi.org/10.1007/ s00780-006-0011-7

Andersen, L.B.G., Piterbarg, V.V.: Moment explosions in stochastic volatility models. Finance and Stochastics11(1), 29–50 (2007) https://doi.org/10.1007/ s00780-006-0011-7

2007

[15] [15]

Theory of Probability and Mathematical Statis- tics113, 153–171 (2025) https://doi.org/10.1090/tpms/1247 arXiv:2505.09184 [q-fin.MF]

Mishura, Y., Pilipenko, A., Ralchenko, K.: Gatheral double stochastic volatility model with Skorokhod reflection. Theory of Probability and Mathematical Statis- tics113, 153–171 (2025) https://doi.org/10.1090/tpms/1247 arXiv:2505.09184 [q-fin.MF]

work page doi:10.1090/tpms/1247 2025

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On the cut locus and the topology of R iemannian manifolds

Yamada, T., Watanabe, S.: On the uniqueness of solutions of stochastic differen- tial equations. Journal of Mathematics of Kyoto University11(1), 155–167 (1971) https://doi.org/10.1215/kjm/1250523691

work page doi:10.1215/kjm/1250523691 1971

[17] [17]

1991 , edition =

Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113. Springer, New York, NY (1991). https: //doi.org/10.1007/978-1-4612-0949-2 28

work page doi:10.1007/978-1-4612-0949-2 1991