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arxiv: 2508.14813 · v2 · submitted 2025-08-20 · 💱 q-fin.MF · math.PR· q-fin.CP

Pricing Options on Forwards in Function-Valued Affine Stochastic Volatility Models

Jian He , Sven Karbach , Asma Khedher This is my paper

Pith reviewed 2026-05-18 22:04 UTC · model grok-4.3

classification 💱 q-fin.MF math.PRq-fin.CP
keywords option pricingforward price curvesaffine stochastic volatilityHJM-Musiela frameworkFourier pricingWishart processinfinite-dimensional modelsstochastic partial differential equations
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The pith

Affine stochastic volatility models on forward price curves yield semi-closed Fourier pricing formulas for vanilla options.

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

The paper shows how to price European call and put options written directly on forward price curves when those curves evolve according to a stochastic partial differential equation driven by an affine stochastic volatility process. Two concrete cases are treated: one where volatility follows a finite-rank Wishart process and one where it follows a pure-jump extension with state-dependent jumps. In both cases the authors first find parameter conditions that guarantee the existence of exponential moments and then exploit the resulting exponential-affine characteristic function to obtain Fourier inversion formulas for the option prices. This matters for forward markets because the infinite-dimensional setup can distinguish risks that differ by maturity while still keeping the pricing calculation tractable.

Core claim

Within the Heath-Jarrow-Morton-Musiela framework the forward price curve is modeled as the solution of a stochastic partial differential equation modulated by either a finite-rank Wishart process or a pure-jump affine process with state-dependent jumps in the covariance. Under explicit conditions ensuring the existence of exponential moments, the characteristic function of the logarithm of the forward price at any fixed maturity remains of exponential-affine form; this property directly produces semi-closed Fourier-based pricing expressions for vanilla call and put options written on the forward price curve.

What carries the argument

The exponential-affine form of the characteristic function that is preserved by the affine structure of the modulated forward-curve SPDE under the risk-neutral measure.

If this is right

  • Pricing remains computationally feasible even when the model contains infinitely many risk factors.
  • The same Fourier approach covers both the continuous Gaussian Wishart case and the discontinuous pure-jump extension.
  • Maturity-specific risks along the forward curve can be priced without reducing the model to a finite-factor approximation.
  • The derived moment conditions delineate the precise parameter region in which the pricing formulas are valid.

Where Pith is reading between the lines

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

  • The same affine construction could be reused to price other curve-dependent claims such as swaptions or caps.
  • Market data on options written on different segments of the forward curve could be used to test whether infinite-factor affine specifications outperform low-dimensional alternatives.
  • Calibration routines that exploit the closed-form characteristic function might allow joint estimation of volatility dynamics and curve shape parameters from observed option surfaces.

Load-bearing premise

The forward price curve dynamics must retain an affine structure under the risk-neutral measure once they are modulated by the chosen stochastic volatility process.

What would settle it

A direct numerical check that the characteristic function of the log-forward price at a chosen maturity deviates from the exponential-affine expression given by the model for any admissible parameter set would invalidate the Fourier pricing formulas.

Figures

Figures reproduced from arXiv: 2508.14813 by Asma Khedher, Jian He, Sven Karbach.

Figure 1
Figure 1. Figure 1: Moment generation function: Monte Carlo v.s. analytical formula, with h0 = 1, α = 0.1, T0 = 1/365, N = 5. 4.3. Pricing in a pure-jump Lévy driven stochastic volatility model. As a second example we consider an affine pure-jump Lévy–type stochastic volatility specification in which the instantaneous covariance is an operator–valued Lévy subordinator. Let (fn)n∈N be the orthonormal basis of Hw from (4.2), an… view at source ↗
Figure 2
Figure 2. Figure 2: Monte Carlo v.s. analytical formula, with Y0 = D, h0 = 1, α = 0.1, T0 = 1, N = 5, strike K = 1 and the intensity β = 1. 4.4. Pricing in the operator-valued Barndorff–Nielsen–Shephard stochastic volatility model. Let A, D, (Wft)t≥0, and (Jt)t≥0 be as in Example 4.3, with Jt = PNt i=1 D a compound Poisson process of intensity β > 0, independent of Wf. Let Hw be the space of self-adjoint Hilbert–Schmidt opera… view at source ↗
Figure 3
Figure 3. Figure 3: Monte Carlo v.s. analytical formula, with Y0 = D, h0 = 1, α = 0.1, T0 = 1, N = 5, strike K = 1 and the intensity β = 1. 4.5. Running time analysis [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
read the original abstract

We study the pricing of European-style options written on forward contracts within function-valued infinite-dimensional affine stochastic volatility models. The dynamics of the underlying forward price curves are modeled within the Heath-Jarrow-Morton-Musiela framework as solution to a stochastic partial differential equation modulated by a stochastic volatility process. We analyze two classes of affine stochastic volatility models: (i) a Gaussian model governed by a finite-rank Wishart process, and (ii) a pure-jump affine model extending the Barndorff--Nielsen--Shephard framework with state-dependent jumps in the covariance component. For both models, we derive conditions for the existence of exponential moments and develop semi-closed Fourier-based pricing formulas for vanilla call and put options written on forward price curves. Our approach allows for tractable pricing in models with infinitely many risk factors, thereby capturing maturity-specific and term structure risk essential in forward markets.

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

3 major / 3 minor

Summary. The manuscript develops pricing formulas for European call and put options written on forward price curves in infinite-dimensional affine stochastic volatility models. Forward curve dynamics are specified as solutions to an SPDE in the Heath-Jarrow-Morton-Musiela framework modulated by either a finite-rank Wishart process or a state-dependent pure-jump covariance process extending the Barndorff-Nielsen-Shephard framework. For both classes the authors derive conditions ensuring existence of exponential moments and obtain semi-closed Fourier pricing formulas.

Significance. If the central derivations hold, the work supplies a tractable route to option pricing in models with infinitely many risk factors, thereby addressing maturity-specific and term-structure risks that are material in forward markets. The explicit moment conditions and Fourier representations constitute a concrete advance over purely numerical approaches in high-dimensional settings.

major comments (3)
  1. [§2] §2 (HJM-Musiela SPDE setup): the claim that the forward curve remains exponentially affine after insertion of the risk-neutral drift (fixed by the volatility operator to enforce the martingale property) is load-bearing for the entire pricing strategy. The manuscript must display the explicit commutation relations between the volatility operators and the shift semigroup that close the Riccati system without residual non-affine terms; the abstract states the modeling choice but does not contain this verification.
  2. [§3.2] §3.2 (finite-rank Wishart case): the conditions for exponential moments are stated, yet the proof that the finite-rank truncation preserves the affine property under the infinite-dimensional shift operator is only sketched. A concrete counter-example or explicit bound on the rank that guarantees closure of the moment-generating function would strengthen the result.
  3. [§4] §4 (pure-jump extension): the state-dependent jump measure in the covariance component must be shown to keep the generator affine once the HJM drift is imposed. The current derivation assumes without further restriction that the jump compensator commutes with the shift; an explicit integrability condition on the Lévy measure relative to the Musiela parametrization is required.
minor comments (3)
  1. [§1] Notation for the Musiela parametrization (e.g., the precise definition of the shift semigroup acting on the volatility surface) appears only after first use; a short preliminary paragraph would improve readability.
  2. [Abstract] The abstract claims 'semi-closed' formulas; the manuscript should clarify whether the Fourier integral is evaluated numerically or admits further closed-form reduction in special cases.
  3. [§5] A brief comparison table contrasting the moment conditions of the two models would help readers assess the relative restrictiveness of the Wishart versus jump specifications.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments help clarify several technical points in the derivations. We address each major comment below and will revise the manuscript to incorporate the requested explicit verifications and strengthened proofs while preserving the core results on exponential moments and Fourier pricing formulas.

read point-by-point responses

  1. Referee: [§2] §2 (HJM-Musiela SPDE setup): the claim that the forward curve remains exponentially affine after insertion of the risk-neutral drift (fixed by the volatility operator to enforce the martingale property) is load-bearing for the entire pricing strategy. The manuscript must display the explicit commutation relations between the volatility operators and the shift semigroup that close the Riccati system without residual non-affine terms; the abstract states the modeling choice but does not contain this verification.

    Authors: We agree that the commutation relations are central and should be displayed explicitly. In the manuscript the risk-neutral drift is constructed precisely so that the solution to the HJM-Musiela SPDE remains exponentially affine; this follows from the fact that the volatility operators are chosen to commute with the shift semigroup generated by the Musiela operator. We will add a new proposition in Section 2 that states and proves the required commutation relations, together with the resulting closed Riccati system for the moment-generating function. revision: yes

  2. Referee: [§3.2] §3.2 (finite-rank Wishart case): the conditions for exponential moments are stated, yet the proof that the finite-rank truncation preserves the affine property under the infinite-dimensional shift operator is only sketched. A concrete counter-example or explicit bound on the rank that guarantees closure of the moment-generating function would strengthen the result.

    Authors: The finite-rank Wishart process is constructed so that its action on the forward curve factors through a finite number of deterministic curves; the infinite-dimensional shift operator therefore maps the finite-rank structure into itself, preserving affinity. We will expand the proof in Section 3.2 to include the explicit verification that the moment-generating function remains of affine form for any finite rank, together with a concrete bound on admissible rank in terms of the model parameters that guarantees the required exponential moments exist. revision: yes

  3. Referee: [§4] §4 (pure-jump extension): the state-dependent jump measure in the covariance component must be shown to keep the generator affine once the HJM drift is imposed. The current derivation assumes without further restriction that the jump compensator commutes with the shift; an explicit integrability condition on the Lévy measure relative to the Musiela parametrization is required.

    Authors: We accept that the integrability condition on the Lévy measure should be stated explicitly. The state-dependent jump measure is defined so that its compensator commutes with the Musiela shift under the stated assumptions; we will add a precise integrability requirement (involving the growth of the Lévy measure with respect to the Musiela parameter) in Section 4 that guarantees the generator remains affine after the HJM drift is imposed. revision: yes

Circularity Check

0 steps flagged

Affine structure is a modeling assumption; derivations are independent of fitted inputs

full rationale

The paper adopts the affine stochastic volatility structure (finite-rank Wishart or state-dependent jump) as an explicit modeling choice within the HJM-Musiela SPDE framework to ensure the characteristic function remains exponential-affine. This enables subsequent derivation of exponential-moment conditions and Fourier pricing formulas. No step reduces a claimed prediction or first-principles result to a fitted parameter or self-citation by construction. The setup is self-contained against external benchmarks for affine models, yielding a normal low circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central results rest on the affine property of the chosen volatility processes, existence of exponential moments under the risk-neutral measure, and the standard HJM-Musiela SPDE setup; no new entities are postulated.

axioms (2)
  • domain assumption The forward price curve satisfies a stochastic partial differential equation of HJM-Musiela type modulated by an affine stochastic volatility process.
    Invoked in the model setup to guarantee the affine structure needed for Fourier pricing.
  • domain assumption Exponential moments of the forward price exist under the stated parameter restrictions.
    Required for the Fourier integral representation to be well-defined; conditions are derived but the existence itself is an assumption on the parameter domain.

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Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Modeling electricity forward prices using the multivariate normal inverse Gaussian distribution.J

    Andresen, A., Koekebakker, S., and Westgaard, S. Modeling electricity forward prices using the multivariate normal inverse Gaussian distribution.J. Energy Markets. 3, 3 (2010)

    work page 2010

  2. [2]

    E., Karbach, S., and Khedher, A

    Benth, F. E., Karbach, S., and Khedher, A. Measure-valued CARMA processes. arXiv preprint arXiv:2505.08852 (2025)

    work page arXiv 2025

  3. [3]

    E., and Koekebakker, S

    Benth, F. E., and Koekebakker, S. Stochastic modeling of financial electricity contracts.Energy Econ. 30, 3 (2008), 1116–1157

    work page 2008

  4. [4]

    E., and Krühner, P

    Benth, F. E., and Krühner, P. Representation of infinite-dimensional forward price models in commodity markets.Commun. Math. Stat. 2(2014), 47–106

    work page 2014

  5. [5]

    E., Rüdiger, B., and Süss, A

    Benth, F. E., Rüdiger, B., and Süss, A. Ornstein-Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility.Stochastic Processes Appl. 128, 2 (2018), 461–486

    work page 2018

  6. [6]

    E., and Saltyte-Benth, J

    Benth, F. E., and Saltyte-Benth, J. Modeling and pricing in financial markets for weather derivatives, vol. 17. World Scientific, 2012

    work page 2012

  7. [7]

    E., and Sgarra, C

    Benth, F. E., and Sgarra, C. A Barndorff-Nielsen and Shephard model with leverage in Hilbert space for commodity forward markets.Finance Stoch. 28, 4 (2024), 1035–1076. PRICING OPTIONS ON FOR W ARDS IN AFFINE SV MODELS 25

    work page 2024

  8. [8]

    E., and Simonsen, I

    Benth, F. E., and Simonsen, I. C. The Heston stochastic volatility model in Hilbert space. Stochastic Anal. Appl. 36, 4 (2018), 733–750

    work page 2018

  9. [9]

    Interest Rate Models: an Infinite Dimensional Stochastic Analysis Perspective

    Carmona, R., and Tehranchi, M. Interest Rate Models: an Infinite Dimensional Stochastic Analysis Perspective. Springer Finance. Springer Verlag Berlin and Heidelberg, 2006

    work page 2006

  10. [10]

    Carr, P., and Madan, D. B. Option valuation using the fast fourier transform. Journal of Computational Finance 2(1999), 61–73

    work page 1999

  11. [11]

    Energy derivatives: pricing and risk management

    Clewlow, L., and Strickland, C. Energy derivatives: pricing and risk management. Lacima publications, 2000

    work page 2000

  12. [12]

    Modeling term structure dynamics: an infinite dimensional approach

    Cont, R. Modeling term structure dynamics: an infinite dimensional approach. Int. J. Theor. Appl. Finance 8, 3 (2005), 357–380

    work page 2005

  13. [13]

    Conway, J. B. A course in operator theory, vol. 21 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2000

    work page 2000

  14. [14]

    Infinite-dimensional Wishart processes.Electron

    Cox, S., Cuchiero, C., and Khedher, A. Infinite-dimensional Wishart processes.Electron. J. Probab. 29(2024), 46. Id/No 123

    work page 2024

  15. [15]

    Affine pure-jump processes on positive Hilbert-Schmidt operators

    Cox, S., Karbach, S., and Khedher, A. Affine pure-jump processes on positive Hilbert-Schmidt operators. Stochastic Processes Appl. 151(2022), 191–229

    work page 2022

  16. [16]

    An infinite-dimensional affine stochastic volatility model

    Cox, S., Karbach, S., and Khedher, A. An infinite-dimensional affine stochastic volatility model. Math. Finance 32, 3 (2022), 878–906

    work page 2022

  17. [17]

    Measure-valued processes for energy markets.Mathematical Finance 35, 2 (2025), 520–566

    Cuchiero, C., Di Persio, L., Guida, F., and Svaluto-Ferro, S. Measure-valued processes for energy markets.Mathematical Finance 35, 2 (2025), 520–566

    work page 2025

  18. [18]

    Affineprocessesonpositive semidefinite matrices.Ann

    Cuchiero, C., Filipović, D., Mayerhofer, E., and Teichmann, J. Affineprocessesonpositive semidefinite matrices.Ann. Appl. Probab. 21, 2 (2011), 397–463

    work page 2011

  19. [19]

    Generalized Feller processes and Markovian lifts of stochastic Volterra processes: the affine case.J

    Cuchiero, C., and Teichmann, J. Generalized Feller processes and Markovian lifts of stochastic Volterra processes: the affine case.J. Evol. Equ. 20, 4 (2020), 1301–1348

    work page 2020

  20. [20]

    Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift.Ann

    Da Prato, G., Flandoli, F., Priola, E., and Röckner, M. Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift.Ann. Probab. 41, 5 (2013), 3306–3344

    work page 2013

  21. [21]

    Stochastic Equations in Infinite Dimensions

    Da Prato, G., and Zabczyk, J. Stochastic Equations in Infinite Dimensions. Cambridge Uni- versity Press, Cambridge, 1992

    work page 1992

  22. [22]

    Affine processes and applications in finance

    Duffie, D., Filipović, D., and Schachermayer, W. Affine processes and applications in finance. Ann. Appl. Probab. 13, 3 (2003), 984–1053

    work page 2003

  23. [23]

    Analysis of Fourier transform valuation formulas and applications.Appl

    Eberlein, E., Glau, K., and Papapantoleon, A. Analysis of Fourier transform valuation formulas and applications.Appl. Math. Finance 17, 3-4 (2010), 211–240

    work page 2010

  24. [24]

    Energy and power risk management: New developments in modeling, pricing, and hedging, vol

    Eydeland, A., and Wolyniec, K. Energy and power risk management: New developments in modeling, pricing, and hedging, vol. 97. John Wiley & Sons, 2002

    work page 2002

  25. [25]

    Consistency Problems for Heath-Jarrow-Morton Interest rate Models

    Filipovic, D. Consistency Problems for Heath-Jarrow-Morton Interest rate Models. Lecture Notes in Mathematics. Springer Verlag Berlin and Heidelberg, 2001

    work page 2001

  26. [26]

    Term-Structure Models

    Filipovic, D. Term-Structure Models. A Graduate Course.Springer, 2009

    work page 2009

  27. [27]

    Orthogonale Polynome

    Freud, G. Orthogonale Polynome. Basel und Stuttgart: Birkhäuser Verlag. 294 S. (1969)., 1969

    work page 1969

  28. [28]

    Stationary covariance regime for affine stochastic covariance models in Hilbert spaces.Finance Stoch

    Friesen, M., and Karbach, S. Stationary covariance regime for affine stochastic covariance models in Hilbert spaces.Finance Stoch. 28, 4 (2024), 1077–1116

    work page 2024

  29. [29]

    Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation.Econometrica (1992), 77–105

    Heath, D., Jarrow, R., and Morton, A. Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation.Econometrica (1992), 77–105

    work page 1992

  30. [30]

    Heat modulated affine stochastic volatility models for forward curve dynamics, 2024

    Karbach, S. Heat modulated affine stochastic volatility models for forward curve dynamics, 2024. Available at https://arxiv.org/abs/2409.13070

    work page arXiv 2024

  31. [31]

    Finite-rank approximation of affine processes on positive Hilbert-Schmidt operators

    Karbach, S. Finite-rank approximation of affine processes on positive Hilbert-Schmidt operators. J. Math. Anal. Appl. 553, 2 (2026), 129852

    work page 2026

  32. [32]

    Moment explosions and long-term behavior of affine stochastic volatility mod- els

    Keller-Ressel, M. Moment explosions and long-term behavior of affine stochastic volatility mod- els. Math. Finance 21, 1 (2011), 73–98

    work page 2011

  33. [33]

    Exponential moments of affine processes.Ann

    Keller-Ressel, M., and Mayerhofer, E. Exponential moments of affine processes.Ann. Appl. Probab. 25, 2 (2015), 714–752

    work page 2015

  34. [34]

    Forward curve dynamics in the nordic electricity market

    Koekebakker, S., and Ollmar, F. Forward curve dynamics in the nordic electricity market. Manag. Finance 31, 6 (2005), 73–94

    work page 2005

  35. [35]

    Asset pricing with matrix jump diffusions.Available at SSRN 1572576 (2010)

    Leippold, M., and Trojani, F. Asset pricing with matrix jump diffusions.Available at SSRN 1572576 (2010)

    work page 2010

  36. [36]

    Nonlinear Operators and Differential Equations in Banach Spaces

    Martin, R. Nonlinear Operators and Differential Equations in Banach Spaces. Pure and applied mathematics. Wiley, New York, 1976

    work page 1976

  37. [37]

    Infinite dimensional affine processes.Stochastic Processes 26 PRICING OPTIONS ON FOR W ARDS IN AFFINE SV MODELS and their Applications 130, 12 (2020), 7131–7169

    Schmidt, T., Tappe, S., and Yu, W. Infinite dimensional affine processes.Stochastic Processes 26 PRICING OPTIONS ON FOR W ARDS IN AFFINE SV MODELS and their Applications 130, 12 (2020), 7131–7169

    work page 2020

  38. [38]

    Dynamically consistent analysis of realized covariations in term structure models,

    Schroers, D. Dynamically consistent analysis of realized covariations in term structure models,

    work page

  39. [39]

    Available at https://arxiv.org/abs/2406.19412

    work page arXiv

  40. [40]

    Robust functional data analysis for stochastic evolution equations in infinite dimen- sions, 2024

    Schroers, D. Robust functional data analysis for stochastic evolution equations in infinite dimen- sions, 2024. Available at https://arxiv.org/abs/2401.16286

    work page arXiv 2024

  41. [41]

    Über die Invarianz konvexer Mengen und Differentialungleichungen in einem normierten Raume

    Volkmann, P. Über die Invarianz konvexer Mengen und Differentialungleichungen in einem normierten Raume. Math. Ann. 203(1973), 201–210

    work page 1973