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arxiv: 1907.02363 · v1 · pith:X6SOAWWXnew · submitted 2019-07-04 · 🧮 math.PR · q-fin.MF

Existence of affine realizations for L\'evy term structure models

Stefan Tappe This is my paper

Pith reviewed 2026-05-25 09:12 UTC · model grok-4.3

classification 🧮 math.PR q-fin.MF
keywords affine realizationsLévy processesterm structure modelsvolatility restrictionsconstant direction volatilitiesshort rate realizationsjump processes
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The pith

Lévy-driven term structure models require stricter volatility restrictions for affine realizations than diffusion models.

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

This paper investigates the existence of affine realizations for term structure models driven by Lévy processes. It establishes that the jumps inherent to Lévy processes impose more severe restrictions on the volatility structure than those needed in the classical diffusion setting without jumps. The analysis covers general conditions and then specializes to constant direction volatilities and the existence of short rate realizations. A sympathetic reader would care because affine realizations allow the infinite-dimensional model to reduce to a finite-dimensional Markov process, which simplifies analysis and computation.

Core claim

For term structure models driven by Lévy processes, the existence of affine realizations demands stricter conditions on the volatility than in the diffusion case. These additional restrictions arise directly from the jump component of the Lévy process and limit the volatility structures that permit the model to admit an affine realization. The paper derives these conditions and applies them to special cases including constant direction volatilities and short rate realizations.

What carries the argument

The volatility structure conditions that permit an affine realization for a Lévy-driven term structure model.

If this is right

  • Constant direction volatilities admit affine realizations only when they also satisfy the additional Lévy-specific constraints.
  • Short rate realizations exist only under the enhanced volatility restrictions imposed by the jumps.
  • Fewer volatility structures overall qualify for affine realizations once jumps are present.
  • The finite-dimensional reduction of the term structure model becomes harder to achieve in the presence of jumps.

Where Pith is reading between the lines

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

  • Practical Lévy models commonly used in finance may rarely satisfy the conditions, limiting the set of tractable models.
  • Numerical pricing methods that rely on affine structure would apply to a narrower class of jump-driven term structures.
  • The result suggests checking whether popular Lévy specifications such as those with tempered stable jumps meet the stricter criteria.

Load-bearing premise

The term structure dynamics are driven by a Lévy process whose volatility structure is of a form that could potentially satisfy the derived affine realization conditions.

What would settle it

A concrete volatility function that satisfies the classical diffusion conditions for affine realizations but produces non-affine dynamics when the driving process is changed to a Lévy process with jumps.

read the original abstract

We investigate the existence of affine realizations for term structure models driven by L\'evy processes. It turns out that we obtain more severe restrictions on the volatility than in the classical diffusion case without jumps. As special cases, we study constant direction volatilities and the existence of short rate realizations.

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

0 major / 3 minor

Summary. The manuscript investigates the existence of affine realizations for term structure models driven by Lévy processes. It establishes that the presence of jumps imposes stricter restrictions on admissible volatility structures than in the classical diffusion setting. Special cases treated include constant-direction volatilities and the existence of short-rate realizations.

Significance. If the central results hold, the work supplies a precise extension of the affine term-structure theory to Lévy-driven dynamics, identifying jump-induced constraints on volatility that are absent in the diffusion case. The special-case analyses are likely to be useful for model calibration and for understanding when finite-dimensional realizations remain possible.

minor comments (3)
  1. [§2] §2 (model setup): the precise definition of the volatility operator and its domain should be stated explicitly before the main existence theorems, as the jump measure interacts with this operator in the subsequent restrictions.
  2. [Theorem 3.1] Theorem 3.1 (or equivalent main result): the statement of the volatility restriction would benefit from an explicit comparison (e.g., a remark or corollary) to the corresponding diffusion condition, so that the claimed severity is immediately verifiable.
  3. [§2] The notation for the short-rate process and the affine realization map is introduced gradually; a consolidated table or list of symbols at the end of §2 would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The manuscript shows that the presence of jumps in Lévy-driven term structure models imposes stricter volatility restrictions for affine realizations than in the diffusion case, with special cases for constant-direction volatilities and short-rate realizations.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a mathematical existence proof deriving conditions for affine realizations in Lévy-driven term structure models, yielding stricter volatility restrictions than the diffusion case. No load-bearing steps reduce by construction to inputs, self-citations, or fitted parameters; the derivation relies on standard analysis of the Lévy measure, volatility operator, and short-rate dynamics, making the result self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5554 in / 805 out tokens · 25500 ms · 2026-05-25T09:12:10.787576+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    (2005): Hypoellipticity in infinite dimensions and an application in interest rate theory.Annals of Applied Probability15(3), 1765–1777

    Baudoin, F., Teichmann, J. (2005): Hypoellipticity in infinite dimensions and an application in interest rate theory.Annals of Applied Probability15(3), 1765–1777

    work page 2005

  2. [2]

    The European Journal of Finance3, 1–26

    Bhar,R.,Chiarella,C.(1997):TransformationofHeath–Jarrow–MortonmodelstoMarkovian systems. The European Journal of Finance3, 1–26

    work page 1997

  3. [3]

    (2003): On the geometry of interest rate models

    Björk, T. (2003): On the geometry of interest rate models. Paris-Princeton Lectures on Mathematical Finance, 133–215

    work page 2003

  4. [4]

    (1999): Interest rate dynamics and consistent forward rate curves

    Björk, T., Christensen, C. (1999): Interest rate dynamics and consistent forward rate curves. Mathematical Finance 9(4), 323–348

    work page 1999

  5. [5]

    (1997): Towards a general theory of bond markets.Finance and Stochastics1(2), 141–174

    Björk, T., Di Masi, G., Kabanov, Y., Runggaldier, W. (1997): Towards a general theory of bond markets.Finance and Stochastics1(2), 141–174

    work page 1997

  6. [6]

    (1999): Minimal realizations of interest rate models.Finance and Stochastics 3(4), 413–432

    Björk, T., Gombani, A. (1999): Minimal realizations of interest rate models.Finance and Stochastics 3(4), 413–432

    work page 1999

  7. [7]

    (1997): Bond market structure in the presence of marked point processes.Mathematical Finance 7(2), 211–239

    Björk, T., Kabanov, Y., Runggaldier, W. (1997): Bond market structure in the presence of marked point processes.Mathematical Finance 7(2), 211–239

    work page 1997

  8. [8]

    (2002): On the construction of finite dimensional realizations for non- linear forward rate models.Finance and Stochastics6(3), 303–331

    Björk, T., Landén, C. (2002): On the construction of finite dimensional realizations for non- linear forward rate models.Finance and Stochastics6(3), 303–331

    work page 2002

  9. [9]

    (2001): On the existence of finite dimensional realizations for nonlinear forward rate models.Mathematical Finance 11(2), 205–243

    Björk, T., Svensson, L. (2001): On the existence of finite dimensional realizations for nonlinear forward rate models.Mathematical Finance 11(2), 205–243

    work page 2001

  10. [10]

    (2006):Interest rate models: An infinite dimensional stochastic analysis perspective.Springer, Berlin

    Carmona, R., Tehranchi, M. (2006):Interest rate models: An infinite dimensional stochastic analysis perspective.Springer, Berlin

    work page 2006

  11. [11]

    Chiarella, C., Kwon, O. K. (2001): Forward rate dependent Markovian transformations of the Heath–Jarrow–Morton term structure model.Finance and Stochastics5(2), 237–257

    work page 2001

  12. [12]

    Chiarella, C., Kwon, O. K. (2003): Finite dimensional affine realizations of HJM models in terms of forward rates and yields.Review of Derivatives Research6(3), 129–155

    work page 2003

  13. [13]

    (1994): A general version of the fundamental theorem of asset pricing.Mathematische Annalen 300, 463–520

    Delbaen, F., Schachermayer, W. (1994): A general version of the fundamental theorem of asset pricing.Mathematische Annalen 300, 463–520

    work page 1994

  14. [14]

    (1996): A yield-factor model of interest rates.Mathematical Finance6(4), 379–406

    Duffie, D., Kan, R. (1996): A yield-factor model of interest rates.Mathematical Finance6(4), 379–406

    work page 1996

  15. [15]

    (2005): Lévy term structure models: No-arbitrage and completeness

    Eberlein, E., Jacod, J., Raible, S. (2005): Lévy term structure models: No-arbitrage and completeness. Finance and Stochastics9(1), 67–88

    work page 2005

  16. [16]

    (2006): Exact pricing formulae for caps and swaptions in a Lévy term structure model.Journal of Computational Finance9(2), 99–125

    Eberlein, E., Kluge, W. (2006): Exact pricing formulae for caps and swaptions in a Lévy term structure model.Journal of Computational Finance9(2), 99–125

    work page 2006

  17. [17]

    (2006): Valuation of floating range notes in Lévy term structure models

    Eberlein, E., Kluge, W. (2006): Valuation of floating range notes in Lévy term structure models. Mathematical Finance 16(2), 237–254

    work page 2006

  18. [18]

    (2007): Calibration of Lévy term structure models

    Eberlein, E., Kluge, W. (2007): Calibration of Lévy term structure models. InAdvances in Mathematical Finance:In Honor of Dilip Madan, M. Fu, R. A. Jarrow, J.-Y. Yen, and R. J. Elliott (Eds.), Birkhäuser, pp. 155–180

    work page 2007

  19. [19]

    (2003): The defaultable Lévy term structure: Ratings and restructur- ing

    Eberlein, E., Özkan, F. (2003): The defaultable Lévy term structure: Ratings and restructur- ing. Mathematical Finance 13(2), 277–300

    work page 2003

  20. [20]

    (1999): Term structure models driven by general Lévy processes

    Eberlein, E., Raible, S. (1999): Term structure models driven by general Lévy processes. Mathematical Finance 9(1), 31–53

    work page 1999

  21. [21]

    (2001):Consistency problems for Heath–Jarrow–Morton interest rate models

    Filipović, D. (2001):Consistency problems for Heath–Jarrow–Morton interest rate models. Springer, Berlin

    work page 2001

  22. [22]

    (2008): Existence of Lévy term structure models

    Filipović, D., Tappe, S. (2008): Existence of Lévy term structure models. Finance and Stochastics 12(1), 83–115

    work page 2008

  23. [23]

    (2003): Existence of invariant manifolds for stochastic equations in infinite dimension.Journal of Functional Analysis197(2), 398–432

    Filipović, D., Teichmann, J. (2003): Existence of invariant manifolds for stochastic equations in infinite dimension.Journal of Functional Analysis197(2), 398–432

    work page 2003

  24. [24]

    (2004): On the geometry of the term structure of interest rates

    Filipović, D., Teichmann, J. (2004): On the geometry of the term structure of interest rates. Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engi- neering Sciences 460(2041), 129–167

    work page 2004

  25. [25]

    (2011):Analysis 1.Vieweg, Wiesbaden

    Forster, O. (2011):Analysis 1.Vieweg, Wiesbaden

    work page 2011

  26. [26]

    (2006): On Markovian short rates in term structure models driven by jump-diffusion processes.Statistics and Decisions24(2), 255–271

    Gapeev, P., Küchler, U. (2006): On Markovian short rates in term structure models driven by jump-diffusion processes.Statistics and Decisions24(2), 255–271

    work page 2006

  27. [27]

    (1992): Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation.Econometrica 60(1), 77–105

    Heath, D., Jarrow, R., Morton, A. (1992): Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation.Econometrica 60(1), 77–105

    work page 1992

  28. [28]

    (2000): Affine term structures and short-rate realizations of forward rate models driven by jump-diffusion processes

    Hyll, M. (2000): Affine term structures and short-rate realizations of forward rate models driven by jump-diffusion processes. InEssays on the term structure of interest ratesPhD thesis, Stockholm School of Economics

    work page 2000

  29. [29]

    (1998): A Markovian framework in multi-factor Heath–Jarrow–Morton models

    Inui, K., Kijima, M. (1998): A Markovian framework in multi-factor Heath–Jarrow–Morton models. Journal of Financial and Quantitative Analysis33(3), 423–440. 16 STEF AN TAPPE

    work page 1998

  30. [30]

    Jarrow, A., Madan, D. B. (1995): Option pricing using the term structure of interest rates to hedge systematic discontinuities in asset returns.Mathematical Finance 5(4), 311–336

    work page 1995

  31. [31]

    (1995): Single factor Heath–Jarrow–Morton term structure models based on Markov spot interest rate dynamics.Journal of Financial and Quantitative Analysis30(4), 619–642

    Jeffrey, A. (1995): Single factor Heath–Jarrow–Morton term structure models based on Markov spot interest rate dynamics.Journal of Financial and Quantitative Analysis30(4), 619–642

    work page 1995

  32. [32]

    (2003): Markovian short rates in a forward rate model with a general class of Lévy processes

    Küchler, U., Naumann, E. (2003): Markovian short rates in a forward rate model with a general class of Lévy processes. Discussion Paper 6, Sonderforschungsbereich 373, Humboldt University Berlin

    work page 2003

  33. [33]

    G., Pardoux, E., Protter, P

    Kurtz, T. G., Pardoux, E., Protter, P. (1995): Stratonovich stochastic differential equations driven by general semimartingales. Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 31(2), 351–377

    work page 1995

  34. [34]

    (2010): Local well-posedness of Musiela’s SPDE with Lévy noise.Mathematical Finance 20(3), 341–363

    Marinelli, C. (2010): Local well-posedness of Musiela’s SPDE with Lévy noise.Mathematical Finance 20(3), 341–363

    work page 2010

  35. [35]

    Markus, S. I. (1978): Modeling and analysis of stochastic differential equations driven by point processes.IEEE Transactions on Information Theory24(2), 164–172

    work page 1978

  36. [36]

    Markus, S. I. (1981): Modeling and approximation of stochastic differential equations driven by semimartingales.Stochastics 4(3), 223–245

    work page 1981

  37. [37]

    (1993): Stochastic PDEs and term structure models.Journées Internationales de Finance, IGR-AFFI, La Baule

    Musiela, M. (1993): Stochastic PDEs and term structure models.Journées Internationales de Finance, IGR-AFFI, La Baule

    work page 1993

  38. [38]

    (1983):Semigroups of linear operators and applications to partial differential equa- tions

    Pazy, A. (1983):Semigroups of linear operators and applications to partial differential equa- tions. Springer, New York

    work page 1983

  39. [39]

    (2007):Stochastic partial differential equations with Lévy noise

    Peszat, S., Zabczyk, J. (2007):Stochastic partial differential equations with Lévy noise. Cam- bridge University Press, Cambridge

    work page 2007

  40. [40]

    (2007): Heath-Jarrow-Morton-Musiela equation of bond market

    Peszat, S., Zabczyk, J. (2007): Heath-Jarrow-Morton-Musiela equation of bond market. Preprint IMPAN 677, Warsaw.(www.impan.gov.pl/EN/Preprints/index.html)

    work page 2007

  41. [41]

    (1995): Volatility structures of forward rates and the dynamics of the term structure.Mathematical Finance 5(1), 55–72

    Ritchken, P., Sankarasubramanian, L. (1995): Volatility structures of forward rates and the dynamics of the term structure.Mathematical Finance 5(1), 55–72

    work page 1995

  42. [42]

    (1991): Interest rate option pricing with Poisson-Gaussian forward rate curve processes

    Shirakawa, H. (1991): Interest rate option pricing with Poisson-Gaussian forward rate curve processes. Mathematical Finance 1(4), 77–94

    work page 1991

  43. [43]

    (2010): An alternative approach on the existence of affine realizations for HJM term structure models.Proceedings of The Royal Society of London

    Tappe, S. (2010): An alternative approach on the existence of affine realizations for HJM term structure models.Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences466(2122), 3033–3060. Leibniz Universität Hannover, Institut für Mathematische Stochastik, Welfen- garten 1, 30167 Hannover, Germany E-mail addres...

    work page 2010