Volatility Modulated Volterra Processes

  • Ole E. Barndorff-Nielsen
  • Fred Espen Benth
  • Almut E. D. Veraart
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 88)


This chapter introduces the class of volatility modulated Volterra processes. We define these processes and discuss basic probabilistic properties with focus on the temporal dependency structure. Several examples are introduced, with particular emphasis on Brownian semistationary processes having generalised hyperbolic marginal distribution. Apart from examples of stochastic volatility processes, we also discuss time change as a tool for volatility modulation.


  1. Alos, E., Leon, J. A. & Vives, J. (2007), ‘On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility’, Finance & Stochastics 11, 571–589.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Applebaum, D. (2009), Lévy processes and stochastic calculus, Vol. 116 of Cambridge Studies in Advanced Mathematics, second edn, Cambridge University Press, Cambridge. Reprinted 2011 with corrections.Google Scholar
  3. Barndorff-Nielsen, O. E. (2001), ‘Superposition of Ornstein–Uhlenbeck type processes’, Theory of Probability and its Applications 45, 175–194.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Barndorff-Nielsen, O. E. (2012), ‘Notes on the gamma kernel’. Thiele Centre Research Report No. 03, May 2012.Google Scholar
  5. Barndorff-Nielsen, O. E. (2016), Gamma kernels and BSS/LSS processes, in ‘Advanced Modelling in Mathematical Finance: In Honour of Ernst Eberlein’, Springer Proceedings in Mathematics & Statistics Volume 189, Springer, pp. 41–61.Google Scholar
  6. Barndorff-Nielsen, O. E., Benth, F. E. & Veraart, A. E. D. (2013a), ‘Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes’, Bernoulli 19(3), 803–845.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Barndorff-Nielsen, O. E., Corcuera, J. & Podolskij, M. (2011b), ‘Multipower variation for Brownian semistationary processes’, Bernoulli 17(4), 1159–1194.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Barndorff-Nielsen, O. E., Pedersen, J. & Sato, K. (2001), ‘Multivariate subordination, self-decomposability and stability’, Advances in Applied Probability 33(1), 160–187.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Barndorff-Nielsen, O. E., Pérez-Abreu, V. & Thorbjørnsen, S. (2013c), ‘Lévy mixing’, ALEA. Latin American Journal of Probability and Mathematical Statistics 10(2), 1013–1062.MathSciNetzbMATHGoogle Scholar
  10. Barndorff-Nielsen, O. E., Sauri, O. & Szozda, B. (2015c), ‘Selfdecomposable fields’, Journal of Theoretical Probability pp. 1–35.Google Scholar
  11. Barndorff-Nielsen, O. E. & Schmiegel, J. (2008), ‘A stochastic differential equation framework for the timewise dynamics of turbulent velocities’, Theory of Probability and its Applications 52(3), 372–388.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Barndorff-Nielsen, O. E. & Schmiegel, J. (2009), Brownian semistationary processes and volatility/intermittency, in H. Albrecher, W. Rungaldier & W. Schachermeyer, eds, ‘Advanced Financial Modelling’, Radon Series on Computational and Applied Mathematics 8, W. de Gruyter, Berlin, pp. 1–26.zbMATHGoogle Scholar
  13. Barndorff-Nielsen, O. E. & Shephard, N. (2001), ‘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(2), 167–241.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Barndorff-Nielsen, O. E. & Shiryaev, A. (2015), Change of time and change of measure, Vol. 21 of Advanced Series on Statistical Science and Applied Probability, second edn, World Scientific.Google Scholar
  15. Barndorff-Nielsen, O. E. & Stelzer, R. (2011), ‘Multivariate supOU processes’, Annals of Applied Probability 21(1), 140–182.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Barndorff-Nielsen, O. E. & Veraart, A. E. D. (2013), ‘Stochastic volatility of volatility and variance risk premia’, Journal of Financial Econometrics 11, 1–46.CrossRefGoogle Scholar
  17. Basse, A. (2008), ‘Gaussian moving averages and semimartingales’, Electronic Journal of Probability 13, 1140–1165.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Basse, A. & Pedersen, J. (2009), ‘Lévy driven moving averages and semimartingales’, Stochastic Processes and their Applications 119(9), 2970–2991.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Basse-O’Connor, A., Graversen, S.-E. & Pedersen, J. (2014), ‘Stochastic integration on the real line’, Theory of Probability and Its Applications 58(2), 193–215.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Basse-O’Connor, A. & Rosinski, J. (2016), ‘On infinitely divisible semimartingales’, Probability Theory and Related Fields 164(1), 133–163.MathSciNetzbMATHCrossRefGoogle Scholar
  21. Bayer, C., Friz, P. & Gatheral, J. (2016), ‘Pricing under rough volatility’, Quantitative Finance 16(6), 887–904.MathSciNetCrossRefGoogle Scholar
  22. Benassi, A., Cohen, S. & Istas, J. (2004), ‘On roughness indices for fractional fields’, Bernoulli 10(2), 357–373.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Bender, C., Lindner, A. & Schicks, M. (2012), ‘Finite variation of fractional Lévy processes’, Journal of Theoretical Probability 25(2), 594–612.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Bender, C. & Marquardt, T. (2009), ‘Integrating volatility clustering into exponential Lévy models’, Journal of Applied Probability 46(3), 609–628.MathSciNetzbMATHCrossRefGoogle Scholar
  25. Bennedsen, M., Lunde, A. & Pakkanen, M. S. (2016), ‘Decoupling the short- and long-term behavior of stochastic volatility’, arXiv:1610.00332.Google Scholar
  26. Bertoin, J. (1996), Lévy processes, Vol. 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge.Google Scholar
  27. Bjerksund, P., Rasmussen, H. & Stensland, G. (2010), Valuation and risk management in the Nordic electricity market, in P. M. P. E. Bjørndal, M. Bjørndal & M. Rønnqvist, eds, ‘Energy, Natural Resources and Environmental Economics’, Springer Verlag, pp. 167–185.Google Scholar
  28. Brockwell, P. (2001a), Continuous–time ARMA processes, in D. Shanbhag & C. Rao, eds, ‘Handbook of Statistics’, Vol. 19 of Stochastic Processes: Theory and Methods, Elsevier, Amsterdam, pp. 249–275.Google Scholar
  29. Brockwell, P. (2001b), ‘Lévy–driven CARMA processes’, Annals of the Institute of Statistical Mathematics 53, 113–124.MathSciNetzbMATHCrossRefGoogle Scholar
  30. Brockwell, P. (2004), ‘Representations of continuous–time ARMA processes’, Journal of Applied Probability 41(A), 375–382.MathSciNetzbMATHCrossRefGoogle Scholar
  31. Brockwell, P. J. (2009), Lévy-driven continuous-time ARMA processes, in T. Mikosch, J.-P. Kreiß, R. A. Davis & T. G. Andersen, eds, ‘Handbook of financial time series’, Springer, Berlin, pp. 457–480.zbMATHCrossRefGoogle Scholar
  32. Cherny, A. & Shiryaev, A. (2005), On stochastic integrals up to infinity and predictable criteria for integrability, in ‘Séminaire de Probabilités XXXVIII’, Vol. 1857 of Lecture Notes in Mathematics, Springer.Google Scholar
  33. Comte, F. & Renault, E. (1998), ‘Long memory in continuous-time stochastic volatility models’, Mathematical Finance 8(4), 291–323.MathSciNetzbMATHCrossRefGoogle Scholar
  34. Cont, R. & Tankov, P. (2004), Financial modelling with jump processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
  35. Cox, J. C., Ingersoll, Jr., J. E. & Ross, S. A. (1985), ‘A theory of the term structure of interest rates’, Econometrica 53(2), 385–407.MathSciNetzbMATHCrossRefGoogle Scholar
  36. Cramér, H. & Leadbetter, M. R. (1967), Stationary and related stochastic processes. Sample function properties and their applications, John Wiley & Sons, Inc., New York-London-Sydney.zbMATHGoogle Scholar
  37. Dubins, L. E. & Schwarz, G. (1965), ‘On continuous martingales’, Proceedings of the National Academy of Sciences of the United States of America 53, 913–916.MathSciNetzbMATHCrossRefGoogle Scholar
  38. El Euch, O., Fukasawa, M. & Rosenbaum, M. (2018), ‘The microstructural foundations of leverage effect and rough volatility’, Finance & Stochastics 22, 241–280.MathSciNetzbMATHCrossRefGoogle Scholar
  39. El Euch, O. & Rosenbaum, M. (2018+), ‘The characteristic function of rough Heston models’, To appear in Mathematical Finance.Google Scholar
  40. Feller, W. (1951), ‘Two singular diffusion problems’, Annals of Mathematics. Second Series 54, 173–182.MathSciNetzbMATHCrossRefGoogle Scholar
  41. Folland, G. B. (1984), Real Analysis – Modern Techniques and their Applications, John Wiley & Sons.zbMATHGoogle Scholar
  42. Gasquet, C. & Witomski, P. (1999), Fourier analysis and applications: Filtering, numerical computation, wavelets, Vol. 30 of Texts in Applied Mathematics, Springer-Verlag, New York.Google Scholar
  43. Gatheral, J., Jaisson, T. & Rosenbaum, M. (2018), ‘Volatility is rough’. Quantitative Finance 18(6), 933–949.MathSciNetzbMATHCrossRefGoogle Scholar
  44. Halgreen, C. (1979), ‘Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions’, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 47, 13–17.MathSciNetzbMATHCrossRefGoogle Scholar
  45. Heston, S. L. (1993), ‘A closed-form solution for options with stochastic volatility with applications to bond and currency options’, The Review of Financial Studies 6(2), 327–343.MathSciNetzbMATHCrossRefGoogle Scholar
  46. Jaisson, T. & Rosenbaum, M. (2016), ‘Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes’, The Annals of Applied Probability 26(5), 2860– 2882.MathSciNetzbMATHCrossRefGoogle Scholar
  47. Jurek, Z. J. & Vervaat, W. (1983), ‘An integral representation for self-decomposable Banach space valued random variables’, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 62, 247–262.MathSciNetzbMATHCrossRefGoogle Scholar
  48. Knight, F. (1992), Foundations of the Prediction Process, Clarendon Press.zbMATHGoogle Scholar
  49. Marquardt, T. (2006), ‘Fractional Lévy processes with an application to long memory moving average processes’, Bernoulli 12(6), 1099–1126.MathSciNetzbMATHCrossRefGoogle Scholar
  50. Monroe, I. (1978), ‘Processes that can be embedded in Brownian motion’, The Annals of Probability 6(1), 42–56.MathSciNetzbMATHCrossRefGoogle Scholar
  51. Pedersen, J. & Sauri, O. (2015), On Lévy semistationary processes with a gamma kernel, in R. H. Mena, J. C. Pardo, V. Rivero & G. U. Bravo, eds, ‘XI Symposium of Probability and Stochastic Processes: CIMAT, Mexico, November 18–22, 2013’, Vol. 69 of Progress in Probability, Springer, pp. 217–239.Google Scholar
  52. Podolskij, M. (2015), Ambit fields: survey and new challenges, in R. H. Mena, J. C. Pardo, V. Rivero & G. U. Bravo, eds, ‘XI Symposium of Probability and Stochastic Processes: CIMAT, Mexico, November 18–22, 2013’, Vol. 69 of Progress in Probability, Springer, pp. 241–279.Google Scholar
  53. Protter, P. E. (2005), Stochastic integration and differential equations, Vol. 21 of Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin. Second edition. Version 2.1, Corrected third printing.Google Scholar
  54. Rajput, B. S. & Rosiński, J. (1989), ‘Spectral representations of infinitely divisible processes’, Probability Theory and Related Fields 82(3), 451–487.MathSciNetzbMATHCrossRefGoogle Scholar
  55. Rodriguez, D. M. (1971), ‘Processes obtainable from Brownian motion by means of a random time change’, Annals of Mathematical Statistics 42, 115–129.MathSciNetzbMATHCrossRefGoogle Scholar
  56. Sato, K. (1999), Lévy processes and infinitely divisible distributions, Vol. 68 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge. Translated from the 1990 Japanese original, Revised by the author.Google Scholar
  57. Shephard, N., ed. (2005), Stochastic Volatility: Selected Readings, Advanced Texts in Econometrics, Oxford University Press, Oxford, UK.Google Scholar
  58. Sørensen, E. H. L. (2012), Stochastic modelling of turbulence: With applications to wind energy, PhD thesis, Aarhus University: Department of Mathematics.Google Scholar
  59. Todorov, V. & Tauchen, G. (2006), ‘Simulation methods for Lévy-driven continuous-time autoregressive moving average (CARMA) stochastic volatility models’, Journal of Business & Economic Statistics 24(4), 455–469.MathSciNetCrossRefGoogle Scholar
  60. Tsai, H. & Chan, K. S. (2005), ‘A note on non-negative continuous time processes’, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67(4), 589–597.MathSciNetzbMATHCrossRefGoogle Scholar
  61. Veraart, A. E. D. (2015b), ‘Stationary and multi-self-similar random fields with stochastic volatility’, Stochastics 87(5), 848–870.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Ole E. Barndorff-Nielsen
    • 1
  • Fred Espen Benth
    • 2
  • Almut E. D. Veraart
    • 3
  1. 1.Department of MathematicsUniversity of AarhusAarhusDenmark
  2. 2.Department of MathematicsUniversity of OsloOsloNorway
  3. 3.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations