Volatility Uncertainty Quantification in a Stochastic Control Problem Applied to Energy

  • Francisco Bernal
  • Emmanuel GobetEmail author
  • Jacques Printems


This work designs a methodology to quantify the uncertainty of a volatility parameter in a stochastic control problem arising in energy management. The difficulty lies in the non-linearity of the underlying scalar Hamilton-Jacobi-Bellman equation. We proceed by decomposing the unknown solution on a Hermite polynomial basis (of the unknown volatility), whose different coefficients are solutions to a system of second order parabolic non-linear PDEs. Numerical tests show that computing the first basis elements may be enough to get an accurate approximation with respect to the uncertain volatility parameter. We provide an example of the methodology in the context of a swing contract (energy contract with flexibility in purchasing energy power), this allows us to introduce the concept of Uncertainty Value Adjustment (UVA), whose aim is to value the risk of misspecification of the volatility model.


Chaos expansion Uncertainty quantification Stochastic control Stochastic programming Swing options Monte Carlo simulations 

Mathematics Subject Classification (2010)

93Exx 62L20 41A10 90C15 49L20 


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This research is part of the Chair Financial Risks of the Risk Foundation, the Finance for Energy Market Research Centre (FiME) and the ANR project CAESARS (ANR-15-CE05-0024).


  1. Basel Committee on Banking Supervision (2015) Review of the credit valuation adjustment risk framework. Bank for International SettlementsGoogle Scholar
  2. Bardou O, Bouthemy S, Pages G (2009) Optimal quantization for the pricing of swing options. Appl Math Finance 16(2):183–217MathSciNetzbMATHGoogle Scholar
  3. Barrera-Esteve C, Bergeret F, Dossal C, Gobet E, Meziou A, Munos R, Reboul-Salze D (2006) Numerical methods for the pricing of swing options: a stochastic control approach. Methodol Comput Appl Probab 8(4):517–540MathSciNetzbMATHGoogle Scholar
  4. Benkert K, Fischer R (2007) An efficient implementation of the Thomas algorithm for block pentadiagonal systems on vector computers. In: Shi Y, van Albada GD, Dongarra J, Sloot PMA (eds) Proceedings of the 7th International Conference on Computer Science, ICCS, pp 144–151Google Scholar
  5. Briand P, Labart C (2014) Simulation of BSDEs by Wiener Chaos Expansion. Ann Appl Probab 24(3):1129–1171MathSciNetzbMATHGoogle Scholar
  6. Brennan MJ (1991) The price of convenience and the valuation of commodity contingent claims. Stochastic Models and Option Values 200(22–71)Google Scholar
  7. Barles G, Souganidis PE (1991) Convergence of approximation schemes for fully nonlinear second order equations. Asymptot Anal 4:271–283MathSciNetzbMATHGoogle Scholar
  8. Clewlow L, Strickland C, Kaminski V (2001) Valuation of swing contracts in trees. Energy Power Risk Manag 6(4):33–34Google Scholar
  9. Gobet E (2002) LAN Property for ergodic diffusion with discrete observations. Ann Inst H Poincaré, Probab Statist 38(5):711–737MathSciNetzbMATHGoogle Scholar
  10. Gerritsma M, van der Steen JB, Vos P, Karniadakis GE (2010) Time-dependent generalized polynomial chaos. J Comput Phys 229(22):8333–8363MathSciNetzbMATHGoogle Scholar
  11. Harvey AC (1989) Forecasting, Structural Time Series Analysis, and the Kalman Filter. CambridgeGoogle Scholar
  12. Huschto T, Sager S (2014) Solving stochastic optimal control problems by a Wiener chaos approach. Vietnam J Math 42(1):83–113MathSciNetzbMATHGoogle Scholar
  13. Jaillet P, Ronn EI, Tompaidis S (2004) Valuation of commodity-based swing options. Manag Sci 50:909–921zbMATHGoogle Scholar
  14. Keppo J (2004) Pricing of electricity swing options. J Derivatives 11(3):26–43Google Scholar
  15. Kleiber M, Hien TD (1992) The stochastic finite element method (basic perturbation technique and computer implementation). Wiley, ChichesterzbMATHGoogle Scholar
  16. Le Maître O, Knio OM (2010) Spectral methods for uncertainty quantification: with applications to computational fluid dynamics. Springer Science & Business Media, New YorkzbMATHGoogle Scholar
  17. Loh WL (1996) On latin hypercube sampling. Ann Stat 24(5):2058–2080MathSciNetzbMATHGoogle Scholar
  18. Liu W, Yang Y, Lu G (2003) Viscosity solutions of fully nonlinear parabolic systems. J Math Anal Appl 281(1):362–381MathSciNetzbMATHGoogle Scholar
  19. Mikulevicius R, Rozovskii B (1998) Linear parabolic stochastic PDE and Wiener chaos. SIAM J Math Anal 29(2):452–480MathSciNetzbMATHGoogle Scholar
  20. Mikulevicius R, Rozovskii B (2005) Global L2-solutions of stochastic navier-Stokes equations. Ann Probab 33(1):137–176MathSciNetzbMATHGoogle Scholar
  21. Manoliu M, Tompaidis S (2002) Energy futures prices: term structure models with Kalman filter estimation. Appl Math Finance 9(1):21–43zbMATHGoogle Scholar
  22. Niederreiter H (1992) Random number generation and quasi-Monte-Carlo methods, volume 63 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, PhiladelphiaGoogle Scholar
  23. Prakasa Rao BLS (1999) Statistical inference for diffusion type processes Kendall’s Library of statistics, vol 8. Edward Arnold. Oxford University Press, LondonGoogle Scholar
  24. Schwartz E (1997) The stochastic behavior of commodity prices: implications for valuation and hedging. J Finance 52(3):923–973Google Scholar
  25. Schwartz E, Smith JE (2000) Short-term variations and long-term dynamics in commodity prices. Manag Sci 46(7):893–911Google Scholar
  26. Tourin A (2013) An introduction to finite difference methods for PDEs in finance. In: Touzi N (ed) Optimal stochastic target problems and backward SDE, Fields Institute Monographs. SpringerGoogle Scholar
  27. Teckentrup AL, Scheichl R, Giles MB, Ullmann E (2013) Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients. Numer Math 125(3):569–600MathSciNetzbMATHGoogle Scholar
  28. Warin X (2016) Some non monotone schemes for time dependent Hamilton-Jacobi-Bellman equations in stochastic control. J Sci Comp 66(3):1122–1147MathSciNetzbMATHGoogle Scholar
  29. Xiu D (2009) Fast numerical methods for stochastic computations: a review. Commun Comput Phys 5(2-4):242–272MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Francisco Bernal
    • 1
  • Emmanuel Gobet
    • 1
    Email author
  • Jacques Printems
    • 2
  1. 1.CMAPEcole PolytechniquePalaiseauFrance
  2. 2.LAMAUniversité Paris-Est CréteilCréteil CedexFrance

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