Skip to main content
Log in

Lookback option pricing under the double Heston model using a deep learning algorithm

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

To price floating strike lookback options, we obtain a partial differential equation (PDE) according to the double Heston model. To solve the PDE, we employ a deep learning algorithm called the deep Galerkin method (DGM), which is well-suited for high-dimensional PDEs. Finally, we compare the obtained results from mentioned method with the option price under the Monte Carlo simulation method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  • Albrecher H, Mayer P, Schoutens W, Tistaert J (2007) The little Heston trap. Wilmott 1:83–92

    Google Scholar 

  • Buchen P, Konstandatos O (2005) A new method of pricing lookback options. Math Finance Int J Math Stat Financ Econ 15(2):245–259

    MathSciNet  MATH  Google Scholar 

  • Christoffersen P, Heston S, Jacobs K (2009) The shape and term structure of the index option smirk: why multifactor stochastic volatility models work so well. Manag Sci 55(12):1914–1932

    Article  MATH  Google Scholar 

  • Conze A, Viswanathan R (1991) Path dependent options: the case of lookback options. J Finance 46(5):1893–1907

    Article  Google Scholar 

  • Deng G (2020) Pricing perpetual American floating strike lookback option under multiscale stochastic volatility model. Chaos Solitons Fract 141:110411

    Article  MathSciNet  MATH  Google Scholar 

  • Dokuchaev GZ, Wang MS (2022) A modification of Galerkin’s method for option pricing. J Ind Manag Optim 18(4):2483

    Article  MathSciNet  MATH  Google Scholar 

  • Duffy DJ (2013) Finite difference methods in financial engineering: a partial differential equation approach. Wiley, New York

    Google Scholar 

  • Fallah S, Mehrdoust F (2019) On the existence and uniqueness of the solution to the double Heston model equation and valuing lookback option. J Comput Appl Math 350:412–422

    Article  MathSciNet  MATH  Google Scholar 

  • Forsyth P, Vetzal K, Zvan R (1999) A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Appl Math Finance 6(2):87–106

    Article  MATH  Google Scholar 

  • Gauthier P, Possamaï D (2010) Efficient simulation of the double Heston model, Available at SSRN 1434853

  • Goldman MB, Sosin HB, Gatto MA (1979) Path dependent options: “buy at the low, sell at the high’’. J Financ 34(5):1111–1127

    Google Scholar 

  • Han J, Jentzen A, Weinan E (2018) Solving high-dimensional partial differential equations using deep learning. Proc Natl Acad Sci 115(34):8505–8510

    Article  MathSciNet  MATH  Google Scholar 

  • Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6(2):327–343

    Article  MathSciNet  MATH  Google Scholar 

  • Hilber N, Reichmann O, Schwab C, Winter C (2013) Computational methods for quantitative finance: finite element methods for derivative pricing. Springer Science & Business Media, Berlin

    Book  MATH  Google Scholar 

  • Hochreiter S, Schmidhuber J (1997) Long short-term memory. Neural Comput 9(8):1735–1780

    Article  Google Scholar 

  • Lagaris IE, Likas A, Fotiadis DI (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw 9(5):987–1000

    Article  Google Scholar 

  • Lee M-K (2017) Pricing perpetual American lookback options under stochastic volatility. Comput Econ 53(3):1265–1277

    Article  Google Scholar 

  • Lee H, Kang IS (1990) Neural algorithm for solving differential equations. J Comput Phys 91(1):110–131

    Article  MathSciNet  MATH  Google Scholar 

  • Leung KS (2013) An analytic pricing formula for lookback options under stochastic volatility. Appl Math Lett 26(1):145–149

    Article  MathSciNet  MATH  Google Scholar 

  • Malek A, Beidokhti RS (2006) Numerical solution for high order differential equations using a hybrid neural network-optimization method. Appl Math Comput 183(1):260–271

    Article  MathSciNet  MATH  Google Scholar 

  • Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686–707

    Article  MathSciNet  MATH  Google Scholar 

  • Shreve SE (2004) Stochastic calculus for finance II: continuous-time models, vol 11. Springer Science & Business Media, Berlin

    Book  MATH  Google Scholar 

  • Sirignano J, Spiliopoulos K (2018) Dgm: a deep learning algorithm for solving partial differential equations. J Comput Phys 375:1339–1364

    Article  MathSciNet  MATH  Google Scholar 

  • Srivastava RK, Greff K, Schmidhuber J (2015) Training very deep networks, arXiv preprint arXiv:1507.06228

  • Weinan E, Han J, Jentzen A (2017) Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun Math Stat 5(4):349–380

    Article  MathSciNet  MATH  Google Scholar 

  • Wilmott P (2013) Paul Wilmott on quantitative finance. Wiley, New York

    MATH  Google Scholar 

  • Wong HY, Chan CM (2007) Lookback options and dynamic fund protection under multiscale stochastic volatility. Insur Math Econ 40(3):357–385

    Article  MathSciNet  MATH  Google Scholar 

  • Yu B (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun Math Stat 6:1–12

    Article  MathSciNet  MATH  Google Scholar 

  • Zang Y, Bao G, Ye X, Zhou H (2020) Weak adversarial networks for high-dimensional partial differential equations. J Comput Phys 411:109409

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu Y, Zabaras N, Koutsourelakis P-S, Perdikaris P (2019) Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. J Comput Phys 394:56–81

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Contributions

All authors contributed to the study conception and design.

Corresponding author

Correspondence to Farshid Mehrdoust.

Ethics declarations

Conflict of interest

We declare that we have no conflict of interest.

Additional information

Communicated by Silvana Manuela Pesenti.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Motameni, M., Mehrdoust, F. & Najafi, A.R. Lookback option pricing under the double Heston model using a deep learning algorithm. Comp. Appl. Math. 41, 378 (2022). https://doi.org/10.1007/s40314-022-02098-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-02098-5

Keywords

Mathematics Subject Classification

Navigation