Top

Finance and Stochastics

Published in:

24-11-2023

Pricing options on flow forwards by neural networks in a Hilbert space

Authors: Fred Espen Benth, Nils Detering, Luca Galimberti

Published in: Finance and Stochastics | Issue 1/2024

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

We propose a new methodology for pricing options on flow forwards by applying infinite-dimensional neural networks. We recast the pricing problem as an optimisation problem in a Hilbert space of real-valued functions on the positive real line, which is the state space for the term structure dynamics. This optimisation problem is solved by using a feedforward neural network architecture designed for approximating continuous functions on the state space. The proposed neural network is built upon the basis of the Hilbert space. We provide case studies that show its numerical efficiency, with superior performance over that of a classical neural network trained on sampling the term structure curves.

previous article Dispersion-constrained martingale Schrödinger problems and the exact joint S&P 500/VIX smile calibration puzzle

next article Optimal investment and consumption for financial markets with jumps under transaction costs

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

https://github.com/ncdetering/FlowForwardsNumerics

Barron, A.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inf. Theory 39, 930–945 (1993) MathSciNetCrossRef

Bayer, C., Stemper, B.: Deep calibration of rough stochastic volatility models. Preprint (2018). https://arxiv.org/abs/1810.03399

Beck, C., Becker, S., Grohs, P., Jaafari, N., Jentzen, A.: Solving the Kolmogorov PDE by means of deep learning. J. Sci. Comput. 88, 73-1–28 (2021) MathSciNetCrossRef

Beck, C., Hutzenthaler, M., Jentzen, A., Kuckuck, B.: An overview on deep learning-based approximation methods for partial differential equations. Discrete Contin. Dyn. Syst., Ser. B 28, 3697–3746 (2023) MathSciNetCrossRef

Benth, F.E., Benth, J.S., Koekebakker, S.: Stochastic Modelling of Electricity and Related Markets. World Scientific, Singapore (2008) CrossRef

Benth, F.E., Detering, N., Galimberti, L.: Neural networks in Fréchet spaces. Ann. Math. Artif. Intell. 91, 75–103 (2023) MathSciNetCrossRef

Benth, F.E., Detering, N., Krühner, P.: Stochastic Volterra integral equations and a class of first order stochastic partial differential equations. Stochastics 94, 1054–1076 (2022) MathSciNetCrossRef

Benth, F.E., Detering, N., Lavagnini, S.: Accuracy of deep learning in calibrating HJM forward curves. Digit. Finance 3, 209–248 (2021) CrossRef

Benth, F.E., Eyjolfsson, H., Veraart, A.: Approximating Lévy semistationary processes via Fourier methods in the context of power markets. SIAM J. Financ. Math. 5, 71–98 (2014) CrossRef

10.

Benth, F.E., Koekebakker, S.: Stochastic modelling of financial electricity contracts. Energy Econ. 30, 1116–1157 (2008) CrossRef

11.

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

12.

Benth, F.E., Krühner, P.: Derivatives pricing in energy markets: an infinite dimensional approach. SIAM J. Financ. Math. 6, 825–869 (2015) MathSciNetCrossRef

13.

Benth, F.E., Krühner, P.: Stochastic Models for Prices Dynamics in Energy and Commodity Markets – an Infinite-Dimensional Perspective. Springer, Berlin (2023) CrossRef

14.

Björk, T.: Arbitrage Theory in Continuous Time, 2nd edn. Oxford University Press, Oxford (1998) CrossRef

15.

Buehler, H., Gonon, L., Teichmann, J., Wood, B.: Deep hedging. Quant. Finance 19, 1271–1291 (2019) MathSciNetCrossRef

16.

Chen, T., Chen, H.: Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Trans. Neural Netw. 6, 911–917 (1995) CrossRef

17.

Clewlow, L., Strickland, C.: Energy Derivatives – Pricing and Risk Management. Lacima Publications, London (2000)

18.

Cuchiero, C., Larsson, M., Teichmann, J.: Deep neural networks, generic universal interpolation, and controlled ODEs. SIAM J. Math. Data Sci. 2, 901–919 (2020) MathSciNetCrossRef

19.

Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. 2, 303–314 (1989) MathSciNetCrossRef

20.

Duffie, D.: Dynamic Asset Pricing Theory. Princeton University Press, Princeton (1992)

21.

E, W.: A proposal on machine learning via dynamical systems. Commun. Math. Stat. 5, 1–11 (2017) MathSciNetCrossRef

22.

Eydeland, A., Wolyniec, K.: Energy and Power Risk Management. Wiley, Hoboken (2003)

23.

Filipović, D.: Consistency Problems for Heath–Jarrow–Morton Interest Rate Models. Lecture Notes in Mathematics, vol. 1760. Springer, Berlin (2001) CrossRef

24.

Filipović, D., Tappe, S., Teichmann, J.: Jump-diffusions in Hilbert spaces: existence, stability and numerics. Stochastics 82, 475–520 (2010) MathSciNetCrossRef

25.

Geman, H.: Commodities and Commodity Derivatives. Wiley, Chichester (2005)

26.

Gonon, L., Schwab, C.: Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models. Finance Stoch. 25, 615–657 (2021) MathSciNetCrossRef

27.

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

28.

Hernandez, A.: Model calibration with neural networks. Risk, June 2017 (2017). Versions available at https://www.risk.net/media/download/952386/download or https://ssrn.com/abstract=2812140

29.

Hornik, K., Stinchcombe, M., White, H.: Multilayer feedforward networks are universal approximators. Neural Netw. 2, 359–366 (1989) CrossRef

30.

Hornik, K., Stinchcombe, M., White, H.: Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks. Neural Netw. 3, 551–560 (1990) CrossRef

31.

Horvath, B., Muguruza, A., Tomas, M.: Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models. Quant. Finance 21, 1–17 (2020) MathSciNet

32.

Hutzenthaler, M., Jentzen, A., Kruse, T., Nguyen, T.A.: A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations. SN Part. Differ. Equ. Appl. 1, 10-1–34 (2020) MathSciNetCrossRef

33.

Peszat, D., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise. Cambridge University Press, Cambridge (2007) CrossRef

34.

Ruf, J., Wang, W.: Neural networks for option pricing and hedging: a literature review. J. Comput. Finance 24(1), 1–45 (2020)

35.

Skov, J.B., Skovmand, D.: Dynamic term structure models for SOFR futures. J. Futures Mark. 41, 1520–1544 (2021) CrossRef

36.

Stinchcombe, M.: Neural network approximation of continuous functionals and continuous functions on compactifications. Neural Netw. 12, 467–477 (1999) CrossRef

Title: Pricing options on flow forwards by neural networks in a Hilbert space
Authors: Fred Espen Benth
Nils Detering
Luca Galimberti
Publication date: 24-11-2023
Publisher: Springer Berlin Heidelberg
Published in: Finance and Stochastics / Issue 1/2024
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-023-00520-2

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 1/2024

Investment–consumption–insurance optimisation problem with multiple habit formation and non-exponential discounting

Dispersion-constrained martingale Schrödinger problems and the exact joint S&P 500/VIX smile calibration puzzle

Arbitrage problems with reflected geometric Brownian motion

Faking Brownian motion with continuous Markov martingales

Optimal investment and consumption for financial markets with jumps under transaction costs

A càdlàg rough path foundation for robust finance