Robust Pricing and Hedging via Neural SDEs

34 Pages Posted: 5 Aug 2020

See all articles by Patryk Gierjatowicz

Patryk Gierjatowicz

University of Edinburgh - School of Mathematics

Marc Sabate-Vidales

University of Edinburgh - School of Mathematics

David Siska

University of Edinburgh - School of Mathematics; Vega

Lukasz Szpruch

University of Edinburgh - School of Mathematics

Zan Zuric

Imperial College London - Department of Mathematics

Date Written: July 8, 2020

Abstract

Mathematical modelling is ubiquitous in the financial industry and drives key decision processes. Any given model provides only a crude approximation to reality and the risk of using an inadequate model is hard to detect and quantify. By contrast, modern data science techniques are opening the door to more robust and data-driven model selection mechanisms. However, most machine learning models are “black-boxes” as individual parameters do not have meaningful interpretation. The aim of this paper is to combine the above approaches achieving the best of both worlds. Combining neural networks with risk models based on classical stochastic differential equations (SDEs), we find robust bounds for prices of derivatives and the corresponding hedging strategies while incorporating relevant market data. The resulting model called neural SDE is an instantiation of generative models and is closely linked with the theory of causal optimal transport. Neural SDEs allow consistent calibration under both the risk-neutral and the real-world measures. Thus the model can be used to simulate market scenarios needed for assessing risk profiles and hedging strategies. We develop and analyse novel algorithms needed for efficient use of neural SDEs. We validate our approach with numerical experiments using both local and stochastic volatility models.

Keywords: Stochastic Differential Equations, Deep Neural Network, Derivative Pricing, Stochastic Gradient Descent

JEL Classification: G12, C45, C60, C63

Suggested Citation

Gierjatowicz, Patryk and Sabate-Vidales, Marc and Siska, David and Szpruch, Lukasz and Zuric, Zan, Robust Pricing and Hedging via Neural SDEs (July 8, 2020). Available at SSRN: https://ssrn.com/abstract=3646241 or http://dx.doi.org/10.2139/ssrn.3646241

Patryk Gierjatowicz

University of Edinburgh - School of Mathematics ( email )

James Clerk Maxwell Building
Peter Guthrie Tait Rd
Edinburgh, EH9 3FD
United Kingdom

Marc Sabate-Vidales

University of Edinburgh - School of Mathematics ( email )

James Clerk Maxwell Building
Peter Guthrie Tait Rd
Edinburgh, EH9 3FD
United Kingdom

David Siska

University of Edinburgh - School of Mathematics ( email )

United Kingdom

HOME PAGE: http://https://www.maths.ed.ac.uk/~dsiska/

Vega ( email )

Vega Holdings Limited
Suite 23 Portland House, Glacis Road
Gibraltar, GX11 1AA
Gibraltar

HOME PAGE: http://vega.xyz/

Lukasz Szpruch (Contact Author)

University of Edinburgh - School of Mathematics ( email )

James Clerk Maxwell Building
Peter Guthrie Tait Rd
Edinburgh, EH9 3FD
United Kingdom

Zan Zuric

Imperial College London - Department of Mathematics ( email )

South Kensington Campus
Imperial College
LONDON, SW7 2AZ
United Kingdom

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