Deep Hedging under Rough Volatility

Blanka Horvath,Zan Zuric,Josef Teichmann

doi:10.2139/ssrn.3778043

Abstract

We investigate the performance of the Deep Hedging framework under training paths beyond the (finite dimensional) Markovian setup. In particular we analyse the hedging performance of the original architecture under rough volatility models with view to existing theoretical results for those. Furthermore, we suggest parsimonious but suitable network architectures capable of capturing the non-Markoviantity of time-series. Secondly, we analyse the hedging behaviour in these models in terms of P\&L distributions and draw comparisons to jump diffusion models if the the rebalancing frequency is realistically small.

Highlights

Deep learning has undoubtedly had a major impact on financial modelling in the past years and has pushed the boundaries of the challenges that can be tackled: can existing problems be solved faster and more efficiently (Bayer et al 2019; Benth et al 2020; Cuchiero et al 2020; Gierjatowicz et al 2020; Hernandez 2016; Horvath et al 2021; Liu et al 2019; Ruf and Wang 2020), but deep learning allows us to derive solutions to optimisations problems (Buehler et al 2019), where classical solutions have so far been limited in scope and generality
In the current paper we go a step further than just presenting an ad hoc well-chosen market simulator: we investigate a situation where the relevant data are structurally so different from the original Markovian setup that it calls for an adjustment of the model architecture itself
We analyse the perfect hedge for the rBergomi model from (Viens and Zhang 2019) and its performance against the deep hedging scheme in (Buehler et al 2019), which had to be adapted to a non-Markovian framework

Summary

Introduction

Deep learning has undoubtedly had a major impact on financial modelling in the past years and has pushed the boundaries of the challenges that can be tackled: can existing problems be solved faster and more efficiently (Bayer et al 2019; Benth et al 2020; Cuchiero et al 2020; Gierjatowicz et al 2020; Hernandez 2016; Horvath et al 2021; Liu et al 2019; Ruf and Wang 2020), but deep learning allows us to derive (approximative) solutions to optimisations problems (Buehler et al 2019), where classical solutions have so far been limited in scope and generality These approaches are fundamentally data driven, which makes them attractive from business perspectives. The main difference is parametrized by the so-called Hurst parameter H ∈ (0, 1) in the volatility process, which models the “memoryness” and the roughness of the driving fractional Brownian motion These models reflect much more closely the stylized facts and essential properties of the financial markets.

Setup and Notation

Implications on the Network Architecture

Proposed Fully Recurrent Architecture

Relation to the Literature

Conclusions