Deep hedging under rough volatility

Horvath, Blanka Nora; Teichmann, Josef; Žuric, Žan Article Deep hedging under rough volatility Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Horvath, Blanka Nora; Teichmann, Josef; Žuric, Žan (2021) : Deep hedging under rough volatility, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 9, Iss. 7, pp. 1-20, https://doi.org/10.3390/risks9070138 This Version is available at: https://hdl.handle.net/10419/258222 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article Deep Hedging under Rough Volatility Blanka Horvath 1, Josef Teichmann 2and Žan Žuriˇc 3,*   Citation: Horvath, Blanka, Josef Teichmann, and Žan Žuriˇc. 2021. Deep Hedging under Rough Volatility. Risks 9: 138. https://doi.org/ 10.3390/risks9070138 Academic Editor: Alexander Szimayer Received: 25 February 2021 Accepted: 8 June 2021 Published: 20 July 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Technische Universität München, King’s College London and The Alan Turing Institute, London WC2R 2LS, UK; [email protected] 2ETH Zürich, 8092 Zürich, Switzerland; [email protected] 3Faculty of Natural Science, Imperial College London, London SW7 2AZ, UK *Correspondence: [email protected] 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 in view of existing theoretical results for those. Furthermore, we suggest parsimonious but suitable network architectures capable of capturing the non-Markoviantity of time-series. We also analyse the hedging behaviour in these models in terms of Profit and Loss (P&L) distributions and draw comparisons to jump diffusion models if the rebalancing frequency is realistically small. Keywords: deep learning; rough volatility; hedging 1. 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: not only 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 also 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. Additionally these approaches are fundamentally data driven, which makes them particularly attractive from business perspectives. It comes as no surprise that the more similar (or “representative”) the data presented to the network in the training phase is to the (unseen) test data that the network is later applied to, the better is the performance of the hedging network on real data in terms of Profit and Loss (P&L). It is also unsurprising that, as markets shift sufficiently far away from a presented regime into new, previously unseen territories, the hedging networks may have to be retrained to adapt to the new environment. In the current paper we go a step further than just presenting an ad hoc well-chosen market simulator (see Buehler et al. 2020b,2020a;Henry-Labordere 2019;Wiese et al. 2019, 2020;Cuchiero et al. 2020;Kondratyev and Schwarz 2019;Xu et al. 2020): 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. In a well-controlled synthetic data environment, we study the behaviour of the hedging engine as relevant properties of the data change. A good candidate class of models comprises the rough volatility models, which have experienced a surge of research interest in recent years starting with the seminal papers of (Alòs et al. 2007;Bayer et al. 2015;Fukasawa 2010;Gatheral et al. 2018), where a crucial difference was introduced in comparison to classical stochastic models, due to their nonMarkovianity. 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 Risks 2021,9, 138. https://doi.org/10.3390/risks9070138 https://www.mdpi.com/journal/risks Risks 2021,9, 138 2 of 20 stylized facts and essential properties of the financial markets. Therefore, now, after several years of research of their asymptotic and numerical behaviour, they provide ideal synthetic data sets to test different machine learning models. More specifically, in our case, we used synthetic data generated from a rough volatility model with varying levels of the Hurst parameter. In its initial setup, we set the Hurst parameter to H= 1 / 2, which reflects a classical (finite dimensional) Markovian case, which is well-aligned with the majority of the most popular classical financial market models, such as, e.g., the Heston model, which the initial version of the deep hedging results were demonstrated on. We then gradually altered the level of the Hurst parameter to (rough) levels around H≈ 0.1, which more realistically reflects market reality as observed in (Alòs et al. 2007;Bolko et al. 2020;Fukasawa 2010;Gatheral et al. 2018;Livieri et al. 2018), thereby introducing a non-Markovian memory into the volatility process. Since rough volatility models are known to reflect the reality of financial markets (as well as the stylised statistical facts) better than classical, finite-dimensional Markovian models do, our findings also give an indication of how a naive application of model architectures to real data could lead to substantial errors. With this, our study allows us to make a number of interesting observations about deep hedging and the data that it is applied to: apart from drawing parallels between discretely observed rough volatility models and jump processes, our findings highlight the need to rethink (or carefully design) risk management frameworks of deep learning models as significant structural shifts in the data occur. To recap, we compare different methods for hedging under rough volatility models. More precisely, 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. We are particularly interested in the dependence of the P&L on the Hurst parameter and conclude that the deep hedge with our fRNN architecture performs better than the discretised perfect hedge for all H . Moreover, we find that the hedging P&L distributions for low H are highly left-skewed and have significant mass in the left tail under the model hedge as well as the deep hedge. Increasing hedging frequency was explored to mitigate the heavy losses in cases when H is close to zero. Intriguingly, slow response to increased hedging frequency and left-skewed P&L distribution under the rough Bergomi are also characteristic for delta hedges under jump diffusion models (Sepp 2012). The paper is organised as follows: Section 2recalls the setup of the original deep hedging framework used in (Buehler et al. 2019). Section 3gives a brief reminder on hedging under rough volatility models and compares the performance of (feed-forward) hedging network on a rough Bergomi model compared to a theoretically derived model hedge. In Sections 3.3 and 3.4, we draw conclusions with respect to the model architecture, and in Section 3.5, we propose a new architecture that is better suited to the data. Section 4 lays out the hedging under the new architecture and draws conclusions to the existing literature, which outlines some parallels between (continuous) rough volatility models and jump processes in this setting, while Section 5summarizes our conclusions. 2. Setup and Notation We adopt the setting in (Buehler et al. 2019) and consider a discrete finite-time financial market with time horizon [ 0, T] for some T∈( 0, ∞) and a finite number of trading dates 0 =t0<t1<··· <tn=T , n∈N . We work on a discrete probability space (Ω , F , P) , with Ω={ω1, . . . , ωN} and a probability measure P for which P[{ωi}]> 0 for all i∈ { 1, . . . , N} and N∈N . Additionally, we fix the notation X:={X:Ω→R} for the set of all R - valued random variables on Ω . Later we shall consider a continuous setup (Ω , F , P) , of which the discrete setup will be a precise approximation via Monte Carlo sampling and time discretisation. The filtration F=Fkk=0,...,n is generated by the Rr -valued information process (Ik)k=0,...,n for some n , r∈N . For any k∈ { 0, . . . , n} , the variable Ik denotes all available Risks 2021,9, 138 3 of 20 new market information at time tk and Fk represents all available market information up to time tk. The market contains d∈N financial instruments, which can be used for hedging, with mid-prices given by an Rd -valued F -adapted stochastic process S= (Sk)k=0,...,n . In order to hedge a claim Z:Ω→R , we may trade in S according to Rd -valued F -adapted processes (strategies), which we denote by δ:= (δk)k=0,...,n−1 with δ−1=δn:= 0 for notational convenience, where δk= (δ1 k , . . . , δd k) . Here, δi k denotes the agent’s holdings of the i -th asset at time tk. We denote the initial cash injected at time t0by p0>0. Furthermore, in order to allow for proportional trading costs, for every time tk and change in position s∈Rd we consider costs ck:s7→ c∈[ 0, ∞) , where ck is Fk -adapted, upper semi continuous and for which ck( 0 ) = 0 for all k∈ { 0, . . . , n} . The total costs up to time T , when trading according to a trading strategy δ are denoted by CT(δ):= ∑n k=0cksk−1(δk−δk−1) again with the notation s−1:= 0. Finally, we denote by H a set of all admissible trading strategies. We consider optimality of hedging under convex risk measures (see, e.g., Föllmer and Schied (2016) for the definition) as in (Buehler et al. 2019;Ilhan et al. 2009;Xu 2005). Let us for a moment consider the following problem. Say the agent’s terminal portfolio value for a contingent claim Z∈ X at Tis given as P&L(p0,δ,Z;T):=−Z+p0+ (δ·S)T−CT(δ), (1) where (δ·S)T:=∑n−1 k=0δk·(Sk+1−Sk) is the discrete stochastic integral and p0> 0 denotes the expectation of Z with respect to Q , i.e., the given risk-neutral price. The price is exogenously given, meaning either quoted in the market or calculated using some other pricing method, separate from our methodology. In complete markets, there exists δ∈ H such that P&L(p0 , δ , Z ; T) = 0 for any Z∈ X ; however, this cannot be said for incomplete markets with frictions. In this setting, the agent has to accept some risk and specify an optimality criterion. Now in the case of no trading costs, there is an alternative view point of variance optimal hedging (Schweizer 1995), which will be taken in this paper. Consider an equivalent pricing measure Q of our financial market; then, we can also minimise the variance π(Z) = inf δ∈HEh(−Z+(δ·S)T+p0)2i, (2) In the rest of this paper, the above optimisation (2) problem and corresponding optimisers are considered in terms of their numerical approximation in the framework of hedging in a neural network setting as formulated in (Buehler et al. 2019). In the remainder of this section, we recall the notation and definitions to formulate this approximation property and the conditions that ensure its validity. Definition 1 (Set of Neural Networks (NNs) with a fixed activation function) . We denote by NNσ ∞,d0,d1 the set of all NNs mapping from Rd0→Rd1 with a fixed activation function σ . The set {NNσ M,d0,d1}M∈N is then a sequence of subsets in NN∞,d0,d1 for which NNσ M,d0,d1={Fθ:θ∈ ΘM,d0,d1}with ΘM,d0,d1⊂Rqfor some q(M),M∈N. Definition 2. We call HM⊂ H the set of unconstrained neural network trading strategies: HM=n(δk)k=0,...,n−1∈ H :δk=Fk(I0, . . . , δk−1),Fk∈ NNM,r(k+1)+d,do =n(δk)k=0,...,n−1∈ H :δk=Fθk k(I0, . . . , δk−1),θk∈ΘM,r(k+1)+d,do(3) Risks 2021,9, 138 4 of 20 We now replace the set H in (2) by the finite subset HM⊂ H . The optimisation problem then becomes πM(Z):=inf δ∈HM Eh(−Z+(δ·S)T+p0)2i =inf θ∈ΘM Eh(−Z+ (δθ·S)T+p0)2i(4) where ΘM=∏n−1 k=0ΘM,r(k+1)+d,d denotes the network parameters from Definition 2. With (3), (4) and Remark 1, the potentially infinite-dimensional problem of finding an optimal hedging strategy is therefore reduced to a finite-dimension and corresponds to finding the optimal NN parameters for the problem (4) . Notice that in general, i.e., before time and space discretisation, the problem is inherently infinitely dimensional. We shall always consider appropriate discretisations, which are sufficiently close to the continuous time model by results from numerical analysis, on which we apply the above theory. Remark 1. Note that in the above, we do not assume that S is an (F , P) -Markov process and that the contingent claim is of the form Z:=g(ST) for a pay-off function g:Rd→R . This would allow us to write the optimal strategy δk=fk(Ik,δk−1)for some fk:Rr+d→Rd. The next proposition, which is a direct use of Lemma 4.4 in (Buehler et al. 2019), recalls the central approximation property, which states that the optimal trading strategy δ∗∈ H that minimizes (2) can be approximated by a semi-recurrent neural network of the form Figure 1in the sense that the functional πM(Z)converges to π(Z)as Mbecomes large. Input Layer ∈ ℝ⁴ Hidden Layer ∈ ℝ¹² Hidden Layer ∈ ℝ¹² Output Layer ∈ ℝ²Input Layer ∈ ℝ⁴ Hidden Layer ∈ ℝ¹² Hidden Layer ∈ ℝ¹² Output Layer ∈ ℝ² St δS t−1 δV t−1 St+1 δS t δV t Vt+1 δS t δV t δS t+1 δV t+1 tt+1 Output Figure 1. Original Network Archiecture. Proposition 1. Define HMas in (3)and πMas in (4). Then for any Z ∈ X lim M→∞πM(Z) = π(Z), where π(Z)denotes the optimal solution of the original optimisaton problem (2). Risks 2021,9, 138 5 of 20 Corollary 1. As a direct consequence, the optimal neural network trading strategy δθ∗ that minimises (4)converges to the minimiser of (2)in the sense that lim M→∞δθ∗ M=δ∗, where θ∗ M∈ΘMare the optimal neural network parameters for M ∈N. In (Buehler et al. 2019), this approximation property is demonstrated for Black–Scholes and Heston models both in their original form and in variants including market frictions such as transaction costs. These results demonstrate how deep hedging, which allows us to take a leap beyond the classical type, results in scenarios where the Markovian structure is preserved. A natural question to ask is how the approximation property of the neural network is affected if the assumption of Markovian structure of the underlying process is no longer satisfied. Rough volatility models (Alòs et al. 2007;Bayer et al. 2015;Fukasawa 2010; Gatheral et al. 2018) represent such a class of non-Markovian models. It is also wellestablished in a series of recent articles, including the aforementioned works, that rough volatility dynamics are superior to standard Markovian models (such as Black–Scholes and Heston) in terms of reflecting market reality and also that rough volatility models are superior to a number of in terms of allowing close fits to market data. By taking hedging behaviour under rough volatility models under a closer examination, we gain insight into the non-Markovian aspects of markets in a controlled numerical setting: varying the Hurst parameter H∈( 0,1 ) of the process (see Gatheral et al. (2018)), which governs the deviation from the Markovian setting in a general fractional (or rough) volatility framework, enables us to control for the influence of the Markovianity assumption on the hedging performance of the deep neural network. Therefore, in this work, we investigate the effect of the loss of Markovianity property of the underlying stochastic process by considering market dynamics that are governed in a rough volatility setting. With this in mind, by applying the original feedforward network architecture to a more realistic model class (represented by rough volatility models), we in particular demonstrate how the choice of the network architecture may affect the performance of the deep hedging framework and how it could potentially break down on real-life data. We also note in passing that the approach we take can be applied as a simple routine sanity check for model governance of deep learning models on real data: • Take a well-understood model class that generalises the modelling to more realistic market scenarios, but where the generalisation no longer satisfies assumptions made in the original architecture. • Test the robustness of the method if the assumption is violated by controlling for the error as the deviation from the assumption increases. • Modify the network architecture accordingly if necessary. 3. Hedging and Network Architectures for Rough Volatility 3.1. Hedging under Rough Volatility Let us now consider the problem of hedging under rough volatility models in general, with an aim to present a theoretical solution, which we use as a benchmark of the deep hedging approach. In this section, we introduce a new filtered probability space (Ω , F , Q , F) , where F:={Ft}0≤t≤T is now a continuous filtration. 1 We know that for a Markovian process of the form ˜ Xt=x+Zt 0b(r,Xr)dr +Zt 0σ(r,Xr)dWr, Risks 2021,9, 138 6 of 20 where b and σ satisfy suitable conditions. Then the price of a contingent claim ˜ Zt:= E[g(˜ XT)|Ft]can be written as ˜ Zt=u(t,˜ Xt), where u solves a parabolic PDE by the Feynman–Kac formula (Kac 1949). However, it was shown in (Bayer et al. 2015) that rough volatility models are not finite-dimensional Markovian, and we therefore have to consider a more general process X and assume it to be a solution to the d-dimensional Volterra SDE: Xt=x+Zt 0b(t;r,X.)dr +Zt 0σ(t;r,X.)dWr,t∈[0, T], (5) where W is a m -dimensional standard Brownian motion, b∈Rd and σ∈Rm×d . Both are adapted in a sense that for ϕ=b,σit holds ϕ(t;r,X.) = ϕ(t;r,Xr∧.). In this general non-Markovian framework, the contingent claim in the form Zt:= E[g(XT)|Ft] will depend on the entire history of the process X:= (Xt)t≥0 up to time t and not just on the value of the process at that time, i.e., Zt=u(t,X[0,t])with notation X[0,t]:={Xr}r∈[0,t], where u this time solves a Path-dependent PDE (PPDE). The setting where X is a semimartingale has already been explored in, e.g., (Cont and Fournié 2013;Dupire 2019). Be that as it may, we know that fBm is not a semi-martingale in general, and as a consequence, the volatility process is not a semi-martingale. Viens and Zhang (2019) are able to cast the problem back in to the semi-martingale framework by rewriting Xt as a orthogonal decomposition to an auxiliary process Θt and a process It , which is independent of the filtration Xs=x+Zt 0b(s;r,X.)dr +Zt 0σ(s;r,X.)dWr +Zs tb(s;r,X.)dr +Zs tσ(s;r,X.)dWr (6) =:x+Θt s+It s(7) for 0 ≤t≤s . By exploiting the semi-martingale property of Θ , they go on to show that the contingent claim can be expressed as a solution of a PPDE Zt=u(t,X[0,t)⊗tΘt [t,T]), (8) where ⊗t denotes concatenation of a path at time t . Moreover, they develop an Itôtype formula for a general non-Markovian process Xt from (5) , which we present in the Appendix B. We have to consider different additional hedging instruments to complete such markets. Most convenient from a theoretical as well as from a practical perspective are variance swaps with different maturities, which are, according to the Breeden–Litzenberger formula, just a linear superposition of plain vanilla European calls. 3.2. The Rough Bergomi Model (rBergomi) As an example we consider the rBergomi with a constant initial forward variance curve ξ0(t) = V0: St=S0+Zt 0Sr√Vrq1−ρ2dBr+ρdWr(9a) Vt=V0E√2HνZt 0(t−r)H−1 2dWr,V0=v0>0, (9b) Risks 2021,9, 138 7 of 20 The model fits into the affine structure of our Volterra SDE in (5) after a simple log-transformation of the volatility process. In this case, we take our auxiliary process to be Θt s=√2HνZt 0(s−r)H−1 2dWr,t<s. (10) It is easy to check that Θt s is a true martingale for a fixed s . The option price dynamics are obtained by using the Functional Itô formula in (A3) . From this, the perfect hedge in terms of a forward variance ˆ Θt Twith maturity Tand a stock Stfollows: dZt=∂xu(t,St,Θt [t,T])dSt+(T−t)1 2−H ˆ Θt TD∂ωu(t,St,Θt [t,T]),atEdˆ Θt T(11) with at s= (s−t)H−1 2 . The path-wise derivative in (11) is the Gateaux derivative along the direction at . For more details and discretisation of the Gateaux derivative, see Appendix A. 3.3. Performance of the Deep Hedging Scheme (with the Original Feedforward Architecture) Compared to the Model Hedge under rBergomi We choose to hedge a plain vanilla called option ZT:=max(ST−K ,0 ) with K= 100 and a monthly maturity T= 30 / 365. The hedging portfolio consists of a stock S with S0= 100 and a forward variance with maturity TFwd = 45 / 365 and is rebalanced daily. For the rBergomi model, forward variance is equal to ˆ Θt TFwd :=EQZTFwd 0VsdsFt=V0expΘt TFwd +1 2ν2h(TFwd −t)2H−T2H Fwdi, (12) with Θt TFwd defined as in (10) . Applying classical Itô’s Lemma to ˆ Θt TFwd =ˆ Θt TFwd (t , Θt TFwd ) yields the dynamics of the forward variance under the rough Bergomi dˆ Θt TFwd =ˆ Θt TFwd √2Hν(TFwd −t)H−1 2dWt, (13) which is well defined for t∈[ 0, TFwd) . Therefore, choosing the maturity of the forward variance to be longer than the option maturity allows us to avoid the singularity as t→T . In practice, this would correspond to hedging with a forward variance with a slightly longer maturity than that of the option. For the simulation of the forward variance, we used the Euler–Mayurama method, whereas paths of the volatility process were simulated with the “turbo-charged” version of the hybrid scheme proposed in (Bennedsen et al. 2017;McCrickerd and Pakkanen 2018). The parameters were chosen such that they describe a typical market scenario with a flat forward variance: ξ0= 0.235 × 0.235, ν= 1.9 and ρ=− 0.7. We were particularly interested in the dependence of the hedging loss on the Hurst parameter. Finally, a quadratic loss function was chosen, and the minimising objective was therefore π(Z) = inf δθ∈HM Eh(−Z+p0+ (δθ·S)T)2i where price p0 was obtained with a Monte-Carlo simulation (e.g., for H= 0.10, p0= 2.39). Next we implement the perfect hedge from (11) (the details of the discretisation of the Gateaux derivative are presented in Appendix C). For evaluation of the option price, we once again use Monte-Carlo, this time with the generating parameters. In practice, we would calibrate the parameters to the market data. Perfect hedge was implemented on the sample of 10 3 different paths for the same parameters as in the deep hedging case. The results of both hedges under quadratic loss for different Hurst parameters are shown in Table 1. We also take a closer look at the P&L distributions of the deep hedge as well as the model hedge for H= 0.10 in Figure 2. Curiously enough, the distributions are very similar to each other. The deep hedge seems to have slightly thinner tails, which is interesting, Risks 2021,9, 138 8 of 20 considering the semi-recurrent architecture makes a strong assumption of Markovianity of the underlying process. Table 1. Comparison of the quadratic loss between model and deep hedges trained 75 epochs for different H. Quadratic Hedging Loss HModel Hedge Deep Hedge 0.10 1.45 1.16 (*1.12) 0.20 0.52 0.67 0.30 0.34 0.46 0.40 0.24 0.36 *—on 200 epochs. Figure 2. rBergomi model hedge (blue) compared to the deep hedge (red) trained on 75 epochs on rBergomi paths with H= 0.10. Note that the option price is only p0= 2.39 and that such a hedge can result in a substantial loss. Indicators that the assumption of finite dimensional Markovianity is violated might be the heavy left tail of the P&L distribution as well as relatively high hedging losses. This prompted us to question the semi-recurrent architecture and devise a way to relax the Markov assumption on the underlying. Note that the heavy tails of these distributions may also imply a link to jump diffusion models. We expand on this in Section 4.3. 3.4. Implications on the Network Architecture As discussed before, in (Buehler et al. 2019) authors heavily rely on Remark 1, where they use the Markov property of the underlying process in order to write the trading strategy at time tk as a function of the information process at tk and trading strategy in the previous time step k− 1. Of course, in the case of rough volatility models, one would have to include the entire history of the information process up to tk in order to get the hedge at that time. However, this would result in numerically infeasible scheme. To illustrate this, take for example a single vanilla call option with maturity T= 30 / 365, where we hedge daily under say the rough Bergomi model. In the 30th time step, the number of input nodes of the NN cell Fθ 30 would be 30 × 2 + 2 = 62 or if we hedged twice a day 30 × 2 × 2 + 2 = 122. Obviously, this scheme quickly becomes very computationally expensive even for a single option with a short maturity. The fBm in (9b) can be written as a linear functional of a Markov process, albeit an infinite-dimensional one. Therefore, if the original Markovian-based architecture can be applied to this setting, we would expect to recover the Hurst parameter also from a Markovian-based sampling procedure, justifying the continued use of the original feed Risks 2021,9, 138 15 of 20 Figure 8. Profit and loss distributions for delta hedging under jump diffusion models (JDM) from (Sepp 2012). This gives us the idea to treat the discretised rough models as jump models. In this case, the market is incomplete, and it is not possible to perfectly hedge a contingent claim with a portfolio containing a finite number of instruments (He et al. 2007). In practice, traders try to come as close as possible to the perfect hedge by trading a number of different options. Unfortunately, when trying to implement the hedge approximation, we are quickly faced with the absence of analytical pricing formulas and limitations of the slow MonteCarlo scheme. In order for us to train the deep hedge, we would have to calculate option prices on every time step of each sample path. In a typical application, we would need around 10 options with different strikes and at least 105sample paths. 5. Conclusions In this work, we presented and compared different methods for hedging under rough volatility models. More specifically, we analysed and implemented the perfect hedge for the rBergomi model from (Viens and Zhang 2019) and used the deep hedging scheme from (Buehler et al. 2019), which had to be adapted to a non-Markovian framework. We were particularly interested in the dependence of the P&L on the Hurst parameter. We conclude that the deep hedge with the proposed architecture performs better than the discretised perfect hedge for all H . We also find that the hedging P&L distributions for low H are highly left-skewed and have a lot of mass in the left tail under the model hedge as well as the deep hedge. To mitigate the heavy losses in cases when H is close to zero, we explored increasing the hedging frequency up to four times a day. The loss did improve and the P&L distribution became less leptocurtic, however only slightly. Intriguingly, slow response to increased hedging frequency and left-skewed P&L distribution are characteristic of delta hedges under jump diffusion models (Sepp 2012). We therefore observe that in terms of hedging, Risks 2021,9, 138 16 of 20 there is a relation between jump diffusion models and rough models. In accordance with the literature, we find that the price process, despite being a continuous martingale, exhibits jump-like behaviour (McCrickerd and Pakkanen 2018). We believe this is an excellent illustration of rough volatility models dynamics. Explosive, almost jump-like patterns in the stock price might be the reason why they can fit the short end of implied volatility so well. In our view, it is crucial to take into account the jump aspect, when looking for an optimal hedge in discretised rough volatility models. Our suggestion for future research is adapting the objective function in the deep hedge scheme for jump risk optimisation. The first step would be optimisation of the expected shortfall risk measure. Next, more appropriate jump risk measures for discretised rough models can be developed. These risk measures cannot be completely analogous to the risk measures in (Sepp 2012), since rough models themselves do not feature jumps. Author Contributions: All authors contributed to the conception of the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: We gratefully acknowledge support by ETH Foundation and SNF project 179114. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. Appendix A. Path Derivatives Denote by D0 a càdlàg space and by Dt and Ct the space of càdlàg functions on [t , T] and the space of continuous functions on [t , T] , respectively. Additionally, we denote by ω the sample paths on [0, T], with ωtas its value at time tand define Λ:= [0, T]×C([0, T],Rd),¯ Λ:=n(t,ω)∈[0, T]×D0:ω|[t,T]∈ Co; kωkT:=sup t∈[0,T]|ωt|,d(t,ω),(t0,ω0):=|t−t0|+kω−ω0kT. Furthermore, we denote the set of all d -continuous functions u:¯ Λ→R by C(¯ Λ) . Define the usual horizontal time derivative for u∈ C(¯ Λ)as in (Dupire 2019): ∂tu(t,ω):=lim δ↓0 u(t+δ,ω)−u(t,ω) δfor all (t,ω)∈¯ Λ, (A1) requiring of course that the limit exists. For the spatial derivative with respect to ω , however, we use the definition of the Gateaux derivative for any (t,ω)∈¯ Λ: h∂ωu(t,ω),ηi=lim ε→0 u(t,ω+εη 1 [t,T])−u(t,ω) εfor any η∈ Ct. (A2) Note that the function u(t , ·) in the definition of the derivative is “lifted” only on [t , T] and not on [0, t). Hence, the convention we follow is actually h∂ωu(t,ω),ηi:=D∂ωu(t,ω),η 1 [t,T]Efor any s<tand η∈ Cs. The definition of Gateaux derivative is clearly also equal to h∂ωu(t,ω),ηi=d dεu(t,ω+εη 1 [t,T])e=0. Risks 2021,9, 138 17 of 20 Remark A1. We remark that our definition of the spatial derivative is different from the one in (Cont and Fournié 2013;Dupire 2019), where functional derivative quantifies the sensitivity of the functional to the variation solely in the end point of the path, i.e., ωt . While in our definition, the perturbation takes place throughout the whole interval [t,T]. We define two more spaces necessary for our analysis: C1,2(¯ Λ):=nu∈ C(¯ Λ):ϕ∈ C(¯ Λ)for ϕ∈ {∂tu,∂ωu,∂2 ωωu}o, C1,2 +(¯ Λ):=nu∈ C(¯ Λ):ϕhas polynomial growth for ϕ∈ {∂tu,∂ωu,∂2 ωωu}and D∂2 ωωu,(η,η)Eis locally uniformly continuous in ωwith polynomial growtho. Appendix B. Functional Itô Formula We have to differentiate two cases. The regular case where H∈(1 2 ,1 ) and the singular case where the coefficients b , σ explode, because of the power-kernel in Riemann–Liouville fractional Brownian motion whenever the Hurst exponent H lies in ( 0, 1 2) . In the singular case, the coefficients b , σ/∈ Ct , and thus they cannot serve as the test function in the right side of (A2) , since Gateaux derivative would not make sense any more. In order to develop an Itô formula for the singular case, definitions need to be slightly amended. Nonetheless, Viens et al. show that both cases yield a similar functional Itô formula. Assumption A1. i The SDE (5)admits a weak solution (X,W). ii Ehsupt∈[0,T]|Xt|pi<∞for all p ≥1. Assumption A2. i (Regular case) For any r∈[ 0, T] , ∂tb(t ; r , ·) , ∂tσ(t ; r , ·) exist for t∈[r , T] and for ϕ= b,σ,∂tb,∂tσ, |ϕ(t;r,ω)| ≤ C0(1+kωkκ0 T)C0,κ0>0. ii (Singular case) For any r∈[ 0, T] , ∂t(t ; r , ·) exists for t∈(r , T] with ϕ=b , σ . There exists H∈(0, 1 2)s.t., for some C0,κ0>0 |ϕ(t;r,ω)| ≤ C0(1+kωkκ0 T)(t−r)H−1 2and |ϕt(t;r,ω)| ≤ C0(1+kωkκ0 T)(t−r)H−3 2 Theorem A1 (Functional Itô formula) . Let X be a weak solution to the SDE (5) for which Ehsupt∈[0,T]|Xt|pi<∞for all p ≥1and Assumption A2 hold. Then du(t,X⊗tΘt) = ∂tu(t,X⊗tΘt)dt +1 2D∂ωωu(t,X⊗tΘt),(σt,X,σt,X)Edt+ D∂ωu(t,X⊗tΘt),bt,XEdt +D∂ωu(t,X⊗tΘt),σt,XEdWt,P-a.s. (A3) for u∈C1,2 +(Λ) in the regular case and u∈C1,2 +,β(Λ) with regularised Gateaux derivative for the singular case. For ϕ=b , σ the notation ϕt,ω s:=ϕ(s ; t , ω) only emphasises the dependence on s∈[t , T] . For the definition of C1,2 +,β(Λ) and precise statement of the theorem in the singular case, see Theorem 3.17 and Theorem 3.10 in (Viens and Zhang 2019) Appendix C. Discretisation of the Gateaux Derivative It can be easily shown that ˆ Θt s=f(Θt s) for some f:R→R . Therefore, we have a direct relation between the auxiliary process Θ and the forward variance ˆ Θ , which allows us to write the option price as the function of the entire forward variance curve ˆ Θt [t,T] at Risks 2021,9, 138 18 of 20 time t∈[ 0, T] , namely u(t,St,Θt [t,T]) = ˜ u(t,St,ˆ Θt [t,T]) . This is important, when performing Monte-Carlo, since in the rough Bergomi model, the forward variance curve is directly modelled in the variance process with ξt(·) = ˆ Θt ·. Let us suppose that we are able to trade at times 0 =t0<t1<··· <tn=T . In order to get the hedging weights at trading times ti , we have to discretise the derivatives. The Gateaux derivative with respect to the stock simplifies to the usual derivative, and the discretisation is straightforward: ∂x˜ u(t,St,ˆ Θt [t,T])≈˜ u(t,St+ε,ˆ Θt [t,T])−˜ u(t,St,ˆ Θt [t,T]) εfor small ε>0. (A4) For the path-wise derivative, the discretisation is not immediately obvious, especially because of the dependence of the option price ˜ u at time t on functional over the whole interval [t , T] , more precisely since u:[ 0, T]×[ 0, ∞)×C([ 0, T]→R) . First, we remind ourselves of the definition of the Gateaux derivative on a path ω: h∂ωu(t,ω),ηi=lim ε→0 u(t,ω+εη 1 [t,T])−u(t,ω) εfor any η∈Ωt. We proceed as in (Jacquier and Oumgari 2019) by approximating ˆ Θt [t,T] as a piecewise constant function ˆ Θt s≈∑ i∈I ˆ Θt i 1 [ti,ti+1)(s)at i=at s 1 [ti,ti+1)(s), (A5) where I:={i∈N:t≤ti≤T} . We introduce the following approximations of the path derivatives along the direction at: D∂ω˜ u(t,St,ˆ Θt [t,T]),atE≈∂ε˜ u t,St,∑ i∈I (ˆ Θt i+εat) 1 [ti,ti+1)(s)s∈[t,T]!ε=0 =∂εˆ ut,St,ˆ Θt i+εat ii∈Iε=0 =∑ i∈I ∂ˆ Θt iˆ u(t,St,θt)at i, with θt:= ( ˆ Θt i)i∈I and ˆ u acts on [ 0, T]×[ 0, ∞)×R#I . Further discretising the derivative, we have for the flat forward variance ξ(t) = ξ0: ∂ω˜ u(t,St,ξ0),at≈˜ u(t,St,ξ0+ε)−˜ u(t,St,ξ0) εatfor small ε>0. The option prices ˜ u can now be evaluated using Monte-Carlo at each time step to get the hedging weights. Note that the discretisation of the Gateaux derivative is purely heuristic and that a rigorous proof of the convergence to the true derivative is out of scope of this work. For more details we refer to (Jacquier and Oumgari 2019). Notes 1 For the numerical implementation of the resulting strategies that we consider in the following sections, we naturally consider again the discrete filtration introduced above in Section 2, representing a discretisation of the continuous market. 2 Several estimation procedures of the Hurst parameter were used; see, e.g., (Di Matteo 2007;Di Matteo et al. 2005). Estimations of the paths simulated with the hybrid scheme (Bennedsen et al. 2017;McCrickerd and Pakkanen 2018) were on the other hand in alignment with the input parameter. 3 Note that by completely recurrent we do not mean the same network is used at each time step, but that the hidden state is passed on to the cell in the next time step along with current portfolio positions. 4 For Heston parameters α= 1, b= 0.04, σ= 0.8, V0= 0.04, S0= 100 and ρ=− 0.7 the quadratic losses were 0.20 under original architecture and 0.162 under the fully recurrent one. Both training times were fairly similar as well. 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