Hedging derivatives under market frictions using deep learning techniques

Transcription

1 Hedging derivatives under market frictions using deep learning techniques Lukas Gonon joint work with Hans Bühler (JP Morgan Chase), Josef Teichmann (ETHZ) and Ben Wood (JP Morgan Chase) ETH Zürich and Universität Sankt Gallen Risk Day 2018 In honour of Paul Embrechts ETH Zürich September 14, 2018 Deep Hedging

2 Risk management of a portfolio of derivatives Outline 1 Risk management of a portfolio of derivatives 2 Formal framework 3 Solution by deep learning techniques 4 Numerical results 5 Conclusion and next steps Lukas Gonon Deep Hedging 1/31

3 Risk management of a portfolio of derivatives Suppose we sell an additional derivative (possibly path-dependent, several underlyings,...) on top of our existing portfolio. How to hedge the overall position? Classical approach: 1 Assume a stochastic model (Heston, local volatility,...) for the future evolution of the price of the underlyings. 2 Calibrate the model to market prices of liquidly traded options. 3 Calculate greeks (sensitivity of model price w.r.t. changes of the underlyings,...) and trade accordingly (in underlying and possibly also options). Rationale: in an idealized model world this strategy allows to perfectly hedge the risk. Consequence: each derivative can be treated separately - linear pricing and hedging is possible. Lukas Gonon Deep Hedging 2/31

4 Risk management of a portfolio of derivatives In practice: In most cases each derivative is handled separately, although e.g. transaction costs would make it favourable to hedge at portfolio level. Traders do not fully believe the models: they recalibrate the model at each time-step, calculate greeks with respect to parameters (and not only state variables), overlay their own intuitive adjustments. Market frictions (transaction costs, liquidity constraints,...) are only taken into account by such adjustments. Problem: Adding market frictions even to simple models signicantly increases complexity. Example: price calculation in one-dimensional Black-Scholes model with transaction costs requires to solve multidimensional PDE with free boundary. Key requirement for classical approach: computational eciency. Lukas Gonon Deep Hedging 3/31

5 Risk management of a portfolio of derivatives In practice: In most cases each derivative is handled separately, although e.g. transaction costs would make it favourable to hedge at portfolio level. Traders do not fully believe the models: they recalibrate the model at each time-step, calculate greeks with respect to parameters (and not only state variables), overlay their own intuitive adjustments. Market frictions (transaction costs, liquidity constraints,...) are only taken into account by such adjustments. Problem: Adding market frictions even to simple models signicantly increases complexity. Example: price calculation in one-dimensional Black-Scholes model with transaction costs requires to solve multidimensional PDE with free boundary. Key requirement for classical approach: computational eciency. Lukas Gonon Deep Hedging 3/31

6 Risk management of a portfolio of derivatives Deep Hedging: a summary Preprint available at and Instead of deriving a hedging strategy in a sophisticated parametric model, we take a step back and work directly with a simple discrete-time sample-based model. This allows to incorporate transaction costs, liquidity restrictions,... Of course, there is no hope for any closed-form solution to the resulting optimization problem, but by parametrizing hedging strategies by neural networks and by leveraging computational advances the optimization is feasible. Note: There are no samples of an optimal hedging strategy, so this is neither supervised nor unsupervised learning. Lukas Gonon Deep Hedging 4/31

7 Risk management of a portfolio of derivatives Deep Hedging: a summary Preprint available at and Instead of deriving a hedging strategy in a sophisticated parametric model, we take a step back and work directly with a simple discrete-time sample-based model. This allows to incorporate transaction costs, liquidity restrictions,... Of course, there is no hope for any closed-form solution to the resulting optimization problem, but by parametrizing hedging strategies by neural networks and by leveraging computational advances the optimization is feasible. Note: There are no samples of an optimal hedging strategy, so this is neither supervised nor unsupervised learning. Lukas Gonon Deep Hedging 4/31

8 Formal framework Outline 1 Risk management of a portfolio of derivatives 2 Formal framework 3 Solution by deep learning techniques 4 Numerical results 5 Conclusion and next steps Lukas Gonon Deep Hedging 5/31

9 Formal framework Discrete-time market model with frictions The bank is given a portfolio of securities, all of which are assumed to expire before some maturity T > 0. Positive values are in favour of the clients. We assume interest rates are zero for convenience. The total return of the portfolio, i.e. the sum of all cash ows at T is denoted by Z. From today to maturity the bank observes a range of liquid instruments (equity, options,...). The bank may trade in these d instruments in order to hedge its portfolio. Trading is possible at time points t 0 = 0 < t 1 <... < t n = T. Prices are modeled by a stochastic process (S tk ) k=0,...,n in R d. For each time t k the bank takes a position δt i k in hedging instrument i. Note that δ tk = (δt 1 k,..., δt d k ) may only depend on information up to time t k (i.e. it is a function of (S t0,..., S tk ) or more general market information process at t k, see paper). Formally we work on a nite probability space (Ω, F, P). Lukas Gonon Deep Hedging 6/31

10 Formal framework Setup and problem formulation in detail Charging price p 0 and hedging according to δ, terminal prot and loss is PLT (Z, p 0, δ) := }{{} Z + p 0 liability }{{} price + (δ S) }{{ T } trading gains C T (δ) }{{} cumulative transaction costs (δ S) T = n 1 k=0 δ t k (S tk+1 S tk ). C T (δ) = n k=0 c k(δ tk δ tk 1, S t0,..., S tk ) with δ t 1 := 0, δ tn := 0. Example: transaction costs proportional to transaction amount, i.e. c k (δ tk δ tk 1, S t0,..., S tk ) = d i=1 ε i δ i t k δ i t k 1 S i t k. Costs can be complex, in particular across instruments.. (1) Note: PLT (Z, p 0, δ) 0 represents a gain for the writer (the bank). How to optimally choose a hedging strategy? Lukas Gonon Deep Hedging 7/31

11 Formal framework Setup and problem formulation in detail Charging price p 0 and hedging according to δ, terminal prot and loss is PLT (Z, p 0, δ) := }{{} Z + p 0 liability }{{} price + (δ S) }{{ T } trading gains C T (δ) }{{} cumulative transaction costs (δ S) T = n 1 k=0 δ t k (S tk+1 S tk ). C T (δ) = n k=0 c k(δ tk δ tk 1, S t0,..., S tk ) with δ t 1 := 0, δ tn := 0. Example: transaction costs proportional to transaction amount, i.e. c k (δ tk δ tk 1, S t0,..., S tk ) = d i=1 ε i δ i t k δ i t k 1 S i t k. Costs can be complex, in particular across instruments.. (1) Note: PLT (Z, p 0, δ) 0 represents a gain for the writer (the bank). How to optimally choose a hedging strategy? Lukas Gonon Deep Hedging 7/31

12 Formal framework Indierence pricing and optimal hedging: Following e.g. Föllmer, Klöppel, Leukert, Schweizer, Xu,... : Describe risk-preferences by a convex risk measure ρ (see e.g. Föllmer and Schied (2016) or Chapter 8 in McNeil, Frey and Embrechts (2015) for an introduction). Key examples: Entropic risk: ρ(x ) := 1 λ log E [ e ] λx for λ > 0. } CVar / Expected Shortfall: ρ(x ) := inf w R {w α E[( X w)+ ] for α [0, 1). Denote H the set of available hedging strategies. The indierence price is the (unique) solution p(z) to inf ρ δ H (PL T (Z, p(z), δ)) = inf ρ δ H (PL T (0, 0, δ)). (2) p(z) is the amount of money the bank needs to allocate to its portfolio to be indierent between holding Z and not doing so. An optimal hedging strategy is a minimizer δ (if it exists) in the left hand side of (2). Numerical calculation of p(z) and δ? Lukas Gonon Deep Hedging 8/31

13 Formal framework Indierence pricing and optimal hedging: Following e.g. Föllmer, Klöppel, Leukert, Schweizer, Xu,... : Describe risk-preferences by a convex risk measure ρ (see e.g. Föllmer and Schied (2016) or Chapter 8 in McNeil, Frey and Embrechts (2015) for an introduction). Key examples: Entropic risk: ρ(x ) := 1 λ log E [ e ] λx for λ > 0. } CVar / Expected Shortfall: ρ(x ) := inf w R {w α E[( X w)+ ] for α [0, 1). Denote H the set of available hedging strategies. The indierence price is the (unique) solution p(z) to inf ρ δ H (PL T (Z, p(z), δ)) = inf ρ δ H (PL T (0, 0, δ)). (2) p(z) is the amount of money the bank needs to allocate to its portfolio to be indierent between holding Z and not doing so. An optimal hedging strategy is a minimizer δ (if it exists) in the left hand side of (2). Numerical calculation of p(z) and δ? Lukas Gonon Deep Hedging 8/31

14 Formal framework Numerical calculation of p(z) and δ : Even in continuous-time limit: closed-form solution only known for very specic parametric models such as Black-Scholes without transaction costs. Even in a one-dimensional Black-Scholes model: with transaction costs one needs to solve multidimensional PDE with free boundary. Higher-dimensional models? Scalable computational method? Lukas Gonon Deep Hedging 9/31

15 Solution by deep learning techniques Outline 1 Risk management of a portfolio of derivatives 2 Formal framework 3 Solution by deep learning techniques 4 Numerical results 5 Conclusion and next steps Lukas Gonon Deep Hedging 10/31

16 Solution by deep learning techniques Our contribution We show: now approximate calculation is feasible thanks to modern deep learning techniques (exploited for other problems in nance e.g. in works by E, Han, Jentzen, Cheridito, Becker,...). Approach: consider only hedging strategies δ = (δ tk ) k=0,...,n 1 of the form δ tk = F θ k (S tk, δ tk 1 ), k = 0,..., n 1 where F θ k is a neural network with weights parametrized by θ k. Key point 1: neural networks are surprisingly ecient at approximating multivariate functions (see works by Bölcskei, Grohs, Kutyniok, Petersen, Shaham, Cloninger, Coifman,...). Key point 2: ecient machine learning optimization algorithms (stochastic gradient-type and backpropagation) and implementations (Tensorow, Theano, Torch,...) are available. Our method is sample-based and highly exible: the same algorithm and implementation can handle wide range of market specications. Lukas Gonon Deep Hedging 11/31

17 Solution by deep learning techniques Approximate indierence price By cash-invariance, the indierence price p(z) is given as p(z) := π( Z) π(0), where π(x ) := inf δ H ρ(x + (δ S) T C T (δ)). Suppose ρ is an optimized certainty equivalent, i.e. ρ(x ) := inf {w + E[l( X w)]} w R for a suitable l: R R. This includes entropic risk and CVar. For suitable parameter set Θ M R r and δt θ k = F θ k (S tk, δt θ k 1 ) as above, approximate π(x ) by ( ) π M (X ) : = inf ρ X + (δ θ S) T C T (δ θ ) θ Θ M { } = inf inf θ Θ M w R w + E[l(Z (δ θ S) T + C T (δ θ ) w)]. amenable to machine learning optimization algorithms (stochastic gradient descent and backpropagation). Lukas Gonon Deep Hedging 12/31

18 Solution by deep learning techniques Main results Theoretically: Establish convergence (as complexity of neural network increases) based on universal approximation theorem. Exploit dual representation of convex risk measures to show that objective is amenable to machine learning techniques very generally. Numerical experiments (for discretized Heston model): Model hedge can be learnt very well using e.g. 0.5-expected shortfall. Impact of risk-preferences on hedging. Price asymptotics for proportional transaction costs. Scalable to higher dimensions. This talk: Numerical results. Lukas Gonon Deep Hedging 13/31

19 Solution by deep learning techniques Main results Theoretically: Establish convergence (as complexity of neural network increases) based on universal approximation theorem. Exploit dual representation of convex risk measures to show that objective is amenable to machine learning techniques very generally. Numerical experiments (for discretized Heston model): Model hedge can be learnt very well using e.g. 0.5-expected shortfall. Impact of risk-preferences on hedging. Price asymptotics for proportional transaction costs. Scalable to higher dimensions. This talk: Numerical results. Lukas Gonon Deep Hedging 13/31

20 Solution by deep learning techniques Main results Theoretically: Establish convergence (as complexity of neural network increases) based on universal approximation theorem. Exploit dual representation of convex risk measures to show that objective is amenable to machine learning techniques very generally. Numerical experiments (for discretized Heston model): Model hedge can be learnt very well using e.g. 0.5-expected shortfall. Impact of risk-preferences on hedging. Price asymptotics for proportional transaction costs. Scalable to higher dimensions. This talk: Numerical results. Lukas Gonon Deep Hedging 13/31

21 Numerical results Outline 1 Risk management of a portfolio of derivatives 2 Formal framework 3 Solution by deep learning techniques 4 Numerical results 5 Conclusion and next steps Lukas Gonon Deep Hedging 14/31

22 Numerical results Example Study Heston Model ds (1) t = V t S (1) t db t, S (1) 0 = s 0 dv t = α(b V t )dt + σ V t dw t, V 0 = v 0 B and W are Brownian motions with d B, W = ρdt (α, b, ρ, σ, v 0, s 0 ) = (1, 0.04, 0.7, 2, 0.04, 100) Payo and Hedging Payo: (S (1) T K)+ with K = 100, T = 30 days / call spread with K 1 = 100, K 2 = 101. Hedging instruments: Trade in S (1) and (idealized) variance swap. Trading: Daily rebalancing of hedging strategy. Risk measure: α-cvar (expected shortfall), { ρ(x ) := inf w + 1 } w R 1 α E[( X w)+ ]. Lukas Gonon Deep Hedging 15/31

23 Numerical results Network Architecture Training For each trading date t k, δ tk is approximated by a 3-layer FFNN. Input: log S (1) t k, variance swap price at t k and δ tk 1. Two hidden layers with 15 nodes each and ReLU activation function (x x + ). Output layer: Two dimensional (amount to be held in S (1) and variance swap), activation function is identity (x x). In total 30 ( ) = parameters. Sample training data from Heston model. Use Adam (rened stochastic gradient descent) with batch size 256 to nd optimal neural network hedging strategy (for dierent criteria). (Out-of-sample) testing: Simulate sample paths and compare dierent hedging strategies. Lukas Gonon Deep Hedging 16/31

24 Numerical results Zoom into recurrent network Input layer at ti Hidden layer I at ti Hidden layer II at ti Output layer at ti log(sti ) vti δ (s) ti Input layer at ti+1 δ (v) ti Input layer at ti+1 Output layer at ti 1 δ (s) ti 1 Output layer at ti 1 δ (v) ti 1 Lukas Gonon Deep Hedging 17/31

25 Numerical results Main results Theoretically: Establish convergence (as complexity of neural network increases) based on universal approximation theorem. Exploit dual representation of convex risk measures to show that objective is amenable to machine learning techniques very generally. Numerical experiments (for discretized Heston model): Model hedge can be learnt very well using e.g. 0.5-expected shortfall. Impact of risk-preferences on hedging. Price asymptotics for proportional transaction costs. Scalable to higher dimensions. Lukas Gonon Deep Hedging 18/31

26 Numerical results Out-of-sample P& L histogram for ATM call option Model hedge vs 50%-expected shortfall deep hedge CVar Model hedge Neural network reproduces model hedge accurately. Lukas Gonon Deep Hedging 19/31

27 Numerical results Out-of-sample P& L histogram for ATM call option Model hedge vs 50%-expected shortfall deep hedge CVar Model hedge Neural network reproduces model hedge accurately. Lukas Gonon Deep Hedging 19/31

28 Numerical results Main results Theoretically: Establish convergence (as complexity of neural network increases) based on universal approximation theorem. Exploit dual representation of convex risk measures to show that objective is amenable to machine learning techniques very generally. Numerical experiments (for discretized Heston model): Model hedge can be learnt very well using e.g. 0.5-expected shortfall. Impact of risk-preferences on hedging. Price asymptotics for proportional transaction costs. Scalable to higher dimensions. Lukas Gonon Deep Hedging 20/31

29 Numerical results Deep hedge for 99%-CVar vs 50%-CVar, centered at p 50%(Z) CVar 0.5-CVar Mean Loss Realized 0.99-CVar Realized 0.5-CVar 0.99-CVar CVar %-CVar strategy avoids extreme loss (smaller realized 99%-CVar value), 50%-CVar strategy minimizes average loss (smaller mean loss). Lukas Gonon Deep Hedging 21/31

30 Numerical results Call spread δ (s) t as a function of (s t, v t ) for t = 15 and α = 0.5, 0.95, 0.99: Model Spread Delta % Spread Delta % Spread Delta Model Delta 50% CVar Delta Higher risk-aversion barrier shift in line with heuristics employed in manual trading. Lukas Gonon Deep Hedging 22/31

31 Numerical results Call spread δ (s) t as a function of (s t, v t ) for t = 15 and α = 0.5, 0.95, 0.99: Model Spread Delta % Spread Delta % Spread Delta Model Delta 50% CVar Delta Higher risk-aversion barrier shift in line with heuristics employed in manual trading. Lukas Gonon Deep Hedging 22/31

32 Numerical results Main results Theoretically: Establish convergence (as complexity of neural network increases) based on universal approximation theorem. Exploit dual representation of convex risk measures to show that objective is amenable to machine learning techniques very generally. Numerical experiments (for discretized Heston model): Model hedge can be learnt very well using e.g. 0.5-expected shortfall. Impact of risk-preferences on hedging. Price asymptotics for proportional transaction costs. Scalable to higher dimensions. Lukas Gonon Deep Hedging 23/31

33 Numerical results Price asymptotics: proportional transaction costs p ε = p ε (Z) is the exponential utility indierence price of Z for proportional transaction costs of size ε. For continuous-time models with d = 1 (e.g. Whalley and Wilmott, Kallsen and Muhle-Karbe): p ε p 0 = O(ε 2/3 ), as ε 0. (3) Rate 2/3 also emerges in related problems with proportional transaction costs, see e.g. works by Davis, Panas, Zariphopoulou, Rogers, Soner, Shreve, Cvitani,... Our methodology: reproduces (3) in a Heston model with d = 2 hedging instruments. For this case (or any other model with d > 1) no results on (3) have been available previously (neither theoretical nor numerical). Lukas Gonon Deep Hedging 24/31

34 Numerical results Rate of convergence is 0.71 log(p p0) log( ) Unreasonable eectiveness of neural networks research questions. Lukas Gonon Deep Hedging 25/31

35 Numerical results Rate of convergence is 0.71 log(p p0) log( ) Unreasonable eectiveness of neural networks research questions. Lukas Gonon Deep Hedging 25/31

36 Numerical results Main results Theoretically: Establish convergence (as complexity of neural network increases) based on universal approximation theorem. Exploit dual representation of convex risk measures to show that objective is amenable to machine learning techniques very generally. Numerical experiments (for discretized Heston model): Model hedge can be learnt very well using e.g. 0.5-expected shortfall. Impact of risk-preferences on hedging. Price asymptotics for proportional transaction costs. Scalable to higher dimensions. Lukas Gonon Deep Hedging 26/31

37 Numerical results Out-of-sample P& L histogram for multiple Heston models Consider a market built from 5 independent Heston models (S (n), V (n) ), n = 1,..., 5, each of them with parameters as above. (S (n) Payo: Z = 5 n=1 T K)+. 10 underlyings, due to independence, hedging this payo amounts to solving 5 separate problems at once allow 5 more hidden nodes, training time increases by a factor of Deep hedge Model hedge Scalable to higher dimensions Lukas Gonon Deep Hedging 27/31

38 Numerical results Out-of-sample P& L histogram for multiple Heston models Consider a market built from 5 independent Heston models (S (n), V (n) ), n = 1,..., 5, each of them with parameters as above. (S (n) Payo: Z = 5 n=1 T K)+. 10 underlyings, due to independence, hedging this payo amounts to solving 5 separate problems at once allow 5 more hidden nodes, training time increases by a factor of Deep hedge Model hedge Scalable to higher dimensions Lukas Gonon Deep Hedging 27/31

39 Conclusion and next steps Outline 1 Risk management of a portfolio of derivatives 2 Formal framework 3 Solution by deep learning techniques 4 Numerical results 5 Conclusion and next steps Lukas Gonon Deep Hedging 28/31

40 Conclusion and next steps Conclusion Deep learning techniques yield a (feasible and scalable) numerical method for risk management of derivatives under market frictions at portfolio level. Key feature 1: very generic approach allows to incorporate frictions, is highly exible in terms of products,... Key feature 2: separates the problem of nding a realistic model from nding an optimal hedging strategy. Next steps Fully statistical markets: in progress. Theoretical insights: optimal machine learning paradigm / network architecture in a dynamic setting (reservoir computing,...)? Lukas Gonon Deep Hedging 29/31

41 Some relevant literature I Becker, S., Cheridito, P. and Jentzen, A., Deep optimal stopping. Preprint arxiv: , Bölcskei, H., Grohs, P., Kutyniok, G. and Petersen, P., Optimal Approximation with Sparsely Connected Deep Neural Networks. Preprint arxiv: , Bühler, H., Gonon, L., Teichmann, J. and Wood, B., Deep hedging. Preprint arxiv: , Davis, M.H.A., Panas, V.G. and Zariphopoulou, T., European Option Pricing with Transaction Costs. SIAM Journal on Control and Optimization, 1993, 31, E, W., Han, J. and Jentzen, A., Deep learning-based numerical methods for high-dimensional parabolic partial dierential equations and backward stochastic dierential equations. Communications in Mathematics and Statistics, 2017, 5, Lukas Gonon Deep Hedging

42 Some relevant literature II Föllmer, H. and Leukert, P., Ecient Hedging: Cost Versus Shortfall Risk. Finance and Stochastics, 2000, 4, Föllmer, H. and Schied, A., Stochastic nance: An introduction in discrete time, 2016, De Gruyter. Hodges, S. and Neuberger, A., Optimal Replication of Contingent Claims Under Transaction Costs. The Review of Futures Markets, 1989, 8, Kallsen, J. and Muhle-Karbe, J., Option Pricing and Hedging with Small Transaction Costs. Mathematical Finance, 2015, 25, Klöppel, S. and Schweizer, M., Dynamic Indierence Valuation via Convex Risk Measures. Mathematical Finance, 2007, 17, McNeil, A., Frey, R. and Embrechts, P., Quantitative Risk Management, revised ed., 2015, Princeton University Press. Lukas Gonon Deep Hedging

43 Some relevant literature III Rogers, L.C.G., Why is the eect of proportional transaction costs O(δ 2/3 ). In Mathematics of Finance, edited by G. Yin and Q. Zhang, pp , 2004 (American Mathematical Society: Providence, RI). Shaham, U., Cloninger, A. and Coifman, R.R., Provable approximation properties for deep neural networks. Applied and Computational Harmonic Analysis, 2018, 44, Soner, H.M., Shreve, S.E. and Cvitani, J., There is no Nontrivial Hedging Portfolio for Option Pricing with Transaction Costs. The Annals of Applied Probability, 1995, 5, Whalley, A.E. and Wilmott, P., An Asymptotic Analysis of an Optimal Hedging Model for Option Pricing with Transaction Costs. Mathematical Finance, 1997, 7, Xu, M., Risk measure pricing and hedging in incomplete markets. Annals of Finance, 2006, 2, Lukas Gonon Deep Hedging