Deep Learning Volatility

Blanka Horvath; Aitor Muguruza; Mehdi Tomas

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Deep Learning Volatility

Blanka Horvath, Aitor Muguruza, Mehdi Tomas

arXiv:1901.09647v2Jan 28, 2019q-fin.MF

TL;DR

Slow numerical pricing makes calibration a bottleneck for many stochastic-volatility models, particularly rough-volatility models. The paper learns pricing or implied-volatility maps offline with deep neural networks and uses them for online calibration, achieving millisecond-scale calibration and broad model applicability while relying on numerical pricing approximations and non-convex optimization conditions.

Problem
Slow numerical pricing can make stochastic-volatility models, especially rough-volatility models, impractical for industrial calibration.
Method
The paper trains deep neural networks offline to approximate pricing or implied-volatility maps, then calibrates the resulting deterministic maps online using grid-based surface information.
Results
A few milliseconds enables online calibration of rough-volatility models, while the learned pricing map retains out-of-sample accuracy within the range of its numerical training approximation.
Takeaways & Limitations
The approach extends fast calibration to a range of volatility models and derivative-contract surfaces, including rough-volatility models.
Takeaways & Limitations
The calibration formulation is idealized because most option prices require numerical approximation, and the calibration optimization problem is non-convex with necessary but not sufficient optimality conditions.

Abstract

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We present a neural network based calibration method that performs the calibration task within a few milliseconds for the full implied volatility surface. The framework is consistently applicable throughout a range of volatility models -including the rough volatility family- and a range of derivative contracts. The aim of neural networks in this work is an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. We highlight how this perspective opens new horizons for quantitative modelling: The calibration bottleneck posed by a slow pricing of derivative contracts is lifted. This brings several numerical pricers and model families (such as rough volatility models) within the scope of applicability in industry practice. The form in which information from available data is extracted and stored influences network performance: This approach is inspired by representing the implied volatility and option prices as a collection of pixels. In a number of applications we demonstrate the prowess of this modelling approach regarding accuracy, speed, robustness and generality and also its potentials towards model recognition.

1 Introduction

The paper frames tractable pricing as central to model adoption and proposes deep neural networks as basis-agnostic approximators of pricing functionals. The approach delivers fast implied-volatility evaluation and millisecond-scale rough-volatility calibration.

1 Introduction: Pricing-method applicability depends on stochastic-model regularity, making tractability a decisive factor in model popularity.The introduction contrasts Fourier, PDE, asymptotic, and Monte Carlo methods across models.
1 Introduction: Direct parameter-to-implied-volatility mappings exist in SSVI, eSSVI, and asymptotic approximations, but the paper targets multidimensional pricing functionals.The proposed approach maps traditional model parameters to option prices or implied-volatility surfaces.
1 Introduction: Deep neural networks provide basis-agnostic functional approximations that can be applied consistently across several stochastic models.The network moves computationally expensive numerical approximation offline.
1 Introduction: Within milliseconds, the trained network outputs implied volatilities across a large range of maturities and strikes for any model-parameter choice.Out-of-sample price-approximation accuracy remains within the accuracy range of the numerical approximation used for training.
1 Introduction: A few milliseconds is sufficient for online calibration of rough-volatility models, including rough Bergomi with a general forward variance curve.Experiments include simulated-data calibration and historical calibration on 10 years of SPX data.

2 A neural network perspective on model calibration

The paper formalizes calibration as selecting model parameters whose prices are close to observed market prices. Because exact pricing is usually unavailable, practical calibration replaces the pricing map with a numerical approximation used as the benchmark.

2 A neural network perspective on model calibration: Calibration chooses parameters so model prices are as close as possible to observed market data for specified financial contracts.The framework represents models by parameterized families and prices by a pricing map.
2 A neural network perspective on model calibration: The ideal calibration problem minimizes a metric δ between model-generated prices and observed market prices.The metric is chosen for the financial contract under consideration.
2 A neural network perspective on model calibration: For most financial models, option prices lack analytical formulas and must be computed with numerical approximation schemes.This makes the ideal calibration formulation an abstraction of practical calibration.
2 A neural network perspective on model calibration: Approximate calibration substitutes a numerical pricing approximation eP for the exact pricing map P.The approximate problem retains the same parameter-fitting objective while using computable prices.
2 A neural network perspective on model calibration: The numerical pricing approximation eP serves as the available-truth benchmark for generating synthetic training samples for neural-network pricing maps.The paper notes that network quality depends on the quality of the original numerical approximation.

2.1 A brief reminder of some (rough) models considered

The framework is presented for classical and rough volatility models, including Heston, Bergomi, rough Bergomi, and SABR. Rough Bergomi uses parameters including the Hurst parameter and a piecewise-constant approximation of the initial forward variance curve.

2.1 A brief reminder of some (rough) models considered: The methodology can in principle be applied to classical or rough volatility models and to parametric implied-volatility models.The stated scope ranges from Black-Scholes and Heston to rough Bergomi and broader rough-volatility classes.
2.1 A brief reminder of some (rough) models considered: Rough Bergomi is parameterized by ξ0, ν, ρ, and H, with H denoting the Hurst parameter.The initial forward variance curve ξ0 is approximated by a piecewise-constant function in the numerical experiments.
2.1 A brief reminder of some (rough) models considered: The Heston model uses parameters a, b, v, and ρ and appears in the paper’s numerical experiments.Its parameter space is represented as a subset of R^4.
2.1 A brief reminder of some (rough) models considered: The Bergomi family is described through correlated Brownian factors and an initial forward variance curve, with one- and two-factor versions considered.The one-factor setup also approximates ξ0 by a piecewise-constant function in the experiments.
2.1 A brief reminder of some (rough) models considered: SABR is included as the stochastic alpha beta rho model with parameters α, β, and ρ.The cited parameter restrictions include α>0 and β∈[0,1].

2.2 Calibration bottlenecks in volatility modelling and deep calibration

Slow numerical pricing can make otherwise attractive volatility models impractical for industrial calibration, especially rough-volatility models. The paper addresses this with a two-step procedure that learns pricing maps offline and calibrates the resulting deterministic approximation online.

2.2 Calibration bottlenecks in volatility modelling and deep calibration: Slow numerical calibration can limit a stochastic-volatility model’s industrial applicability despite other desirable features.Rough volatility is especially affected because rough fractional Brownian dynamics rule out usual Markovian finite-difference pricing methods.
2.2 Calibration bottlenecks in volatility modelling and deep calibration: The paper advocates two-step calibration rather than direct neural-network calibration.The two steps are learning a model and calibrating it to data.
2.2 Calibration bottlenecks in volatility modelling and deep calibration: Offline training learns a neural-network approximation of the pricing or implied-volatility map from stochastic-model parameters to pricing functions.The learned map is stored for subsequent online calibration.
2.2 Calibration bottlenecks in volatility modelling and deep calibration: Online calibration replaces the numerical pricing approximation with a learned deterministic counterpart, making calibration considerably faster.The speedup follows from avoiding repeated numerical simulation of stochastic expected payoffs.
2.2 Calibration bottlenecks in volatility modelling and deep calibration: The paper reports learning errors and calibration times in numerical experiments using the two-step approach.The experiments cover a family of stochastic-volatility models, with corresponding codes made available on GitHub.

2.3 Challenges in neural network approximations of pricing functionals

Numerical pricing evaluations can dominate calibration time when pricing maps lack closed forms. The neural-network replacement is designed to preserve the numerical pricer’s accuracy order while accelerating repeated evaluations.

Slow numerical pricing evaluations can create substantial calibration bottlenecks because calibration repeatedly calls the pricing map.The number of evaluations depends on the optimisation procedure and desired precision.
A neural network approximates the pricing map from the full model-parameter set to contract prices, targeting faster functional evaluations.
The approximation is required to remain within the original numerical pricing discretisation’s error order for every parameter combination.The target is eF(θ) = P(M(θ), ζ) + O(ϵ) whenever the numerical approximation has the same O(ϵ) accuracy.
The training objective treats the available numerical pricing approximation as ground truth and evaluates whether the network reaches its accuracy.The experiments also assess the resulting reduction in functional-evaluation time.

2.4 Motivations for our choice of training setup and features of neural networks as approximators of pricing functionals

The training setup separates offline price approximation from calibration, preserving established model knowledge while using grid-based representations to exploit the structure of implied-volatility surfaces. Deep networks provide flexible multivariate approximation, but architecture and data size remain linked.

Motivation for separating pricing and calibration: Separating pricing from calibration keeps traditional model knowledge and risk-management libraries usable while the network supplies computational enhancement.
Motivation for separating pricing and calibration: Synthetic data generated by traditional numerical methods removes dependence on externally available training data.
Motivation for separating pricing and calibration: Whenever a consistent numerical pricer exists, it can be approximated and replaced by a deep network for fast pricing-map evaluations.
Grid-based implicit training: Grid-based training evaluates implied volatility across 8×11 gridpoints and uses neighbouring surface values to exploit image-like data structure.The paper describes the surface as an image whose pixels are implied-volatility values.
Grid-based implicit training: Evaluating many gridpoints makes distinct parameter combinations less likely to produce identical outputs than pointwise training.
Neural networks as functional approximators: Feed-forward networks learn mappings from inputs to outputs through adjustable weights, with hidden layers and activations determining the composition.In this application, inputs are model parameters and outputs are an implied-volatility grid.
Neural networks as functional approximators: Smooth activation functions are needed when approximating derivatives, whereas ReLU is not smooth beyond order zero.
Neural networks as functional approximators: Increasing network width reduces approximation bias but requires more training data to control variance and overfitting.

3 Pricing and calibration with neural networks: Optimising network and training

The paper compares direct calibration with an image-based implicit-learning approach and evaluates network architectures together with calibration-step optimisers.

The section compares direct calibration to data with image-based implicit learning and motivates the selected objective function.
It also provides network-architecture details and compares optimisers used for the calibration step.

3.1 The objective function

The objective maps model parameters to option prices or implied-volatility surfaces evaluated on prescribed contract grids. The same image-based construction extends from vanilla strikes and maturities to exotic barrier contracts.

General objective: The pricing approximation maps model parameters to contract prices evaluated on a prespecified grid of product attributes.Attributes can include maturity, strike, or barrier level.
Vanilla contracts: For vanilla contracts, calibration compares model-generated and market implied-volatility surfaces on strike–maturity grids.
Vanilla contracts: The proposed image-based pricing map stores implied-volatility surfaces as pixel grids over fixed strikes and maturities.
Exotic payoffs: The framework also covers digital barriers, no-touch barriers, cliquets, and autocallables.
Exotic payoffs: For digital barrier options, the strike grid is replaced by a barrier-level grid over barrier levels and maturities.

3.2 Network architecture and training

The calibration is organized as an implicit two-step procedure: a neural network approximates the implied-volatility pricing map, followed by a calibration layer. The network uses model parameters as inputs and outputs a grid of implied volatilities, with normalization and regularization supporting training.

3.2 Network architecture and training: The implicit approach separates calibration into a deep approximation network and a calibration layer.The approximation network is trained first, after which calibration is performed on top of it.
3.2 Network architecture and training: The network is fully connected with 4 hidden layers and 30 neurons per layer.The architecture is motivated by the paper’s theoretical analysis.
3.2 Network architecture and training: Model parameters enter at the input layer, while the output is an 8 × 11 implied-volatility grid.The grid corresponds to 11 strikes and 8 maturities in this experiment, though the grid can be modified.
3.2 Network architecture and training: Training uses mini-batches of 32, early stopping, and normalization of both model parameters and implied volatilities.Training stops after 25 test-set steps without improvement, and implied-volatility surfaces are standardized using their sample mean and standard deviation.
3.2 Network architecture and training: Regularization techniques are used to speed convergence, avoid overfitting, and improve network performance.

3.3 The calibration step

The calibration step optimizes model parameters using the approximated pricing map and can employ gradient-based or gradient-free methods. Gradient methods rely on smoothness and can converge quickly, but non-convex objectives require special care.

3.3 The calibration step: After approximating the pricing map, calibration solves for model parameters using the network’s implied-volatility approximation.The pricing map is assumed to be at least C1 differentiable with respect to the input parameters.
Gradient-based optimizers: Gradient-based calibration uses first-order optimality conditions and updates parameters along the objective gradient.The paper lists gradient descent, Levenberg-Marquardt, BFGS, L-BFGS-B, and SLSQP as examples.
3.3 The calibration step: Gradient-based optimizers can converge quickly, but first-order conditions are necessary rather than sufficient for optimality in non-convex problems.The paper therefore notes that special care is required for non-convex calibration objectives.
Gradient-based optimizers: Smooth activation functions are chosen so the network pricing-map gradients approximately match the true pricing-map gradients.
Gradient-free optimizers: Gradient-free methods avoid the C1 smoothness assumption and include Nelder-Mead, COBYLA, and Differential Evolution.They are presented for high-dimensional, nonlinear, non-differentiable, or non-convex problems.

4 Numerical experiments

The experiments evaluate neural-network pricing and calibration across rough Bergomi and 1 Factor Bergomi models, including piecewise constant forward variances. The approximation matches Monte Carlo accuracy while enabling fast calibration and generalizes to unseen parameter combinations.

Method: The neural network separates implied-volatility approximation from calibration, replacing expensive Monte Carlo evaluations with a deterministic optimization problem.It maps stochastic-volatility model parameters to implied-volatility surfaces and supports subsequent calibration.
Numerical accuracy: The approximation achieves Monte Carlo-level accuracy, with errors within a few basis points on the full implied-volatility surface.The comparison uses Monte Carlo prices computed with 60,000 paths and 95% confidence intervals.
Numerical accuracy: Average relative error remains below 0.5%, standard deviation below 1%, and maximum relative error reaches 25% across the evaluated parameter combinations.This behavior is reported for both rough Bergomi and 1 Factor Bergomi comparisons.
Calibration speed: Gradient-based optimizers are faster than gradient-free methods, while Levenberg-Marquardt provides the best balance of calibration speed and convergence in these experiments.Gradient-free calibration times are more affected by the dimensionality of the forward-variance parameterization.
Calibration accuracy: For rough Bergomi and Bergomi models with piecewise constant forward variances, the 99% RMSE quantile remains below 1%.The authors interpret this as successful generalization to general forward variances.
Historical calibration: In historical rough Bergomi calibration, neural-network and brute-force Monte Carlo fits differ by less than 0.2% most of the time.The comparison also reports that Differential Evolution outperforms Levenberg-Marquardt in this experiment.

5 Conclusions and outlook: “best-fit” models

The paper uses neural networks to approximate implied-volatility pricing maps on grids, enabling fast evaluations and exploring whether surfaces can identify or combine stochastic models.

Approximation and representation: Grid-based neural networks approximate complex pricing or implied-volatility mappings that are difficult or time-consuming to evaluate by other means.The grid representation is inspired by treating surface values like pixels, while moving numerical approximation offline.
Model recognition: A classifier is trained to identify whether an implied-volatility surface was generated by the Heston, Bergomi, or rough Bergomi model.Each training surface receives a model flag, and the classifier outputs model probabilities through a softmax layer.
Model recognition: The framework extends recognition from single-model surfaces to mixtures formed by convexly combining surfaces from Heston, Bergomi, and rough Bergomi.Mixture coefficients are nonnegative and sum to one, allowing the network to represent combinations of expert models.
Model recognition: A proof-of-concept experiment tests recognition on mixtures of rough Bergomi and Heston surfaces, including mixture coefficients sampled on a finer validation grid.Training uses coefficients from 0 to 1 in increments of 0.1, while validation also includes increments of 0.05.