Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models

Abstract

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.

The Universal Approximation Property

“…Since and i are homeomorphisms and since φ i is continuous then ψ i is well-defined and continuous. Moreover, analogously to (15) we compute that ψ n i (g i ) n∈N is dense in L 2 (R). Since L 2 (R) is a complete separable metric space with no isolated points and ψ i is continuous self-map of L 2 (R) for which there is a vectorg i ∈ L 2 (R) such that the set of iterates {ψ n i (g i )} n∈N is dense in L 2 (R) then Birkhoff Transitivity Theorem, see the formulation of [74,Theorem 1.16], implies that for every pair of non-empty open subsetsŨ,Ṽ ⊆ L 2 (R) there is some nŨ ,Ṽ satisfying φ nŨ ,Ṽ (Ũ) ∩Ṽ = ∅.…”

“…Since then the universal approximation capabilities, of a limited number of neural network architectures, such as the feed-forward, residual, and convolutional neural networks has been solidified as a cornerstone of their approximation success. This, coupled with the numerous hardware advances have led neural networks to find ubiquitous use in a number of areas, ranging from biology, see [7,8], to computer vision and imaging, see [9,10], and to mathematical finance, see [11][12][13][14][15]. As a result, a variety of neural network architectures have emerged with the common thread between them being that they describe an algorithmically generated set of complicated functions built by combining elementary functions in some manner.…”

Accuracy of deep learning in calibrating HJM forward curves

“…Because of the complexity of the model (due to the state-dependent volatility and the infinite-dimensional setting), it is in general not possible to derive closed pricing formulas. To avoid time consuming numerical methods which render calibration almost impossible, in Paper IV we propose a machine-learning approach: we adapt the strategy presented in [68] and train a neural network which approximates option prices as a function of the HJM model parameters. This step is costly but off-line, meaning that no market data is used for training.…”

Deep calibration of financial models: turning theory into practice

“…( 2019 ), Horvath et al. ( 2021 ) and Stone ( 2020 ) . The general idea is the acceleration of the instrument valuation via the application of a neural network.…”