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 second-generation stochastic volatility models and the rough volatility family—and a range of derivative contracts. Neural networks in this work are used in an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. The form in which information from available data is extracted and used influences network performance: The grid-based algorithm used for calibration is inspired by representing the implied volatility and option prices as a collection of pixels. We highlight how this perspective opens new horizons for quantitative modelling. The calibration bottleneck posed by a slow pricing of derivative contracts is lifted, and stochastic volatility models (classical and rough) can be handled in great generality as the framework also allows taking the forward variance curve as an input. We demonstrate the calibration performance both on simulated and historical data, on different derivative contracts and on a number of example models of increasing complexity, and also showcase some of the potentials of this approach towards model recognition. The algorithm and examples are provided in the Github repository GitHub: NN-StochVol-Calibrations.

Keywords:

• Rough volatility
• Volatility modelling
• Volterra process
• Machine learning
• Accurate price approximation
• Calibration
• Model assessment
• Monte Carlo

2010 Mathematics Subject Classifications:

• 60G15
• 60G22
• 91G20
• 91G60
• 91B25

Open Scholarship

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Acknowledgements

The authors are grateful to Jim Gatheral, Ben Wood, Antoine Savine and Ryan McCrickerd for stimulating discussions. MT conducted research within the ‘Econophysique et Systèmes Complexes’ under the aegis of the Fondation du Risque, a joint initiative by the Fondation de l'École Polytechnique, l'École Polytechnique, Capital Fund Management. MT also gratefully acknowledges the financial support of the ERC 679836 Staqamof and the Chair Analytics and Models for Regulation.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For details and an overview on calibration methods, see Goodfellow et al. (2016).

2 Note that the set ${\theta }_1,\dots ,{\theta }_N$ in (9(9) ${\theta }_i\mapsto \tilde{P}\left(\mathcal{M}\right({\theta }_i),\zeta ),{\theta }_i\in \{{\theta }_1,\dots {\theta }_N\}$(9) ) is extended to the full set of possible parameter combinations Θ in (10(10) $\begin{array}{rl}\tilde{F}:& \mathrm{\Theta }⟶\tilde{P}\left(\mathcal{M}\right)\\ & \theta \mapsto \tilde{F}(\theta ,\zeta )\end{array}$(10) ).

3 The network is called feed forward if there are no feedback connections in which outputs of the model are fed back into itself.

4 In our case, y is a $8\times 11$-point grid on the implied volatility surface and x are model parameters $\theta \in \mathrm{\Theta }$, for details see Section 3.

5 For sake of completeness, we introduce the Black–Scholes Call pricing function in terms of log-strike k, initial spot $S_0$, maturity T and volatility σ: $\begin{array}{rl}BS(\sigma ,S_0,k,T)& :=S_0\mathcal{N}\left(d_{+}\right)-K\mathcal{N}\left(d_{-}\right),\\ d_{\pm }& :=\frac{\log \left(S_0\right)-k}{\sqrt{T}\sigma }\pm \frac{\sqrt{T}\sigma }{2},\end{array}$ where $\mathcal{N}(\cdot )$ denotes the Gaussian cumulative distribution function. The implied volatility induced by a Call option pricing function $P(K,T)$ is then given by the unique solution ${\sigma }_{BS}(k,T)$ of the following equation $BS\left({\sigma }_{BS}\right(k,T),S_0,k,T)=P(k,T).$ Precisely, we seek to solve the following calibration problem (15) $\hat{\theta }:=\arg \underset{\theta \in \mathrm{\Theta }}{min}d({\mathrm{\Sigma }}_{BS}^{\mathcal{M}\left(\theta \right)},{\mathrm{\Sigma }}_{BS}^{MKT})$(15) where ${\mathrm{\Sigma }}_{BS}^{\mathcal{M}\left(\theta \right)}:=\left\{{\sigma }_{BS}^{\mathcal{M}\left(\theta \right)}\right(k_i,T_j){\}}_{i=1,\dots ,n,j=1,\dots ,m}$ represents the set of implied volatilities generated by the model pricing function $P\left(\mathcal{M}\right(\theta ),k,T)$ and ${\mathrm{\Sigma }}_{BS}^{MKT}:=\left\{{\sigma }_{BS}^{MKT}\right(k_i,T_j){\}}_{i=1,\dots ,n,j=1,\dots ,m}$ are the corresponding market implied volatilities, for some metric $d:{\mathbb{R}}^{n\times m}\times {\mathbb{R}}^{n\times m}\to {\mathbb{R}}^{+}$.

Additional information

Funding

This work was supported by H2020 European Research Council [grant number ERC 679836] and Chair Analytics and Models for Regulation.