[Paper Review] Deep calibration of rough stochastic volatility models

Christian Bayer, Benjamin Stemper|arXiv (Cornell University)|Oct 8, 2018

Stock Market Forecasting Methods38 citations

TL;DR

This paper proposes a deep learning-enhanced calibration method for rough stochastic volatility models, replacing computationally expensive Monte Carlo simulations with a neural network surrogate to approximate the implied volatility surface. The approach achieves high accuracy and speed by integrating a neural network with a Levenberg-Marquardt optimizer, validated through synthetic and real-market data with Bayesian calibration confirming robust parameter recovery.

ABSTRACT

Sparked by Alòs, León, and Vives (2007); Fukasawa (2011, 2017); Gatheral, Jaisson, and Rosenbaum (2018), so-called rough stochastic volatility models such as the rough Bergomi model by Bayer, Friz, and Gatheral (2016) constitute the latest evolution in option price modeling. Unlike standard bivariate diffusion models such as Heston (1993), these non-Markovian models with fractional volatility drivers allow to parsimoniously recover key stylized facts of market implied volatility surfaces such as the exploding power-law behaviour of the at-the-money volatility skew as time to maturity goes to zero. Standard model calibration routines rely on the repetitive evaluation of the map from model parameters to Black-Scholes implied volatility, rendering calibration of many (rough) stochastic volatility models prohibitively expensive since there the map can often only be approximated by costly Monte Carlo (MC) simulations (Bennedsen, Lunde, & Pakkanen, 2017; McCrickerd & Pakkanen, 2018; Bayer et al., 2016; Horvath, Jacquier, & Muguruza, 2017). As a remedy, we propose to combine a standard Levenberg-Marquardt calibration routine with neural network regression, replacing expensive MC simulations with cheap forward runs of a neural network trained to approximate the implied volatility map. Numerical experiments confirm the high accuracy and speed of our approach.

Motivation & Objective

• To address the high computational cost of calibrating rough stochastic volatility models, which rely on expensive Monte Carlo simulations for implied volatility computation.
• To develop a scalable and efficient calibration framework that maintains accuracy while reducing computation time.
• To validate the neural network surrogate's fidelity in approximating the implied volatility map across both synthetic and real market data.
• To demonstrate the feasibility of combining deep learning with traditional optimization (Levenberg-Marquardt) for financial model calibration.
• To confirm the robustness of the method through Bayesian calibration, assessing posterior distributions of model parameters.

Proposed method

• Train a neural network to learn the mapping from model parameters and option characteristics (moneyness, time to maturity) to Black-Scholes implied volatility, replacing Monte Carlo simulations.
• Use a Levenberg-Marquardt optimization routine to calibrate model parameters, with the neural network providing fast forward evaluations of the implied volatility surface.
• Apply liquidity-weighted nonlinear Bayesian regression to account for market data heteroskedasticity and bid-ask spreads, using inverse spreads as weights.
• Construct a synthetic IV surface using a reference Monte Carlo method to validate the neural network's accuracy before real-data application.
• Perform MCMC-based posterior inference to assess uncertainty and parameter identifiability, visualizing joint and marginal posterior distributions.
• Use a diagonal weight matrix in the likelihood function to reflect liquidity differences across options, with weights derived from mid-price and bid-ask spreads.

Experimental results

Research questions

• RQ1 Can a neural network surrogate accurately approximate the implied volatility surface of rough stochastic volatility models, replacing costly Monte Carlo simulations?
• RQ2 Does the integration of a neural network with a Levenberg-Marquardt optimizer enable fast and accurate calibration of rough volatility models?
• RQ3 How well does the calibrated model recover true parameters on synthetic data when using Bayesian inference with liquidity-weighted likelihoods?
• RQ4 To what extent does the neural network preserve parameter identifiability in the presence of degeneracy (e.g., trade-offs between H, η, ρ)?
• RQ5 Can the method successfully calibrate to real SPX implied volatility data, with posterior distributions aligning with previously reported parameter estimates?

Key findings

• The neural network surrogate achieves high accuracy in approximating the implied volatility surface, enabling fast calibration without sacrificing fidelity.
• Bayesian calibration on synthetic data shows unimodal posterior distributions centered on the true parameters, confirming the method's reliability.
• The posterior distributions exhibit diagonal elliptical contours in parameter pairs (η,H) and (η,ρ), indicating expected parameter degeneracy due to compensatory effects in the model.
• On real SPX data from May 19, 2017, the method successfully recovers parameter regions consistent with prior studies, with posterior peaks near values reported by BFG16.
• Residuals in the Bayesian calibration show that while full degeneracy effects are less pronounced on real data, scatter plots reveal remnants of the trade-off behavior between H, η, and ρ.
• The liquidity-weighted approach effectively handles market microstructure noise, with inverse bid-ask spreads used as weights to improve calibration robustness.