Authors: Chataigner, Cousin, Crepey, Dixon, and Gueye.

If this code is used for research purposes, please cite as:

M. Chataigner, A. Cousin, S. Crepey, M.F. Dixon, and D. Gueye, Beyond Surrogate Modeling: Learning the Local Volatility Via Shape Constraints, working paper, 2020. M. Chataigner, S. Crepey and M. Dixon, Deep Local Volatility, Risks 8(3), 82, Special Issue on Machine Learning in Finance, Eds. Thorsten Schmidt, 2020.

Overview

Run the Master.ipynb or view the Master.html file to see a demo showing the backtesting of the GP, SSVI and Neural Network. The file also compares the backtesting results.

Notebook folder contains experiments from "Deep Local Volatility" paper.

The code folder contains python script for our library :

• BS.py offers some service related to black-scholes model (implied volatility calibration, pricing ...).
• SSVI.py implements arbitrage free calibration of Gatheral, J., & Jacquier, A. (2014). Arbitrage-free SVI volatility surfaces. Quantitative Finance, 14(1), 59-71.
• SSVIUnconstrained.py implements standard SVI calibration. Interpolation of SVI parameters is inspired from Gatheral (2014) but an unconstrained raw SVI parametrization is fitted for each slice.
• backtest.py implements Monte Carlo and PDE repricing in order to assess local volatility.
• bootstrapping.py extracts discount curve and dividend curve.
• dataSetConstruction.py provides some features engineering features.
• loadData.py extracts original data from excel, dat and csv files.
• neuralNetwork.py implements various architectures and run tensorflow models.
• plotTools.py provides plotting and performance diagnosis features.

A subfolder named GP is devoted to Gaussian Process implementation :

• Kriging_vol_surface_scattered_data.m : script used to run the GP approach on bid/ask price of European option. This script all the other functions of this project
• Import_data_scattered_SnP500_18_05_2019 : script used to import the data set
• log_likelihood_unconstr_single_noise_param_scatter : compute the unconstrained log-likelihood given a set of hyper-parameter
• Gamma_decomp : construct the covariance matrix of the 2D GP process at some test points
• HMC_exact : Hamiltonian Monte Carlo sampling of a truncated Gaussian vector given some linear inequality constraints
• Vol_from_putPrice_with_div, VolSurface_from_putPrice_with_div : compute implied volatility by inversion of B&S formula, in the presence of interest rate and dividend yield
• kernel_Gauss_1D : 1D Gaussian kernel
• kernel_Matern_5_2_1D : 1D Matérn 5/2 kernel
• Basis_func_scattered_data, Basis_func_decomp, basis_func, basis_func_vect : compute the value of all basis functions at a set of scattered data

The Matlab code implements the shape constrained GP code and is written by A. Cousin and D. Gueye.

SSVI calibration is inspired from Matlab code Philipp Rindler (2020). Gatherals and Jacquier's Arbitrage-Free SVI Volatility Surfaces (https://www.mathworks.com/matlabcentral/fileexchange/49962-gatherals-and-jacquier-s-arbitrage-free-svi-volatility-surfaces), MATLAB Central File Exchange. Retrieved June 22, 2020.

The BS folder contains some additional Python scripts for implied volatility estimation, using the Bisection algorithm, written by M. Dixon.

These notebooks are fully compatible with a local notebook environment provided that Tensorflow is installed for Python 3 with a version at or above 2.0. For deployment on a Google colab environment, please be aware of files and foldr structure (e.g. variable "workingFolder") in order to properly load data, python scripts and experiment results.

Several days of DAX index option chain data is provided in the data folder. S&P 500 and Eurostoxx 50 option prices are also included for more recent data.

Each notebook indicates which data file you should load for execution.