Transcription
Completing Partial Implied Vol Surfaces withVariational Autoencoders
Previously faculty member at theUniversity of ChicagoMaxime BergeronDirector R&D, RiskfuelResearch on the topology of highdimensional parameter spaces and itsimplications in machine learning
Three pillars of deep learning at Riskfuel1. Fast models (pricing of exotic derivatives)2. Unsupervised learning (variational autoencoders)3. Reinforcement learning (deep hedging, the next steps)
Riskfuel's 1st ProblemDeeply Learning DerivativesRyan Ferguson and Andrew Green, 2018Given: A family of partial differential equations problems A method (slow and noisy) to compute individual solutions (calibration and inference)Can we:Build a fast and accurate mesh-free representation of the solutions operator O ?Yes:Leveraging the slow solver to generate training data for deep neural networks!Results in lightning-fast inference and (when they exist) sensitivities
Real-time PricingBlack-ScholesBermudan Swaptions 157 dimensionshic sunt draconesSwapsFX Barriers 81 dimensionsAutocallables 400 dimensions
FX Double Knock-Out Partial Barrier Option 81 input dimensions Full volatility surface Interest rate curves Trade specificsTrained over wide range of synthetic valuesto accommodate extreme scenariosHandled Covid Crisiswithout a ogy-innovation-of-the-year-scotiabank
Bermudan Swaption Demo 157 input dimensions! Volatility surface Interest rate curve Trade specifics (strike, NC, term,.)pricer.riskfuel.com
Three pillars of deep learning at Riskfuel1. Fast models (pricing of exotic derivatives) 2. Unsupervised learning (variational autoencoders)3. Reinforcement learning (deep hedging, the next steps)Fast pricers need fast good data
Volatility surfaces maintain their mystique All models have flaws Backward looking, good until they break (W) Have bad understanding of what vol surfaces really looks likeHands off: let the data speak for itself!
What we want from a good volatility modelWant the ability to robustly: Complete Compress ExploreWant the model to be: Fast Realistic Interpretable
Complete& Interpolate Complete data in real timefrom partial information Monitor data for outliersand trading opportunities
Complete& Interpolate Complete data in real timefrom partial information Monitor data for outliersand trading opportunities
ExploreWant to understand geometry ofthe space of volatility surfaces Realistic stress testing:surfaces don't move in parallel shifts! Generate synthetic training datafor fast AI models
CompressHow many parameters do weneed to represent anentire surface? Parsimony vs accuracy ofrepresentation Monitor data for outliers andtrading opportunities
Variational Autoencoders: not your grandpa's PCAMonitorCompressCompletePCA (linear) AutoEncoders (AE) better Variational AutoEncoders (VAE) robust robustEmbrace the freedom of non-linearity!SampleExplore
Manifold Learning VAE is trained on real volatility surfaces to encode their distribution in its latent variables Gives a continuous structure to latent variables: they are not just compressions!
Manifold LearningPCA & AE Learn shape of volatility surface VAE Learn shape of the space of volatility surfaces
Trained VAE: 3 things for the price of 11. Encoder 2. Latent Space 3. Decoder
Decoder (Generator)Feed latent variables through decoder half of previously trained VAEAccurate and fast matching of market dataZ1Z2
EncoderFeed full surfaces through encoder half of previously trained VAEGreat for detecting outliers and trade opportunitiesZ1Z2
Explore latent spaceLatent variable values can be perturbed or sampled arbitrarily Resulting surfaces produced by previously trained decoder Can generate "good bad vol surfaces" for stress testing!Z1'Z2'
Case study: FX market data OTC data from 2012-2020 for five currency pairs Each surface 40-point grid of 8 maturities x 5 deltas Data split chronologically: set aside March – December 2020 for validation Train VAE's with 2, 3 and 4 dimensional latent encodings24
Test ability to complete volatility surfaces Observe subset of points on a surface (tight deltas, short maturities) Find latent encodings that generate best fit surface at these locations
Test ability to complete volatility surfaces Observe subset of points on a surface (tight deltas, short maturities) Find latent encodings that generate best fit surface at these locations
A typicalday in March VAE has seen days like this before Nails the fit using only a handfulof observed points in the first fewmaturities Mean Absolute Error in bps
Error distribution in bps across the AUD/USD validation setusing a VAE with four latent variables to complete partially observed surfaces.meanbid-askspreadHow bad do things get?March 24th, 2020
A wild day inMarch The VAE (trained on data from2012 to January 2020) was neverexposed to true crisis conditions! Mean Absolute Error in bps.
What's happening under the hood? Focus on 3 dimensional VAE for full visualization of latent space This means three numbers parametrize an entire surface What did the VAE do with these three numbers?
Skew and Wings
Term Structure
Volatility level
A knot in latent space
What's next? Equity indexes Single equities Time filtrations
Three pillars of deep learning at Riskfuel 1. Fast models (pricing of exotic derivatives) 2. Unsupervised learning (variational autoencoders) 3. Reinforcement learning (deep hedging, the next steps)
