πͺοΈ VIX and Variance Derivatives: Theory, Pricing, and Replication
π EPFL β Master in Financial Engineering, Year 1 (2025)
π₯ Team: Matthias Wyss, William Jallot, Antoine Garin, Amine Bengelloun
π Final Report: Report
π GitHub Repository: GitHub
This project investigates the pricing, replication, and calibration of VIX and variance-linked derivatives using theoretical and numerical tools. It builds on three major axes:
- π Carr-Madan replication of arbitrary payoffs using options, including log and power payoffs
- π Model-free VIX approximation and interpretation as a replicating portfolio of OTM options
- π Affine models for VIX and variance futures pricing, including sensitivity to volatility parameters
We detail the mathematical foundations of these products and implement:
- Static replication strategies
- VIX futures pricing using Laplace transform and Gaussian integral representations
- Calibration of variance process parameters (Ξ», ΞΈ, ΞΎ)
- Replication of SPX options using VIX and variance futures
We conclude by discussing the arbitrage relation between spot VIX and newly issued variance futures, and propose a delta- and vega-neutral hedging strategy using futures.
π Tools & Libraries:
- Python
- LaTeX (for derivations and report)
π Techniques:
- Risk-Neutral Valuation
- Carr-Madan Option Replication
- VIX Derivation from Market Data
- Affine Transform and Laplace Techniques
- Calibration via Optimization (L-BFGS-B)
- Volatility Surface Approximation
- Static and Dynamic Hedging