Joint Implied Willow Tree: An Approach for Joint S&P 500/VIX Calibration

Abstract

Since the inception of Volatility Index (VIX) options trading, academic literature has persistently sought accurate methods for jointly calibrating the prices of the S&P 500 index (SPX) and VIX options. This study introduces a novel nonparametric approach, called the joint implied willow tree (JIWT) method, aimed at resolving this joint calibration challenge. The resulting willow tree adheres to the martingale constraint for the SPX and ensures that the VIX is derived as the implied volatility of a 30-day log contract on the SPX. A notable advantage of our method is its ability to recover not only the unconditional probabilities for a fixed maturity but also the conditional probabilities across different maturities. Consequently, we reconstruct the entire term structure of the SPX, aligning it with market information from both SPX and VIX options. Numerical and empirical analyses demonstrate that the JIWT method excels in accurately capturing the volatility smile of SPX and VIX across various maturities.