The term structure of implied correlations between S&amp;P and VIX markets

Laura Ballotta; Ernst Eberlein; Grégory Rayée

doi:10.3934/fmf.2025005

Article Contents

June 2025, Volume 5, 73-93. Doi: 10.3934/fmf.2025005 open access

This volume Previous Article Portfolio optimization in a multivariate jump-diffusion model Next Article Risk premium and rough volatility

The term structure of implied correlations between S&P and VIX markets

1.
Faculty of Finance, Bayes Business School (formerly Cass), City St George's, University of London, UK
2.
Department of Mathematical Stochastics, University of Freiburg, Germany
3.
ALM Analytics, Belfius Banque & Assurances, Brussels, Belgium

^*Corresponding author: Laura Ballotta
^*Corresponding author: Laura Ballotta

Received: January 2025

Revised: February 2025

Published online: March 20, 2025

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Abstract

We develop a joint model for the S&P500 and the VIX indices with the aim of extracting forward looking information on the correlation between the two markets. We achieve this by building the model on time changed Lévy processes, deriving closed analytical expressions for relevant quantities directly from the joint characteristic function, and exploiting the market quotes of options on both indices. We perform a piecewise joint calibration to the option prices to ensure the highest level of precision within the limits of the availability of quotes in the dataset and their liquidity. Using the calibrated parameters, we are able to quantify the leverage effect along the term structure of the VIX options and corresponding VIX futures. We illustrate the model using market data on S&P500 options and both futures and options on the VIX.

Keywords:

Mathematics Subject Classification: 91G15, 91G20, 91G70, 60E10, 60G51.

Citation:

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Figure 1. Correlation between $ X(t) $ and $ v(t) $: sensitivity analysis with respect to the 'loading' coefficients $ \sigma_{1} $, $ \eta_{1} $ and $ \eta_{2} $. Correlation calculated using equations (20)–(22) for a specific choice of the remaining parameters

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Figure 2. A sample of market data. Top-panels: implied volatilities from the S&P500 market (left-hand side panel), and the VIX market (right-hand side panel). Bottom panel: VIX futures prices for maturities corresponding to the ones of VIX options. Source: CBOE. Observation date: May $ 3^{rd} $ 2023

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Figure 3. Joint calibration with piecewise approach. Left-hand-side panel: implied volatility of S&P500 options expiring at $ \tau_{j}^{SPX} = \tau_{j}^{VIX} $ (up to 2 days). Centre panel: VIX futures price (vertical line) and implied volatility of VIX options expiring at $ \tau_{j}^{VIX} $. Right-hand-side panel: implied volatility of S&P500 options expiring at $ \tau_{j}^{VIX}+\Delta_{\tau} $

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Figure 4. Term structure of implied correlations $ \mathbb{C}orr(X(\tau), v(\tau)) = \mathbb{C}orr(X(\tau), V(\tau, \tau+\Delta_{\tau})^2) $ obtained from equations (20)-(22) and the calibrated parameters

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Table 1. Joint calibration with piecewise approach. Performance measures as defined in eqs. (23)–(25)

Maturity interval	[7, 37]	[14, 44]	[21, 51]	[28, 58]	[49, 79]	[77, 107]
$ \epsilon_1 $	5.90%	2.54%	3.65%	3.61%	4.07%	2.64%
$ \epsilon_2 $	2.07%	1.09%	2.49%	1.43%	2.57%	2.95%
$ \epsilon_F $	6.65E-03	1.48E-04	1.71E-03	1.43E-03	0.35%	3.93E-03
$ \epsilon $	5.42%	2.30%	3.43%	3.39%	3.76%	2.66%

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