Keeping Up with the Correlations: Stochastic Spot/Volatility Correlation and Exotic Pricing

TL;DR

This paper extends the Heston framework by introducing stochastic, mean-reverting spot/volatility correlation within a three-factor affine structure (two variance components with shared $\beta$ and $\alpha$ but different correlations). It derives closed-form characteristic functions for European options, enabling fast pricing, and quantifies how stochastic correlation amplifies the implied volatility smile and affects exotic derivatives in FX markets, particularly barrier options, one-touch options, knockout options, and volatility swaps. The authors connect the model to empirical risk-reversal dynamics through the risk reversal beta, propose an empirical method to estimate $\eta$, calibrate the model on a single expiration tenor, and show that price impacts from stochastic correlation are often comparable to or larger than market bid/ask spreads. The results imply market-makers using the standard Heston framework may underprice several exotics, motivating calibration to stochastic correlation dynamics and future work on term-structure extensions and broader delta coverage for vol surfaces.

Abstract

We consider a novel use case for the Double Heston model (Christoffersen et al,, 2009), where the two Heston sub-variances have different spot/volatility correlations but the same volatility of volatility and mean reversion speed. This parameterization generalizes the traditional Heston stochastic volatility model (Heston, 1993) to include stochastic spot/volatility correlation. It is an affine model, allowing European options to be priced efficiently by numerically integrating over a closed-form characteristic function. This model incorporates a key dynamic relevant for pricing barrier derivatives in the foreign exchange markets: a positive correlation between moves in implied volatility skew and moves in the spot price. We analyze that correlation and its impact on both barrier option pricing and volatility swap pricing. Those price impacts are comparable to or larger than the bid/ask spreads for these products. Adding stochastic spot/volatility correlation increases the prices of out-of-the-money knockout options and one touch options, assuming that the model is calibrated to market vanilla option prices. It also increases the fair strike of volatility swaps compared to the Heston model.

Figures (12)

• Figure 1: Implied Volatility Impact of Adding Stochastic Spot/Volatility Correlation
• Figure 2: Calibrated Parameters vs Volatility of Correlation
• Figure 3: EURUSD 3m-Expiration 25-Delta Risk Reversals and EURUSD Spot
• Figure 4: USDJPY 3m-Expiration 25-Delta Risk Reversals and USDJPY Spot
• Figure 5: EURUSD 3m-Expiration Risk Reversal Beta