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

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.

💡 Research Summary

The paper proposes a novel extension of the Double Heston framework that introduces stochastic spot‑volatility correlation while preserving the affine structure that enables efficient pricing of European options via characteristic functions. In the standard Heston model the spot‑volatility correlation ρ is a constant, but market evidence—especially in foreign‑exchange (FX) markets—shows that this correlation varies over time and is positively correlated with spot moves. To capture this, the authors split the instantaneous variance vₜ into two CIR‑type sub‑variances vₜ⁺ and vₜ⁻. Both sub‑variances share the same mean‑reversion speed β and volatility‑of‑variance α, but they are driven by Brownian motions that have distinct correlations with the spot Brownian motion: ρ̄ + η for v⁺ and ρ̄ − η for v⁻. The parameter η therefore controls the width of the admissible correlation band