What Is Kappa Der (∂V/∂κ)?
Last reviewed: by Options Analysis Suite Research.
Kappa Der (∂V/∂κ) is the first derivative of option value with respect to the Heston mean-reversion speed parameter (kappa). The Heston variance process follows dv = kappa(theta - v)dt + nu sqrt(v) dW; kappa controls how quickly variance reverts to its long-run level theta. Kappa Der is the structural sensitivity to mean-reversion speed and matters for vol-term-structure trading and Heston calibration analysis.
What Is Heston Kappa-Der?
Kappa Der quantifies how option value changes when the Heston mean-reversion speed shifts. A high kappa means variance reverts quickly to its long-run level (mean-reverting fast); a low kappa means variance is persistent (mean-reverting slowly). The same option will be priced differently under high-kappa vs low-kappa regimes because the implied path of variance over the option's life differs.
Two intuitions. First, Kappa Der is the term-structure-sensitivity Greek - high kappa flattens the implied volatility term structure (vol regimes converge fast), low kappa preserves a sloped term structure. Second, Kappa Der is the analytical signal of how Heston-calibrated prices depend on the mean-reversion assumption, which is rarely directly observable from market prices.
Why Does Kappa Der Matter?
Three contexts where Kappa Der is operationally relevant. First, Heston calibration stability. When calibrating Heston to market prices, the kappa parameter is often weakly identified - the data may admit a range of kappa values that produce similar smile fits. Kappa Der tells you how sensitive your calibrated price is to kappa uncertainty.
Second, vol term-structure trading. Trades that bet on term-structure flattening or steepening are essentially trades on kappa. Kappa Der is the analytical Greek that quantifies the position's sensitivity to kappa shifts.
Third, regime-detection analytics. Cross-time changes in fitted kappa indicate regime shifts in vol persistence. Aggregating Kappa Der across positions reveals how exposed a book is to kappa-driven regime changes.
How Heston Computes Kappa Der
Kappa Der is computed by central finite difference on the Heston pricer: bump kappa by 1% (multiplicatively), re-price, take the difference. Closed-form Kappa Der from differentiating the Heston characteristic function exists but is rarely used in production due to its complexity.
Related Greeks
Kappa Der is one of four core Heston-parameter Greeks. The siblings are Theta Param (∂V/∂θ), Vol of Vol (∂V/∂ξ), and Rho Der (∂V/∂ρ). Together they describe the full first-order parameter sensitivity structure of Heston pricing.
Related Concepts
Heston Model · Theta Param · Vol of Vol Greek · Rho Der · Term Structure · All 17 Greeks
References & Further Reading
- Heston, S. L. (1993). "A Closed-Form Solution for Options with Stochastic Volatility." Review of Financial Studies, 6(2), 327-343.
- Gatheral, J. (2006). The Volatility Surface. Wiley.
- Cui, Y., del Bano Rollin, S., and Germano, G. (2017). "Full and fast calibration of the Heston stochastic volatility model." European Journal of Operational Research, 263(2), 625-638.
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