What Is Theta Param (∂V/∂θ)?
Last reviewed: by Options Analysis Suite Research.
Theta Param (∂V/∂θ) is the first derivative of option value with respect to the Heston long-run variance parameter (theta). Important: this is structurally distinct from the time-decay Greek theta - same Greek letter, completely different meaning. Heston's theta is the variance level toward which the variance process mean-reverts under the dynamics dv = kappa(theta - v)dt + nu sqrt(v) dW.
What Is Heston Theta-Param?
Theta Param quantifies how option value changes when the Heston long-run variance level shifts. A higher long-run variance means the variance process mean-reverts to a higher target, raising expected vol over the option's life and increasing option value. Theta Param is positive for long options (calls and puts gain value when long-run vol rises).
Two intuitions. First, Theta Param is the long-tenor vega - it tells you sensitivity to the structural vol regime, distinct from sensitivity to current vol level (which is captured by the v0-derivative, essentially Heston vega). Second, Theta Param is the parameter most closely linked to the long-end of the IV term structure: long-dated ATM IV converges to sqrt(theta) in the Heston framework, so changes in theta map directly to long-tenor IV changes.
Why Does Theta Param Matter?
Three operational contexts. First, long-tenor vega risk. A book holding 1-year+ options is materially exposed to long-run-variance changes; Theta Param is the appropriate Greek for that exposure rather than the standard short-tenor vega.
Second, vol term-structure trading. Trades that express views on the long-end of the vol curve (calendar spreads, term-structure flatteners/steepeners) have aggregate Theta Param exposure that tells you the long-end-vol P&L direction.
Third, Heston calibration. Theta and v0 together pin the variance process; their separate identification from market prices depends on the time-series of vol levels. Theta Param tells you how sensitive a given calibration is to the long-run-variance estimate.
Naming Disambiguation
The Heston theta parameter (long-run variance level) shares its symbol with the Black-Scholes theta Greek (time decay of value). They are conceptually unrelated. The -param suffix in this slug exists specifically to disambiguate. Trading desks that work in Heston use phrases like "Heston theta", "long-run variance", or "θ-mean" to avoid confusion with time-decay theta.
How Heston Computes Theta Param
Theta Param is computed by central finite difference on the Heston pricer: bump theta by a small amount (e.g., from 0.04 to 0.0405 for 1% relative bump), re-price, take the difference.
Related Greeks
Theta Param is one of four Heston-parameter Greeks. Siblings: Kappa Der, Vol of Vol Greek, Rho Der.
Related Concepts
Theta (time decay) · Heston Model · Kappa Der · Vol of Vol Greek · 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.
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This page is part of the 17 Greeks reference covering every options Greek with formula, intuition, worked example, and how each pricing model computes it. Browse the full documentation.