What Is RhoR (Heston)?
Last reviewed: by Options Analysis Suite Research.
RhoR is the first derivative of option value with respect to the risk-free interest rate, computed under the Heston stochastic-volatility model rather than Black-Scholes. It is the Heston-context analog of standard rho. The two coincide for short-tenor ATM options and diverge for long-tenor or deep-OTM options where the Heston vol dynamics meaningfully affect the rate-discounted payoff distribution.
What Is Heston RhoR?
RhoR captures the rate-sensitivity of an option priced under Heston. Mathematically it is partial VHeston / partial r, computed by differentiating the Heston Fourier pricing formula with respect to the rate input. The structural interpretation is identical to standard rho - rate moves affect the present-value of the strike payment - but the magnitude differs because Heston's stochastic-vol dynamics produce a slightly different payoff distribution than BS, which in turn changes how rate moves flow through to value.
For ATM short-tenor options, RhoR and standard rho coincide to within fractions of a cent because the Heston payoff distribution closely matches the BS log-normal at the money. For deep-OTM long-tenor options, the gap can be 5-15% because Heston's fatter tails (driven by vol-of-vol and spot-vol correlation) materially shift the rate-discounting effect on payoff probability.
How Heston Computes RhoR
Heston pricing uses the Carr-Madan Fourier-inversion approach: option price equals an integral of the Heston characteristic function. Differentiating with respect to r gives RhoR by interchanging the derivative and the integral. The resulting formula has the same structure as Heston pricing but with an extra factor reflecting the rate's role in both the discount factor and the risk-neutral drift.
In production, RhoR is typically computed by central finite difference on the Heston pricer: bump r by 1bp, re-price, take the difference. This is robust and matches closed-form to several decimal places for typical Heston parameter values.
Why Does RhoR Matter?
For desks running Heston-priced books (institutional vol arbitrage, structured-product market makers), RhoR is the operational rate-Greek used in hedging. The reason for using RhoR rather than BS rho is internal consistency: a book priced under Heston should be hedged using Heston Greeks. Mixing BS rho with Heston pricing produces hedge errors that compound over time.
For long-dated structured products (warrants, convertibles, multi-year structured notes), RhoR can differ from BS rho by hundreds of dollars per contract on large-size positions. Risk teams running multi-year structured-product books explicitly track RhoR-vs-rho dispersion as a calibration-quality metric.
Related Greeks
RhoR pairs with RhoQ (Heston dividend-yield sensitivity) and the standard rate-and-yield Greeks rho, epsilon, and phi. Together they describe the carry-and-discount sensitivities under Heston pricing.
Related Concepts
Rho · RhoQ · Vol of Vol Greek · Heston Model · 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.
- Carr, P. and Madan, D. (1999). "Option Valuation Using the Fast Fourier Transform." Journal of Computational Finance, 2(4), 61-73.
- Gatheral, J. (2006). The Volatility Surface. Wiley.
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