Theta and Rho Conventions for Interest Rate Derivatives
Summary
The document explains why interest rate traders often define rho differently from the Black–Scholes sensitivity to a single interest rate. For options on forwards priced with the Black formula, the rate does not appear in the option formula itself; discounting is a separate source of price sensitivity. In multi-curve settings, discounting risk is commonly measured separately from risk to forwarding curves, with aggregation depending on practice.
Theta depends on what market inputs are held fixed as time advances. A baseline convention holds the forward curve constant, which also keeps the Black forward rate fixed and isolates the option’s passage-of-time effect. Other measures, such as rolldown or carry, may instead hold the spot curve fixed. The document also cautions that the volatility surface or smile assumptions must be specified. It gives conceptual conventions rather than a calculation recipe, and does not prescribe a universal treatment for every instrument or desk.
Key ideas
- Interest rate rho is often treated as discounting risk rather than Black–Scholes sensitivity to a single rate.
- In multi-curve pricing, discounting and forwarding-curve sensitivities may be reported separately.
- A baseline theta convention advances time while holding the forward curve and Black forward rate constant.
- Alternative theta measures depend on which curves and volatility inputs are held fixed.
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Full text
# How to calculate theta/rho for interest rate derivatives? # How to calculate theta/rho for interest rate derivatives? For interest rate derivatives priced with the Black model, we calculate some sort of forward rate that can be inserted into the Black formula. Calculating the greeks of the Black formula is easy enough, but what I don't understand is how to calculate theta and rho. Both time and interest rates are used to determine the forward rate, so surely we also need to differentiate the forward-rate calculation itself (a highly non-standard calculation) with respect to time and derivatives and then use the chain-rule? And yet that does not seem to be the convention, what I am seeing is that people just calculate the standard rho with respect to the Black formula? ## Answer by piterbarg (score 8) https://quant.stackexchange.com/a/59308 Interest rate traders/quants do not really talk about rho, as in the sensitivity of the Black Scholes price to $r$. The reason, I guess, being that we use Black (not Black-Scholes) formula for options on forwards and there is no $r$ in that formula except in the discounting factor multiplying the undiscounted option value. Rho in interest rate models is then closely associated with the so-called "discounting risk" which is the sensitivity of prices to the curve you use for discounting. In the multi-curve world we live in it is typically calculated separately from risk to other curves (like Libor) and may, or may not, be aggregated with other sensitivities to the same curve. Theta is a whole different story and it can get very complicated. A standard base-level definition of Theta in interest rates models keeps the forward curve constant when moving time forward. Almost by definition that means that you should keep the forward rate that goes into the Black formula constant when calculating Theta, so the time change only affects time value of the option. Traders do want to look at "other" thetas (and often give them other names such as rolldown, carry etc) such as when the spot curve is kept constant, etc. also it needs to be defined exactly which volatilities or, indeed, volatility smiles should be kept constant when calculating theta.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.