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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?

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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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    $\begingroup$ Are the real Vladimir Piterbarg? :) $\endgroup$ Nov 11 '20 at 17:12
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    $\begingroup$ It would seem so, judging from the quality of the answers... $\endgroup$
    – noob2
    Nov 11 '20 at 20:30

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