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In Hull's book (9th edition), on page 420, in table 19.6, it says rho of a European call on an asset with yield $q$ is $$KTe^{-rT}N(d_2)$$ Below it says we can compute greeks of European options on futures by setting $q=r$. But then it says the rho for a call futures option is $-cT$. I am a bit confused here. The price of a futures call option is $$c=e^{-rT}(F_0N(d_1)-KN(d_2))$$ where the futures price $F_0=e^{rT}S_0$ when there is no dividend and interest rate is constant. Then wouldn't the rho of this option be $$\frac{\partial c}{\partial r}=KTe^{-rT}N(d_2)$$ rather than $$-cT=KTe^{-rT}N(d_2)-TS_0e^{-rT}N(d_1)?$$ Are we not treating $F_0$ as a function of $r$? What am I missing here?

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  • $\begingroup$ I think you should treat the futures price as an exogeneous. The futures price is arrived at from supply and demand. Depending on your application, you could model the futures price as well, of course; for example when modelling bond futures options and such. $\endgroup$ – Kermittfrog Dec 16 '20 at 9:07
  • $\begingroup$ That makes sense, thanks! $\endgroup$ – Xiaohuolong Dec 16 '20 at 17:17
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Your option has exposure to interest rates for two different reasons:

  • The discounting of (expected) terminal payoff.
  • The forward (~cost of financing of the delta hedging).

Mathematically, if you note your call as a function of rates and of the forward (itself a function of rates) $C(r, F(r))$, you have by chain rule:

$$\frac{dC}{dr} = \frac{\partial C}{\partial r} + \frac{\partial C}{\partial F}\frac{\partial F}{\partial r}$$

The first RHS term $\frac{\partial C}{\partial r}$ comes from discounting the expected terminal payoff (i.e. the call present value). The second RHS term comes from financing the call's replicating strategy. To replicate the call, you must hold a certain quantity of forward (which is the delta of the call $\frac{\partial C}{\partial F}$. And that forward has a sensitivity to interest rates $\frac{\partial F}{\partial r}$ which is the cost of funding the underlying asset.

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