Hull's book - Futures option's rho

Question

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?

Answer 1

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.