What should be the sign of greek letter $\rho$?

Question

I'm reading the book Risk Management and Shareholders Value in Banking by Resti & Sironi. I quote a paragraph from the book (Chapter 5, appendix):

The derivative of an option’s value with respect to the interest rate is the rho ($\rho$) coefficient. From a logical point of view, the relation between these two quantities is dual. On the one hand, a higher interest rate decreases the present value of the option’s expected final value (payoff at maturity): this means that a higher interest rate has a negative effect on the value of any option. On the other hand, a higher interest rate implies a higher forward price of the underlying asset, so a higher value for call options and a lower value for put options. The two effects therefore act in the same direction in the case of put options (making the total effect negative), while they act in a conflicting way in the case of call options, for which, in any case, the first effect usually predominates. So, in brief:
$$\rho_{c}=XTe^{-iT}N(d_2)<0$$ $$\rho_{p}=XTe^{-iT}N(d_2)<0$$.

I think the formulae are wrong. They should be:

$$\rho_{c}=XTe^{-iT}N(d_2)>0$$

$$\rho_{p}=-XTe^{-iT}N(-d_2)<0$$

Furthermore, even the explanation they give for the sign of $\rho_c$ is wrong: they say the negative effect predominates, which is not true.

I would like to know if the formulae and the signs are correct, and I would also like to have more detail on the statement "a higher interest rate implies a higher forward price of the underlying asset". I don't understand how this comes into play.

Answer 1

Yes, you are right. It appears to be a trivial typographical error in the book. I checked the formulas on Wikipedia https://en.wikipedia.org/wiki/Greeks_%28finance%29 and they agree with yours. The signs are obvious also since N(.) is between 0 and 1, i.e. non-negative.

Now, about the reasoning starting with "from a logical point of view". Are you familiar with the binomial model? Imagine running it twice, once with a low interest rate, once with a high one (all other things the same). In the model the stock price grows at the rate r. So in the high interest rate case $S_T$ (what he calls the forward price of the stock) will have a higher expected value, which will increase $E(S_T-K)^+$ and decrease $E(K-S_T)^+$. But then we have to discount these two values to the present to find the call value and the put value, and with high interest rate that lowers both values. So for calls the total effect seems ambiguous, for puts there is an overall decrease in value (negative $\rho_P$).