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:
I think the formulae are wrong. They should be:
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.