Rho

The only parameter of the BS formula that we did not consider so far is the interest rate. In a world where the interest rates change in a stochastic way, one can manage the exposure towards interest rate changes in the bond markets, without dealing in the derivative markets. Mainly for completeness, we define the rho of a portfolio as the sensitivity of its value with respect to interest rate changes. Formally

$$\varrho _{\Pi }=\frac{\partial \Pi }{\partial r}$$.

Achieving rho neutrality using traded derivatives is done in a fashion similar to the one that achieves gamma and/or vega neutrality. For a European call the value of rho is given by

$$\varrho _{C}=KTe^{-rT}\mathcal{N}\left( d_{2}\right)$$   ,

which is illustrated in figure 8.6

Figure 8.6: Behavior of a Call option Rho. Part (a) gives the behavior of the rho on a three options with specifications $(K,\tau,r,\sigma)$ equal to $(1,0.2,0,0.5)$ for the solid line, $(1,0.2,0,1)$ for the thin line, and $(1,0.5,0,0.5)$ for the dashed line. Part (b) gives the behavior of the rho for a contract which is in-the-money [solid line], at-the-money [thin line], and out-of-the-money [dashed line].