I am working on an Index and I am trying to price Call options on it. I work with the 3 Months LIBOR as Cash.
I use the following Black-Scholes formula $$C_{t} = S_{t}e^{-q_{t}(T-t)}\mbox{N}[d_{1}(t)] - K e^{-r_{t}(T-t)}\mbox{N}[d_{0}(t)] $$ with the usual notations. $r_{t}$ is the LIBOR rate and $q_{t}$ is the dividend of my Index. I cannot use the classical formula $R(t,T) = \frac{1}{T-t} \int_{t}^{T} r_{s} ds$ since the rate are not deterministic.
I implemented a classical delta-hedging. My delta-hedging works well, except for too deep In the money call options, where the P&L behaves exactly like the interest rate.
This happens only for Call with maturities of several years and strikes far in the money. The conclusion of my manager is that I have to hedge against the stochasticity of the interest rate, using a zero-coupon bond.
I looked at some documentations on zero-coupon, and saw that we can have a closed formula under the Hull-White model. However, I am not so sure how to calibrate it as my only inputs are the LIBOR rates.
Besides, I do not know which amount I should invest in my zero-coupon bond : should I invest the $\rho$ ( $\frac{\partial C_{t}}{\partial r_{t}}$) ?
I do not really know where to start so any help would be appreciated :)
Thank you!