Heging against stochastic interest rate

Question

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!

Answer 1

Question : Could you please precise the $d_{1}(t)$ and $d_{0}(t)$ that you use in your formula?

That is if you are using the same $d_{1}$ and $d_{2}$ as in classical B&S Theory, your formula is incorrect as it only holds with constant deterministic rates.

I believe there are modified versions of Black Scholes that take into account stochastic rates, but then the formulas are different from classic B&S and are model dependent.

$r_{t}$ in the Hull White model doesn’t really represent a real rate (it has no maturity), it is only used for modeling purpose and corresponds to an instantaneous borrowing/lending transaction starting and ending at date t and which also corresponds to the limit of the instantaneous forward rate $f(t, T) $ defined as $-\frac{\partial}{\partial T} logP(t, T) = f(t, T)\underset{T \rightarrow t}{\rightarrow} r_{t}$ where $P(t,T)$ is the $t$ price of a Zero Coupon maturing at $T$, hence using $r_{t}$ as a LIBOR rate would be pretty inaccurate.

Concerning Hull White calibration, it is usually done by using swaptions prices and the Jamshidian trick (cf Hull White Calibration and Jamshidian Swaption Formula Fine Tuned), I guess you could always regress your LIBOR fixings and consider that these correspond to $r_{t}$’s, but then again this would give you Zero Coupon prices that are very far from market prices.

Finally, regarding the $\rho$ hedging in an equity world, (although i'm no expert), i think it is done using swaps, so you only need to have a proper discount curve and a forward curve in order to price your swaps (cf Bianchetti & Ametrano 2013) (eventually if you are dealing with callable products you'll need to price callable swaps, which can be seen as a combination of a swap and a bermuda swaption, and in this case you'll need a model to price your berms).

Regards

Answer 2

you might want to take a look at "The Pricing of Risky Debt when Interest Rates are Stochastic" by David C. Shimko, Naohiko Tejima and Donald R. van Deventer from 1993. They use a BS model with stochastic interest rates that you may find inspiration in.