meaning of discount term in FRA value

Question

Consider a forward rate agreement on LIBOR (say), which starts 2 months from now, expires after 3 months and has strike $K$, and is based on $3M$ LIBOR -- $FRA_{2\times 5}$. Now the present value of this contract is,

\begin{equation} \frac{\alpha\cdot (L_{3m}(2m) - K)}{1 + \alpha\cdot L_{3m}(2m)} \end{equation}

where $\alpha$ is the relevant day count fraction, $L_{3m}(2m)$ refers to the 3 month LIBOR rate in 2 months time (assume unit notional for ease).

I see the term $1 + \alpha\cdot L_{3m}(2m)$ in the above equation as a discount rate applied to the pay out $L_{3m}(2m) - K$.

Why do we apply the $L_{3m}(2m)$ rate for the discounting? Why not apply a short term money market rate or any other rate for that matter -- I guess I am also asking how we choose the rate that we apply for the discounting?

Answer 1

A very good and up-to-date question.

Whether you use the LIBOR-rate or any other rate for discounting depends on what you decide to be the fundamental rates in the market.

Before the crisis LIBOR-rates were mostly seen as the fundamental market rates (or the "risk-neutral" rates). After the crisis it turned out that these rates were not completely free of default risk. This is why noawadys an increasing number of banks has started to use the OIS-curve as the base curve.

In the paper LIBOR vs. OIS: The Derivatives Discounting Dilemma Hull and White present a solid case in favor of OIS discounting.

For the concrete application to forward-contracts this entails that one might end up using a different rate than LIBOR for discounting - e.g. ones base on overnight indexed swaps (OIS). Thus you will have to work with two curves instead of one (LIBOR + the curve you use for discounting)

As always Fabio Mercurio already did most of the work again. You will find an approah on how to deal with forward-contracts in a multi-curve setting in his paper: LIBOR Market Models with Stochastic Basis

An easy rule of thumb when to use LIBOR and when the e.g. OIS-curve: When the transactions involved in the product are collaterized $\to$ use OIS or an equivalent. For non-collaterized transaction discounting with LIBOR can be used.