How can we write swap as a chain of FRA's

For the rest of my question I use the notation from Brigo. The discounted payoff of a receiver interest rate swap (RFS) at $t<T_{\alpha}$, where $T_{\alpha}$ is the first resetting date, is given by

$$(RFS)=\sum^\beta_{i=\alpha +1 }D(t,T_i)N\tau_i(K-L(T_{i-1},T_i))$$

where

• $D(t,T_i)$ the discount factor at $t$ of time $T_i$
• N some notional
• $\tau_i$, general daycount convention for between $T_{i-1}$ and $T_i$
• $L(T_{i-1},T_i)$ market rate between $T_{i-1}$ and $T_i$
• strike rate $K$

My first question: Brigo says one can write the above value as a chain of foward rate agreements:

$$(RFS)= \sum^\beta_{i=\alpha +1 }NP(t,T_i)\tau_i(K-F(t;T_{i-1},T_i))$$

However I do not see how this is true by using

$$P(t,T)(1+\tau(t,T)L(t,T))=1$$ and $$F(t;T,S):=\frac{1}{\tau(T,S}\left(\frac{P(t,T}{P(t,S)}-1\right)$$

• The first equation you want to use is needed only to replicate the FRA payoff and compute the FRA rate. About the equivalence between a swap and many FRAs, just substitute the equation for F in the RFS equation: everything will be equal except for the discount factor. However, the discount factor and the zero-coupon bond price have been the same thing in a world free of default risk. – Arrigo Jan 12 '15 at 19:54

t         τ----T


A FRA from $\tau$ to $T$ pays the difference between the fixed rate and the actual fixing (Libor), discounted from $T$ back to $\tau$ at the Libor rate. This is from when that was a good measure of the risk free rate, with the idea that you would receive this and invest at Libor from $\tau$ to $T$. Thus the cash flow at $\tau$ is:

$$C(\tau) = f(\tau,T).[K-L(\tau)].D(L(\tau),\tau,T)$$

Where $f(\tau,T)$ is the year fraction of the FRA period, $K$ is the FRA fixed rate, $L(\tau)$ is the appropriate fixing for $\tau$.

A swap's float leg involves some very similar payments, but the main difference is that they pay just the difference between the rates at the end of the roll rather than the beginning:

$$C'(T) = f(\tau,T).[K-L(\tau)]$$

Apart from that detail, the exposure from a FRA is equivalent to the exposure from one roll of a floating note, i.e. to $L(\tau)$. So since your swap consists of a series of Libor fixed rolls, you can reconstruct the float leg value from the FRAs.

In reality, this is a bit more fraught these days - FRAs are not commonly traded for terms longer than 2y, and in this part of the curve the dominant instrument is 3m IR Futures (for the major currencies). So usually you would use Futures rather than FRAs to hedge a swap.

But 2015! Clearing rules are fey: for USD, Futures clear on CME but most swaps clear on LCH.Clearnet, so hedging your swap with futures leaves you with a clearing spread equivalent to a float/float between the two venues. For EUR and GBP the fork is Eurex and LCH. I suspect most of the spread there is down to the differences in the curves rather than realistic expectations of either defaulting; there is a valuation risk which could leave the hedger forked between margin calls on both. Welcome to the money market.