Follow-up: Multi-curve Modeling

In [Mercurio 2009], two different approaches are proposed as attempts to resolve the new post-crisis market norm. (Recap: the issues include rate dependence on contract tenor, failure of textbook no-arb relationships between different rate securities)

The first approach focuses on the difference in counterparty risk for various contracts. For example, regarding the textbook case of replicating an FRA using a long and a short LIBOR deposits, we can introduce hazard rate and default time into the replication argument. That way, we end up with a forward LIBOR that is higher than the comparable FRA.

This approach, however, cannot be easily utilized in pricing, and more importantly the industry seems to favour 'segmenting market rates,' i.e. having multiple curves in existence simultaneously.

Digression: Bootstrapping
This is a description of how zero curve bootstrapping is done in general setting. (i.e. not limited to post-crisis multi-curve method)

A single kind of contract is unlikely to cover the entire range of maturities, hence different securities are required for different sections of the curve. What actually got to be chosen for curve construction depends on currency, but in general:

• The shortest end is built from cash/money market deposit. Since there is only a single bullet payment in a deposit transaction, the 'bootstrapping' is trivially done by day count (and perhaps compounding) adjustment.
• The middle section is built out of FRA or ED future contracts. Once again, since there is only one cash flow the bootstrapping is straight forward. Note however that if future contracts are used, a convexity adjustment is required so that the adjusted rate is comparable to that of a forward contract. (the convexity comes from the relative advantage of holding an interest rate future over forward, because MTM profit/loss can be reinvested/covered at a higher lower/rate)
• Finally, the far end of the curve is built out of swap. In this case we have to proceed in an order of ascending maturity because there are more than one payments in a swap. Suppose the rates are known up to (t-1). Then, with the market swap rate, S(t), and the discount factor up to (t-1), P(0,T_i), the discount factor and hence the rate for t can be found using the swap rate formula.

End of Digression

In the multi curve case, the only difference from the general procedure above is that we would only use one family of securities in building a curve of a certain tenor. (e.g. 3m deposit, 3m FRA with various maturities and 3m LIBOR fixing swaps with various number of payments to build the 3m tenor curve)

Finally, as an aside, note that curve interpolation is a tricky business, especially if you want to also calculate the instantaneous forward curve which is very sensitive to local curve fluctuations.