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.

Multi-curve Pricing: Short Notes

Issue:
After Mid 2007, a lot of the good-o textbook wisdom failed to hold, mainly in two aspects:

1. Some rates use to match one another almost exactly (i.e. zero spread), for example deposit rate vs overnight swap rate; now the spread is much larger.
2. Swaps with different settlement frequencies have very large rate spread, i.e. the size of the swap rate depends on the fixing.

Proposed remedies:

1. Use separate curves for discounting and forwarding
2. Treat LIBOR's with different fixings as independent underlyings

Questions:
Even in the good-o textbook context, shall we expect LIBOR's with different tenors to have zero spread? Or is it just that tenor used to be irrelevant in the old narrative?

Reference:
Henrard 2009, "Irony in Discounting: The Crisis"
Bianchetti 2009, "Two Curves, One Price"
Mercurio 2009, "Interest Rates and the Credit Crunch: New Formulas and Market Models"

Credit Flattener and Steepener: Duration and Convexity Exposures

Flattener/Steepener are long/short strategies that bet on the relative movements of long vs. short maturity ends on a curve. The curve can be interest rate, variance or credit spread. For the sake of discussion, we consider a credit spread flattener.

The three main exposures are TIME, CURVE SHIFT and DEFAULT. Here we focus on CURVE SHIFT. Surely, we can mitigate the risk by duration hedging (even so we will be left with convexity risk, see below). However, notional matching has the advantage of zero default exposure (at least until the shorter leg expires).

In a notional-matched flattener, the investor speculate on flattening credit curve by selling (buying) CDS protection at the long (short) end. If the curve steepens instead, we would of course have a loss. What if the curve tightens by a parallel shift?

The mark-to-market as the spread moves is calculated by multiplying spread change to risky DV01, which, to recap, is the CDS spread numeraire. Obviously, DV01 of longer leg > DV01 of shorter leg. Hence a parallel upward shift (widening) of the curve affects the long leg more. But widening spread is bad for protection seller. Therefore a flattener position is harmed by widening spread (because the benefit on the short end < the loss on the long end).

The previous paragraph assumes that DV01 remains constant when the shift is small enough, and it considers only the relative sizes of DV01's. Not surprisingly, DV01 itself does not stay constant as the curve shifts further and hence there is convexity effect, which concerns the relative sensitivities of DV01's. Turns out DV01 decreases as spread widens, and the DV01 of the long end is more sensitive to curve shift than the short end. It follows then, that a flattener has positive convexity (when spread widens => DV01 drops => bad effect at long end diminishes => good for investor) while steepener has negative convexity.

Reference: JP Morgan 'Credit Derivatives Handbook'