Ebook Interest Rates and The Credit Crunch: New Formulas and Market Models
Before the credit crunch of 2007, the interest rates quoted in the market showed typical consistencies that we learned on books. We knew that a floating rate bond, where rates are set at the beginning of their application period and paid at the end, is always worth par at inception, irrespectively of the length of the underlying rate (as soon as the payment schedule is re-adjusted accordingly). For instance, Hull (2002) recites: “The floating-rate bond underlying the swap pays LIBOR. As a result, the value of this bond equals the swap principal.” We also knew that a forward rate agreement (FRA) could be replicated by going long a deposit and selling short another with maturities equal to the FRA’s maturity and reset time.
These consistencies between rates allowed the construction of a well-defined zero-coupon curve, typically using bootstrapping techniques in conjunction with interpolation methods. Differences between similar rates were present in the market, but generally regarded as negligible. For instance, deposit rates and OIS (EONIA) rates for the same maturity would chase each other, but keeping a safety distance (the basis) of a few basis points. Similarly, swap rates with the same maturity, but based on different lengths for the underlying floating rates, would be quoted at a non-zero (but again negligible) spread.
Then, August 2007 arrived, and our convictions became to weaver. The liquidity crisis widened the basis, so that market rates that were consistent with each other suddenly revealed a degree of incompatibility that worsened as time passed by. For instance, the forward rates implied by two consecutive deposits became different than the quoted FRA rates or the forward rates implied by OIS (EONIA) quotes. Remarkably, this divergence in values does not create arbitrage opportunities when credit or liquidity issues are taken into account. As an example, a swap rate based on semiannual payments of the six-month LIBOR rate can be different (and higher) than the same-maturity swap rate based on quarterly payments of the three-month LIBOR rate.
These stylized facts suggest that the consistent construction of a yield curve is possible only thanks to credit and liquidity theories justifying the simultaneous existence of different values for same-tenor market rates. Morini (2008) is, to our knowledge, the first to design a theoretical framework that motivates the divergence in value of such rates. To this end, he introduces a stochastic default probability and, assuming no liquidity risk and that the risk in the FRA contract exceeds that in the LIBOR rates, obtains patterns similar to the market’s. However, while waiting for a combined credit liquidity theory to be produced and become effective, practitioners seem to agree on an empirical approach, which is based on the construction of as many curves as possible rate lengths (e.g. 1m, 3m, 6m, 1y). Future cash flows are thus generated through the curves associated to the underlying rates and then discounted by another curve, which we term “discount curve”.
Assuming different curves for different rate lengths, however, immediately invalidates the classic pricing approaches, which were built on the cornerstone of a unique, and fully consistent, zero-coupon curve, used both in the generation of future cash flows and in the calculation of their present value. This paper shows how to generalize the main (interest rate) market models so as to account for the new market practice of using multiple curves for each single currency.
The valuation of interest rate derivatives under different curves for generating future rates and for discounting received little attention in the (non-credit related) financial literature, and mainly concerning the valuation of cross currency swaps, see Fruchard et al. (1995), Boenkost and Schmidt (2005) and Kijima et al. (2008). To our knowledge, Bianchetti (2008) is the first to apply the methodology to the single currency case. In this article, we start from the approach proposed by Kijima et al. (2008), and show how to extend accordingly the (single currency) LIBOR market model (LMM).
Our extended version of the LMM is based on the joint evolution of FRA rates, namely of the fixed rates that give zero value to the related forward rate agreements. In the single-curve case, an FRA rate can be defined by the expectation of the corresponding LIBOR rate under a given forward measure, see e.g. Brigo and Mercurio (2006). In our multi-curve setting, an analogous definition applies, but with the complication that the LIBOR rate and the forward measure belong, in general, to different curves. FRA rates thus become different objects than the LIBOR rates they originate from, and as such can be modeled with their own dynamics.
In fact, FRA rates are martingales under the associated forward measure for the discount curve, but modeling their joint evolution is not equivalent to defining their instantaneous covariation structure. In this article, we will start by considering the basic example of lognormal dynamics and then introduce general stochastic volatility processes. The dynamics of FRA rates under non-canonical measures will be shown to be similar to those in the classic LMM. The main difference is given by the drift rates that depend on the relevant forward rates for the discount curve, rather then the other FRA rates in the considered family.
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