| Abstract: |
| Before the 2007-08 financial crisis, the term structure of forward rates of different tenors are linked by the discount curve. As a result, interest-rate modeling could be carried out with the forward-rate curve of a particular tenor, say, the three-month tenor. Such a linkage, however, broke down during the financial crisis. Nowadays, for pricing purposes, the term structure of forward rates of different tenors are modeled separately, which is coined the multi-curve modeling and has become the new norm of LIBOR derivatives modeling. The majority of multi-curve modeling approaches, however, are at odd with the stylized pattern of basis swap curves: smooth and monotonically decreasing in terms (or maturities), which cannot be retained if forward rates of different tenors were driven by different random factors in any usual way. The multi-curve modeling has served to legitimize, undesirably, sector segregation in pricing and hedging. In this article, we decompose a LIBOR rate into an OIS forward rate and an ``discrete loss rate, which represent the risk-free component and the default-risk component, respectively, and model them simultaneously using some popular dynamics for interest rates. In particular, we adopt the lognormal and CEV dynamics with stochastic volatility and establish the dual-curve versions of the LIBOR market model and the SABR model, respectively. Closed-form pricing formulae are developed for caplets and swaptions under the dual-curve SABR model, along the approach of heat kernel expansion. |
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