# Abstracts

**An HJM approach for multiple yield curves**

*Christa Cuchiero (Vienna University of Technology, Austria)*

Tuesday June 3, 14:00-14:30 | session 2.4 | Interest Rates | room K

We consider modeling of multiple yield curves, which emerged in the course of the financial crisis due to credit and liquidity risk of the interbank sector. More precisely, we provide an HJM approach using a general semimartingale setup to model the term structure of multiplicative spreads between forward prices of OIS bonds and forward prices of risky bonds derived from Libor rates. We specify an HJM drift and consistency condition ensuring absence of arbitrage. In addition we show how to construct models such that the multiplicative spreads are greater than 1 and ordered for different tenors. We also provide an analogy to foreign exchange rate modeling relating the considered spreads to forward exchange rates associated with OIS bonds and risky bonds. This HJM framework allows to unify and extend several approaches which have been proposed in literature in the context of multiple curve modeling. For instance, the Lévy driven HJM model studied by Crepey et al. (2012, 2013) corresponds to a particular specification in our setup. Beyond that also short rate models as for example considered by Kenyon (2010) or the multi-curve affine Libor model introduced by Grbac et al. (2013) can be covered. When the driving process of both the instantaneous forward rates obtained from OIS rates and the spreads are specified to be affine, we obtain a Markovian structure which is very flexible and allows at the same time for simple pricing formulas for Libor interest rate derivatives.

**Linear-Rational Term Structure Models**

*Damir Filipovic (EPFL and Swiss Finance Institute, Switzerland)*

Tuesday June 3, 14:30-15:00 | session 2.4 | Interest Rates | room K

We introduce the class of linear-rational term structure models, where the state price density is modeled such that bond prices become linear-rational functions of the current state. This class is highly tractable with several distinct advantages: i) ensures non-negative interest rates, ii) easily accommodates unspanned factors aff ecting volatility and risk premia, and iii) admits analytical solutions to swaptions. For comparison, affi ne term structure models can match either i) or ii), but not both simultaneously, and never iii). A parsimonious speci fication of the model with three term structure factors and one, or possibly two, unspanned factors has a very good fit to both interest rate swaps and swaptions since 1997. In particular, the model captures well the dynamics of the term structure and volatility during the recent period of near-zero interest rates.

**Default Times Not Avoiding Stopping Times - Defaultable Term Structure Modelling Beyond the Intensity Paradigm**

*Thorsten Schmidt (Chemnitz University of Technology, Germany)*

Tuesday June 3, 15:00-15:30 | session 2.4 | Interest Rates | room K

Credit risk is typically either treated with a structural viewpoint pioneered in Merton (1974) or the reduced-form framework as in Jarrow \& Turnbull (1995). The goal of this article is to provide a unified view on both approaches in a sufficiently generalized setup. The considered model allows for default times which have an intensity \emph{or} occur at a fixed time with positive probability, henceforth not avoiding stopping times. A forteriori, the considered default times do not fall in the class of so-called intensity-based models.

The setup is intimately related to the theory of enlargement of filtrations, to which we provide a new class of examples where the Az\`ema supermartingale contains jumps, both at predictable and totally inaccessible stopping times. In the second part of the paper dynamic term structures prone to default risk in the framework of Heath-Jarrow-Morton (1992). It turns out, that previously considered models in this framework lead to arbitrage possibilities. We propose a suitable, arbitrage-free generalization of this class with an additional stochastic integral containing atoms at predictable stopping time.

**Swaption Pricing and Hedging with Default Risk**

*Dirk Bangert (Denmarks Technical University, Denmark)*

Tuesday June 3, 15:30-16:00 | session 2.4 | Interest Rates | room K

The possibility of a counterparty to a financial derivatives contract failing to perform its obligations is a scenario of great practical importance to traders, risk managers and regulators. We show how credit risk can be included in the valuation of an interest rate swaption contract, and the effect on hedging. Our modelling approach combines market customary interest rate short rate models, and a reduced form default intensity model. Numerical results are generated using a modified finite difference implementation of a generalized Gaussian short rate model. The inclusion of credit risk in the modelling of swaptions changes value and hedging strategy significantly. The decision to exercise a swaption is becoming a function of the credit worthiness of the counterparties in addition to the interest rate market scenario. Settlement features of the swaption contract become much more prominent when credit risk is included. The effect of credit risk on physically settled swaptions is greater than on cash settled swaptions. Such physically settled swaptions are an important contract type and frequently traded between banks and their corporate customers.