# Abstracts

**Social Discounting and the Long Rate of Interest**

*Lane Hughston (Brunel University London, UK)*

Wednesday June 4, 16:30-17:00 | session 6.4 | Interest Rates | room K

The well-known theorem of Dybvig, Ingersoll and Ross shows that the long zero-coupon rate can never fall. This result, whichâ€”although undoubtedly correctâ€”has been regarded by many as counterintuitive and even pathological, stems from the implicit assumption that the long-term discount function has an exponential tail. We revisit the problem in the setting of modern interest rate theory, and show that if the long simple interest rate (or Libor rate) is finite, then this rate (unlike the zero-coupon rate) acts viably as a state variable, the value of which can fluctuate randomly in line with other economic indicators. New interest rate models are constructed, under this hypothesis, that illustrate explicitly the good asymptotic behaviour of the resulting discount bond system. The conditions necessary for the existence of such hyperbolic long rates turn out to be those of so-called social discounting, which allow for long-term cash flows to be treated as broadly just as important as those of the short or medium term. As a consequence, we are able to provide a consistent arbitrage-free valuation framework for the cost-benefit analysis and risk management of long-term social projects, such as those associated with sustainable energy, resource conservation, and climate change. The model is also applicable to long-term issues arising in connection with (a) insurance claims reserving and (b) the financing of pension funds.

**Multi-curve framework with collateral**

*Marc Henrard (OpenGamma, UK)*

Wednesday June 4, 17:00-17:30 | session 6.4 | Interest Rates | room K

The multi-curve framework has become the standard interest rate derivative pricing framework; collateral discounting is becoming a standard in presence of collateral agreements. We described a generalised collateral framework that covers cash collateral, foreign currency collateral, collateral by assets, collateral with haircut and their combinations. The formulas obtained is similar to the standard discounting with collateral cash account formulas but with a generalised meaning of rate. The new rate includes repo rate of collateral assets, haircut factors and interest actually paid. Generalised definitions of pseudo-discount factors are introduced.

The collateral framework is applied to interest rate derivatives, leading to a unified multi-curve and collateral framework. The pricing of standard interest rate instruments based on Ibor-like indexes are presented in the framework. The exact details under which the framework can be applied are analysed, in particular we explicit the exact meaning of ``OIS discounting''. A generic approach to curve calibration in the framework is proposed. The market instruments containing informations about the theoretical quantities like pseudo-discount factors and collateral index forward rates are detailed.

The collateral forward rates depend of the collateral agreement. In practice it is important to be able to compute the forward rate for different collateral agreements from the forward rates associated to the standard collateral rule. Ways to achieve those conversions using market instruments are proposed.

In the last part, HJM-like dynamic of the collateral curves and the main results of the model are described. In that model the pricing of STIR futures under stochastic spread between the collateral and the forward curves is presented.

Based on the HJM dynamic, we also present convexity adjustments for forward rates resulting from change of collateral. Those theoretical results are important to estimate the magnitude of the change when the market is not providing enough information to estimate them directly.

**Polynomial preserving diffusions and models of the term structure**

*Martin Larsson (Ecole Polytechnique Federal de Lausanne, Switzerland)*

Wednesday June 4, 17:30-18:00 | session 6.4 | Interest Rates | room K

Polynomial preserving processes are multivariate Markov processes that extend the important class of affine processes. They are defined by the property that the semigroup leaves the space of polynomials of degree at most $n$ invariant, for each $n$, which lends significant tractability to models based on these processes. In this talk I will discuss existence and uniqueness of polynomial preserving diffusions, a task which is made nontrivial due to degenerate and non-Lipschitz diffusion coefficients, as well as a complicated geometric structure of the state space. I will also describe how polynomial preserving processes can be used to build term structure models that accommodate three features that are otherwise difficult to combine: nonnegative short rates, tractable swaption pricing, and unspanned factors affecting volatility and risk premia.