# Abstracts

**Heat Kernel Framework for Asset Pricing in Finite Time**

*Andrea Macrina (University College London, UK)*

Wednesday June 4, 11:30-12:00 | session 4.4 | Interest Rates | room K

A heat kernel approach is proposed for the development of a flexible and mathematically tractable asset pricing framework in finite time. The pricing kernel, giving rise to the price system in an incomplete market, is modelled by weighted heat kernels which are driven by multivariate Markov processes and which provide enough degrees of freedom in order to calibrate to relevant data, e. g. to the term structure of bond prices. It is shown how, for a class of models, the prices of bonds, caplets, and swaptions can be computed in closed form. The dynamical equations for the price processes are derived, and explicit formulae are obtained for the short rate of interest, the risk premium, and for the stochastic volatility of prices. Several of the closed-form asset price models presented in this paper are driven by combinations of Markovian jump processes with different probability laws. Such models provide a rich basis for consistent applications in several sectors of a financial market including equity, fixed-income, commodities, and insurance. The flexible, multidimensional and multivariate structure, on which the asset price models are constructed, lends itself well to the transparent modelling of dependence across asset classes. As an illustration, the impact on prices by spiralling debt, a typical feature of a financial crisis, is modelled explicitly, and contagion effects are readily observed in the dynamics of asset returns. (Link to paper: ssrn.com/abstract=2254771)

**Extrapolating the term structure of interest rates with parameter uncertainty**

*Anne Balter (Maastricht University, The Netherlands)*

Wednesday June 4, 12:00-12:30 | session 4.4 | Interest Rates | room K

Market consistent pricing of very long-dated liabilities is difficult due to the limited liquidity or absence of very long-dated market instruments. Life-insurance or pension fund liabilities can be as long as 100 years, whereas the available liquid instruments in the market have maturities of 20 years or less.

The purpose of the paper is to extrapolate the yield curve. Taking data from time series of observed yield curves up to 20 years, what are model implied yields for longer maturities? We use a standard Gaussian essentially affine model that we estimate by Bayesian methods. Our data consists of Euro swap rates from 2002 to September 2013. From the full posterior density of the model parameters we obtain the predictive density for all maturities conditional on the initial maturities up to 20 years.

With an uninformative prior we find that longer term yields beyond the 20 years point very quickly become highly uncertain. The posterior mean for such yields can easily be negative with a very large credible interval. The problem is due to the very low estimate of the mean reversion parameter in the risk neutral density. Credible intervals become smaller and more sensible when we use an informative prior based on a zero lower bound for nominal interest rates. Since the mean reversion parameter of the level factor remains very small under this prior, convergence of the yield curve to a constant ultimate forward rate is very slow and does not occur before maturities of 100 years.

Our extrapolation method uses a financial no-arbitrage model. When we compare our method to curve fitting methods such as Smith-Wilson extrapolation and Nelson-Siegel calibration, we find large differences. Since our Bayesian procedure also estimates the uncertainty around the mean estimate of the extrapolation, the alternatives are, however, mostly within our estimated credibility regions.

**Ramsey rules and yield curves dynamics**

*Caroline Hillairet (Ecole Polytechnique, France)*

Wednesday June 4, 12:30-13:00 | session 4.4 | Interest Rates | room K

For financing of ecological projects reducing global warming, for longevity issues or any other investment with a long term impact, it is necessary to model accurately long run interest rates. The answer cannot find in financial market, since for longer maturities, the bond market becomes highly illiquid and standard financial interest rates models cannot be easily extended. In general, these issues are addressed at macroeconomic level, where long-run interest rates has not necessary the same meaning than in financial market.They are called socially efficient or economic interest rates, because they would be only affected by structural characteristics of the economy, and to be low-sensitive to monetary policy.

The macroeconomics literature typically relates the economic equilibrium rate to the time preference rate and to the average rate of productivity growth. A typical example is the Ramsey rule. In our financial point of view, the representative agent may invest in a financial market in addition to the money market. We consider an arbitrage approach with exogenously given interest rate, instead of an equilibrium approach that determines them endogenously.

In a first step, considering a classical portfolio optimization, we give a financial interpretation of the equilibrium yield curve given by the Ramsey Rule. In a second step, we introduce dynamic utility functions that allow to get rid of the dependency on the maturity of the classical backward optimization problem and thus gives time consistency for the optimal choices. Besides, as dynamic utility functions take into account that the preferences and risk aversion of investor may change with time, they are also more accurate. Indeed, in the presence of generalized long term uncertainty, the decision scheme must evolve. Finally, to give more precise properties of the marginal utility yield curve, we study forward and backward power utilities, in the example of log-normal market and affine market.