# Abstracts

**Bayesian Inference on Multi-Factor Affine Term Structure Models**

*Marcel Rindisbacher (Boston University, USA)*

Wednesday June 4, 14:30-15:00 | session 5.4 | Interest Rates | room K

In this paper, we develop Markov Chain Monte Carlo (MCMC) algorithms to conduct a Bayesian inference analysis for multi-factor affine term structure models (ATSM).

We propose efficient sampling algorithms for the posterior distribution of parameters and latent states via the MCMC methods. The MCMC algorithms allow us to remove the stringent assumption imposed to break the stochastic singularity, and help us to decompose the high-dimensional inference problem into iterations of univariate sampling problems. With this sampling scheme, we conduct a full-fledged Bayesian analysis on two market data sets covering different economic regimes. The Bayesian inference delivers promising results in in-sample fitting, out-of-sample forecasting and model comparison.

First, we show the strength of the MCMC methods in reconstructing yield observations under the no arbitrage condition. The fitting errors are smaller than those obtained by the inversion-MLE method and the model-free method. We are also able to reconstruct the short rates, which is documented as a challenging task. The inferred short rates from the MCMC algorithm closely resemble the one month yield data.

Second, with the MCMC sampling method, we forecast future yield levels with satisfactory precision. We are able to forecast the twelve weeks ahead yield levels with out-of-sample errors within a couple of basis points. We run a horse-race among several conventional prediction methods. The Bayesian forecast performance of the three-factor model with one restricted variable dominates the ordinary least square prediction and frequentist-type prediction for all maturities. It also dominates the random walk prediction for all maturities greater or equal to one year.

At last, with the MCMC method, we conduct a Bayesian model comparison for different ATSMs. We find that the ranking of the models by the Bayesian model selection criteria is consistent with both in-sample fitting and out-of-sample forecast performances. We apply the model comparison analysis on the two data sets. The first data has the feature of non-normality and a humped shape of the volatility curve. It supports the three-factor model with one restricted state variable. The second data has the feature of non-normality, but has downward shape of volatility curve. It supports the three-factor model with Gaussian dynamics.

**When roll-overs do not qualify as numéraire: bond markets beyond short rate paradigms**

*Josef Teichmann (ETH Zurich, Switzerland)*

Wednesday June 4, 15:00-15:30 | session 5.4 | Interest Rates | room K

We investigate default-free bond markets where the standard relationship between a possibly existing bank account process and the term structure of bond prices is broken, i.e. the bank account process is not necessarily a valid numéraire. We argue that this feature is not the exception but rather the rule in bond markets when starting with, e.g., terminal bonds as numéraires.

Our setting are general càdlàg processes as bond prices, where we employ directly methods from large financial markets. Moreover, we do not restrict price process to be semimartingales, which allows for example to consider markets driven by fractional Brownian motion. In the core of the article we relate the appropriate no arbitrage assumptions (NAFL), i.e. no asymptotic free lunch, to the existence of an equivalent local martingale measure with respect to the terminal bond as numéraire, and no arbitrage opportunities of the first kind (NAA1) to the existence of a supermartingale deflator, respectively. In all settings we obtain existence of a generalized bank account as a limit of convex combinations of roll-over bonds.

Additionally we provide an alternative definition of the concept of a numéraire, leading to a possibly interesting connection to bubbles. If we can construct a bank account process through roll-overs, we can relate the impossibility of taking the bank account as numéraire to liquidity effects. Here we enter endogenously the arena of multiple yield curves. The theory is illustrated by several examples.

**Model Risk in Pricing Interest Rate Derivatives**

*Koichi Matsumoto (Kyushu University, Japan)*

Wednesday June 4, 15:30-16:00 | session 5.4 | Interest Rates | room K

Many financial institutions use various models for various purposes such as investment, risk management and pricing derivatives. Usually a model includes some parameters but it is difficult or impossible to know all true parameters perfectly. We call the risk caused by not knowing the true parameters, the model risk. In the classical theory, the model risk is neglected but it is one of the most important risks. Recently some problems in the financial world are studied with consideration for the model risk.

We are interested in a mean reverting interest rated model. A mean reverting interest rate model is suitable for practical use. We can calculate prices of complicated derivatives on interest rate by a mean reverting model. A mean reverting model can be represented by both a continuous time model and a discrete time model. These models include some parameters and then there is the model risk.

In this study, we consider a mean reverting interest rate model whose volatility process is uncertain. We consider an investor who knows the volatility process moves between two deterministic processes but he does not know the true volatility process. Most of prices of interest rate derivatives cannot be determined uniquely, based on this interest rate model. Our problem is to find the price bounds of a derivative, that is, we want to find the maximum and minimum prices. To solve the problem, we propose a trinomial model and show how to calculate the price bounds using the dynamic programming method. Our trinomial model is related with the uncertain volatility model proposed by Avellaneda, Levy and Paras (1995) and the Hull White interest rate model in Hull and White (1994, 1996). By the numerical experiments, we study the model risk of options and their portfolios. We show the optimal volatility process which attains the maximum price or the minimum price. Further we study the portfolio effect of the model risk.