Damir Filipović

Equivalent and absolutely continuous measure changes for jump-diffusion processes

We provide explicit sufficient conditions for absolute continuity and equivalence between the distributions of two jump-diffusion processes that can explode and be killed by a potential.

The Term Structure of Interbank Risk

We infer a term structure of interbank risk from spreads between rates on interest rate swaps indexed to the London Interbank Offered Rate (LIBOR) and overnight indexed swaps. We develop a tractable model of interbank risk to decompose the term structure into default and non-default (liquidity) components. From August 2007 to January 2011, the fraction of total interbank risk due to default risk, on average, increases with maturity. At short maturities, the non-default component is important in the first half of the sample period and is correlated with measures of funding and market liquidity. The model also provides a framework for pricing, hedging, and risk management of interest rate swaps in the presence of significant basis risk.

Affine processes on positive semidefinite matrices

This article provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. This analysis has been motivated by a large and growing use of matrix-valued affine processes in finance, including multi-asset option pricing with stochastic volatility and correlation structures, and fixed-income models with stochastically correlated risk factors and default intensities.

A general characterization of one factor affine term structure models

Abstract. We give a complete characterization of affine term structure models based on a general nonnegative Markov short rate process. This applies to the classical CIR model but includes as well short rate processes with jumps. We provide a link to the theory of branching processes and show how CBI-processes naturally enter the field of term structure modelling. Using Markov semigroup theory we exploit the full structure behind an affine term structure model and provide a deeper understanding of some well-known properties of the CIR model. Based on these fundamental results we construct a new short rate model with jumps, which extends the CIR model and still gives closed form expressions for bond options.

The Canonical Model Space For Law-Invariant Convex Risk Measures Is L1

In this paper, we establish a one‐to‐one correspondence between law‐invariant convex risk measures on L∞ and L1. This proves that the canonical model space for the predominant class of law‐invariant convex risk measures is L1.

EXISTENCE OF INVARIANT MANIFOLDS FOR STOCHASTIC EQUATIONS IN INFINITE DIMENSION

We provide a Frobenius type existence result for finite-dimensional invariant submanifolds for stochastic equations in infinite dimension, in the spirit of Da Prato and Zabczyk [5]. We recapture and make use of the convenient calculus on Frechet spaces, as developed by Kriegl and Michor [16]. Our main result is a weak version of the Frobenius theorem on Frechet spaces. As an application we characterize all finite-dimensional realizations for a stochastic equation which describes the evolution of the term structure of interest rates

EXPONENTIAL-POLYNOMIAL FAMILIES AND THE TERM STRUCTURE OF INTEREST RATES

Exponential-polynomial families like the Nelson-Siegel or Svensson family are widely used to estimate the current forward rate curve. We investigate whether these methods go well with inter-temporal modelling. We characterize the consistent Ito processes which have the property to provide an arbitrage free interest rate model when representing the parameters of some bounded exponential-polynomial type function. This includes in particular diffusion processes. We show that there is a strong limitation on their choice. Bounded exponential-polynomial families should rather not be used for modelling the term structure of interest rates.

Optimal capital and risk allocations for law- and cash-invariant convex functions

In this paper we provide a complete solution to the existence and characterization problem of optimal capital and risk allocations for not necessarily monotone, law-invariant convex risk measures on the model space Lp for any p∈[1,∞]. Our main result says that the capital and risk allocation problem always admits a solution via contracts whose payoffs are defined as increasing Lipschitz-continuous functions of the aggregate risk.

ON THE GEOMETRY OF THE TERM STRUCTURE OF INTEREST RATES

We present recently developed geometric methods for the analysis of finite–dimensional term–structure models of the interest rates. This includes an extension of the Frobenius theorem for Fréchet spaces in particular. This approach puts new light on many of the classical models, such as the Hull-White extended Vasícek and Cox–Ingersoll–Ross short–rate models. The notion of a finite–dimensional realization (FDR) is central for this analysis: we motivate it, classify all generic FDRs and provide some new results for the corresponding factor processes, such as hypoellipticity of its generators and the existence of smooth densities. Furthermore, we include finite–dimensional external factors, thus admitting a stochastic volatility structure.

Affine Diffusion Processes: Theory and Applications

We revisit affine diffusion processes on general and on the canonical state space in particular. A detailed study of theoretic and applied aspects of this class of Markov processes is given. In particular, we derive admissibility conditions and provide a full proof of existence and uniqueness through stochastic invariance of the canonical state space. Existence of exponential moments and the full range of validity of the affine transform formula are established. This is applied to the pricing of bond and stock options, which is illustrated for the Vasicek Cox-Ingersoll-Ross and Heston Models.