Abstract. We consider a class of asset pricing models, where the risk-neutral joint process of log-price and its stochastic variance is an affine process in the sense of Duffie, Filipovic, and Schachermayer [2003]. First we obtain conditions for the price process to be conservative and a martingale. Then we present some results on the long-term behavior of the model, including an expression for the invariant distribution of the stochastic variance process. We study moment explosions of the price process, and provide explicit expressions for the time at which a moment of given order becomes infinite. We discuss applications of these results, in particular to the asymptotics of the implied volatility smile, and conclude with some calculations for the Heston model, a model of Bates and the Barndorff-Nielsen-Shephard model.

We introduce a class of Markov processes, called m-polynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as Lévy-driven SDEs with affine vector fields. Thus, many popular models such as exponential Lévy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.

We investigate the maximal domain of the moment generating function of affine processes in the sense of Duffie, Filipovi\'{c} and Schachermayer [Ann. Appl. Probab. 13 (2003) 984-1053], and we show the validity of the affine transform formula that connects exponential moments with the solution of a generalized Riccati differential equation. Our result extends and unifies those preceding it (e.g., Glasserman and Kim [Math. Finance 20 (2010) 1-33], Filipovi\'{c} and Mayerhofer [Radon Ser. Comput. Appl. Math. 8 (2009) 1-40] and Kallsen and Muhle-Karbe [Stochastic Process Appl. 120 (2010) 163-181]) in that it allows processes with very general jump behavior, applies to any convex state space and provides both sufficient and necessary conditions for finiteness of exponential moments.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1009 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

We provide a general and flexible approach to LIBOR modeling based on the class of affine factor processes. Our approach respects the basic economic requirement that LIBOR rates are nonnegative, and the basic requirement from mathematical finance that LIBOR rates are analytically tractable martingales with respect to their own forward measure. Additionally, and most importantly, our approach also leads to analytically tractable expressions of multi-LIBOR payoffs. This approach unifies therefore the advantages of well-known forward price models with those of classical LIBOR rate models. Several examples are added and prototypical volatility smiles are shown. We believe that the CIR process-based LIBOR model might be of particular interest for applications, since closed form valuation formulas for caps and swaptions are derived. KEY WORDS: LIBOR rate models, forward price models, affine processes, analytically tractable models. 627 628 M. KELLER-RESSEL, A. PAPAPANTOLEON, AND J. TEICHMANN THE AFFINE LIBOR MODELS 629 AXIOMSLet us denote by L(t, T) the time-t forward LIBOR rate that is settled at time T and received at time T + δ; here T denotes some finite time horizon. The LIBOR rate is related to the prices of zero coupon bonds, denoted by B(t, T), and the forward price, denoted by F(t, T, T + δ), by the following equations:One postulates that the LIBOR rate should satisfy the following axioms, motivated by economic theory, arbitrage pricing theory, and applications.AXIOM 2.1. The LIBOR rate should be nonnegative, i.e., L(t, T) ≥ 0 for all 0 ≤ t ≤ T.AXIOM 2.2. The LIBOR rate process should be a martingale under the corresponding forward measure, i.e., L(·, T) ∈ M(P T+δ ). AXIOM 2.3. The LIBOR rate process, i.e., the (multivariate) collection of all LIBOR rates, should be analytically tractable with respect to as many forward measures as possible. Minimally, closed-form or semi-analytic valuation formulas should be available for the most liquid interest rate derivatives, i.e., caps and swaptions, so that the model can be calibrated to market data in reasonable time.Furthermore we wish to have rich structural properties: that is, the model should be able to reproduce the observed phenomena in interest rate markets, e.g., the shape of the implied volatility surface in cap markets or the implied correlation structure in swaption markets. EXISTING APPROACHESThere are several approaches to LIBOR modeling developed in the literature attempting to fulfill the axioms and practical requirements discussed in the previous section. We describe later the two main approaches, namely the LIBOR market models and the forward price model, and comment on their ability to fulfill them. We also briefly discuss Markov-functional models. forward LIBOR rate is modeled as an exponential Brownian motion under its corresponding forward measure. This model provides a theoretical justification for the common market practice of pricing caplets according to Black's futures formula (Black 1976), i.e., assuming that the forward LIBOR rate is log-no...

Abstract. We show that stochastically continuous, time-homogeneous affine processes on the canonical state space R m 0 × R n are always regular. In the paper of Duffie, Filipovic, and Schachermayer (2003) regularity was used as a crucial basic assumption. It was left open whether this regularity condition is automatically satisfied, for stochastically continuous affine processes. We now show that the regularity assumption is indeed superfluous, since regularity follows from stochastic continuity and the exponentially affine behavior of the characteristic function. For the proof we combine classic results on the differentiability of transformation semigroups with the method of the moving frame which has been recently found to be useful in the theory of SPDEs.

Abstract. We consider a model for interest rates, where the short rate is given under the risk-neutral measure by a time-homogenous, one-dimensional affine process in the sense of Duffie, Filipović, and Schachermayer. We show that in such a model yield curves can only be normal, inverse or humped (i.e. endowed with a single local maximum). Each case can be characterized by simple conditions on the present short rate rt. We give conditions under which the short rate process will converge to a limit distribution and describe the risk-neutral limit distribution in terms of its cumulant generating function. We apply our results to the Vasiček model, the CIR model, a CIR model with added jumps and a model of Ornstein-Uhlenbeck type.

Motivated by applications in economics and finance, in particular to the modeling of limit order books, we study a class of stochastic second-order PDEs with non-linear Stefan-type boundary interaction. To solve the equation we transform the problem from a moving boundary problem into a stochastic evolution equation with fixed boundary conditions. Using results from interpolation theory we obtain existence and uniqueness of local strong solutions, extending results of Kim, Zheng and Sowers. In addition, we formulate conditions for existence of global solutions and provide a refined analysis of possible blow-up behavior in finite time.Comment: 34 page

Abstract. -We consider affine Markov processes taking values in convex cones. In particular, we characterize all affine processes taking values in an irreducible symmetric cone in terms of certain Lévy-Khintchine triplets. This is the complete classification of affine processes on these conic state spaces, thus extending the theory of Wishart processes on positive semidefinite matrices, as put forward by Bru (1991).