Martin Keller-Ressel

Yield curve shapes and the asymptotic short rate distribution in affine one-factor models

We consider a model for interest rates where the short rate is given under the risk-neutral measure by a time-homogeneous 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 converges 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.

A Stefan-type stochastic moving boundary problem

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.

Geometric Asian option pricing in general affine stochastic volatility models with jumps

In this paper, we present some results on Geometric Asian option valuation for affine stochastic volatility models with jumps. We shall provide a general framework into which several different valuation problems based on some average process can be cast, and we shall obtain closed form solutions for some relevant affine model classes.

Convex Order Properties of Discrete Realized Variance and Applications to Variance Options

We consider a square-integrable semimartingale with conditionally independent increments and symmetric jump measure, and show that its discrete realized variance dominates its quadratic variation in increasing convex order. The result has immediate applications to the pricing of options on realized variance. For a class of models including time-changed Levy models and Sato processes with symmetric jumps our results show that options on variance are typically underpriced, if quadratic variation is substituted for the discretely sampled realized variance.

Affine forward variance models

We introduce the class of affine forward variance (AFV) models of which both the conventional Heston model and the rough Heston model are special cases. We show that AFV models can be characterized by the affine form of their cumulant generating function, which can be obtained as solution of a convolution Riccati equation. We further introduce the class of affine forward order flow intensity (AFI) models, which are structurally similar to AFV models, but driven by jump processes, and which include Hawkes-type models. We show that the cumulant generating function of an AFI model satisfies a generalized convolution Riccati equation and that a high-frequency limit of AFI models converges in distribution to the AFV model.

Correction to: Yield curve shapes and the asymptotic short rate distribution in affine one-factor models

Correction to: Finance Stoch. (2008) 12: 149–172https://doi.org/10.1007/s00780-007-0059-z I should like to thank Ralf Korn for alerting me to an error in the original paper [2]. The error concerns the threshold at which the yield curve in an affine short rate model changes from normal (strictly increasing) to humped (endowed with a single maximum). In particular, it is not true that this threshold is the same for the forward curve and for the yield curve, as claimed in [2]. Below, the correct mathematical expression for the threshold is given, supplemented with a self-contained and corrected proof.