Patrick Cheridito

Arbitrage in fractional Brownian motion models

Abstract. We construct arbitrage strategies for a financial market that consists of a money market account and a stock whose discounted price follows a fractional Brownian motion with drift or an exponential fractional Brownian motion with drift. Then we show how arbitrage can be excluded from these models by restricting the class of trading strategies.

Mixed fractional Brownian motion

We show that the sum of a Brownian motion and a non-trivial multiple of an independent fractional Brownian motion with Hurst parameter H E (0, 1] is not a semimartingale if H E (0, 1) U (, 3], that it is equivalent to a multiple of Brownian motion if H = 1 and equivalent to Brownian motion if H E (, 1]. As an application we discuss the price of a European call option on an asset driven by a linear combination of a Brownian motion and an independent fractional Brownian motion.

Risk Measures on Orlicz Hearts

Coherent, convex, and monetary risk measures were introduced in a setup where uncertain outcomes are modeled by bounded random variables. In this paper, we study such risk measures on Orlicz hearts. This includes coherent, convex, and monetary risk measures on Lp-spaces for 1 ≤ p

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.

COMPOSITION OF TIME-CONSISTENT DYNAMIC MONETARY RISK MEASURES IN DISCRETE TIME

In discrete time, every time-consistent dynamic monetary risk measure can be written as a composition of one-step risk measures. We exploit this structure to give new dual representation results for time-consistent convex monetary risk measures in terms of one-step penalty functions. We first study risk measures for random variables modelling financial positions at a fixed future time. Then we consider the more general case of risk measures that depend on stochastic processes describing the evolution of financial positions or cumulated cash flows. In both cases the new representations allow for a simple composition of one-step risk measures in the dual. We discuss several explicit examples and provide connections to the recently introduced class of dynamic variational preferences.

Equilibrium Pricing in Incomplete Markets Under Translation Invariant Preferences

We propose a general discrete-time framework for deriving equilibrium prices of financial securities. It allows for heterogeneous agents, unspanned random endowments, and convex trading constraints. We give a dual characterization of equilibria and provide general results on their existence and uniqueness. In the special case where all agents have preferences of the same type and in equilibrium, all random endowments are replicable by trading in the financial market, we show that a one-fund theorem holds and give an explicit expression for the equilibrium pricing kernel.

A Note on the Dai–Singleton Canonical Representation of Affine Term Structure Models

Dai and Singleton (2000) study a class of term structure models for interest rates that specify the instantaneous interest rate as an affine combination of the components of an N-dimensional affine diffusion process. Observable quantities of such models are invariant under regular affine transformations of the underlying diffusion process. And in their canonical form, the models in Dai and Singleton (2000) are based on diffusion processes with diagonal diffusion matrices. This motivates the following question: Can the diffusion matrix of an affine diffusion process always be diagonalized by means of a regular affine transformation? We show that if the state space of the diffusion is of the form D = Rm+ x RN - m for integers 0 = N - 1, then there exists a regular affine transformation of D onto itself that diagonalizes the diffusion matrix. On the other hand, we provide examples of affine diffusion processes with state space R2+ x R2 whose diffusion matrices cannot be diagonalized through regular affine transformation.

Multidimensional quadratic and subquadratic BSDEs with special structure

We study multidimensional BSDEs of the formwith bounded terminal conditions and drivers that grow at most quadratically in . We consider three different cases. In the first case, the BSDE is Markovian, and a solution can be obtained from a solution to a related FBSDE. In the second case, the BSDE becomes a one-dimensional quadratic BSDE when projected to a one-dimensional subspace, and a solution can be derived from a solution of the one-dimensional equation. In the third case, the growth of the driver in is strictly subquadratic, and the existence and uniqueness of a solution can be shown by first solving the BSDE on a short time interval and then extending it recursively.

Measuring and Allocating Systemic Risk

In this paper we develop a framework for measuring, allocating and managing systemic risk. SystRisk, our measure of total systemic risk, captures the a priori cost to society for providing tail-risk insurance to the financial system. Our allocation principle distributes the total systemic risk among individual institutions according to their size-shifted marginal contributions. To describe economic shocks and systemic feedback effects we propose a reduced form stochastic model that can be calibrated to historical data. We also discuss systemic risk limits, systemic risk charges and a cap and trade system for systemic risk.

BS

We provide existence results and comparison principles for solutions of backward stochastic difference equations (BSΔEs) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BSΔEs and BSDEs are governed by drivers fN(t,ω,y,z) and f(t,ω,y,z), respectively. The new feature of this paper is that they may be non-Lipschitz in z. For the convergence results it is assumed that the BSΔEs are based on d-dimensional random walks WN approximating the d-dimensional Brownian motion W underlying the BSDE and that fN converges to f. Conditions are given under which for any bounded terminal condition ξ for the BSDE, there exist bounded terminal conditions ξN for the sequence of BSΔEs converging to ξ, such that the corresponding solutions converge to the solution of the limiting BSDE. An important special case is when fN and f are convex in z. We show that in this situation, the solutions of the BSΔEs converge to the solution of the BSDE for every uniformly bounded sequence ξN converging to ξ. As a consequence, one obtains that the BSDE is robust in the sense that if (WN,ξN) is close to (W,ξ) in distribution, then the solution of the Nth BSΔE is close to the solution of the BSDE in distribution too.