The Dynamic Spread of the Forward CDS with General Random Loss

Abstract and Applied Analysis ◽

10.1155/2014/580713 ◽

2014 ◽

Vol 2014 ◽

pp. 1-17

Author(s):

Kun Tian ◽

Dewen Xiong ◽

Zhongxing Ye

Keyword(s):

Interest Rates ◽

Stopping Time ◽

Random Measure ◽

Random Variable ◽

Conditional Density ◽

Default Time ◽

Defaultable Bond ◽

Term Structures ◽

Stochastic Interest ◽

Random Loss

We assume that the filtrationFis generated by ad-dimensional Brownian motionW=(W1,…,Wd)′as well as an integer-valued random measureμ(du,dy). The random variableτ~is the default time andLis the default loss. LetG={Gt;t≥0}be the progressive enlargement ofFby(τ~,L); that is,Gis the smallest filtration includingFsuch thatτ~is aG-stopping time andLisGτ~-measurable. We mainly consider the forward CDS with loss in the framework of stochastic interest rates whose term structures are modeled by the Heath-Jarrow-Morton approach with jumps under the general conditional density hypothesis. We describe the dynamics of the defaultable bond inGand the forward CDS with random loss explicitly by the BSDEs method.

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Two Approaches for a Dividend Maximization Problem under an Ornstein-Uhlenbeck Interest Rate

Mathematics ◽

10.3390/math9182257 ◽

2021 ◽

Vol 9 (18) ◽

pp. 2257

Author(s):

Julia Eisenberg ◽

Stefan Kremsner ◽

Alexander Steinicke

Keyword(s):

Optimal Control ◽

Interest Rate ◽

Interest Rates ◽

Optimal Control Problems ◽

Stopping Time ◽

Maximization Problem ◽

Control Problems ◽

Stochastic Interest Rates ◽

Stochastic Interest ◽

And Behavior

We investigate a dividend maximization problem under stochastic interest rates with Ornstein-Uhlenbeck dynamics. This setup also takes negative rates into account. First a deterministic time is considered, where an explicit separating curve α(t) can be found to determine the optimal strategy at time t. In a second setting, we introduce a strategy-independent stopping time. The properties and behavior of these optimal control problems in both settings are analyzed in an analytical HJB-driven approach, and we also use backward stochastic differential equations.

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A MARKOVIAN DEFAULTABLE TERM STRUCTURE MODEL WITH STATE DEPENDENT VOLATILITIES

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024907004147 ◽

2007 ◽

Vol 10 (01) ◽

pp. 155-202 ◽

Cited By ~ 8

Author(s):

CARL CHIARELLA ◽

CHRISTINA NIKITOPOULOS SKLIBOSIOS ◽

ERIK SCHLÖGL

Keyword(s):

Diffusion Process ◽

Interest Rates ◽

Term Structure ◽

Forward Rate ◽

Term Structure Model ◽

Forward Rates ◽

Defaultable Bond ◽

Term Structures ◽

State Dependent ◽

Path Independence

The defaultable forward rate is modelled as a jump diffusion process within the Schönbucher [26,27] general Heath, Jarrow and Morton [20] framework where jumps in the defaultable term structure fd(t,T) cause jumps and defaults to the defaultable bond prices Pd(t,T). Within this framework, we investigate an appropriate forward rate volatility structure that results in Markovian defaultable spot rate dynamics. In particular, we consider state dependent Wiener volatility functions and time dependent Poisson volatility functions. The corresponding term structures of interest rates are expressed as finite dimensional affine realizations in terms of benchmark defaultable forward rates. In addition, we extend this model to incorporate stochastic spreads by allowing jump intensities to follow a square-root diffusion process. In that case the dynamics become non-Markovian and to restore path independence we propose either an approximate Markovian scheme or, alternatively, constant Poisson volatility functions. We also conduct some numerical simulations to gauge the effect of the stochastic intensity and the distributional implications of various volatility specifications.

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