Term structure modeling under volatility uncertainty

In this paper, we study term structure movements in the spirit of Heath et al. (Econometrica 60(1):77–105, 1992) under volatility uncertainty. We model the instantaneous forward rate as a diffusion process driven by a G-Brownian motion. The G-Brownian motion represents the uncertainty about the volatility. Within this framework, we derive a sufficient condition for the absence of arbitrage, known as the drift condition. In contrast to the traditional model, the drift condition consists of several equations and several market prices, termed market price of risk and market prices of uncertainty, respectively. The drift condition is still consistent with the classical one if there is no volatility uncertainty. Similar to the traditional model, the risk-neutral dynamics of the forward rate are completely determined by its diffusion term. The drift condition allows to construct arbitrage-free term structure models that are completely robust with respect to the volatility. In particular, we obtain robust versions of classical term structure models.

Finite-State-Space Truncations for Infinite Quasi-Birth-Death Processes

For dealing numerically with the infinite-state-space Markov chains, a truncation of the state space is inevitable, that is, an approximation by a finite-state-space Markov chain has to be performed. In this paper, we consider level-dependent quasi-birth-death processes, and we focus on the computation of stationary expectations. In previous literature, efficient methods for computing approximations to these characteristics have been suggested and established. These methods rely on truncating the process at some level N, and for N⟶∞, convergence of the approximation to the desired characteristic is guaranteed. This paper’s main goal is to quantify the speed of convergence. Under the assumption of an f-modulated drift condition, we derive terms for a lower bound and an upper bound on stationary expectations which converge quickly to the same value and which can be efficiently computed.

Forward Rate Models

In this chapter we study the Heath–Jarrow–Morton framework for forward rate models. Building on results from the previous chapter, the HJM drift condition is derived, some examples are studied, and the general Gaussian HJM model is analyzed in detail. The Musiela parameterization of forward rates is introduced and discussed in the context of infinite dimensional SDEs.

Absolute regularity of semi-contractive GARCH-type processes

We prove existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p, q). In contrast to previous work we impose, besides a geometric drift condition, only a semi-contractive condition which allows us to include models which would be ruled out by a fully contractive condition. This results in a subgeometric rather than the more usual geometric decay rate of the mixing coefficients. The proofs are heavily based on a coupling of two versions of the processes.

Irreducibility and continuity assumptions for positive operators with application to threshold GARCH time series models

Suppose that {Xt} is a Markov chain such as the state space model for a threshold GARCH time series. The regularity assumptions for a drift condition approach to establishing the ergodicity of {Xt} typically are ϕ-irreducibility, aperiodicity, and a minorization condition for compact sets. These can be very tedious to verify due to the discontinuous and singular nature of the Markov transition probabilities. We first demonstrate that, for Feller chains, the problem can at least be simplified to focusing on whether the process can reach some neighborhood that satisfies the minorization condition. The results are valid not just for the transition kernels of Markov chains but also for bounded positive kernels, opening the possibility for new ergodic results. More significantly, we show that threshold GARCH time series and related models of interest can often be embedded into Feller chains, allowing us to apply the conclusions above.

Irreducibility and continuity assumptions for positive operators with application to threshold GARCH time series models

Suppose that {X t } is a Markov chain such as the state space model for a threshold GARCH time series. The regularity assumptions for a drift condition approach to establishing the ergodicity of {X t } typically are ϕ-irreducibility, aperiodicity, and a minorization condition for compact sets. These can be very tedious to verify due to the discontinuous and singular nature of the Markov transition probabilities. We first demonstrate that, for Feller chains, the problem can at least be simplified to focusing on whether the process can reach some neighborhood that satisfies the minorization condition. The results are valid not just for the transition kernels of Markov chains but also for bounded positive kernels, opening the possibility for new ergodic results. More significantly, we show that threshold GARCH time series and related models of interest can often be embedded into Feller chains, allowing us to apply the conclusions above.

A Generalization Of Lattice Specifications For Currency Options

<p class="MsoNormal" style="text-align: justify; margin: 0in 0.5in 0pt; mso-pagination: none;"><span style="font-family: Times New Roman; font-size: x-small;">This article revisits the topic of two-state pricing of currency options.<span style="mso-spacerun: yes;">&nbsp; </span>It examines the models developed by Cox, Ross, and Rubinstein, Rendleman and Bartter, and Trigeorgis, and presents two alternative binomial models based on the continuous and discrete time Geometric Brownian Motion processes respectively.<span style="mso-spacerun: yes;">&nbsp; </span>This work generalizes the standard binomial approach incorporating the main existing models as particular cases.<span style="mso-spacerun: yes;">&nbsp; </span>The proposed models are straightforward, flexible, accommodate any drift condition and afford additional insights into binomial trees and lattice models in general.<span style="mso-spacerun: yes;">&nbsp; </span>Further, the alternative parameterizations are free of the negative aspects associated with the Cox, Ross, and Rubinstein model.</span></p>