A Markovian Defaultable Term Structure Model with State Dependent Volatilities

The defaultable forward rate is modeled as a jump diffusion process within the Schonbucher (2000, 2003) general Heath, jarrow and Morton (1992) framework where jumps in the defaultable term structure f d(t, T) cause jumps and defaults to the defaultable bond prices P d(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 realisations 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.defaultable HJM model; strochastic credit spreads; defaultable bond prices

Alternative Defaultable Term Structure Models

The objective of this paper is to consider defaultable term structure models in a general setting beyond standard risk-neutral models. Using as numeraire the growth optimal portfolio, defaultable interest rate derivatives are priced under the real-world probability measure. Therefore, the existence of an equivalent risk-neutral probability measure is not required. In particular, the real-world dynamics of the instantaneous defaultable forward rates under a jump-diffusion extension of a HJM type framework are derived. Thus, by establishing a modelling framework fully under the real-world probability measure, the challenge of reconciling real-world and risk-neutral probabilities of default is deliberately avoided, which provides significant extra modelling freedom. In addition, for certain volatility specifications, finite dimensional Markovian defaultable term structure models are derived. The paper also demonstrates an alternative defaultable term structure model. It provides tractable expressions for the prices of defaultable derivatives under the assumption of independence between the discounted growth optimal portfolio and the default-adjusted short rate. These expressions are then used in a more general model as control variates for Monte Carlo simulations of credit derivatives.defaultable forward rates; jump-diffusion processes; growth optimal portfolio; real-world pricing

Economic determinants of oil futures volatility: a term structure perspective

To assess the economic determinants of oil futures volatility, we firstly develop and estimate a multi-factor oil futures pricing model with stochastic volatility that is able to disentangle long-term, medium-term and short-term variations in commodity markets volatility. The volatility estimates reveal that in line with theory, the volatility factors are unspanned, persistent and carry negative market price of risk, while crude oil markets are becoming more integrated with financial markets. After 2004, short-term volatility is driven by industrial production, term and credit spreads, the S&P 500 and the US dollar index, along with the traditional drivers including hedging pressure and VIX. Medium-term volatility is consistently related to open interest and credit spreads, while after 2004 oil sector variables such as inventory and consumption also impact this part of the term structure. Interest rates mostly matter for long-term futures price volatility

A class of Markovian models for the term structure of interest rates under jump-diffusions

University of Technology, Sydney. Faculty of Business.NO FULL TEXT AVAILABLE. Access is restricted indefinitely. The hardcopy may be available for consultation at the UTS Library.NO FULL TEXT AVAILABLE. Access is restricted indefinitely. ----- Jump-Diffusion processes capture the standardized empirical statistical features of interest rate dynamics, thus providing an improved setting to overcome some of the mispricing of derivative securities that arises with the extensively developed pure diffusion models. A combination of jump-diffusion models with state dependent volatility specifications generates a class of models that accommodates the empirical statistical evidence of jump components and the more general and realistic setting of stochastic volatility. For modelling the term structure of interest rates, the Heath, Jarrow & Morton (1992) (hereafter HJM) framework constitutes the most general and adaptable platform for the study of interest rate dynamics that accommodates, by construction, consistency with the currently observed yield curve within an arbitrage free environment. The HJM model requires two main inputs, the market information of the initial forward curve and the specification of the forward rate volatility. This second requirement of the volatility specification enables the model builder to generate a wide class of models and in particular to derive within the HJM framework a number of the popular interest rate models. However, the general HJM model is Markovian only in the entire yield curve, thus requiring an infinite number of state variables to determine the future evolution of the yield curve. By imposing appropriate conditions on the forward rate volatility, the HJM model can admit finite dimensional Markovian structures, where the generality of the HJM models coexists with the computational tractability of Markovian structures. The main contributions of this thesis include: Markovianisation of jump-diffusion versions of the HJM model-Chapters 2 and 3. Under a specific formulation of state and time dependent forward rate volatility specifications, Markovian representations of a generalised Shirakawa (1991) model are developed. Further, finite dimensional affine realisations of the term structure in terms of forward rates are obtained. Within this framework, some specific classes of jump-diffusion term structure models are examined such as extensions of the Hull & White (1990), (1994) class of models and the Ritchken & Sankarasubramanian (1995) class of models to the jump-diffusion case. Markovianisation of defaultable HJM models - Chapters 4. Suitable state dependent volatility specifications, under deterministic default intensity, lead to Markovian defaultable term structures under the Schonbucher (2000), (2003) general HJM framework. The state variables of this model can be expressed in terms of a finite number of benchmark defaultable forward rates. Moving to the more general setting of stochastic intensity of defaultable term structures, we discuss model limitations and an approximate Markovianisation of the system is proposed. Bond option pricing under jump-diffusions - Chapter 5. Within the jump-diffusion framework, the pricing of interest rate derivative securities is discussed. A tractable Black-Scholes type pricing formula is derived under the assumption of constant jump volatility specifications and a viable control variate method is proposed for pricing by Monte Carlo simulation under more general volatility specifications

Real-world jump-diffusion term structure models

This paper considers interest rate term structure models in a market attracting both continuous and discrete types of uncertainty. The event-driven noise is modelled by a Poisson random measure. Using as numeraire the growth optimal portfolio, interest rate derivatives are priced under the real-world probability measure. In particular, the real-world dynamics of the forward rates are derived and, for specific volatility structures, finite-dimensional Markovian representations are obtained. Furthermore, allowing for a stochastic short rate in a non-Markovian setting, a class of tractable affine term structures is derived where an equivalent risk-neutral probability measure may not exist.Stochastic analysis, Stochastic volatility, Quantitative finance, Numerical simulation,

Alternative Defaultable Term Structure Models

Abstract. The objective of this paper is to consider defaultable term structure models in a general setting beyond standard riskneutral models. Using as numeraire the growth optimal portfolio, defaultable interest rate derivatives are priced under the real-world probability measure. Therefore, the existence of an equivalent risk-neutral probability measure is not required. In particular, the real-world dynamics of the instantaneous defaultable forward rates under a jump-diffusion extension of a HJM type framework are derived. Thus, by establishing a modelling framework fully under the real-world probability measure, the challenge of reconciling real-world and risk-neutral probabilities of default is deliberately avoided, which provides significant extra modelling freedom. In addition, for certain volatility specifications, finite dimensional Markovian defaultable term structure models are derived. The paper also demonstrates an alternative defaultable term structure model. It provides tractable expressions for the prices of defaultable derivatives under the assumption of independence between the discounted growth optimal portfolio and the default-adjusted short rate. These expressions are then used in a more general model as control variates for Monte Carlo simulations of credit derivatives

A Class of Jump-Diffusion Bond Pricing Models within the HJM Framework

This paper considers a class of term structure models that is a parameterisation of the Shirakawa (1991) extension of the Heath et al. (1992) model to the case of jump-diffusions. We consider specific forward rate volatility structures that incorporate state dependent Wiener volatility functions and time dependent Poisson volatility functions. Within this framework, we discuss the Markovianisation issue, and obtain the corresponding affine term structure of interest rates. As a result we are able to obtain a broad tractable class of jump-diffusion term structure models. We relate our approach to the existing class of jump-diffusion term structure models whose starting point is a jump-diffusion process for the spot rate. In particular we obtain natural jump-diffusion versions of the Hull and White (1990, 1994) one-factor and two-factor models and the Ritchken and Sankarasubramanian (1995) model within the HJM framework. We also give some numerical simulations to gauge the effect of the jump-component on yield curves and the implications of various volatility specifications for the spot rate distribution. Copyright Springer Science + Business Media, Inc. 2003jump-diffusions, Markovian HJM model, state dependent volatility,