Bond Market Structure in the Presence of Marked Point Processes (original) (raw)

Bond Markets Where Prices are Driven by a General Marked Point Process

Social Science Research Network, 1995

We investigate the term structure of zero coupon bonds when interest rates are driven by a general marked point process as well as by a Wiener process. Developing a theory which allows for measure-valued trading portfolios we study existence and uniqueness of a martingale measure. We also study completeness and its relation to the uniqueness of a martingale measure. For the case of a finite jump spectrum we give a fairly general completeness result and for a Wiener-Poisson model we prove the existence of a time-independent set of basic bonds. We also give sufficient conditions for the existence of an affine term structure.

Term structure of interest rates: The martingale approach

Advances in Applied Mathematics, 1989

Martingale methods are used to study interest rate risk in a market with two fundamental assets: savings accounts and zero coupon bonds. Discounted prices of bonds have to be a martingale for a risk-neutral probability. Specifications are given when the instantaneous rate of interest is adapted to a Brownian motion or follows a ditTusion. B 1989 Academic Press, Inc. CONTENTS. Introduction. 1. The discounted bond price process as a martingale. 1.1. The instantaneous interest rate and the bond price process. 1.2. Trading strategies and marketed assets. 1.3. Pricing under absence of free lunch. 1.4. Viability of a price system. 1.5. Martingale discounted pricing as condition for viability. 1.6. Reformulation of the martingale property. 2. The case of an instantaneous interest rate process adapted to a Brownian motion. 2.1. Instantaneous interest rates and bond prices adapted to a Brownian motion. 2.2 The bond price process as a stochastic integral. 2.3. Description of the drift term in stochastic differentials, 2.4. Uniqueness results. 2.5. Recovering the instantaneous interest rate process from the bond price process. 3. The case of an instantaneous interest rate process following a d@sion process. 3.1. A sufficient condition for the savings account process to be integrable. 3.2. Condition for a deterministic dependence of (t, P,) on (t, 4). 3.3. The bond price process as a diffusion. The intended audience for this paper is twofold: probabilists interested in application of martingale theory to the field of finance, as well as finance theorists concerned with the valuation of contingent claims related to interest rates risks. Originally we were interested in the pricing of policy loan options in life insurance, and therefore in the pricing of bonds and options on them. It 95

Pricing under the Real-World Probability Measure for Jump-Diffusion Term Structure Models

2007

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, a class of tractable affine term structures is derived where an equivalent risk-neutral probability measure does not exist.

The equivalent martingale measure conditions in a general model for interest rates

Advances in Applied Probability, 2005

Assuming that the forward rates ftu are semimartingales, we give conditions on their components under which the discounted bond prices are martingales. To achieve this, we give sufficient conditions for the integrated processes ftu=∫0uftvdv to be semimartingales, and identify their various components. We recover the no-arbitrage conditions in models well known in the literature and, finally, we formulate a new random field model for interest rates and give its equivalent martingale measure (no-arbitrage) condition.

On the Generalized Brownian Motion and its Applications in Finance

SSRN Electronic Journal, 2008

This paper deals with dynamic term structure models (DTSMs) and proposes a new way to handle the limitation of the classical affine models. In particular, the paper expands the flexibility of the DTSMs by applying generalized Brownian motions with dependent increments as the governing force of the state variables instead of standard Brownian motions. This is a new direction in pricing non defaultable bonds. By extending the theory developed by Dippon & Schiemert (2006a), the paper developes a bond market with memory, and proves the absence of arbitrage. The framework is readily extendable to other markets or multi factors. As a complement the paper shows an example of how to derive the implied bond pricing parameters using the ordinary Kalman filter.

On the use of measure-valued strategies in bond markets

Finance and Stochastics, 2004

We propose here a theory of cylindrical stochastic integration, recently developed by Mikulevicius and Rozovskii, as mathematical background to the theory of bond markets. In this theory, since there is a continuum of securities, it seems natural to define a portfolio as a measure on maturities. However, it turns out that this set of strategies is not complete, and the theory of cylindrical integration allows one to overcome this difficulty. Our approach generalizes the measure-valued strategies: this explains some known results, such as approximate completeness, but at the same time it also shows that either the optimal strategy is based on a finite number of bonds or it is not necessarily a measure-valued process. The first author gratefully acknowledges financial support from the CNR Strategic Project "Modellizzazione matematica di fenomeni economici".

A General Framework for Term Structure Models Driven by Levy Processes

2004

We describe a framework in which to generalize the Heath, Jarrow and Morton model for the term structure of interest rates. We represent the model in terms of the triplet of characteristics of the underlying semimartingales. We state and prove the necessary and sufficient conditions for absence of arbitrage in terms of the characteristics of the price process. The methodology is then extended to find sufficient conditions for absence of arbitrage in the defaultable case.

A theory of stochastic integration for bond markets

The Annals of Applied Probability, 2005

We introduce a theory of stochastic integration with respect to a family of semimartingales depending on a continuous parameter, as a mathematical background to the theory of bond markets. We apply our results to the problem of super-replication and utility maximization from terminal wealth in a bond market. Finally, we compare our approach to those already existing in literature.