Volatility Structures of Forward Rates and the Dynamics of the Term Structure (original) (raw)
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Volatility Structures of Forward Rates and the Dynamics of the Term STRUCTURE1
Mathematical Finance, 1995
For general volatility structures for forward rates, the evolution of interest rates may not be Markovian and the entire path may be necessary to capture the dynamics of the term structure. This article identifies conditions on the volatility structure of forward rates that permit the dynamics of the term structure to be represented by a two‐dimensional state variable Markov process. the permissible set of volatility structures that accomplishes this goal is shown to be quite large and includes many stochastic structures. In general, analytical characterization of the terminal distributions of the two state variables is unlikely, and numerical procedures are required to value claims. Efficient simulation algorithms using control variates are developed to price claims against the 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.
Econometrica, 1992
This paper presents a unifying theory for valuing contingent claims under a stochastic term structure of interest rates. The methodology, based on the equivalent martingale measure technique, takes as given an initial forward rate curve and a family of potential stochastic processeE for its subsequent movements. A no arbitrage condition restricts this family of processes yielding valuation formulae for interest rate sensitive contingent claims which do not explicitly depend on the market prices of risk. Examples are provided to illustrate the key results.
On Finite Dimensional Realizations of Forward Price Term Structure Models
2004
In this paper we study a fairly general Wiener driven model for the term structure of forward prices. The model, under a fixed martingale measure, Q, consists of two infinite dimensional stochastic differential equations (SDEs). The first system is a standard HJM model for (forward) interest rates, driven by a multidimensional Wiener process W. The second system is an infinite
A Markovian Defaultable Term Structure Model with State Dependent Volatilities
Social Science Research Network, 2004
The defaultable forward rate is modeled as a jump diffusion process within the Schönbucher (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.
A Complete Yield Curve Description of a Markov Interest Rate Model
International Journal of Theoretical and Applied Finance, 2003
In the interest-rate market, the forward rate f (t, T ) and the yield Y (t, T ) can be written in terms of the bond price P (t, T ). Conversely, the bond price P (t, T ) can be given in terms of the forward and yield rates. Although, these three descriptions of the yield curve are equivalent, it is not always straightforward to compute them from the short rate and express one in terms of the other. Starting from a simple but very general assumption that the short-term rate r is a function of a continuous time Markov chain, we aim to get explicit analytic solutions to these three yield curve descriptions. Using stochastic flows, the bond price P (t, T ) is derived and thus the formula for the yield rate is immediately obtained. The forward measure is introduced establishing the Unbiased Expectation Hypothesis and this remarkable result is applied to calculate the dynamics of the forward rate. Our results therefore complete a term structure characterization in the study of a Markov interest rate model.
Term Structure Movements and Pricing Interest Rate Contingent Claims
Journal of Finance, 1986
This paper derives an arbitrage-free interest rate movements model (AR model). This model takes the complete term structure as given and derives the subsequent stochastic movement of the term structure such that the movement is arbitrage free. We then show that the AR model can be used to price interest rate contingent claims relative to the observed complete term structure of interest rates. This paper also studies the behavior and the economics of the model. Our approach can be used to price a broad range of interest rate contingent claims, including bond options and callable bonds. INTEREST RATE OPTIONS, CALLABLE bonds, and floating rate notes are a few examples of interest rate contingent claims. They are characterized by their finite lives and their price behavior, which crucially depends 'on the term structure and its stochastic movements. In recent years, with the increase in interest rate volatility and the prevalent use of the contingent claims, the pricing of these securities has become a primary concern in financial research. The purpose of this paper is to present a general methodology to price a broad class of interest rate contingent claims. The crux of the problem in pricing interest rate contingent claims is to model the term structure movements and to relate the movements to the assets' prices. Much academic literature has been devoted to this problem. One earlier attempt is that of Pye [15]. He assumed that the interest rates move according to a (Markov) transition probabilities matrix, and he then used the expectation hypothesis to price the expected cash flow of the asset-in his case, a callable bond (Pye [16]). Recently, investigators have focused more on developing equilibrium models. Cox, Ingersoll, and Ross (CIR) [7] assumed that the short rate follows a meanreverting process. By further assuming that all interest rate contingent claims are priced contingent on only the short rate, using a continuous arbitrage argument they derived an equilibrium pricing model. Brennan and Schwartz (BS) [2] extended the CIR model to incorporate both short and long rates and studied the pricing of a broad range of contingent claims (BS [1, 3]). In these
Continuous-time term structure models: Forward measure approach
Finance and Stochastics, 1997
The problem of term structure of interest rates modelling is considered in a continuous-time framework. The emphasis is on the bond prices, forward bond prices and so-called LIBOR rates, rather than on the instantaneous continuously compounded rates as in most traditional models. Forward and spot probability measures are introduced in this general setup. Two conditions of noarbitrage between bonds and cash are examined. A process of savings account implied by an arbitrage-free family of bond prices is identified by means of a multiplicative decomposition of semimartingales. The uniqueness of an implied savings account is established under fairly general conditions. The notion of a family of forward processes is introduced, and the existence of an associated arbitrage-free family of bond prices is examined. A straightforward construction of a lognormal model of forward LIBOR rates, based on the backward induction, is presented.