Pricing Fixed-Income Securities in an Information-Based Framework

In this paper we introduce a class of information-based models for the pricing of fixed-income securities. We consider a set of continuous- time information processes that describe the flow of information about market factors in a monetary economy. The nominal pricing kernel is at any given time assumed to be given by a function of the values of information processes at that time. By use of a change-of-measure technique we derive explicit expressions for the price processes of nominal discount bonds, and deduce the associated dynamics of the short rate of interest and the market price of risk. The interest rate positivity condition is expressed as a differential inequality. We proceed to the modelling of the price-level, which at any given time is also taken to be a function of the values of the information processes at that time. A simple model for a stochastic monetary economy is introduced in which the prices of nominal discount bonds and inflation-linked notes can be expressed in terms of aggregate consumption and the liquidity benefit generated by the money supply

Discrete-Time Interest Rate Modelling

This paper presents an axiomatic scheme for interest rate models in discrete time. We take a pricing kernel approach, which builds in the arbitrage-free property and provides a link to equilibrium economics. We require that the pricing kernel be consistent with a pair of axioms, one giving the inter-temporal relations for dividend-paying assets, and the other ensuring the existence of a money-market asset. We show that the existence of a positive-return asset implies the existence of a previsible money-market account. A general expression for the price process of a limited-liability asset is derived. This expression includes two terms, one being the discounted risk-adjusted value of the dividend stream, the other characterising retained earnings. The vanishing of the latter is given by a transversality condition. We show (under the assumed axioms) that, in the case of a limited-liability asset with no permanently-retained earnings, the price process is given by the ratio of a pair of potentials. Explicit examples of discrete-time models are provided

On the Representation of General Interest Rate Models as Square Integrable Wiener Functionals

In the setting proposed by Hughston & Rafailidis (2005) we consider general interest rate models in the case of a Brownian market information filtration $(\mathcal{F}_t)_{t\geq0}$. Let $X$ be a square-integrable $\mathcal{F}_\infty$-measurable random variable, and assume the non-degeneracy condition that for all $t<\infty$ the random variable $X$ is not $\mathcal{F}_t$-measurable. Let ${\sigma_t}$ denote the integrand appearing in the representation of $X$ as a stochastic integral, write $\pi_t$ for the conditional variance of $X$ at time $t$, and set $r_t = \sigma^2_t / \pi_t$. Then $\pi_t$ is a potential, and as such can act as a model for a pricing kernel (or state price density), where $r_t$ is the associated interest rate. Under the stated assumptions, we prove the following: (a) that the money market account process defined by $B_t = \exp (\int_0^t r_s \,ds)$ is finite almost surely at all finite times; and (b) that the product of the money-market account and the pricing kernel is a local martingale, and is a martingale provided a certain integrability condition is satisfied. The fact that a martingale is thus obtained shows that from any non-degenerate element of Wiener space satisfying the integrability condition we can construct an associated interest-rate model. The model thereby constructed is valid over an infinite time horizon, with strictly positive interest, and satisfies the relevant intertemporal relations associated with the absence of arbitrage. The results thus stated pave the way for the use of Wiener chaos methods in interest rate modelling, since any such square-integrable Wiener functional admits a chaos expansion, the individual terms of which can be regarded as parametric degrees of freedom in the associated interest rate model to be fixed by calibration to appropriately liquid sectors of the interest rate derivatives markets.Comment: 17 page

Information, Inflation, and Interest

We propose a class of discrete-time stochastic models for the pricing of inflation-linked assets. The paper begins with an axiomatic scheme for asset pricing and interest rate theory in a discrete-time setting. The first axiom introduces a "risk-free" asset, and the second axiom determines the intertemporal pricing relations that hold for dividend-paying assets. The nominal and real pricing kernels, in terms of which the price index can be expressed, are then modelled by introducing a Sidrauski-type utility function depending on (a) the aggregate rate of consumption, and (b) the aggregate rate of real liquidity benefit conferred by the money supply. Consumption and money supply policies are chosen such that the expected joint utility obtained over a specified time horizon is maximised subject to a budget constraint that takes into account the "value" of the liquidity benefit associated with the money supply. For any choice of the bivariate utility function, the resulting model determines a relation between the rate of consumption, the price level, and the money supply. The model also produces explicit expressions for the real and nominal pricing kernels, and hence establishes a basis for the valuation of inflation-linked securities

Quantum states and space-time causality

Space-time symmetries and internal quantum symmetries can be placed on equal footing in a hyperspin geometry. Four-dimensional classical space-time emerges as a result of a decoherence that disentangles the quantum and the space-time degrees of freedom. A map from the quantum space-time to classical space-time that preserves the causality relations of space-time events is necessarily a density matrix.Comment: 9 pages, to appear in the Proceedings of the 2nd International Symposium on Information Geometry and its Application

Pricing Fixed-Income Securities in an Information-Based Framework

In this paper we introduce a class of information-based models for the pricing of fixed-income securities. We consider a set of continuous- time information processes that describe the flow of information about market factors in a monetary economy. The nominal pricing kernel is at any given time assumed to be given by a function of the values of information processes at that time. By use of a change-of-measure technique we derive explicit expressions for the price processes of nom- inal discount bonds, and deduce the associated dynamics of the short rate of interest and the market price of risk. The interest rate positiv- ity condition is expressed as a differential inequality. We proceed to the modelling of the price-level, which at any given time is also taken to be a function of the values of the information processes at that time. A simple model for a stochastic monetary economy is introduced in which the prices of nominal discount bonds and inflation-linked notes can be expressed in terms of aggregate consumption and the liquidity benefit generated by the money supply.Fixed-income securities, interest rate theory, inflation, inflation-linked securities, non-linear filtering, incomplete information

Theory of Quantum Space-Time

A generalised equivalence principle is put forward according to which space-time symmetries and internal quantum symmetries are indistinguishable before symmetry breaking. Based on this principle, a higher-dimensional extension of Minkowski space is proposed and its properties examined. In this scheme the structure of space-time is intrinsically quantum mechanical. It is shown that the causal geometry of such a quantum space-time possesses a rich hierarchical structure. The natural extension of the Poincare group to quantum space-time is investigated. In particular, we prove that the symmetry group of a quantum space-time is generated in general by a system of irreducible Killing tensors. When the symmetries of a quantum space-time are spontaneously broken, then the points of the quantum space-time can be interpreted as space-time valued operators. The generic point of a quantum space-time in the broken symmetry phase thus becomes a Minkowski space-time valued operator. Classical space-time emerges as a map from quantum space-time to Minkowski space. It is shown that the general such map satisfying appropriate causality-preserving conditions ensuring linearity and Poincare invariance is necessarily a density matrix

Geometry of Thermodynamic States

A novel geometric formalism for statistical estimation is applied here to the canonical distribution of classical statistical mechanics. In this scheme thermodynamic states, or equivalently, statistical mechanical states, can be characterised concisely in terms of the geometry of a submanifold ${\cal M}$ of the unit sphere ${\cal S}$ in a real Hilbert space ${\cal H}$. The measurement of a thermodynamic variable then corresponds to the reduction of a state vector in ${\cal H}$ to an eigenstate, where the transition probability is the Boltzmann weight. We derive a set of uncertainty relations for conjugate thermodynamic variables in the equilibrium thermodynamic states. These follow as a consequence of a striking thermodynamic analogue of the Anandan-Aharonov relations in quantum mechanics. As a result we are able to provide a resolution to the controversy surrounding the status of `temperature fluctuations' in the canonical ensemble. By consideration of the curvature of the thermodynamic trajectory in its state space we are then able to derive a series of higher order variance bounds, which we calculate explicitly to second order.Comment: 7 pages, RevTe