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BOND PRICING, INTEREST-RATE PROCESSES, AND THE LIBOR MARKET MODEL

Ser-Huang Poon and Richard Stapleton

in Asset Pricing in Discrete Time: A Complete Markets Approach

‘Bond Pricing, Interest-rate Processes, and the LIBOR Market Model’ uses the complete market, pricing kernel approach to value bonds, given stochastic interest rates. To value interest-rate ... More

‘Bond Pricing, Interest-rate Processes, and the LIBOR Market Model’ uses the complete market, pricing kernel approach to value bonds, given stochastic interest rates. To value interest-rate derivatives, one important and practical problem is to model bond prices and interest rates with the correct drifts. The authors derive here the drift of the bond prices and interest rates under the period-by-period risk-neutral measure. As a special case, they derive the drift of the forward London Interbank Offer Rate (LIBOR) in what is generally known as the LIBOR Market Model.Less

Forward Rate Models

Tomas Björk

in Arbitrage Theory in Continuous Time

The drawbacks associated with short rate models have prompted various authors to propose models that use more than one state variable. In the method proposed by Heath-Jarrow-Morton (HJM), the entire ... More

The drawbacks associated with short rate models have prompted various authors to propose models that use more than one state variable. In the method proposed by Heath-Jarrow-Morton (HJM), the entire forward rate curve is chosen as the (infinite dimensional) state variable. This chapter discusses the HJM framework, martingale modelling, and the Musiela parameterization. Practice exercises are included.Less

Heath‐Jarrow‐Morton Models

Claus Munk

in Fixed Income Modelling

Instead of calibrating a simple diffusion model to match the observed yield curve, a more natural modelling approach is to start from the observed yield curve and then modelling its possible future ... More

Instead of calibrating a simple diffusion model to match the observed yield curve, a more natural modelling approach is to start from the observed yield curve and then modelling its possible future evolution. This is the approach taken by the so-called Heath–Jarrow–Morton (HJM) models studied in this chapter. The HJM models specify the dynamics of all forward interest rates and a key result is a link between the drift and volatilities of the forward rates. The main characteristics, advantages, and drawbacks of this approach are presented as well as several specific HJM models including highly tractable Gaussian models. The relationship between HJM models and the diffusion models discussed in previous chapters is discussed.Less

Forward Rate Models

Tomas Björk

in Arbitrage Theory in Continuous Time

Here we present the Heath–Jarrow–Morton approach of modelling the evolution of the entire forward rate curve.

Analysis of Positively Valued Economic Time Series

Bernt P. Stigum

in Econometrics in a Formal Science of Economics: Theory and the Measurement of Economic Relations

Chapter VII has two purposes. One is to study the methodological problems that arise in analysing positively valued time series in foreign exchange. The other is to contrast the analysis of time ... More

Chapter VII has two purposes. One is to study the methodological problems that arise in analysing positively valued time series in foreign exchange. The other is to contrast the analysis of time series in formal econometrics with the analysis of such data in present-day econometrics. The chapter presents an axiomatic data confrontation of a theory of spot and forward exchange in foreign currency markets. In the formulation of the axioms, actual and auxiliary theory and data variables interact in such a way that the problem that usually arise in the analysis of positively valued time series disappears. The data for the empirical analysis comprise observations on spot and forward exchange rates in the market for Swiss Francs and US Dollars. In the empirical analysis, the given data are analysed, first, with the prescriptions of formal econometrics and, then, with the prescriptions on which present-day econometric time-series analysis insist. The statistical results yield different descriptions of the dynamics of foreign exchange and different inferences about the economics of social reality. In doing that the two contrasting empirical analyses provide interesting ingredients for the discussion of how best to incorporate economic theory in empirical analyses.Less

Valuing and Analyzing Fixed Income Derivatives

Koray D. Simsek and Halil Kiymaz

in Debt Markets and Investments

Derivatives valuation is based on the key principle of no-arbitrage pricing. This chapter presents valuation models for various types of fixed income derivatives, including forward rate agreements ... More

Derivatives valuation is based on the key principle of no-arbitrage pricing. This chapter presents valuation models for various types of fixed income derivatives, including forward rate agreements (FRAs), interest rate swaps, Eurodollar and Treasury bond futures, bond options, caps and floors, swaptions, and options on interest rate futures. Following the financial crisis that began in the summer of 2007, major changes occurred in the practice of fixed income derivatives valuation, particularly regarding the adoption of overnight indexed swaps (OIS) as a source of the risk-free rate. This chapter shows how OIS discounting is implemented in FRA pricing and swap valuation. Traditional approaches such as cost of carry valuation in futures pricing are illustrated. With respect to option valuation, this chapter explains the risk-neutral pricing approach as well as closed-form solutions such as the Black model. The chapter also provides numeric examples to illustrate the practical use of the presented models and formulas.Less

Bonds and Interest Rates

Tomas Björk

in Arbitrage Theory in Continuous Time

In this chapter the reader is introduced to the basic concepts of interest rate theory. Starting with a market for zero coupon bonds we define the relevant interest rates such as the short rate, the ... More

In this chapter the reader is introduced to the basic concepts of interest rate theory. Starting with a market for zero coupon bonds we define the relevant interest rates such as the short rate, the spot rates, and the forward rates. There is an in-depth study of the relations between the dynamics of these rates, and we also discuss some more applied topics as fixed coupon bonds, floating rate bonds, yields, duration, and convexity.Less

Bonds and Interest Rates

Tomas Björk

in Arbitrage Theory in Continuous Time

In this chapter, the reader is introduced to the basic facts and concepts concerning bond markets.

Derivatives Markets

Halil Kiymaz and Koray D. Simsek

in Debt Markets and Investments

Interest rate derivatives markets have enjoyed substantial growth since the late 1990s. This chapter discusses the development of these markets since 2000 and introduces the most popular interest ... More

Interest rate derivatives markets have enjoyed substantial growth since the late 1990s. This chapter discusses the development of these markets since 2000 and introduces the most popular interest rate derivative instruments. Although forward rate agreements and interest rate swaps are important examples of over-the-counter (OTC) products, futures on interest rates and bonds are innovations of organized exchanges. Both OTC interest rate options and exchange-traded options on interest rate futures are discussed to illustrate an overlapping area of both types of derivatives markets. Participants in debt markets are also exposed to both interest rate and credit risk. To mitigate the latter risk, the OTC fixed income derivatives markets provide credit default swaps (CDSs). As credit derivatives are also a subset of fixed income derivatives, CDSs are discussed further.Less

Term Structure Models

Kerry E. Back

in Asset Pricing and Portfolio Choice Theory

Bond yields and forward rates are defined. The fundamental PDE is derived. Affine term strucure models are explained, including the Vasicek model and the Cox‐Ingersoll‐Ross square root model. ... More

Bond yields and forward rates are defined. The fundamental PDE is derived. Affine term strucure models are explained, including the Vasicek model and the Cox‐Ingersoll‐Ross square root model. Gaussian affine models, completely affine models, and multifactor CIR models are explained. Quadratic models are described. The various versions of the expectations hypothesis are explained. We can fit a given yield curve by adding a deterministic function of time to an interest rate model or allowing model parameters to be time varying. Heath‐Jarrow‐Morton models are explained, and it is shown that drifts of forward rates under the risk neutral probability are determined by their volatilities.Less

Molecular Assemblies, Probes, and Proteomics in Flow Cytometry

Steven W. Graves and John P. Nolan

in Flow Cytometry for Biotechnology

The many proteins and nucleic acids encoded in the genome predominantly perform their functions as macromolecular assemblies. In fact, modern biomedical research often targets the interactions of ... More

The many proteins and nucleic acids encoded in the genome predominantly perform their functions as macromolecular assemblies. In fact, modern biomedical research often targets the interactions of individual molecules of these assemblies, usually by disrupting or enhancing specific contacts, to provide treatment for many different diseases. Therefore, efficient pharmaceutical design requires knowledge of how macromolecular assemblies are built and function. To achieve this goal, sensitive and quantitative tools are essential. This chapter will discuss the use of flow cytometry as a general platform for sensitive measurement and quantification of molecular assemblies. First, this chapter will introduce general methods for analysis of molecular interactions along with a comparison of flow cytometry with these methods. Second, an overview of current flow cytometry instrumentation, assay technologies, and applications in molecular assembly analysis will be given. Third, the implementation of the above approaches in molecular assembly will be discussed. Finally, potential future directions of flow cytometry in molecular assembly analysis will be explored. At present, the analysis of macromolecular assemblies is performed by a wide variety of techniques that are chosen for the target molecules under study (proteins, DNA, lipids, etc.), the type of measurement required (kinetic or equilibrium), and whether the assembly of interest needs to be studied in vivo or in vitro. This continuum of techniques can be divided into the heterogeneous assays, which require a separation step to resolve products from reactants, and homogeneous assays, which can measure interactions without a separation step. Heterogeneous assays, in general, use radioisotopes, which are not perturbing; offer excellent sensitivity; and provide accurate quantification. The products are quantified after a separation step such as gel filtration, gel electrophoresis, or centrifugation. Rapid quench methods can provide subsecond kinetic resolution; however, the added separation steps are tedious and make collection of kinetic time courses difficult, as each time point must be separated and measured individually. Furthermore, in the time it takes the separation to occur, the interaction of interest can dissociate, which is a problem specific to low-affinity assemblies. Nonetheless, by using rapid chemical quench techniques, reaction times as short as a few milliseconds can be observed. Homogenous assays can be separated into solution- or surface-based assays. Solutionbased assays measure an optical signal generated by the assembly to quantify an interaction. High component concentrations (micromolar) allow changes in intrinsic molecular properties, such as protein fluorescence or circular dichroism, to be used to study molecular assemblies. For greater sensitivity (nanomole component concentrations), resonance energy transfer or polarization assays using exogenous fluorescent labels can be used. In combination with stopped-flow spectroscopy methodologies, solution-based assays allow reactions to monitored in a continuous fashion with submillisecond dead times. Less

Diffusion, Buffering, and Binding

Christof Koch

in Biophysics of Computation: Information Processing in Single Neurons

In Chap. 9 we introduced calcium ions and alluded to their crucial role in regulating the day-to-day life of neurons. The dynamics of the free intracellular calcium is controlled by a number of ... More

In Chap. 9 we introduced calcium ions and alluded to their crucial role in regulating the day-to-day life of neurons. The dynamics of the free intracellular calcium is controlled by a number of physical and chemical processes, foremost among them diffusion and binding to a host of different proteins, which serve as calcium buffers and as calcium sensors or triggers. Whereas buffers simply bind Ca2+ above some critical concentration, releasing it back into the cytoplasm when [Ca2+]i has been reduced below this level, certain proteins— such as calmodulin—change their conformation when they bind with Ca2+ ions, thereby activating or modulating enzymes, ionic channels, or other proteins. The calcium concentration inside the cell not only determines the degree of activation of calcium-dependent potassium currents but—much more importantly—is relevant for determining the changes in structure expressed in synaptic plasticity. As discussed in Chap. 13, it is these changes that are thought to underlie learning. Given the relevance of second messenger molecules, such as Ca2+, IP3, cyclic AMP and others, for the processes underlying growth, sensory adaptation, and the establishment and maintenance of synaptic plasticity, it is crucial that we have some understanding of the role that diffusion and chemical kinetics play in governing the behavior of these substances. Today, we have unprecedented access to the spatio-temporal dynamics of intracellular calcium in individual neurons using fluorescent calcium dyes, such as fura-2 or fluo-3, in combination with confocal or two-photon microscopy in the visible or in the infrared spectrum (Tsien, 1988; Tank et al., 1988; Hernández-Cruz, Sala, and Adams, 1990; Ghosh and Greenberg, 1995). Less