Algorithms behind Term Structure Models of Interest Rates: I. Valuation and Hedging of Interest Rates Derivatives with the Ho-Lee Model (original) (raw)
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SSRN Electronic Journal, 2000
In this article we implement the trinomial tree of the Hull-White model, which can be easily extended to allow different assumptions about the dynamics of the short rate process. We present the Mathematica algorithm for the extended Vasicek and the Black-Karasinski model. Whenever negative interest rates are generated with a positive probability, we make use of alternative branching processes, which guarantee the positivity of interest rates. Finally we show how to price simple options such as caplets, and compare the convergence of trinomial trees with different geometries.
Modeling the Term Structure of Interest Rates: A Review of the Literature
Foundations and Trends® in Finance, 2010
The last decades have seen the development of a profusion of theoretical models of the term structure of interest rates. The aim of this survey is to provide a comprehensive review of these continuous time modeling techniques of the term structure applicable to value and hedge defaultfree bonds and other interest rate derivatives. The originality of the survey lies in the fact that it provides a unifying framework in which most continuous-time term structure models can be nested and thus related to each other. Thus, we not only present the most important continuous-time term structure models in the literature but also provide a mathematically rigorous and unifying setting in which these models can be compared in terms of their similarities, distinguished in terms of their idiosyncratic features and in which their main contributions and limitations can easily be highlighted.
One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities
The Journal of Financial and Quantitative Analysis, 1993
This paper compares different approaches to developing arbitrage-free models of the term structure. It presents a numerical procedure that can be used to construct a wide range of one-factor models of the short rate that are both Markov and consistent with the initial term structure of interest rates.
Essays on the term structure of interest rates
2000
This volume contains five essays on topics related to interest rate theory.The first essay, Affine Term Structures and Short-Rate Realizations of Forward Rate Models Driven by Jump-Diffusion Processes, examines the problem of determining when a given forward rate model has a short-rate realization, and when a short-rate model gives rise to an affine term structure.The second essay, On the Inversion of the Yield Curve, co-authored with Tomas Bjork, considers a general benchmark short-rate factor model of the term structure of interest rates. It is showed that the benchmark model can be extended so that the implied theoretical term structure can be fitted exactly to an arbitrary initially observed yield curve. A general formula for pricing simple contingent claims in the extended model is also provided.The third essay, An Efficient Series Expansion Approach to a Two-Factor Model of the Term Structure of Interest Rates, presents a two-factor model where both factors follow CIR-type dif...
Dynamic Term Structure Modeling: The Fixed Income Valuation Course
2007
List of Figures. List of Tables. CHAPTER 1. A Simple Introduction to Continuous-Time Stochastic Processes. CHAPTER 2. Arbitrage-Free Valuation. CHAPTER 3. Valuing Interest Rate and Credit Derivatives: Basic Pricing Frameworks. CHAPTER 4. Fundamental and Preference-Free Single-Factor Gaussian Models. CHAPTER 5. Fundamental and Preference-Free Jump-Extended Gaussian Models. CHAPTER 6. The Fundamental Cox, Ingersoll, and Ross Model with Exponential and Lognormal Jumps. CHAPTER 7. Preference-Free CIR and CEV Models with Jumps. CHAPTER 8. Fundamental and Preference-Free Two-Factor Affine Models. CHAPTER 9. Fundamental and Preference-Free Multifactor Affine Models. CHAPTER 10. Fundamental and Preference-Free Quadratic Models. CHAPTER 11. The HJM Forward Rate Model. CHAPTER 12. The LIBOR Market Model. References. About the CD-ROM. Index.
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.
The analysis and valuation of interest rate options
Journal of Banking & Finance, 1993
This paper provides a simple, alternative model for the valuation of European-style interest rate options. The assumption that drives the hedging argument in the model is that the forward prices of bonds follow an arbitrary two-state process. Later, this assumption is made more specific by postulating that the discount on a zero-coupon bond follows a multiplicative binomial process. In contrast to the BlackkScholes assumption applied to zero-coupon bonds, the limiting distribution of this process has the attractive features that the zero-bond price has a natural barrier at unity (thus precluding negative interest rates), and that the bond price is negatively skewed. The model is used to price interest rate options in general. and interest rate caps and floors in particular. The model is then generalized and applied to European-style options on bonds. A relationship is established between options on swaps and options on coupon bonds. The generalized model then provides a computationally simple formula, closely related to the Black-Scholes formula, for the valuation of European-style options on swaps.
A yield-only model for the term structure of interest rates
Annals of Actuarial Science, 2013
This paper develops a term structure model for the UK nominal, real and implied inflation spot zero-coupon rates simultaneously. We start with fitting a descriptive yield curve model proposed by Cairns (1998) to fill the missing values for certain given days at certain maturities in the yield curve data provided by the Bank of England. We compare four different fixed ‘exponential rate’ parameter sets and decide the set of parameters which fits the data best. With the chosen set of parameters we fit the Cairns model to the daily values of the term structures. By applying principal component analysis on the hybrid data (Bank of England data and fitted spot rates for the missing values) we find three principal components, which can be described as ‘level’, ‘slope’ and ‘curvature’, for each of these series. We explore the relation between these principal components to construct a ‘yield-only’ model for actuarial applications. Main contribution of this paper is that the models developed ...