Term-structure models. A graduate course
This text by one of the leading authorities on term-structure and interest-rate models is useful for one semester graduate or advanced honors courses. Alternatively researchers in the field of mathematical finance, who wish to have a rather short reference text, may find this book appealing. The author comes straight to the point, without confusing the reader with to0 much material that is not directly related to the development of the theory. After a short introduction, the author introduces the main objects and instruments that are relevant in interest-rate theory in chapter 2. These include forward and future rates, fixed coupon bonds, interest-rate swaps, caps and floors. The author also discusses the various market conventions such as day count etc. and introduces Black's formula for caplets and swaptions, without proof. In chapter 3 the author discusses various techniques for estimating the term-structure, such as bootstrapping, non-parametric estimation, parametric estimation using various classes of splines and principal component analysis. Chapter 4 contains all necessary material on stochastic calculus. The presentation here is very brief and results are presented without proof, so that prior knowledge of stochastic calculus should be recommended for this text. Nevertheless, the chapter is written in a clear and rigorous way, and does its function in reminding the reader on all relevant techniques from stochastic calculus that are used in modern mathematical finance. In chapter 5 the author discusses short-rate models. He introduces affine term structure models, discusses the problem of inverting the forward curve and its relevance and provides various examples, which include all classical models, ranging from Vasicek and CIR to Hull and White. Chapter 6 is designated to the Heath-Jarrow-Morton framework. A particular emphasis lies on the discussion of absence of arbitrage and its consequences, i.e., the Heath-Jarrow-Morton drift condition, as well as the interplay between the short rate models from chapter 5 and the Heath-Jarrow-Morton framework. Forward measures and their advantageous when it comes to pricing bond options are discussed in chapter 7. The case of a Gaussian Heath-Jarrow-Morton framework as well as the case of a Black-Scholes model with Gaussian interest-rates are discussed in detail and in both cases the corresponding Black-Scholes formula is derived. In a very short chapter 8, the author discusses forwards and futures, while in chapter 9 he focuses on consistent term structure parametrization with a particular focus towards multi-factor models. As special cases affine and polynomial term structures are considered and the maximal degree problem is discussed and solved. Furthermore the Nelson-Siegel and the Svensson families are studied in the context of consistency. Chapter 10 gives a general introduction into affine processes, with a view towards their classification, existence and uniqueness. Applications include bond option pricing in affine models via the technique of Fourier decomposition. The LIBOR market model, its dynamic, implied bond markets and money market accounts, swaption pricing, Monte Carlo simulation, volatility structure and calibration are discussed in chapter 11. The final chapter 12 focuses on default risk, with particular emphasis given to the intensity based approach and the pricing of default risk. All chapters conclude with a section ``notes'', in which the author indicates further relevant literature and work that has been done by other authors. Each chapter also contains a significant number of well chosen exercises.
scientific article; zbMATH DE number 5292023