PDF Ebook Fixed Income Pricing
This chapter surveys the literature on fixed-income pricing models, including dynamic term structure models (DTSMs) and interest rate sensitive, derivative pricing models. This literature is vast with both the academic and practitioner communities having proposed a wide variety of models and model-selection criteria. Central to all pricing models, implicitly or explicitly, are: (i) the identity of the state vector: whether it is latent or observable and, in the latter case, which observable series; (ii) the law of motion (conditional distribution) of the state vector under the pricing measure; and (iii) the functional dependence of the short-term interest rate on this state vector. A primary objective, then, of research on fixed-income pricing has been the selection of these ingredients to capture relevant features of history, given the objectives of the modeler, while maintaining tractability, given available data and computational algorithms. Accordingly, we overview alternative conceptual approaches to fixed-income pricing, highlighting some of the tradeoffs that have emerged in the literature between the complexity of the probability model for the state, data availability, the pricing objective, and the tractability of the resulting model.
A pricing model may be “monolithic” in the sense that it prices both bonds (as functions of a set of underlying state variables or “risk factors” – i.e., is a “term structure model”) and fixed-income derivatives (with pay-offs expressed in terms of the prices or yields on these underlying bonds). Alternatively, a model may be designed to price fixed-income derivatives, taking as given the current shape of the underlying yield curve. The former modeling strategy is certainly more comprehensive than the latter. However, researchers have often found that the latter approach offers more flexibility in calibration and tractability in computation when pricing certain derivatives.
Initially, taking the monolithic approach, we overview a variety of models for pricing default-free bonds and associated derivatives written on these (or portfolios of these) bonds. Basic issues in pricing fixed-income securities (FIS) for the case where the state vector follows a diffusion are discussed in Section 2. “Yield-based” DTSMs are reviewed in Section 3. Extensions of these pricing models to allow for jumps or regime shifts are explored in Sections 4 and 5, respectively.
Then, in Section 6, we turn to the case of defaultable securities. Here we start by considering a quite general framework in which there are multiple credit classes (possibly indexed by rating) and deriving pricing relations for the case where issuers may transition between classes according to a Markov process. Several of the most widely studied models for pricing defaultable bonds are compared by specializing to the case of a single credit class.
The pricing of fixed-income derivatives is overviewed in Section 7. Initially, we continue our discussion of DTSMs and overview recent research on the pricing of derivatives using yield-based term structure models. Then we shift our focus from monolithic models to models for pricing derivatives in which the current yield curve, and possibly the associated yield volatilities, are taken as inputs into the pricing problem. These include models based on forward rates (both for default-free and defaultable securities), and the LIBOR and Swaption Market models.
To keep our overview of the literature manageable we focus, for the most part, on term structure models and fairly standard derivatives on zero-coupon and coupon bonds (both default-free and defaultable), plain-vanilla swaps, caps, and swaptions. In particular, we do not delve deeply into many of the complex structured products that are increasingly being traded. Of particular note, we have chosen to side-step the important issue of pricing securities in which correlated defaults play a central role in valuation. 1 Additionally, we focus almost exclusively on pricing and the associated “pricing measures.” Our companion paper Dai and Singleton [2002] explores in depth the specifications of the market prices of risk that connect the pricing with the actual measures, as well as the empirical goodness-of-fit of models 2 under alternative specifications of the market prices of risks.
Contents
1 Introduction
2 Fixed-income Pricing in a Diffusion Setting
- 2.1 The Term Structure
2.2 FIS with Deterministic Payoffs
2.3 FIS with State-dependent Payoffs
2.4 FIS with Stopping Times
3 DTSMs for Default-free Bonds
- 3.1 One-factor DTSMs
3.2 Multi-factor DTSMs
4 DTSMs with Jump Diffusions
5 DTSMs with Regime Shifts
6 DTSMs with Rating Migrations
- 6.1 Fractional Recovery of Market Value
6.2 Fractional Recovery of Par, Payable at Maturity
6.3 Fractional Recovery of Par, Payable at Default
6.4 Pricing Defaultable Coupon Bonds
6.5 Pricing Eurodollar Swaps
7 Pricing of Fixed-Income Derivatives
- 7.1 Derivatives Pricing using DTSMs
7.2 Derivatives Pricing using Forward Rate Models
7.3 Defaultable Forward Rate Models with Rating Migrations
7.4 The LIBOR Market Model
7.5 The Swaption Market Model
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PDF Ebook Fixed Income Pricing
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