Pricing Fixed-Income Derivatives
TWO QUITE DISTINCT approaches to the pricing of fixed-income derivatives have been pursued in the literature. One approach takes the yield curve as given—essentially the entire yield curve is the current state vector. Then, assuming no arbitrage opportunities, prices for derivative claims with payoffs that depend on the yield curve are derived. Examples of models in this first group are the widely studied “forward-rate” models of Heath et al. (1992), Brace et al. (1997), and Miltersen et al. (1997). Since the yield curve is an Shin put, there is typically no associated DTSM; the model used to price derivatives does not price the underlying bonds. The second approach starts with a DTSM, often in one of the families discussed in Chapter 12, which is used to simultaneously price the underlying fixed-income securities and the derivatives written against those securities, all under the assumption that there are no arbitrage opportunities. With the growing availability of time-series data on the implied volatilities of fixed-income derivatives, both approaches have been pursued in exploring the fits of pricing models to the historical implied volatilities of fixed-income derivatives.
In discussing the pricing of fixed-income derivatives, we place particular emphasis on the formulations of the pricing models underlying recent empirical studies of derivatives pricing models. We begin with a review of pricing with affine DTSMs.1 This is followed by an introduction to pricing with forward-rate-based models. Since these models are being introduced for the first time, we deal with the various pricing measures that have been used to price derivatives in some depth. We then turn to a discussion of some of the more striking empirical challenges that have been raised based
1 See Leippold and Wu (2002) for a discussion of the pricing of fixed-income derivatives in the class of quadratic-Gaussian DTSMs. Many of the solutions discussed subsequently to the pricing problems faced with affine DTSMs carry over, in suitably modified forms, to the QG class of models.