ABSTRACT:

This talk surveys results from the following three papers:

Complex Times: Asset Pricing and Conditional Moments under Non-Affine Diffusions

Many applications in continuous-time financial economics require calculation of conditional moments or contingent claims prices, but such expressions are known in closed form for only a few specific models. Power series approximations (in the time variable) to the solutions are easily derived, but often fail to convergence, even for very short time horizons. We characterize a large class of continuous-time conditional moment and contingent claim pricing problems with solutions that are analytic in the time variable, and that therefore can be approximated by convergent power series. The ability to approximate solutions accurately and in closed form simplifies the estimation of non-affine continuous-time term structure models, since the bond pricing problem must be solved for many different parameter vectors during a typical estimation procedure. When combined with time transformation methods, our technique often allows construction of approximations of contingent claim prices (such as bond prices in term structure models) that converge uniformly in maturity, and are therefore very accurate even with extremely long time horizons.

Changing Times: Accurate Solutions to Pricing and Conditional Moment Problems in Non-Affine Continuous-Time Models

Applications in continuous-time financial economics often require calculation of conditional moments or contingent claims prices, which are known in closed-form only for a very limited class of models. Recent research identifies a large class of models for which such problems can be solved through power series approximation. However, the convergence properties of these power series are often very poor for long time horizons.

We develop a method of time transformation, in which the variable representing calendar time is replaced by a non-affine function of calendar time. With appropriate choice of the time transformation, the convergence properties of power series approximations can often be dramatically improved, sometimes resulting in uniform convergence for all time horizons.

Such approximations are therefore suitable for applications such as bond pricing, in which the time-to-maturity may be many years. The ability to approximate solutions accurately and in closed-form simplifies the estimation of non-affine continuous-time term structure models, since the bond pricing problem must be solved for many different parameter vectors during a typical estimation procedure. We show through a bond pricing example that the approximations are easy to derive and highly accurate over a wide range of interest rate levels for arbitrarily long maturities.

Asset Prices and Conditional Moments in Multifactor Non-Affine Models

In recent research, methods are developed to approximate conditional moments and contingent claims prices in a large class of non-affine continuous-time models. These methods employ series approximations in the time variable, and use change of variable techniques to improve convergence properties. In some cases, convergence is uniform for all time horizons; in other words, the approximation converges as rapidly for time horizons of many years as it does for time horizons of a few days. This strong convergence result makes such approximations useful for applications such as bond pricing in term structure models, in which we must sometimes consider maturities of many years. However, at present, these methods have only been developed for scalar diffusion problems. In some limited cases, multifactor bond pricing problems can be broken into several independent scalar problems; however, the necessary restrictions severely limit the class of models to which these approximation techniques can be applied. We extend this earlier research to allow accurate approximation of contingent claim prices in a much broader class of multifactor models. Our technique proceeds through several stages.

First, a large class of non-affine multifactor pricing problems is shown to be equivalent, after change of variables, to a class of conditional moment problems in a multifactor affine diffusion setting. We then develop a method for approximation of the solution to the transformed problem, such that the approximations often converge uniformly for arbitrary time horizons. Our technique is to embed the path of ``true'' time in a higher-dimensional space of ``artificial'' time. The bond (or other contingent claim) price is meaningful only as a function of ``true'' time; however, extending this function to ``artificial'' time greatly simplifies the problem of constructing series approximations with good convergence properties. The true bond price function is then the restriction of the extended price function to a curve (representing ``true'' time) through the higher-dimensional space of ``artificial'' time. We show through examples, in which the bond price function is known in closed-form, that the approximations are easy to derive and converge very rapidly, in many cases, for arbitrarily long time horizons. We then develop a method for construction of a large class of non-affine multifactor term structure models to which our technique can be applied, resulting in accurate approximations of bond prices and yields (regardless of maturity) even when the true bond price function is not known in closed-form.