Changing Times: The Pricing Problem in Non-Linear Models

Abstract

Finding conditional moments and derivative prices is a common application in continuous-time financial economics, but these quantities are known in closed-form only for a few specific models. Recent research identifies a large class of models for which solutions to such problems have convergent power series, allowing approximation even when not known in closed-form. However, such power series may converge slowly or not at all for long time horizons, limiting their practical use. We develop the method of time transformation, in which the variable representing time is replaced by a non-linear function of itself. With appropriate choice of the time transformation, power series often converge for much longer time horizons, and also much faster, sometimes uniformly for all time horizons. For applications such as bond pricing, in which the time-to-maturity may be many years, rapid convergence is very important for practical application. 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 several examples that the series are easy to derive, and, using term structure models for which bond prices are known explicitly, also show that the series are extremely accurate over a wide range of interest rate levels for arbitrarily long maturities; in some cases, they are many orders of magnitude more accurate than series constructed without time transformations. Other potential applications include pricing of callable bonds and credit derivatives.