THE BOND MARKETS: ANALYTICAL METHODOLOGY 3

The Moments. The moments of the jump-diffusion process offer valuable insights. First, the behavior of options prices may be inferred from a study of the moments. Second, the moments are easily used in method of moments estimation models. In this subsection, the derivation of the moments incorporates two innovations: (i) moments are obtained for any jump distribution, and (ii) the moments are derived without necessarily obtaining the characteristic function in closed form.
w6631-3

w6631-4
Analytical Jump-diffusion Models for Bond Pricing. Equation (2.1) states the statistical process for the interest rate. However, the pricing of interest rate sensitive securities is undertaken by translating this statistical process to a risk-neutral one. Assume that an equivalent martingale measure exists such that the risk-neutral (drift adjusted) interest rate process is

w6631-5
The expression д[£(т)] accommodates any jump distribution provided the moments are finite. In general, the integral equation for A(t) (which is the solution to an ordinary differential equation) does not always admit a closed-form. Specifically, even in the simplest cases, when J is a constant or is distributed Gaussian, no closed-form solution is achievable. Das and Foresi [23] find that in the special case of a jump with a sign based on a Bernoulli distribution and a size based on an exponential distribution, it is possible to obtain a closed-form solution.

Chacko [16],[17] extends the Das-Foresi model to incorporate stochastic volatility and stochastic mean as well by exploiting the facile properties of the Bernoulli-exponential form. When the Bernoulli-exponential form is not availed of, the ordinary differential equation for A(t) is solved numerically by Runge-Kutta methods.

An important aspect of this solution is the fact that the yields [= ^ In (P[r])] are ‘affine’, i.e. linear functions of the short rate. This is a useful property for estimation purposes.

This concludes the development of all the analytics needed for estimating and pricing jump-diffusion based term structure derivative securities. The next section deals with the application of these methods to bond market data.