THE BOND MARKETS: ANALYTICAL METHODOLOGY 2

The Stochastic Process. The following is the mean reverting process for interest rates employed in this paper:
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where 9 is a central tendency parameter for the interest rate r, which reverts at rate k. Therefore the interest rate evolves with mean-reverting drift and two random terms, one a diffusion and the other a Poisson process embodying a random jump J. The variance coefficient of the diffusion is v2 and the arrival of jumps is governed by a Poisson process ж with arrival frequency parameter /1, which denotes the number of jumps per year. The jump size J can be completely general and may be a constant or drawn from a probability distribution. The diffusion and Poisson processes are independent of each other, and each of them is independent of J as well.

The Characteristic Function. Assessing the impact of jumps on pricing interest rate dependent securities requires an analysis of the probability distribution of a jump-diffusion interest rate process, and the moments of this distribution. The characteristic function of the jump-diffusion process offers the raw material with which to derive the density functions as well as the moments.

Assume that we are at time t — 0, and that we are looking ahead to time t = T. We are interested in the distribution of r(T) given the current value of the interest rate 7′(0) = r0 — r. In order to derive the T-interval characteristic function F(r,T\ s) for the process (2.1), (s is the characteristic function parameter) we solve its Kolmogorov backward equation (KBE) subject to the boundary condition that
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The closed-form characteristic function above exploits the fact that we can solve the partial-differential difference equation for the characteristic function. No other solutions are currently known to exist for interest rate processes.