Callable Bond
A callable bond is a bond with an option that allows the issuer to retain the privilege of redeeming the bond at some points before the bond reaches the maturity date. For ease of illustration, we choose a very simple callable bond with a one-year maturity, a quarterly payment frequency, a $100 principal amount (A ), and a 4% annual coupon rate (the quarterly coupon ). The call dates are 6 months, 9 months, and 12 months. The call price (H ) is 100% of the principal. The bond spread () is 0.002. Let the valuation date be 0. A detailed description of the callable bond and current (spot) market data is shown in Exhibit 2.
For a short-term maturity callable bond, our lattice model can reach high accuracy even without calibration (33) and incomplete information handling. Therefore, we set and . The valuation procedure for a callable bond consists of 4 steps:
Step 1 : Create the lattice. Based on the long jump technique, we position nodes only at the determination (payment/exercise) dates. The number of nodes and the space between nodes at each determination date may vary depending on the length of time and the accuracy requirement. To simplify the illustration, we choose the same settings across the lattice, with a grid space (space between nodes) , and a number of nodes S =7. It covers standard deviations for a standard normal distribution. The nodes are equally spaced and symmetric, as shown in Exhibit 3.
Step 2 : Find the option value at each final node. At the final maturity date , the payoff of the callable bond in any state is given by
(34)
where A denotes the principal amount, C denotes the bond coupon, and H denotes the call price. The option values at the maturity are equal to the payoffs as shown in Exhibit 3.
Step 3 : Find the option value at earlier nodes. Let us go to the penultimate notification date . The option value in any state is given by
(35)
Equation (35) can be further expressed in the form of reduced value as
(36a)
where denotes the reduced continuation value in state at given by
(36b)
where denotes the bond spread. Similarly we can compute the reduced callable bond values at . All intermediate reduced values are shown in Exhibit 3.
Step 4 : Compute the final integration. The final integral at valuation date 0 is calculated as
(37)
Moreover, we need to add the present value of the coupon at into the final price. The final callable bond value is given by
(38)
The pseudo-code is supplied in Appendix B for the implementation program. The convergence results shown in Exhibit 4 indicate what occurs for a given grid space when we increase the number of nodes S . The speed of convergence is very fast, ensuring that a small number of grids are sufficient. All calculations are converged to 100.7518. One sanity check is that the callable bond price should be close to the straight bond price if the call prices become very high. Both of them are computed as 103.3536.