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.