Tools › Investing › Bonds › Callable Bond and OAS › Example

Callable bond on a binomial rate tree: straight value, callable value, call option, effective duration and convexity

Tool	Callable Bond and OAS
Source	Black, F., Derman, E., and Toy, W. (1990). "A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options." Financial Analysts Journal, Vol. 46, No. 1, pp. 33–39.
Level	Graduate

Problem

A 10-year bond of face value 100 pays a 6 % coupon twice a year and can be called by the issuer at 100 on any coupon date from year 3. Yields are flat at 5 % and the short rate has a volatility of 15 % a year. Build a lognormal binomial tree of the half-year rate that reprices every zero-coupon bond, value the bond backwards with the issuer's call, and find the value of the call option, the yields, and the effective duration and convexity.

Inputs to Enter

Face value 100, coupon 6 % paid semi-annually, maturity 10 years (20 tree steps); callable at 100 from year 3
Flat yield 5 % (semi-annual compounding), rate volatility 15 %; curve moved by 25 basis points for duration

Open the Callable Bond and OAS tool (loaded with this example)

Answers

Straight bond: 3 · 15.589 + 100 · 0.6103 = 107.795
Tree: ratio of neighbouring rates u = exp(2 · 0.15 · √0.5) = 1.2363; step 0 rate 5.000 %; step 1 rates 4.473 % and 5.530 %; step 2 rates 4.002 %, 4.948 %, 6.118 %
First call date, lowest rate 2.573 %: continuation value 101.692 exceeds the call price, value 100; the bond is called at 101 of the 189 call-date nodes
Callable bond today = (0.5 · (98.703 + 102.940) + 3) / 1.025 = 101.289; call option = 107.795 − 101.289 = 6.505
Yield to maturity 5.828 %, yield to call and to worst 5.528 % at the tree value
Effective duration 4.37 years (straight bond 7.57); effective convexity −118 (straight bond +71)
Advanced: at a market price of 100.50 the option-adjusted spread is 20.1 basis points

Notes

The tool's default inputs are this case; press Submit. The tree is the Black-Derman-Toy lognormal model with one constant volatility. The numbers are the tool's own calculation, checked against an independent numpy / scipy implementation and closed forms (test_callable_bond_oas.php); not a worked example printed in the cited paper. A calculator for education and analysis, not investment advice.