European Options on Bonds
In this chapter, we use the three-factor Heath, Jarrow, and Morton (HJM) yield curve model to value European options on corporate bonds. As we noted in Chapter 19, Jarrow (2004) has summarized the conditions under which a risky bond issuer’s (called ABC Corporation in this chapter) coupon-bearing bond issues can be analyzed as a portfolio of zero-coupon bonds using the formulas of Chapter 16 and Chapter 19. We explained in Chapter 17 that we can create today’s yield curve for ABC Corporation in one of two ways. In method one, we estimate the term structure of default risk using logistic regression for ABC and then apply the techniques of Chapter 17 to estimate what the intersection of supply and demand for credit would be for ABC corporation on that day; this would be the yield curve that we would use to price ABC securities and derivatives on them. The second method is to collect data on existing ABC bond prices and apply the yield curve smoothing techniques of Chapter 5 to extract a smooth, continuous yield curve.
As an example, we looked at the net present value (“dirty price,” which is price plus accrued interest) on 77 bonds for a prestigious U.S. bond issuer on August 2, 2004. We smoothed the U.S. Treasury curve using the maximum smoothness forward rate approach of Chapter 5. We then solved for the maximum smoothness forward credit spread curve that minimized the sum of squared pricing errors on the bonds of this firm, which we call ABC ...