American Fixed Income Options
In Chapters 21 through 26, we have emphasized the valuation of securities where the option embedded in the security was exercisable at only one date. These European options, as we have shown, generally have explicit analytical solutions in the Vasicek family of term structure models. As we saw in Chapter 3, however, one-factor interest rate models are very inaccurate models of interest rate movements. In this chapter, we continue to emphasize numerical solutions using the example of the three-factor Heath, Jarrow, and Morton (HJM) model from Chapter 9. In previous chapters, we have frequently remarked that an HJM Monte Carlo simulation was the preferred valuation methodology, with an HJM bushy tree used only for expositional purposes. In this chapter, the situation is reversed. Using the bushy tree approach is the most accurate technique for valuing American fixed income options. Monte Carlo simulation, even using the HJM approach, is one notch lower in accuracy for reasons we explain. Please note, however, that, even in the case of a one-factor term structure model, numerical methods are necessary for the accurate valuation of American fixed income options. This is even truer when the issuer of the security has a positive probability of default. The interaction of the security’s payoffs with the credit risk of the issuer has interesting and important implications.
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