Exam 27: Factor Models of the Term Structure

Assume annual compounding. The one-year and two-year zero-coupon rates in the BDT model are 6% and 7%. The volatility is given to be $\sigma = 0.30$ . What are the one-year rates (up and down) after one year?

In the Cox-Ingersoll-Ross (CIR 1985) model, you are given that $d r _ { t } = \kappa \left( \theta - r _ { t } \right) d t + \sigma \sqrt { r _ { t } } d W _ { t }$ where $x = 0.5$ , $\theta = 0.06$ , $\sigma = 0.10$ , and the current short rate of interest is $r _ { 0 } = 0.08$ . What is the expected short rate of interest one year hence?

The Ho & Lee (1986) model directly models the following on a binomial tree:

In the Ho & Lee (1986) model, assume that the initial curve of zero-coupon rates for one and two years is 6% and 7%, respectively. Assume that the probability of an upshift in discount functions is equal to that of a downshift. If the parameter $\delta = 0.95$ , then the price of a one-year maturity call option on a two-year $100 face value zero-coupon bond in the up node after one year at a strike of $92 will be