Suppose that the one-year and two-year zero-coupon rates are 6% and 7%, respectively (assume continuous compounding). After one year, let the one-year zero-coupon rates move down to $r _ { d }$ or up to $r _ { u } = 1.2 r _ { d }$ , with equal probability. The rate $r _ { u }$ that is arbitrage-free under these conditions is
A) 5.92%
B) 6.63%
C) 7.27%
D) 8.73%

Question

Quizplus · Accepted Answer

To find the arbitrage-free rate $r_u$, we use the formula for the forward rate: $f = \frac{(1 + r_2)^2}{(1 + r_1)} - 1$. With continuous compounding, this becomes $f = e^{2 \times 0.07 - 1 \times 0.06} - 1$. Solving gives $f \approx 0.0873$ or 8.73%.

Suppose That the One-Year and Two-Year Zero-Coupon Rates Are 6 rdr _ { d }rd​

Suppose That the One-Year and Two-Year Zero-Coupon Rates Are 6 $r _ { d }$