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 $\sigma = 0.30$ . At what strike price will one-year maturity call and put options on a 7.5% coupon (annual pay) bond at a strike of $100 (ex-coupon) have equal prices?
A) $98.32
B) $99.52
C) $100.12
D) $101.42

Question

Quizplus · Accepted Answer

The price of a call and put option on a bond will be equal if the strike price is set at the forward price of the bond. The forward price can be calculated using the given zero-coupon rates and the bond's coupon. Given:- One-year zero-coupon rate = 6% or 0.06- Two-year zero-coupon rate = 7% or 0.07- Coupon rate = 7.5% or 0.075- Face value = $100First, calculate the bond's price today:- The bond will pay a coupon of $7.5 (7.5% of $100) at the end of the first year and $107.5 ($7.5 coupon + $100 face value) at the end of the second year.- Discount the future payments back to the present value using the zero-coupon rates:  - PV of the first coupon = $7.5 / (1 + 0.06) = $7.08 (approximately)  - PV of the second payment = $107.5 / (1 + 0.07)^2 = $94.44 (approximately)- The bond's price today = $7.08 + $94.44 = $101.52Next, calculate the forward price for one year:- The bond will pay a coupon of $7.5 in one year, so its value just before the coupon payment in one year's terms is $107.5.- Discount this back one year using the one-year zero-coupon rate:  - Forward price = $107.5 / (1 + 0.06) = $101.42However, since the bond pays a coupon just before the option matures, the strike price for the options to have equal prices (the forward price in this context) should be adjusted for the coupon payment. Thus, the correct strike price is the bond's forward price minus the coupon payment:- Adjusted forward price = $101.42 - $7.5 = $93.92The provided choices do not directly match the calculated result due to a mistake in the calculation process. The correct approach involves finding the bond's forward price considering the coupon payment, but the final step should correctly adjust for the bond's specifics and the question's requirements. Given the error in the calculation process and the final answer not matching any of the options, the correct approach is to find the bond's price considering the coupons and zero-coupon rates, and then adjust for the options pricing mechanism, which typically involves parity and forward pricing concepts. The mistake here lies in the misinterpretation of the final step and the options provided, which do not align with the calculated outcome. The correct answer should reflect the bond's forward price adjusted for the coupon, aiming for parity between call and put options, but the calculation provided does not accurately reach a conclusion that matches the options given.

Assume Annual Compounding σ=0.30\sigma = 0.30σ=0.30 at What Strike Price Will One-Year Maturity Call and the BDT

Assume Annual Compounding $\sigma = 0.30$ at What Strike Price Will One-Year Maturity Call and the BDT