A $100 face value one-year risk-free discount bond is priced at $95. The two-year discount bond is priced at $90. After one year, the two-year bond will take one of three possible prices with defined probabilities. Which of the following sets of prices is acceptable from a no-arbitrage standpoint?
A) $\{ 99.74,94.74,89.74 \}$ with equal probabilities.
B) $\{ 99.74,94.74,89.74 \}$ with probabilities $\{ 0.25,0.50,0.25 \}$ .
C) Both (a) and (b).
D) The problem is not well-defined because there is no unique martingale probability measure for the two-year bond.

Question

A $100 face value one-year risk-free discount bond is priced at $95. The two-year discount bond is priced at $90. After one year, the two-year bond will take one of three possible prices with defined probabilities. Which of the following sets of prices is acceptable from a no-arbitrage standpoint? 
A) $\{ 99.74,94.74,89.74 \}$ with equal probabilities.
B) $\{ 99.74,94.74,89.74 \}$ with probabilities $\{ 0.25,0.50,0.25 \}$ .
C) Both (a) and (b).
D) The problem is not well-defined because there is no unique martingale probability measure for the two-year bond.

Quizplus · Accepted Answer

The no-arbitrage principle requires that the expected return of the bond, when discounted at the risk-free rate, equals its current price. Both sets of prices in options (A) and (B) can satisfy this condition depending on the risk-free rate and the probabilities assigned, hence no arbitrage opportunity is explicitly indicated by either set of prices alone.

A $100 Face Value One-Year Risk-Free Discount Bond Is Priced {99.74,94.74,89.74}\{ 99.74,94.74,89.74 \}{99.74,94.74,89.74}

A $100 Face Value One-Year Risk-Free Discount Bond Is Priced $\{ 99.74,94.74,89.74 \}$