A default-free coupon bond maturing in 6 months, that pays a coupon of 2.00 after 3 months and makes a final payment of 102.00 (the last coupon and the principal), trades at 101.00 today. Moreover, a 6-month default-free zero-coupon bond is traded at 97.1, and pays 100.00 at maturity.
Enter the price of the 3-month default-free zero-coupon bond that pays 100 at maturity, such that there are no arbitrage opportunities:
To find the price of the 3-month default-free zero-coupon bond, we can use the principle of no arbitrage. The concept of no arbitrage states that in a well-functioning market, there should be no risk-free opportunity for profit.
Let's assume that the price of the 3-month default-free zero-coupon bond is x.
Given:
- Price of the 6-month default-free coupon bond = 101.00
- Price of the 6-month default-free zero-coupon bond = 97.1
- Coupon payment after 3 months = 2.00
- Final payment after 6 months (last coupon and principal) = 102.00
- Payment at maturity for the 3-month zero-coupon bond = 100.00
We can create an arbitrage opportunity by constructing a synthetic position using the 6-month coupon bond and the 6-month zero-coupon bond.
The synthetic position will involve:
- Buying the 6-month zero-coupon bond at a price of 97.1
- Buying the 3-month zero-coupon bond at a price of x
- Borrowing the face value of the coupon bond for 3 months
- Receiving the coupon payment of 2.00 after 3 months
- Receiving the final payment of 102.00 after 6 months
To simplify the calculations, let's assume that the face value of the coupon bond is 100.00.
The initial cost of the synthetic position will be:
- Cost of buying the 6-month zero-coupon bond = 97.1
- Cost of buying the 3-month zero-coupon bond = x
- Borrowing the face value of the coupon bond for 3 months = 100*(1 + (2/100)*(3/12)) = 100.50 (assuming an annual coupon rate of 2%)
The final proceeds from the synthetic position will be:
- Receiving the coupon payment of 2.00 after 3 months
- Receiving the final payment of 102.00 after 6 months
Now, since there are no arbitrage opportunities in the market, the initial cost of the synthetic position should be equal to the final proceeds.
Therefore, we can set up the equation:
97.1 + x = 100.50 + 2 + 102
To find the value of x, we solve the equation:
97.1 + x = 100.50 + 2 + 102
x = 100.50 + 2 + 102 - 97.1
x = 100.50 + 2 + 102 - 97.1
x = 3.4
Hence, the price of the 3-month default-free zero-coupon bond that pays 100 at maturity, such that there are no arbitrage opportunities, is $3.4.