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 3-month default-free zero-coupon bond is traded at 99, and pays 100.00 at maturity.
Enter the price of the 6-month default-free zero-coupon bond that pays 100 at maturity, such that there are no arbitrage opportunities:
99.009732-correct
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The one-year and two-year risk-free rates (yields) are 1% and 1.025%, respectively. Our model of the term structure says that one year from now the one-year interest rate will be one of the following two values: 0.01 or 0.01*u, where u is the up factor. Here, the rates are the effective annual rates, so that one dollar invested in a T-bond returns (1+r)^T dollars, where T is measured in years. The model also says that the risk-neutral probabilities of these two possibilities are the same, equal to 1/2.
Enter the price of the one-year European put option written on the two-year risk-free zero coupon bond paying 100 at maturity, with strike price 98.95
Can you show workings for the answer above as I cannot obtain it?
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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 3-month default-free zero-coupon bond is traded at 99, and pays 100.00 at maturity.
Enter the price of the 6-month default-free zero-coupon bond that pays 100 at maturity, such that there are no arbitrage opportunities:
To determine the price of the 6-month default-free zero-coupon bond, we can use the concept of no-arbitrage pricing. The idea is that there should be no opportunity to make risk-free profits by exploiting discrepancies in prices.
In this case, we have two pieces of information. First, the price of the 3-month default-free zero-coupon bond is traded at 99 and pays 100 at maturity. Second, the price of the default-free coupon bond, which matures in 6 months, is traded at 101 and pays a coupon of 2 after 3 months and makes a final payment of 102.
To calculate the price of the 6-month default-free zero-coupon bond, we can break it down into two parts:
1. The initial price of the bond: Since it is a zero-coupon bond, the initial price will be less than its maturity value. Let's assume the price is denoted as P. We don't know the exact price yet; we need to solve for it.
2. The future value of the bond: At maturity, the bond pays 100.
Now, let's consider two scenarios:
Scenario 1: Hold the 3-month default-free zero-coupon bond and the 6-month default-free zero-coupon bond until maturity.
- At the end of 3 months, we receive a coupon payment of 2 from the 6-month bond.
- At the end of 6 months, we receive the final payment of 102 from the 6-month bond or 100 from the 3-month bond.
From scenario 1, we can derive the following equation in terms of P (the price of the 6-month bond):
99 + 2 = P + 102
Simplifying the equation gives:
P = 99 - 2 + 102 = 199
Therefore, the price of the 6-month default-free zero-coupon bond that pays 100 at maturity, such that there are no arbitrage opportunities, is 199.