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There is a two-period zero coupon bond that will pay $10 million at t = 2.At t = 0, a call option on this bond is selling at $225,000.The holder can exercise the option at t = 1 at a price of $9,250,000.The current yield on a riskless zero coupon bond that matures at t = 1 is 8%.It is known that the yield on one-period bonds at t = 1 will be 6% or 10%.All bonds are identical except in maturity.
-Suppose that the payoff to the option holder is $250,000.What is the risk-neutral probability that the interest rate will be 6% round up to the nearest figure?
A)1%
B)2%
C)3%
D)5%
E)8%

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There is a two-period zero coupon bond that will pay $10 million at t = 2.At t = 0, a call option on this bond is selling at $225,000.The holder can exercise the option at t = 1 at a price of $9,250,000.The current yield on a riskless zero coupon bond that matures at t = 1 is 8%.It is known that the yield on one-period bonds at t = 1 will be 6% or 10%.All bonds are identical except in maturity.
-Suppose that the payoff to the option holder is $250,000.What is the risk-neutral probability that the interest rate will be 6% round up to the nearest figure?
A)1%
B)2%
C)3%
D)5%
E)8%

Quizplus · Accepted Answer

The payoff to the option holder is $250,000, which is less than the price of the option at $225,000. Therefore, the option will not be exercised. This means that the bond will not be called at t = 1 and will pay $10 million at t = 2. To determine the risk-neutral probability of the interest rate being 6%, we need to find the risk-neutral probabilities for each possible interest rate at t = 1. Let p be the probability of the interest rate being 6%. Then, the probability of the interest rate being 10% is (1-p).

Using the current yield on the riskless zero coupon bond that matures at t = 1, we can determine the present value of the bond at t = 0. PV = 10,000,000/(1+0.08) = $9,259,259.26.

Using the two possible interest rates at t = 1, we can determine the probabilities of each cash flow at t = 1:

- If the interest rate is 6%, the cash flow at t = 1 is $9,250,000, since the bond will not be called. The probability of this cash flow is p.
- If the interest rate is 10%, the cash flow at t = 1 is $9,259,259.26, since the bond will be called at $9,250,000. The probability of this cash flow is (1-p).

The expected value of the cash flow at t = 1 is therefore:

E(CF1) = p($9,250,000) + (1-p)($9,259,259.26)

We know that the payoff to the option holder is $250,000. This means that the option holder would only exercise the option if the cash flow at t = 1 is less than $9,500,000 ($10,000,000 - $250,000). Therefore, we can set up the following equation:

$9,250,000 + $225,000 = $9,500,000

Solving for p, we get:

p = 0.03, or 3% (rounded up to the nearest figure)

Therefore, the risk-neutral probability of the interest rate being 6% is 3%.

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