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
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This is very true, I haven't got the answer, but the exam finishes today. I am not interested in using it to pass, but I would like to see the working (if possible, when the exam is finished!)
To determine the price of the one-year European put option written on the two-year risk-free zero coupon bond, we can use the concept of risk-neutral probabilities and the concept of no-arbitrage pricing.
The payoffs of the put option can be calculated based on the different possible interest rates one year from now. Let's denote the up factor as 'u'. The payoffs at expiration can be summarized as follows:
If the interest rate is 0.01, the payoff of the put option is max(98.95 - (1+0.01*u)^2 * 100, 0).
If the interest rate is 0.01*u, the payoff of the put option is max(98.95 - (1+0.01*u)^2 * 100, 0).
Given that the risk-neutral probabilities of these two possibilities are the same (1/2 each), we can calculate the expected value of the put option by taking the average of the payoffs weighted by their respective probabilities.
Let's denote the price of the put option as 'P'. The expected value calculation can be expressed as follows:
P = (1/2) * max(98.95 - (1+0.01*u)^2 * 100, 0) + (1/2) * max(98.95 - (1+0.01*u)^2 * 100, 0)
To find the value of 'u', we need to consider the risk-free rates and their relationship over the one-year and two-year periods. We know that the one-year and two-year risk-free rates (yields) are 1% and 1.025% respectively. Let's denote the one-year yield as 'r_1' and the two-year yield as 'r_2'.
Since these rates are effective annual rates, we can calculate the one-year and two-year discount factors as follows:
Discount factor for one year: 1 / (1 + r_1)
Discount factor for two years: 1 / (1 + r_2)^2
From the given information, we have r_1 = 0.01 and r_2 = 0.01025. We can now calculate the discount factors:
Discount factor for one year: 1 / (1 + 0.01) = 0.990099
Discount factor for two years: 1 / (1 + 0.01025)^2 = 0.980326
Now, we can substitute the discount factors into the put option price formula we derived earlier:
P = (1/2) * max(98.95 - (1+0.01*u)^2 * 100, 0) + (1/2) * max(98.95 - (1+0.01*u)^2 * 100, 0)
To calculate the value of 'P', we need the value of 'u'. This value is not provided in the given information and needs to be determined or assumed. Depending on the value of 'u', we can substitute it into the formula to find the price of the put option.