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
Help me with this:
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 probability 1−p of the one-year interest rate being 0.01u equals to 2/3.
Enter the price of the one-year European call option written on the two-year risk-free zero coupon bond paying 100 at maturity, with strike price 98.97:
did you ever get this? would be awesome if you could help me out with it
To calculate the price of the one-year European put option written on the two-year risk-free zero coupon bond, we can use the risk-neutral probabilities and a binomial tree model.
Step 1: Set up the binomial tree
- The tree will have two time periods: the current time (0) and one year (1).
- At time 0, the bond can either go up to a value of 100 or down to a value of 100*u.
- At time 1, the bond can either go up to a value of 100 or down to a value of 100*u^2.
Step 2: Calculate the payoffs at maturity (time 1)
- The payoff of the put option at maturity is given by max(strike price - bond value, 0).
- If the bond value is 100 at time 1, the payoff is max(98.95 - 100, 0) = 0.
- If the bond value is 100*u^2 at time 1, the payoff is max(98.95 - 100*u^2, 0).
Step 3: Calculate the option prices at time 0
- Discount the payoffs at time 1 to time 0 using the one-year risk-free rate.
- For the bond value of 100, the discounted payoff is 0.
- For the bond value of 100*u^2, the discounted payoff is [max(98.95 - 100*u^2, 0)] / (1 + one-year risk-free rate).
Step 4: Calculate the option price at time 0
- The option price at time 0 is the risk-neutral expected value of the discounted payoffs at time 1.
- Using the risk-neutral probabilities of 1/2 for each possibility,
the option price at time 0 is (1/2) * 0 + (1/2) * [max(98.95 - 100*u^2, 0)] / (1 + one-year risk-free rate).
Step 5: Substitute the given values and solve
- Substitute the given values: one-year risk-free rate = 1%, strike price = 98.95.
- Solve for the option price using the formula derived in Step 4.
- The price of the one-year European put option is (1/2) * [max(98.95 - 100*u^2, 0)] / (1 + 1%).
Note: To solve for the specific option price, the value of the up factor (u) needs to be provided.
To calculate the price of the one-year European put option written on the two-year risk-free zero coupon bond, we can use the risk-neutral pricing formula. The formula for the price of a European put option is:
Put Price = e^(-r*T) * [K * P(S_T <= K) - S_0 * P(S_T <= S_0)]
Where:
- r is the risk-free interest rate
- T is the time to expiration in years
- K is the strike price of the option
- P(S_T <= K) is the risk-neutral probability of the bond's price being below the strike price at expiration
- S_0 is the current price of the bond
Using the given information, we have:
- r = 1.025% = 0.01025 (two-year risk-free rate)
- T = 1 year (time to expiration)
- K = 98.95 (strike price)
- S_0 = 100 (current price of the bond)
To calculate P(S_T <= K), we need to consider the two possible values for the one-year interest rate, 0.01 and 0.01*u (where u is the up factor). The risk-neutral probabilities for these two outcomes are equal, so each possibility has a probability of 1/2.
Let's calculate the probabilities for each scenario:
1. When the one-year interest rate is 0.01:
P(S_T <= K) = P(Bond price at expiration <= 98.95)
P(S_T <= K) = P(e^(r*T) <= 98.95) [since S_T = e^(r*T)]
P(S_T <= K) = P(e^(0.01025*1) <= 98.95)
P(S_T <= K) = P(e^0.01025 <= 98.95)
P(S_T <= K) = P(1.010324 <= 98.95)
P(S_T <= K) = 1 [since 1.010324 <= 98.95 is always true]
2. When the one-year interest rate is 0.01*u:
P(S_T <= K) = P(Bond price at expiration <= 98.95)
P(S_T <= K) = P(e^((0.010*u)*1) <= 98.95)
P(S_T <= K) = P(e^(0.010*u) <= 98.95)
P(S_T <= K) = P(1.010^u <= 98.95)
Now, we need to compare the two possible probabilities and calculate the put option price using the risk-neutral pricing formula:
Put Price = e^(-r*T) * [K * P(S_T <= K) - S_0 * P(S_T <= S_0)]
Put Price = e^(-0.01025*1) * [98.95 * 1 - 100 * P(1.010^u <= 98.95)]
Since we don't have the exact value of u, we cannot directly calculate the put option price. However, the formula provided gives you the framework to calculate the price once you have the value for u.