Introduction to Derivative Securities
1. Term structure of interest rates and swap valuation
Suppose the current term structure of interest rates, assuming annual compounding, is as follows:
s1 s2 s3 s4 s5 s6
7.0% 7.3% 7.7% 8.1% 8.4% 8.8%
•What is the discount rate d(0,4)? (Recall that interest rates are always quoted on an annual basis unless stated otherwise.)
2. Swap Rates
Suppose a 6-year swap with a notional principal of $10 million is being configured. What is the fixed rate of interest that will make the value of the swap equal to zero. (You should use the term structure of interest rates from Question 1.)
Submit answer as a percentage rounded to two decimal places. So for example, if your answer is 4.567% or equivalently 0.04567, then you should submit an answer of 4.57.
3. Hedging using futures
Suppose a farmer is expecting that her crop of oranges will be ready for harvest and sale as 150,000 pounds of orange juice in 3 months time. Suppose each orange juice futures contract is for 15,000 pounds of orange juice, and the current futures price is F0=118.65 cents-per-pound.
Assuming that the farmer has enough cash liquidity to fund any margin calls, what is the risk-free price that she can guarantee herself.
Please submit your answer in cents-per-pound rounded to two decimal places. So for example, if your answer is 123.456, then you should submit an answer of 123.47.
4. Minimum variance hedge
Suppose a farmer is expecting that her crop of grapefruit will be ready for harvest and sale as 150,000 pounds of grapefruit juice in 3
months time. She would like to use futures to hedge her risk but unfortunately there are no futures contracts on grapefruit juice.
Instead she will use orange juice futures.
Suppose each orange juice futures contract is for 15,000 pounds of orange juice and the current futures price is F0=118.65 cents-per-pound.
The volatility, i.e. the standard deviation, of the prices of orange juice and grape fruit juice is 20% and 25%, respectively, and the correlation coefficient is 0.7. What is the approximate number of contracts she should purchase to minimize the variance of her payoff?
Submit answer rounded to the nearest integer. So for example, if your calculations result in 10.78 contracts you should submit an answer of 11.
5. Call Options
Consider a 1-period binomial model with R=1.02, S0=100, u=1/d=1.05. Compute the value of a European call option on the stock with strike K=102. The stock does not pay dividends.
Submit your answer rounded to two decimal places. So for example, if your answer is 3.4567 then you should submit an answer of 3.46.
6. Call Options II
When you construct the replicating portfolio for the option in the previous question how many dollars do you need to invest in the cash account?
Submit answer rounded to three decimal places. So for example, if your answer is -43.4567 then you should submit an answer of -43.457.