## IEOR 4700 – Term structure of interest rates and swap valuation

1. Term structure of interest rates and swap valuationSuppose the current term structure of interest rates, assuming annual compounding, is as follows:s1s2s3s4s5s67.0%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 unlessstated otherwise.)Please submit your answer rounded to three decimal places. So for example, if your answer is 0.4567 then you should submit an answer of 0.457.2. Swap RatesSuppose a 6-year swap with a notional principal of $10 million is beingconfigured. What is the fixed rate of interest that will make the valueof the swap equal to zero. (You should use the term structure of interest rates from Question 1.)Please submit your 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 futuresSuppose a farmer is expecting that her crop of oranges will be ready forharvest and sale as 150,000 pounds of orange juice in 3months time. Suppose each orange juice futures contract is for 15,000pounds of orange juice, and the current futures price is F0=118.65 cents-per-pound.Assuming that the farmer has enough cash liquidity to fundany 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 hedgeSuppose a farmer is expecting that her crop of grapefruit will be ready forharvest and sale as 150,000 pounds of grapefruit juice in 3months time. She would like to use futures to hedge her risk but unfortunately thereare no futures contracts on grapefruit juice. Instead she will use orange juice futures.Suppose each orange juice futures contract is for 15,000pounds 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 oforange juice and grape fruit juice is 20% and 25%, respectively,and the correlation coefficient is 0.7. What is the approximate numberof contracts she should purchase to minimize the variance of her payoff?Please submit your 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 OptionsConsider 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 stockwith strike K=102. The stock does not pay dividends.Please 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 IIWhen you construct the replicating portfolio for the option in the previous question how many dollars do you need to invest in the cash account?Please submit your answer rounded to three decimal places. So for example, if your answer is ?43.4567 then you should submit an answer of ?43.457.