Explain the main characteristics of indifference curves

1 The current annually compounded risk-free yield curve is given as follows: Term Spot Rate Term Spot Rate (years) "wears 0.5 0.65 4.5 0.95 1.0 0.67 5.0 1.03 1.5 0.70 5.5 1.11 2.0 0.72 6.0 1.20 2.5 0.75 6.5 1.30 3.0 0.78 7.0 1.39 3.5 0.82 7.5 1.49 4.0 0.88 8.0 1.58 Interest rate swaps have semi-annual payments. (1) Show that the current forward swap rate for a three year swap starting in five years' time is 2.48% per annum. [4] (ii) Calculate the value of a European five year payer swaption to enter into a three year swap with lm notional and a 2% per annum swap rate, using a Black volatility of 20% per annum. [4] (iii) Determine the value of the equivalent five year receiver swaption using put-call parity. [2] A trader is offering to buy or sell a derivative with a payout in five years' time equal to flm multiplied by the three year swap rate on that date. He is quoting a price of f1,000,000 * 2.48% x 1.0103-3 = f23,561. (iv) Explain (without calculation) the arbitrage opportunity that this presents. [2]Consider an interest rate derivative that pays f at time 7 in a world where the instantaneous short rate / is stochastic, following a random walk. (1) Show that the value of the derivative at time / is: where E means "expected value in the risk-neutral world", and F is the average short rate in the time interval between / and I. [4] [ Hint: consider a mongy market account as the momneraire. ] (ii) Hence show that the entire term structure of interest rates can be derived from the risk-neutral process for r. [2]In the Vasicek model, r, is governed by the stochastic differential equation: de, = a(b -7)di + o dW. where a and b are positive constants and W, is a standard Brownian motion. (iii) Solve this differential equation to find the exact distribution followed by r. [5] [Hint: consider the process s, =e"r,.] [Total 11] 3 The European Space Agency (ESA) will be launching a manned rocket to Mars and is expecting to order 100,000 barrels of rocket fuel in three months' time. It is concerned about the risk that fluctuating fuel prices pose to the project's total cost. ESA has decided to hedge this risk using crude oil futures, for which a contract is available which expires in three months' time. You are given the following information: . The spot price of one barrel of crude oil is $101.2. . The relevant futures price for delivery of one barrel of crude oil is $106.3. . The standard deviation of changes in the spot price of a barrel of rocket fuel over the next three months is $4.2. The standard deviation of changes in the relevant futures price for crude oil over the next three months is $2.5. . The coefficient of correlation between changes in crude oil futures prices and rocket fuel spot prices is 0.75. . One crude oil future represents 1,000 barrels of crude oil.(1) Define the following terms: (a) cross hedging (b) hedge ratio (c) minimum variance hedge ratio (d) tailing the hedge [4] (ii) Calculate the number of crude oil futures that ESA should buy to minimise the variance of the project's total costs. [3] (iii) Discuss in general terms what can be deduced about crude oil storage costs from crude oil futures prices. [3] [Total 10]