莫纳什大学期货期权第四周练习题BFF5915.pdf

1 AFF9150 Week 4 Problem Set Pricing of Forwards and Futures Q1 True or false The theoretical forward price decreases with maturity That is for example the theoretical price of a three month forward must be greater than the theoretical price of a six month forward Q2 A security is currently trading at 97 It will pay a coupon of 5 in two months No other payouts are expected in the next six months a If the term structure is at 12 what should the be forward price on the security for delivery in six months b If the actual forward price is 92 explain how an arbitrage may be created Q3 Consider a three month forward contract on pound sterling Suppose the spot exchange rate is 1 40 the three month interest rate on the dollar is 5 and the three month interest rate on the pound is 5 5 If the forward price is given to be 1 41 identify whether there are any arbitrage opportunities and how you would take advantage of them Q4 The current level of a stock index is 450 The dividend yield on the index is 4 in continuously compounded terms and the risk free rate of interest is 8 for six month investments A six month futures contract on the index is trading for 465 Identify the arbitrage opportunities in this setting and explain how you would exploit them Q5 Suppose there is an active lease market for gold in which arbitrageurs can short or lend out gold at a lease rate of c 1 p a this is the convenience yield mentioned in the lecture Assume gold has no other costs benefits of carry Consider a three month forward contract on gold a If the spot price of gold is 360 oz and the three month interest rate is 4 what is the arbitrage free forward price of gold b Suppose the actual forward price is given to be 366 oz Is there an arbitrage opportunity If so how can it be exploited Q6 A trader owns gold as part of a long term investment portfolio The trader can buy gold for 450 per ounce and sell it for 449 per ounce The trader can borrow funds at 6 per year and invest funds at 5 5 per year both interest rates are expressed with annual compounding For what range of one year forward prices of gold does the trader have no arbitrage opportunities Assume there is no bid ask spread for forward prices Hint Arbitrage exists when the forward is either overpriced or underpriced Design an arbitrage transaction for each scenario and derive the no arbitrage price range accordingly 2 Suggested Solution Q1 False When the holding benefits are greater than the holding costs the forward price will be less than the spot price In the case of a forward futures contract based on cost of carry when there are no holding benefits then as maturity increases the value of the forward contract will also increase If there are holding benefits such as dividends or interest the price of the forward may decrease with maturity Q2 We have that S 97 and the PV of holding benefits is 5 exp 0 12x2 12 4 9010 Thus the forward price should be 97 4 9010 exp 0 12 x 6 12 97 794 Since the forward price is 92 it is mispriced under priced The arbitrage is as follows At inception Buy forward at 92 Sell short spot at 97 Invest PV 5 4 901 for three months at 12 Invest 97 PV 5 92 099 for six months at 12 In three months use the cash inflow of 5 from the investment to pay the coupon due on the shorted security In six months receive the cash from the six month investment Pay the delivery price of 97 on the forward and receive unit of the security Use this to close the short spot position The initial and interim cash flows are zero and the final cash flow is positive as the following table shows Cash flows Trade Intial Interim Final Long forward 0 ST 92 00 Short spot 97 5 ST 3 month investment 4 901 5 6 month investment 92 099 97 794 Net CF 0 0 5 794 Q3 3 We are given the information that S 1 40 r 0 05 and d 0 055 From this data the arbitrage free forward price of a three month forward contract should be F e r d T S e 0 05 0 055 1 4 1 40 1 3983 Thus at the given forward price of 1 41 the forward contract is overvalued relative to spot To take advantage of the opportunity we should sell forward buy spot and borrow to finance the spot purchase Specifically Enter into a short forward contract to deliver pounds in three months at 1 41 Buy e dT 0 9863 pounds spot at the spot price of 1 40 Cost 1 40 0 9863 1 3809 Invest the 0 9863 for three months at 5 5 Amount received after three months GBP 1 Borrow 1 3809 for three months at 5 Amount due in three months e 0 05 1 4 1 3809 1 3983 In 3 months time receive GBP1 from the pound investment use it to deliver under the short forward contract and receive 1 41 Use this dollar receipt to pay for the USD loan now due at 1 3983 The net profit is 1 41 1 3983 0 0117 This is the arbitrage profit for each trade involving 1 pound A trade of the size of say GBP 1M means profit of 11 700 of course before trading costs Q4 We are given S 450 F 465 d 0 04 r 0 08 and T 1 2 Using the continuous dividend formula the forward price should be e r d TS 459 09 At the given price of 465 the forward is overvalued relative to spot To make an arbitrage profit we should sell forward buy spot and borrow More specifically At time 0 Buy e dT 0 9802 units of spot finance the spot purchase by borrowing Se dT 450 0 9802 441 09 for repayment in six months enter into a short forward position for delivery in six months Net cash flow 441 09 to buy spot 441 09 from borrowing 0 Between time 0 and time T Invest all dividends into buying more units of the spot asset Net cash flow 0 At time T Under the given strategy the total holdings of the spot asset at time T is 1 unit Use this unit to deliver to the forward position and receive a cash inflow of 465 Repay the borrowing amount due 441 09 erT 441 09 1 0408 459 09 Net cash flow 465 459 09 5 91 With no net cash outflows and a positive cash inflow at maturity this strategy is clearly an arbitrage 4 Q5 a The forward price of gold will be given by the formula F e r c T So F 360 exp 0 04 0 01 x3 12 362 71 b If the quoted forward price is 366 then there is an arbitrage since the true price is 362 71 In order to construct the arbitrage we do the following Sell 1 oz of gold forward at F 366 Buy e cT 0 9975 oz of gold spot Cost 359 10 Lease the gold out for 3 months at 1 lease rate Amount received at end of the lease 1 oz Borrow 359 10 for 3 months at 4 Amount owed at maturity 362 71 At maturity deliver the 1 oz of gold received from t