莫纳什大学期货期权第四周练习题BFF5915

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.)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:

flows

Cash

Final Trade Intial

Interim

Long forward 0 S T -92.00

Short spot +97 -5 -S T

3-month investment -4.901 +5

6-month investment -92.099 +97.794

Net CF 0 0 +5.794

Q3)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)T S = 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)(e rT ) =(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.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 the lessee to the forward contract and receive F = 366. Repay 362.71 on the borrowing. Net cash flow: +3.29.

Q 6

Arbitrage opportunities exist in one of following two situations

1.Forward is overpriced, in which one would short forward, long gold spot and

borrow to finance the long position, for a net CF=0 at time 0. The cashflow in 1 year’s time would be F0 -450(1+.06)=F0 -477. Arbitrage is not possible if F0 -477<=0, or F0 <=477.

2.Forward is underpriced, , in which one would long forward, short gold spot and

lend the proceed, for a net CF=0 at time 0. The cashflow in 1 year’s time would be 449(1+5.5%)-F0 =473.7-F0 . Arbitrage is not possible if 473.7-F0 <=0, or F0 >=473.7

Combining (1) and (2), the trader has no arbitrage opportunities if 473.7<=F0 <=477