1.1 Slide 19

A company wishes to buy \(200\) ounces of gold in \(3\) months’ time. The interest rate is \(4.9\%\) p.a. continuously compounded.

1. If the spot price for gold of this purity is \(\$700/\)ounce, find the \(3-\)month forward price of gold.
2. \(1\) month later, the spot price is \(\$701.5/\)ounce. If the interest rate is unchanged, find, on this date, the \(2-\)month forward price of gold.

Solution.

1. \(F_{3/12}=700\cdot e^{4.9\%\cdot3/12}\approx708.63.\)
2. \(F_{2/12}=701.5\cdot e^{4.9\%\cdot2/12}\approx707.25.\)

1.2 Slide 20

Consider a \(4-\)month forward contract to buy a zero-coupon bond, with face value \(\$1{,}000\), that will mature \(1\) year from today (this means that the bond will have \(8\) months to go when the forward contract matures). The current price of the bond is \(\$930.\) We assume that the risk-free rate, continuously compounded, is \(6\%\) p.a. What is the delivery price of the forward contract?

Solution. The delivery price is \[F=930\cdot e^{6\%\cdot4/12}\approx948.79.\]

1.3 Slide 29

The price of a share is £\(6.75.\) The share pays no dividend. An investor enters a forward contract to sell \(1\) share in \(14\) months’ time. He can borrow money at \(4.5\%\) p.a. with interest compounded annually. What is the delivery price on this contract? Is the delivery higher or lower when the share paying dividends? Why?

Solution. We set up a portfolio \(V\) as follows:

1. Today \((t=0):\)

• Borrow £\(6.75\) to buy the share.
• Enter the contract to sell the share (at maturity) for £\(K.\)
• Additional fees: £\(0.\)
• Contract value: \(V_0=6.75-6.75+0=0.\)

2. At maturity \((t=14/12):\)

• Repay the loan with £\(6.75\cdot(1+4.5\%)^{14/12}\approx\) £\(7.106.\)
• Hand over the share and receive the delivery price £\(K.\)
• Additional fees: £\(0.\)
• Contract value: \(V_{14/12}=-7.106+K+0=K-7.106.\)

3. Under the assumption of no arbitrage, \(V_{14/12}=0\Rightarrow K=7.106.\)

4. The delivery price is, therefore, £\(7.106.\)

1.4 Slide 30

The price of a share is £\(8.69.\) The share is paying a dividend of \(5\%\) p.a., every \(6\) months, with the \(1^{\textrm{st}}\) dividend due in the next \(2\) months. An investor enters a forward contract to sell \(5{,}000\) shares in \(18\) months’ time. He can borrow money at \(7.1\%\) p.a. with semi-annual compounding. Calculate the \(18-\)month forward price of the \(5{,}000\) shares.

Solution. The price of \(5{,}000\) shares is £\(8.69\cdot5{,}000=\) £\(43{,}450.\) The present values of the dividend payments are \[D_1=\frac{1{,}086.25}{(1+7.1\%/2)^{2/6}}= 1{,}073.69,\] \[D_2=\frac{1{,}086.25}{(1+7.1\%/2)^{8/6}}= 1{,}036.88,\] \[D_3=\frac{1{,}086.25}{(1+7.1\%/2)^{14/6}}= 1{,}001.335.\]

We set up a portfolio \(V\) as follows:

1. Today \((t=0):\)

• Borrow £\(43{,}450\) to buy \(5{,}000\) shares as follows:
  • First loan: borrow £\(1{,}073.69\) for \(2\) months.
  • Second loan: borrow £\(1{,}036.88\) for \(8\) months.
  • Third loan: borrow £\(1{,}001.335\) for \(14\) months.
  • Fourth loan: borrow £\(40{,}338.095\) for \(18\) months.
• Enter the contract to sell \(5{,}000\) shares (at maturity) for £\(K.\)
• Additional fees: £\(0.\)
• Contract value: \(V_0=43{,}450-43{,}450+0=0.\)

2. At \(t=2/12:\) use the \(1^{\textrm{st}}\) dividend payment to pay the \(1^{\textrm{st}}\) loan.
3. At \(t=8/12:\) use the \(2^{\textrm{nd}}\) dividend payment to pay the \(2^{\textrm{nd}}\) loan.
4. At \(t=14/12:\) use the \(3^{\textrm{rd}}\) dividend payment to pay the \(3^{\textrm{rd}}\) loan.
5. At maturity \((t=18/12):\)

• Repay the loan with £\(40{,}338.095\cdot(1+7.1\%/2)^{18/6}\approx\) £\(44{,}788.415.\)
• Hand over \(5{,}000\) shares and receive the delivery price £\(K.\)
• Additional fees: £\(0.\)
• Contract value: \(V_{18/12}=-44{,}788.415+K+0=K-44{,}788.415.\)

3. Under the assumption of no arbitrage, \(V_{18/12}=0\Rightarrow K=44{,}788.415.\)

4. The delivery price is, therefore, £\(44{,}788.415.\)

1.5 Slide 31 - P1

Consider a \(10-\)month forward contract on a stock when the stock price is \(\$50.\) We assume that the risk-free rate, continuously compounded, is \(8\%\) p.a. for all maturities. We also assume that dividends of \(\$0.75\) per share are expected after \(3\) months, \(6\) months, and \(9\) months. What is the delivery of the forward contract?

Solution. Solution. We have \[D=\frac{0.75}{e^{8\%\cdot3/12}}+\frac{0.75}{e^{8\%\cdot6/12}}+\frac{0.75}{e^{8\%\cdot9/12}}\approx2.162\] and hence \[K=F_{10/12}=(50-D)\cdot e^{8\%\cdot10/12}\approx51.14.\]

1.6 Slide 31 - P2

Consider a \(6-\)month forward contract on an asset that is expected to provide income equal to \(2\%\) of the asset price once during a \(6-\)month period. The risk-free rate of interest, with continuous compounding, is \(10\%\) p.a. The asset price is \(\$25.\) What is the delivery of the forward contract?

Solution. We have \((1+4\%/2)^2=e^Q\Rightarrow Q\approx3.961\%\) which implies \[F=S_0\cdot e^{(R-Q)\cdot t}=25\cdot e^{(10\%-3.961\%)\cdot6/12}\approx25.766.\]

1.7 Slide 40

A long forward contract on a non-dividend-paying stock was entered into some time ago. It currently has \(6\) months to maturity, with the delivery price is \(\$24.\) The risk-free rate of interest, with continuous compounding, is \(10\%\) p.a. and the stock price is \(\$25.\) What is the current value of the forward contract?

Solution. \(F=S_0-Ke^{-rt}=25-24\cdot e^{-10\%\cdot6/12}\approx2.17.\)

1.8 Slide 42

Suppose an asset is currently worth \(\$20\) and the \(6-\)month futures price of this asset is \(\$22.5.\) By assuming the stock does not pay any dividends and the risk-free interest rate \(r,\) compounded continuously, is the same for all maturities, calculate the \(1-\)year futures price of this asset.

Solution. Note that \[22.5=F_{6/12}=20\cdot e^{r\cdot6/12}\] hence \(r\approx23.56\%\) which implies \[F_1=20\cdot e^{r}=20\cdot e^{23.56\%}\approx25.313.\]

1.9 Slide 43

Let the current price of a stock be \(\$12.75\) that pays a continuous dividend yield \(\delta.\) Suppose the risk-free interest rate is \(6\%\) p.a. and the price of a \(6-\)month forward contract is \(\$13.25.\) Find \(\delta.\)

Solution. The equation \[13.25=F_{0,T}(S)=e^{(r-\delta)\cdot T}S_0=e^{(6\%-\delta)\cdot6/12}\cdot12.75\] implies \[\frac{6\%-\delta}{2}=\ln\left(\frac{13.25}{12.75}\right)\Rightarrow\delta\approx-0.0169.\]

1.10 Slide 44 - P1

The interest rate is \(7.3\%\) compounded annually. A stock paying a dividend of £\(2.5\) every \(6\) months (the next dividend payment is due in \(3\) months’ time) is valued at £\(35.\) The \(6-\)month forward price on the stock is £\(34.\) Is this an arbitrage opportunity? If so, how precisely would you respond?

Solution. We have \[D=\frac{2.5}{(1+7.3\%)^{3/12}}\approx2.456\] and hence the delivery price for a forward contract is \[K=(35-D)\cdot(1+7.3\%)^{6/12}\approx33.711<34\] hence it is an arbitrage opportunity: we can make profit by enters a short position \(6-\)month forward contract as follows:

1. Today:

• Borrow \(D\) for three months and \(35-D\) for \(6\) months.
• Buy a share.
• Enter a short forward contract on this share with delivery price £\(34.\)

2. At maturity:

• Pay £\(33.711\) for the loan.
• Hand over the share and receive £\(34.\)
• Profit: £\(34-\) £\(33.711=\) £\(0.289.\)

1.11 Slide 44 - P2

The interest rate in London is \(4.35\%\) p.a. compounded annually. In Cyprus, the interest rate is \(6.16\%\) compounded annually. 1 CYP = £1.1981. In 4 months’ time, how many pounds might I expect to buy for 10,000 CYP?

Solution. We set up a portfolio \(V\) as follows:

1. Today \((t=0):\)

• Buy \(10000\cdot(1+6.16\%)^{-4/12}\approx9802.715\) CYP by borrowing \(9802.715\cdot£1.1981\approx£11744.633.\)
• Enter the contract to sell 10,000 CYP (at maturity) for \(£K.\)
• Additional fees: £0.
• Contract value: \(V_0=11744.633-11744.633+0=0.\)

2. At maturity \((t=4/12):\)

• Repay the loan with \(£11744.633\cdot(1+4.35\%)^{4/12}\approx£11912.519.\)
• Hand over 10,000 CYP and receive the delivery price \(£K.\)
• Additional fees: £0.
• Contract value: \(V_{6/12}=-11912.519+K+0=K-11912.519.\)

3. Under the assumption of no arbitrage, \(V_{4/12}=0\Rightarrow K=11912.519.\)

4. The delivery price is, therefore, £11,912.519.

1.12 Slide 44 - P3

The interest rate in the UK is 4.87% compounded annually. Today, £1 = 3,767.676 BYR. The 6-month forward price on the BYR is £1 = 3,925.831 BYR. Estimate the interest rate \(r,\) annually compounded, in Belarus.

Solution. The equality \[3925.831=1\cdot(1+4.87\%)^{-6/12}\cdot3767.676\cdot(1+r)^{6/12}\] implies \(r\approx13.859\%.\)

2.1 Problem 2

A bushel of corn is valued today at $3.49. The annual safe rate is 6.3% compounded annually. A forward contract is entered to sell the corn in 4 months’ time. What is the delivery price \(\$K\) in the contract?

Solution. We set up a portfolio \(V\) as follows:

1. Today \((t=0):\)

• Borrow $3.49 to buy one bushel of corn.
• Enter the contract to sell one bushel of corn (at maturity) for \(\$K.\)
• Additional fees: $0.
• Contract value: \(V_0=3.49-3.49+0=0.\)

2. At maturity \((t=4/12):\)

• Repay the loan with \(\$3.49\cdot(1+6.3\%)^{4/12}\approx\$3.562.\)
• Hand over one bushel of corn and receive the delivery price \(\$K.\)
• Additional fees: $0.
• Contract value: \(V_{4/12}=-3.562+K+0=K-3.562.\)

3. Under the assumption of no arbitrage, \(V_{4/12}=0\Rightarrow K=3.562.\)

4. The delivery price is, therefore, $3.562.

2.2 Problem 3

An ounce of gold today is priced at $110.7. The interest rate is 5.8%, compounded twice yearly. A forward contract is entered to buy 20 ounces of gold in 3 months’ time. What is the delivery price in this contract?

Solution. We set up a portfolio \(V\) as follows:

1. Today \((t=0):\)

• Borrow \(\$110.7\cdot20=\$2214\) to buy 20 ounces of gold.
• Enter the contract to sell 20 ounces of gold (at maturity) for \(\$K.\)
• Additional fees: $0.
• Contract value: \(V_0=2214-2214+0=0.\)

2. At maturity \((t=3/6):\)

• Repay the loan with \(\$2214\cdot(1+5.8\%/2)^{3/6}\approx\$2245.874.\)
• Hand over one bushel of corn and receive the delivery price \(\$K.\)
• Additional fees: $0.
• Contract value: \(V_{3/6}=-2245.874+K+0=K-2245.874.\)

3. Under the assumption of no arbitrage, \(V_{3/6}=0\Rightarrow K=2245.874.\)

4. The delivery price is, therefore, $2,245.874.

2.3 Problem 5

An asset is valued today at $20. What is meant by the 3-month forward price of the asset? How is the 3-month forward price calculated?

Solution. Suppose the risk-free interest rate is \(r\) p.a., then the delivery price \(F\) of the contract is the future value of $20. corresponding to \(r:\)

1. If \(r\) compounds \(k\) times p.a., \(F=20\cdot(1+r/k)^{3k/12}.\)
2. If \(r\) compounds continuously, \(F=20\cdot e^{3r/12}.\)

2.4 Problem 6

A Bebop share is valued today at £10.5. The interest rate is 7.2%, compounded continuously.

1. Find the 4-month forward price on a Bebop share.
2. Find the 6-month forward price on a Bebop share.
3. I enter a 6-month forward contract on Bebop shares. What is the delivery price in this contract?

Solution.

1. \(F_{4/12}=10.5\cdot e^{7.2\%\cdot4/12}\approx10.755.\)
2. \(F_{4/12}=10.5\cdot e^{7.2\%\cdot6/12}\approx10.885.\)
3. Same as 2.

2.5 Problem 7

A football team is valued at £35 million. Things have not been going too well and the owner decides to sell the team at the end of the season, 4 months away. He enters a forward contract to sell the team in 4 months’ time. The interest rate is 4.8% p.a., compounded semi-annually.

1. What is the delivery price on this forward contract? What is the 4-month forward price on the team?
2. 1 month later, the team is valued at £32.5 million. What is the delivery price in the forward contract in part 1? What is the 3-month forward price of the team?

Solution.

1. \(K=3.5\cdot10^7\cdot(1+4.8\%/2)^{4/6}\approx3.556\cdot10^6=F_{4/6}.\)
2. \(K=3.556\cdot10^6\) unchanged, \(F_{3/6}=3.25\cdot10^7\cdot(1+4.8\%/2)^{3/6}\approx3.289\cdot10^6.\)

2.6 Problem 8

A 1-year forward contract to buy a non-dividend-paying stock is entered when the price of the stock is £15. The risk-free interest rate is 7.5% p.a. with continuous compounding.

1. What is the initial value of the forward contract?
2. What is the 1-year forward price on the stock?
3. In 6 months’ time, the stock is trading at £17.7 and the interest rate is unchanged. What is the delivery price on the original forward contract? What is the 6-month forward price on the stock?

Solution.

1. \(V_0=0.\)
2. \(F_1=15\cdot e^{7.5\%}\approx16.168.\)
3. \(K=F_1=16.168,F_{6/12}=17.7\cdot e^{7.5\%\cdot6/12}\approx18.376.\)

2.7 Problem 9

A 6-month forward contract to buy a non-dividend-paying stock (a long forward contract) is entered when the stock sells at £18.5. The current safe interest rate is 8.7% p.a. compounded annually.

1. What is the 6-month forward price on the stock?
2. After 1 month, the stock is valued at £19.98. What is the 5-month forward price? What, today, is the value of the forward contract?
3. 1 further month later, the stock was selling at £19.5. Calculate the 4-month forward price. What is now the value of the forward contract?

Solution.

1. \(F_{6/12}=18.5\cdot(1+8.7\%)^{6/12}\approx19.288=K\)
2. \(F_{5/12}=19.98\cdot(1+8.7\%)^{5/12}\approx20.687\) and \[f_{1/12}=\frac{F_{5/12}-K}{(1+8.7\%)^{5/12}}\approx1.351.\]
3. \(F_{4/12}=19.5\cdot(1+8.7\%)^{4/12}\approx20.05\) and \[f_{2/12}=\frac{F_{4/12}-K}{(1+8.7\%)^{4/12}}\approx0.741.\]

2.8 Problem 10

An asset pays a dividend of £5.5 twice each year. The \(1^{\textrm{st}}\) dividend payment is due in 1 month’s time. The asset is valued at £150. If the interest rate is 6% p.a., compounded annually, find the delivery price in a forward contract with delivery date in 1 year’s time.

Solution. We have \[D=\frac{5.5}{(1+6\%)^{1/12}}+\frac{5.5}{(1+6\%)^{7/12}}\approx10.79\] and hence \[K=F_1=(150-D)\cdot(1+6\%)\approx147.563.\]

2.9 Problem 11

A share is priced today at £10. The share pays a dividend of 5% p.a., in 2 equal installments: the \(1^{\textrm{st}}\) dividend payment is due in 4 months’ time. The interest rate is 6% p.a., continuously compounded. I enter a forward contract on the share with delivery date in 14 months’ time. Find the delivery price.

Solution. We have \[D=\frac{10\cdot5\%/2}{e^{6\%\cdot4/12}}+\frac{10\cdot5\%/2}{e^{6\%\cdot10/12}}\approx0.483\] and hence \[K=F_{14/12}=(10-D)\cdot e^{6\%\cdot14/12}\approx10.207.\]

2.10 Problem 12

An asset is valued today at $20. The asset pays a dividend of $1 twice each year. The \(1^{\textrm{st}}\) dividend is due in 3 months’ time. The interest rate is 6.5% p.a. compounded continuously. A 10-month short forward contract is entered.

1. What is the 10-month forward price and what, today, is the value of the contract?
2. 1 month later, the value of the stock has fallen to $17. What is the 9-month forward price on the stock and what is now the value of the forward contract?
3. After a further month, the stock is selling at $25. What is the 8-month forward price and what is the value of the forward contract?

Solution.

1. We have \(V_0=0\) and \[D=\frac{1}{e^{6.5\%\cdot3/12}}+\frac{1}{e^{6.5\%\cdot9/12}}\approx1.936\] and hence \[F_{10/12}=(20-D)\cdot e^{6.5\%\cdot10/12}\approx19.069=K.\]
2. We have \[D=\frac{1}{e^{6.5\%\cdot2/12}}+\frac{1}{e^{6.5\%\cdot8/12}}\approx1.947\] and hence \[F_{9/12}=(20-D)\cdot e^{6.5\%\cdot9/12}\approx18.955\] so \[f_{1/12}=\frac{K-F_{9/12}}{e^{6.5\%\cdot9/12}}\approx0.109.\]
3. We have \[D=\frac{1}{e^{6.5\%\cdot1/12}}+\frac{1}{e^{6.5\%\cdot7/12}}\approx1.957\] and hence \[F_{8/12}=(20-D)\cdot e^{6.5\%\cdot8/12}\approx18.842\] so \[f_{1/12}=\frac{K-F_{8/12}}{e^{6.5\%\cdot8/12}}\approx0.217.\]

2.11 Problem 13

The interest rate is 7.3% compounded annually. A stock paying a dividend of £2.5 every 6 months (the next dividend payment is due in 3 months’ time) is valued at £35. The 6-month forward price on the stock is £34. Is this an arbitrage opportunity? If so, how precisely would you respond?

Solution. We have \[D=\frac{2.5}{(1+7.3\%)^{3/12}}\approx2.456\] and hence the delivery price for a forward contract is \[K=(35-D)\cdot(1+7.3\%)^{6/12}\approx33.711<34\] hence it is an arbitrage opportunity: we can make profit by enters a long position 6-month forward contract.

2.12 Problem 14

The interest rate in the UK is 4.8% p.a. compounded continuously. The interest rate in the US is 5.3% p.a., compounded continuously. Today, £1 = $1.87. Calculate, in pounds, the 6-month forward price of the dollar. Why is the 6-month forward price of the dollar less than the spot price?

Solution. Note that $1 = £0.535, we set up a portfolio \(V\) as follows:

1. Today \((t=0):\)

• Buy \(1\cdot e^{5.3\%\cdot(-6/12)}\approx0.974\) dollar by borrowing \(0.974\cdot£0.535\approx£0.521.\)
• Enter the contract to sell 1 dollar (at maturity) for \(£K.\)
• Additional fees: £0.
• Contract value: \(V_0=0.521-0.521+0=0.\)

2. At maturity \((t=6/12):\)

• Repay the loan with \(£0.521\cdot e^{4.8\%\cdot6/12}\approx£0.534.\)
• Hand over 1 dollar and receive the delivery price \(£K.\)
• Additional fees: £0.
• Contract value: \(V_{6/12}=-0.534+K+0=K-0.534.\)

3. Under the assumption of no arbitrage, \(V_{6/12}=0\Rightarrow K=0.534.\)

4. The delivery price is, therefore, £0.534, less than the spot price of 1 dollar since the interest rate in UK is smaller than in US.

2.13 Problem 16

The interest rate in London is 4.35% p.a. compounded annually. In Cyprus, the interest rate is 6.16% compounded annually. 1 CYP = £1.1981. In 4 months’ time, how many pounds might I expect to buy for 10,000 CYP?

Solution. We set up a portfolio \(V\) as follows:

1. Today \((t=0):\)

• Buy \(10000\cdot(1+6.16\%)^{-4/12}\approx9802.715\) CYP by borrowing \(9802.715\cdot£1.1981\approx£11744.633.\)
• Enter the contract to sell 1 dollar (at maturity) for \(£K.\)
• Additional fees: £0.
• Contract value: \(V_0=11744.633-11744.633+0=0.\)

2. At maturity \((t=4/12):\)

• Repay the loan with \(£11744.633\cdot(1+4.35\%)^{4/12}\approx£11912.519.\)
• Hand over 1 dollar and receive the delivery price \(£K.\)
• Additional fees: £0.
• Contract value: \(V_{6/12}=-11912.519+K+0=K-11912.519.\)

3. Under the assumption of no arbitrage, \(V_{4/12}=0\Rightarrow K=11912.519.\)

4. The delivery price is, therefore, £11,912.519.

2.14 Problem 17

In Australia, the interest rate is 4.23% compounded continuously. The interest rate in the US is 4.78% compounded continuously. Today, 1 AUD = $0.7632. Describe carefully what a currency dealer in New York might do if the 6-month forward price of the AUD was:

1. $0.77.
2. $0.76.

Solution. The delivery price is \[1\cdot e^{4.23\%\cdot(-6)/12}\cdot0.7632\cdot e^{4.78\%\cdot6/12}\approx0.765\] hence if the 6-month forward price of the AUD was $0.77 ($0.76) then the dealer should enter a long (short) forward contract.

2.15 Problem 18

BBR is a company which was a late arrival on the BBQ scene. The BBR share price today is £5.7 and the company will pay a dividend of 2% of today’s share price in 3 months and again in 9 months. The interest rate is 4.5% compounded quarterly.

1. Calculate the equivalent continuously compounded annual rate for the safe interest rate (call the continuously compounded rate R) and the two dividend payments (call this continuously compounded rate Q).
2. Use the formula \(F=S_0\cdot e^{(R-Q)T}\) to calculate the 1-year forward price of the BBR share.

Solution.

1. We have \((1+4.5\%/4)^4=e^R\Rightarrow R\approx4.475\%\) and \[(1+2\%)^2=e^Q\Rightarrow Q\approx3.961\%.\]
2. \(F=5.7\cdot e^{(4.475\%-3.961\%)\cdot1}\approx5.729.\)

2.16 Problem 19

Polyfew chemical shares cost, today, £15.5. The shares pay an annual dividend of 6% in 2 6-monthly dividend payments. The next dividend is to be paid in 4 months’ time. It is thought that next year the annual dividend will be 6.1%. Assume the share price next year remains at £15.5. The interest rate is 5.5% compounded annually. Use the formulas

1. \((S_0-D)(1+R)^T\) for discrete dividend payments, and
2. \(S_0\cdot e^{(R-Q)T}\) with equivalent continuously compounded rates,

to calculate the 18-month forward price on the shares.

Solution.

1. We have \[D=\frac{15.5\cdot6\%/2}{(1+5.5\%)^{4/12}}+\frac{15.5\cdot6\%/2}{(1+5.5\%)^{10/12}}+\frac{15.5\cdot6.1\%/2}{(1+5.5\%)^{16/12}}\approx1.342\] hence \[F=(15.5-D)\cdot(1+5.5\%)^{18/12}\approx15.342.\]
2. We have \(1+5.5\%=e^R\Rightarrow R\approx5.354\%\) and \[(1+6\%/2)^2\cdot(1+6.1\%/2)=e^{3Q/2}\Rightarrow Q\approx5.944\%\] and hence \[F=15.5\cdot e^{(5.354\%-5.944\%)\cdot18/12}\approx15.363.\]

2.17 Problem 20

The interest rate in the UK is 4.87% compounded annually. Today, £1 = 3,767.676 BYR. The 6-month forward price on the BYR is £1 = 3,925.831 BYR. Estimate the interest rate \(r,\) annually compounded, in Belarus.

Solution. The equality \[3925.831=1\cdot(1+4.87\%)^{-6/12}\cdot3767.676\cdot(1+r)^{6/12}\] implies \(r\approx13.859\%.\)

2.18 Problem 22

An asset is valued today at £55. The asset pays a dividend of £2.5 every 6 months, with the \(1^{\textrm{st}}\) dividend due in 2 months’ time. The interest rate is 5%, compounded continuously.

1. Calculate the 12-month forward price of the asset. A long 12-month forward contract on the asset (the delivery price is the 12-month forward price) is entered today. What is the value of this forward contract today?
2. The value of the asset in 1 month’s time, 2 months’ time and so on are given in the table below. Calculate the value of the forward contract described in part 1 on each of these dates. Draw a graph of the value of the forward contract against asset value.

Solution.

1. We have \[D=\frac{2.5}{e^{5\%\cdot2/12}}+\frac{2.5}{e^{5\%\cdot8/12}}\approx4.897\] and hence \[F=(55-D)\cdot e^{5\%}\approx52.672.\]

3.1 Slide 8

An investor enters into a short position in a gold futures contract at 294.2 USD. Each futures contract controls 100 troy ounces. The initial margin is USD 3,200, and the maintenance margin is USD 2,900. At the end of the first day, the futures price drops to 286.6 USD. What is the profit or loss at the end of the first trading day?

Solution. Since the price drops and the entered contract is in short position, the investor earns an amount of \[(294.2-286.6)\cdot100=760.\]

3.2 Slide 11

The below table shows the results of 9 days’ trading in 2 identical long futures contracts on VN30 index. Unit of prices is 1,000 (VND). Complete the table and find the overall gain or loss over the 9 days’ trading in the 4 contracts.

Solution.

• Margin Ratio: \((15000/103502)\cdot100 = 14.4925\%.\)
• Maintenance ratio: \((12000/15000)\cdot100 = 80\%.\)
• Overall Gain/Loss: \(2926000\cdot2=5852000\)(VND).
• Rate of Return: \(2926/15000 = 19.51\%.\)

3.3 Slide 16

It is \(1^{\textrm{st}}\) of July. A company knows it will have to buy 1,000 metric tons of gas oil on \(1^{\textrm{st}}\) of October. The spot price per metric ton on \(1^{\textrm{st}}\) of July is $214.65. A long October futures contract in gas oil is priced at $219.70 per metric ton. The size of each futures contract is 100 metric tons.

1. What is the number of futures contracts bought for perfect hedging?
2. How much money should the company pay to buy 1,000 metric tons of gas on \(1^{\textrm{st}}\) October?

Solution.

1. The number of futures contracts bought for perfect hedging is: 1000/100=10 contracts
2. On 1 October, there are two possibilities (all prices are per metric ton).

• The spot price on 1 October is greater than or equal to $219.70. Suppose the spot price is $220.50.

Since, at maturity, the spot price and the futures price converge, the futures price on this day will be close to $220.50. The futures contracts are to buy the gas oil for $219.70, so by closing out its futures contracts, the company will make a profit of (approx) $(220.50 − 219.70). The company will pay $220.50 for the gas oil. Hence, the company will be paying $220.50 − $(220.50 − 219.70) = $219.70 per metric ton, as hoped for.

• The spot price on 1 October is less than $219.70.

Suppose the spot price is $218.30. By closing out the futures contracts, the company loses (approx) $(219.70 − 218.30). In the purchase, the company pays out $218.30. So in total, the company pays out (approximately) $218.30 + $(219.70 − 218.30) = $219.70.

So, money should the company pay to buy 1000 metric tons of gas on \(1^{\textrm{st}}\) October is:$ 219.7 =$219700.$

4.1 Problem 5

On 7 September, Martin enters three long futures contracts in corn with December delivery. One contract is to buy 5,000 bushels of corn and today the price of a December futures contract in corn is 268 cents per bushel. Initial margin is set at $1,300 per contract. Maintenance margin is 75% of the initial margin. The futures price on successive days is:

Martin closes out the contract at the close of trading on Day 7. Complete the table. What is Martin’s overall gain or loss?

Solution.

Overall Gain/Loss: -$1,650.

4.2 Problem 6

The details of 9 days’ trading in a long futures contract are shown. There is something wrong with this table. What is it?

Solution.

On Day 6, the margin account balance should be $21600. The calculation performs a margin call when the margin account falls below the level of the initial margin. A margin call should be made when the margin account falls below the maintenance margin.

4.3 Problem 7

The details of 8 days’ trading in a short futures contract are shown. Prices are in pounds. Complete the table. What was the overall gain or loss over the 8 days of trading?

Solution.

Overall Gain/Loss: -$19,000.

4.4 Problem 8

The results of 8 days’ trading in a long futures contract are shown. Prices are in dollars. Complete the table. Find the overall gain or loss over the 8 days.

Solution.

Overall Gain/Loss: $840 per contract or $2520 on the three contracts.

4.5 Problem 9

The figures from 7 days’ trading in a short futures contract can be seen. Prices are in pounds. Complete the table. What was the overall gain or loss over the 7 days’ trading?

Solution.

Overall Gain/Loss: £1700 per contract or £5100 on the three contracts.