Problem:

A farmer grows and freezes organically cultivated peas. He is worried that the price of peas might fall. The current price for one unit of frozen peas is $3. Today, the farmer enters a forward contract to sell 10,000 units of frozen peas on the next 8 months. The risk-free interest rate is 6.5% per year, compounded continuously. What should be the delivery price (per unit) of the forward contract?

Solution: To determine the delivery price (per unit) of the forward contract, we use the formula for the forward price in a risk-free context. The formula for the forward price \(F\) is given by:

\[F= S_{0}\times e^{rT}= 3\times e^{0.065\times\frac{2}{3}}= 3.13\]

Problem 1:

The price of a share is $7. The share will pay a dividend of $0.4 after four months and after ten months. An investor enters a forward contract to sell one share in 14 months’ time. The risk-free interest rate is 7% per year with interest compounded continuously. What is the delivery price on this contract?

Calculate the present value of the dividends:

• The present value of the first dividend \(D_{1}\): \[PV_{1}= 0.4\times e^{-0.07\times\frac{4}{12}}\]

• The present value of the second dividend \(D_{2}\): \[PV_{2}= 0.4\times e^{-0.07\times\frac{10}{12}}\]

• The present value of dividend: \[PV= 0.4\times e^{-0.07\times\frac{4}{12}}+0.4\times e^{-0.07\times\frac{10}{12}}= 0.7681\]

The formula for the forward price \(F\) when the underlying asset pays discrete dividends is: \[F= (S_{0}-PV)\times e^{rT}= (7-0.7681)\times e^{0.07\times \frac{7}{6}}= 6.7622\]

Problem 2:

The price of a share is $7. The share pays no dividend. An investor enters a forward contract to sell one share in 14 months’ time. The risk-free interest rate is 7% per year with interest compounded continuously.

a)What is the delivery price on this contract? \[F= S_{0}\times e^{rT}= 7\times e^{0.07\times\frac{14}{12}}= 7.5957\]

b)Is the delivery price higher or lower when the share paying dividends? Why?

The delivery price is lower when the share pays dividends due to the subtraction of the present value of expected dividends from the current spot price in the forward price calculation.

Problem 3:

Suppose the risk-free interest rate is 10% per year compounded continuously. A stock paying a dividend of 7,000 (VND) every 6 months (the first dividend payment is due in 2 months’ time) is valued at 115,000 (VND). The 15-month forward price on the stock is 120,000 (VND) per share.

a)Is this an arbitrage opportunity? Why?

• The present value of dividend: \[PV= 7000\times e^{-0.1\times\frac{2}{12}}+7000\times e^{-0.1\times\frac{8}{12}}+ 7000\times e^{-0.1\times\frac{14}{12}}= 19662\]

The formula for the forward price \(F\) when the underlying asset pays discrete dividends is: \[F= (S_{0}-PV)\times e^{rT}= (115000−19662)\times e^{0.10\times \frac{15}{12}}= 108014\]

b)If so, how precisely would you respond to exploit the arbitrage opportunity to earn at least 10 million VND?

Problem 1

A company knows it will have to buy 100,000 barrels of crude oil, in the market, in one month’s time. The situation in the Middle East is turbulent and the company fears that the price of crude oil could rise considerably. The market price, today, for one barrel of crude oil is $68.05. Assume the interest rate is r = 7.85% per year compounded monthly. The company enters a long one-month forward contract on 100,000 barrels of oil.

a)What is the delivery price (per barrel) of this contract?

Solution: The delivery price of contract

\[ F_{\frac{4}{4}} = K = 68.05 \times (1+\frac{7.85%}{12})^1 = 68.4952 \]

b)After one week, the price of one barrel of oil has risen to $69.75. What is the three-week forward price for one barrel of this oil?

Solution: 3-weeks forward price of oil

\[ F_{\frac{3}{4}} = K = 69.75 \times (1+\frac{7.85%}{12})^\frac{3}{4} = 70.0919 \]

c)What is the value of the contract (per barrel) after one week?

Solution: Value of the contract after one week

\[ V = \frac{F_{\frac{3}{4}}-F_{\frac{4}{4}}}{(1+\frac{7.85%}{12})^\frac{3}{4}}= \frac{70.0919-68.4952}{(1+\frac{7.85%}{12})^\frac{3}{4}} = 1.5889 \]

Problem 2

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

a)If the spot price for gold of this purity is $700 per ounce, find the three-month forward price (per ounce) of gold.

Solution: 3-month forward price of golds is:

\[ K = 700 \times e^{0.049\times \frac{3}{12}} = 708.628\]

b)One month later, the spot price is $701.5 per ounce. If the interest rate is unchanged, find, on this date, the two-month forward price (per ounce) of gold.

Solution: 2-month forward price (per ounce) of gold:

\[ K = 701.5 \times e^{0.049\times \frac{2}{12}} = 707.252\]

c)What is the value of the contract (per barrel) after one month?

Solution: Value of the contract after one month

\[ V = \frac{707.252 - 708.628}{e^{0.049\times \frac{2}{12}}} = -1.365\]

Problem 3

Today is 8 March. ULS shares are selling today at $10.75. The share pays a dividend of 6% per year in two equal dividend payments, made on 8 April and 8 October. Stella wants to enter a forward contract to buy 1000 shares on 8 November. The risk-free interest rate is 5% per year, compounded continuously

a)Find the eight-month forward price (per share) on ULS shares.

Solutions: 8-month forward price on ULS shares

\[ F_0 = 10.75e^{0.05 \times \frac{9}{12}} - 10.75\times\frac{0.06}{2}(e^{0.05 \times \frac{7}{12}}+e^{0.05 \times \frac{3}{12}}) = 10.51\] b)On 8 May, the share price has fallen to $9.25. On this day, what is the value (per share) of Stella’s forward contract?

Solution: The value (per share) of Stella’s forward contract

\[F_1 = 9.25e^{0.05 \times \frac{6}{12}}-9.25\frac{0.06}{2}e^{0.05 \times \frac{1}{12}} = 9.21\] \[\to PV_1 = \frac{F_1-F_0}{e^{0.05 \times \frac{1}{12}}} = -1.268\]

Problem 4

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% per annum and the stock price is $25. What is the current value of the forward contract?

Solution: Current value of the forward contract

\[\frac{F_1-F_0}{e^{0.5\times0.1}}= 25-24\times e^{0.5\times0.1} = 2.17 \]