Question

The current price (in June) of gold is $1,900 per ounce. Daphne wants to write a long forward on an ounce of gold for 3 months from now (in September). The current continuously compounded risk-free interest rate is 4% per annum.

In the following calculations, ignore any tax, transaction costs, and other fees.

a) Calculate the arbitrage-free forward price. [2 marks]

b) Suppose the current market forward price is $1,925. Describe all the steps you would take to make an arbitrage profit. [3 marks]

c) Daphne entered into the long forward at the price calculated in part a). One month passes (it is now July), and the current price of gold is $1,950. The continuously compounded risk-free interest rate is now 5% per annum.

Eddie, another investor, is interested in taking over Daphne’s position in her forward contract. Perform the relevant calculations for the amount of money that would have to change hands. Also, clearly state which person would have to pay the other. [4 marks]

d) Roz was the counterparty to Daphne’s original forward contract. State the value of Roz’s position in July.

Answer 1

a.

Arbitrage free forward price In case of continuously comounding-

F= S* e ^rt

Where F= forward price

S= spot price

r= risk free interest

t = period

e=2.71828

Hence in the current case

F= $1900 * e^4*3/12

=>F = $1900 * e^0.01

=>F = $1900*1.0100 (Note = e^0.01= 1.0100)

=> F = $1919

----------------------------------------------------------------

B.

Current markwt forward price = $ 1925

Fair forward price or arbitrage free forward price = $ 1919

The arbitrage gain can be obtained by entering into the following transactions-

A. Borrow $1900@ 4% pa interest rate for 3 month and buy one ounce of Gold today @$ 1900	$1900
B. Enter into contract to sell gold forward after 3 month @ $1925	$1925
C. after 3 month $1900 borrowed amount to be repaid along with interest =$1900 * e^4%*3/12 =$1900*1.01 =$1919	$1919
C. Sell the gold ounces as per the forward contract after 3 month @$1925 and pay the $ loan along with the interest
D. Arbitrage profit after 3 month - Receive---------------------$1925 Pay towards loan---------$1919 Profit = $1925-$1919 = $6 per ounces	$6 per ounces

----------------------------------------------------------------------

C.

S= $1950

t= 2months

r= 5% pa

Fair Forward price or F in the July month= S * e ^ r*t

=>F=$1950 * (2.71828 ^5%*2/12)

=>F= $1950 * 2.71828 ^0.833%

=>F= $1950 *1.0083 = $1966.185

Actual forward price = $ 1919

It means if Ediie takes the forward position today from Daphne then She will earn profit of $47.185 ( $1966.185-$1919) after 2 months.

Hence Eddie should pay the present value of the profit to Daphne and Daphne is foregoing the profit.

Hence payment should be made by Ediie = $47.185 / 1.0083 = $46.79

-----------------------------------------------------------------------------

d.

Roz being the counter party has a SHORT FORWARD POSITION in july.

For Roz in july the Loss position will be = $ 1950-$1919 = $31 as on july