Question

Forward prices of a generic asset The purpose of these problem is to guide you and introduce you the “no-arbitrage” condition required to compute forward prices. For the following problems, assume the following information: There is an asset A. The price of the asset today, denoted by ?0, is ?0 = $100. The CCIR (yearly) is 6%.

Problem 4: No storage cost, and a convenience yield. Assume that asset A has no storage cost and there is a convenience yield. Every 9 months, the holder of the asset receives 5% of the asset value as a payment (you can call that a dividend). Suppose that someone is willing to enter a forward contract of Asset A for delivery in one year from now at ?0,1 = $115

a. Suppose that in 9 months, the price of the asset is ? 9 12 . How much money does one receive at t=9 months from holding the asset? Assume you have ? shares.

b. If you spend the dividend to buy more of asset A, how many more units of A can you buy (assume you can buy fractions of the asset)

c. We don’t know a priori if there is a mispricing. Compute an arbitrage portfolio to exploit the potential mispricing. Hint: start by borrowing today to buy $100 1+5% units of asset A and reinvest the dividend to buy more of A to get one unit at t=1 year.

d. What would be the forward price that makes the profit in a) zero?

e. Now try to find the general pricing formula. Suppose that the rate is ?, the spot price is ?0 and someone is willing to enter a forward at a forward price of ?0,? for delivery at time t=T. Replicate your portfolio/strategy in a) using this new information. What is the no-arbitrage forward price? Assume that a dividend %? is paid at ?1,?2,?3, … ,?? < T

Answer 1

the general formula for calculating forward price is -
F1= M0 * e^ (r-c+u)T
Where-
r = the risk-free rate that applies to the life of the forward contract
t = the delivery date in years
c = convenience yield
u = storage cost

Lets calcuate forward price for following parameters-
r	6%
c	5%	for 9 months
c	6.67%	for 12 months
M0	100
T	1
F1=	100* e^((6%-6.67%)*1)
Forward price 1 year from now	99.33555
	= no arbitrage price
A
F9=	100* e^((6%-6.67%)*9/12)
Forward price 9 months from now	99.50125
	= no arbitrage price
F9' (without dividend)	100* e^((6%-0)*9/12)
Forward price 9 months from now	104.6028
Dividend received	5.230139	=104.603*5%
Total money received	104.7314	=5.23+99.50

for each shared priced 100 today, the forward contract will be for 99.50 which is the money received for forward contract. Additionally the holder will receive 5% dividend of the future value at 9 months which is 5.23. Hence the total amount received is $104.73

B
Dividend received after 9 months	5.230139
Value of share after 9 months after dividend is paid	99.50125
no of shares that can be purchased from dividend after it is paid	0.052564	=5.23/99.50

C
Arbitrage in potential mispricing
Today-
1. Borrow $100 at 6% interest
2. Buy the asset at $100
3. Enter into forward contract to sell the asset after 1 year a forward price - dividend of 99.3355
9 months later
4. receive dividend of $5.23 and buy more 5.25% of partial share
1 year later-
5. Receive the forward price for the commodity
6. Pay off the borrowed $100 + interest which is = $106.18
7. Settle the forward contract at 99.3355
Since arbitrage will work and make profit at a forward contract lower than the forward price of 99.3355

D
No arbitrage price of 99.3355 will give a arbitrage profit of 0 in the above trade

E
The general formula is as below-
F1= M0 * e^ (r-c+u)T
Where-
r = the risk-free rate that applies to the life of the forward contract
t = the delivery date in years
c = convenience yield
u = storage cost