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 2: Storage cost, and no convenience yield. Assume that asset A has a storage cost and there is no convenience yield. Every 7 months, the holder of the asset needs to pay $10 dollars for storage. Suppose that someone is willing to enter a forward contract of Asset A for delivery in one year from now at ?0,1 = $120

a. 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 $100

b. Now suppose that someone is willing to enter a forward contract of Asset A for delivery in one year from now at ?0,1 = $103 . Compute an arbitrage portfolio to exploit the potential mispricing. Hint: start by short-selling the asset

c. What would be the forward price that makes the profit in a) and b) zero?

d. 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 storage cost $? has to be paid ?1,?2,?3, … ,?? < �

Answer 1

The general formula for calculating forward price is -

M1= M0 * e^ rT

Where-

r = the risk-free rate that applies to the life of the forward contract

t = the delivery date in years

Lets calcuate forward price for following parameters-
r	6%
M0	100
T	1
M1=	100* e^(6%*1)
	106.1837
Storage cost for 7 months	10
Storage cost for 12 months	17.143	=10/7*12
Adding storage cost to M1
F1	123.327
hence, no arbitrage forward contract price	123.327

A

Arbitrage in potential mispricing

Today-

1. Borrow $100 at 6% interest

2. Buy the commodity at $100

3. Enter into forward contract to sell the commodity after 1 year a forward price + storage cost

1 year later-

4. Receive the forward price for the commodity and repay storage cost

5. Pay off the borrowed $100 + interest which is = $106.18

6. Any income from forward contract above the storage cost + repayment cost i.e. above $123.327 is profit from arbitrage.

B

Forward contract at $103

Today-

1. Sell the asset short at $100 and receive the money

2. Invest the $100 received from short at 6% interest

3. Enter into forward contract with the counterparty to buy the asset at $103 1 year from now

1 year later-

4. Receive the $100 capital plus interest which is = $106.18
5. Close the forward contract at $103

6. Profit is the difference between money received from investment and paid to close forward contract i.e. $106.18- $103 = $3.18

Additionally, if we could enter into a separate forward contract to sell the asset 1 year from now at the no arbitrage price, we could make a profit equal to difference between the 2 forward contract prices after 1 year when we square off the 2 contracts. This is possible in forward contracts since they are not traded on the exchange and there is no standardization. Hence, there are gaps in the market which allow for such mispricing for the same security.

C

The above transaction in A and B will have no arbitrage profit if the forward contract is traded at the derived forward contract price of $123.327.

D

The general formula for calculating forward price is calculated above. If we add storage cost in the form of % of asset value, the formula becomes-

M1= M0 * e^(r+u)*T

Where-

u = storage cost