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 1: No storage cost, and no convenience yield. Assume that asset A has no storage cost and there is no convenience yield. Suppose that someone is willing to enter a forward contract of Asset A for delivery in one year from now at ?0,1 = $109

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?

Answer 1

a.

Borrow $100 and buy the asset in the spot market (Price of the asset today is $100). At the same time, enter into the forward contract to sell this asset one year from now at a price of $109.

One year from now, we would have to pay back the $100 which would have compounded at a 6% annual rate. The amount that must be paid back, assuming continuous compounding, can be calculate using the formula:

............(1)

where,

• e is a constant equal to 2.718
• r is the annual interest rate
• T is the number of years

So, the amount that must be paid back is:

Also at the end of one year, we would receive $109 as per the forward contract. So on a net basis you end up with a riskless profit of $2.82 ($109 - $106.18).

b.

In this case short sell the asset right now. Short selling is when you borrow the asset from a broker and sell it today with the obligation of buying that asset in the future to return it back to the broker. (This strategy is used when you expect the price of an asset to go down).

So, you short sell and in return you get $100 today, which you invest at a CCIR (continuously compounded interest rate) of 6%. At the same time you enter into a forward to buy the asset after a year at a price of $103.

Now the $100, after a year, becomes $106.18 (as computed above). You use $103 out of this $106.18 to buy the asset as per the forward contract and return it to the broker. So, net you pocket $3.18 ($106.18 - $103).

c.

So, it is clear that for there to be no arbitrage opportunity, the forward price has to be $106.18, which is the value of the asset today, compounded over the duration of the forward contract by the spot CCIR.

d.

The general pricing formula can therefore be written on the lines of equation (1) as:

Where

• is the forward price
• is the spot price
• r is the CCIR
• T is time period till the forward contract matures

This equation computes a forward price that leaves no room for a riskless profit to be made using strategies discussed above.