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)(continuously compounded interest rate) is 6%.

Problem 3: 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 $13 dollars (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. 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 = $80 . 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 dividend $? is paid at ?1,?2,?3, … ,?? < T

Answer 1

For spot Price = M0, CICR = r = 6%, storage costs = S, Convenience Yield = C, Maturity of forward contract = T

Forward Contract for delivery at time

Here, PV(X) = Present Value of X, S = 0, C = $13, T = 1

PV(S) = 0, PV(C)   

Therefore,

a.) So, if the forward contract is available for $115 which is more than no-arbitrage value of $92.9839. Thus, we have an arbitrage opportunities. To take advantage of an arbitrage opportunity we always do the same thing

"Buy low, Sell High"

Thus, we enter into forward contract as a seller as it has higher value and buy the asset today as it has lower value.

To buy the asset, we borrow $100 today and using that $100 we buy the asset for $100 today. After 1 year when the contract matures, we sell the asset A and receive $115 as per the forward contract. During the contract, we receive $13 in the 9th month. After selling the asset for $115. We pay $106 back to the lender ($100 principal + $6 interest). Thus, our overall cost taking Time value of money into account is the no-arbitrage value $92.9839 and we sell for $115. Thus, we made a riskless profit of $22.0161 ($115 - $92.9839). This is arbitrage.

b.) If forward contract is available for $80, it is priced low, so we enter into the contract as a buyer and short-sell the higher priced asset. (Short-selling is selling an asset you do not own, you borrow it from your broker)

Thus, We sell the asset for $100 today, lend this $100 in the market for 6% interest. receive $106 after 1 year, out of this $106 we use $80 to buy asset A as per the contract. The downside is we miss the $13 we were supposed to get if we held onto the asset.

Thus our profit out of this arbitrage after taking time-value of money is $12.9839 ($92.9839 - $80).

However, notice for a.) and b.), our profits are as on the day of maturity of contract. If asked to find profits today, find the present value of profit.

c.) The forward price that makes the profits in a.) and b.) zero is the no-arbitrage price of $92.9839.

d.) General pricing formula for Forward price is  

if a dividend $d is paid at t1,t2,t3,.....,tn<T, then we find the present value of all these dividends.

(Note, convenience yield and dividend are treated similarly, so we take that present value of dividends is included in PV(C).

PV(dn) =

where, dn is the $d received at time tn

So total value of all dividends becomes .