Quiz-1 (Forward pricing)

First of all, sorry to all of you, for the delayed new posting! But I will try to post them regularly.

Which of the following options is most accurate for measuring the forward price of a commodity or an investment asset, assuming that the investment value of the asset grows continuously at risk-free rate of interest?
a.      S₀ x e^rt
b.      S₀ + e^rt
c.       S₀ / e^rt
d.      S₀ - e^rt

Assuming that that the value of investment asset grows at discrete rate of interest, which of the following formulae correctly explain relationship between the current spot price and forward price of an asset?
a.      S₀ (1+r)^t
b.      S₀ +(1+r)^t
c.       S₀ - (1+r)^t
d.      S₀ /(1+r)^t

A trader invested $520 in an investment asset, with an intention to sell it after 6 months. If the risk-free rate of interest is 4.5% per annum and the value of the asset grows continuously, the forward price of the asset is equal to:

a.      530.70
b.      531.83
c.       531.95
d.      532.05

A trader invested $520 in an investment asset, with an intention to sell it after 6 months. If the risk-free rate of interest is 4.5% per annum and the value of the asset grows on discrete basis, the forward price of the asset is equal to:

a.      530.70
b.      531.83
c.       531.95
d.      532.05

Assume that current spot price of an investment asset is $60 and risk-free rate of interest is 4.0% per annum. The forward price of an investment asset, growing at risk-free rate of interest (continuously) is calculated at time- T_{0 ,}  by the following theoretical forward price formula:

F₀ = S₀ x e^rt

When the traders identify that the above formula does not hold good, they attempt for arbitrage opportunities. If the market quoted forward price does not match the theoretical forward price, then there exists an arbitrage opportunity.

Which of the following statements about the arbitrage opportunities is correct, when the market determined forward price is higher than the theoretical forward price for a maturity period of 6 months?

Answer:

a. Borrow at risk-free rate, buy asset, sell forward

b. Lend at risk-free rate, buy asset, sell forward

c. Borrow at risk-free rate, sell asset and buy forward

d. Lend at risk-free rate, sell asset and buy forward

Frequently asked conceptual queries...

What is lease rate in commodity markets?

That is the rate charged by the lender of commodity, for lending the commodity. In the above case, (discussed in my previous blog) the lease rate is 6%.

What is E₀(S_T)?

We assume that a commodity will reach certain price in future. The assumption or expectation is made today. That is what E₀ refers.  That is the price of the commodity at time T (in future)

What is F_(0,T)?

That is theoretical forward price of the commodity.

Can there be a difference between F_(0,T) and E₀(S_T)?

Yes, there can be.  E₀(S_T) is based on expectations and F_(0,T) is based on mathematical calculation formula of : S₀ x e^rT

What will happen when there is a difference between F_(0,T) and E₀(S_T)?

There might be an arbitrage opportunity, when the theoretical prices deviate from the expected prices. Traders generally generate arbitrage opportunities due to this difference.

Can we discuss two examples of simple arbitrages, due to the difference between F_(0,T) and E₀(S_T)?

Yes, let us discuss these two examples to understand two arbitrages due to the difference between F_(0,T) and E₀(S_T).

S0 (Spot price)	$5.00
Price quotation	Cents per bushel
Contractual commodity	Corn
Time to maturity of the contract	1 year
Risk-free rate	6%
Theoretical forward price formula	S0 x erT
Theoretical forward price	$5.00 x e0.06 x 1 year = $5.31
What the trader can expect?	The trader expects that the market prices will deviate from theoretical prices (due to many reasons) and the trader can expect the prices after one year be either greater than $5.31 or lower than $5.31

Arbitrage example one: When the trader expects the realistic prices do not match with the theoretical prices and realistic prices will be higher than $5.31; Let us say the trader identified another trader who is willing to buy at $5.35.

Some steps are initiated at Time T0	Some steps are initiated at Time TT (on maturity of contract)
Borrow $5.00 @ risk-free rate	Sell  the commodity at $5.35
Buy the commodity in the spot market for $5.00	Repay the loan with interest: -$5.31
Enter into a forward contract to sell at $5.35 (Short forward contract) after one year.	Risk-free profit  due to arbitrage is: $5.35 - $5.31 = $0.04 per bushel
The arbitrage ceases to exist, when either the theoretical prices increase or when the market prices come down. When all the traders start doing the same activity, the price would come down to $5.31 Or, when the risk-free interest increases, the forward price may go up and arbitrage ceases to exist. This is a cash and carry arbitrage (borrow money, buy the asset and sell forward)

Arbitrage example two: When the trader expects the realistic prices do not match with the theoretical prices and realistic prices will be less than $5.31; Let us say the trader identified another trader who is willing to sell at $5.25 after one year.

Some steps are initiated at Time T0	Some steps are initiated at Time TT (on maturity of contract)
Sell the asset today in the cash market at the price of S0	At time T0: We get $5.00 by selling the commodity in cash market.
Invest the proceeds at the risk-free rate of 6%	Invest at 6%; After one year, we will get $5.31 from the investment.
Buy the asset in future through long forward contract at a price of $5.25	Buy the asset $5.25 in the forward market. Create an arbitrage opportunity of $0.06 per bushel.
This is an inverse cash and carry arbitrage: Sell today, invest the proceeds; buy in future or long forward