Question

Assume no storage costs in this problem. Also, assume continuous compounding.
Consider the following forward prices on an investment asset

Forward prices (cents/unit of the asset) as of February

Mar 270.25

May 275.25

Use the above information on March and May forward prices to determine the spot price of the asset and risk-free rate in February at which there should be no profitable arbitrage opportunities.

Now let us assume that the observed risk-free rate is different from your answer to the above question (i.e. either higher or lower than the risk-free rate you obtained above). Can you clearly outline a trading strategy in February to exploit any arbitrage opportunities?

Answer 1

Let the spot price be S and the risk free rate be r (per month)

Forward price (March) = F1 = 270.25 and Forward prie (May) = F2 = 275.25

Therefore, by the cost of carry model:

S x e^(r x 1) = F1 = 270.25 - (a)

S x e^(r x 3) = F2 = 275.25 - (b)

Dividing Equation (b) with Equation (a) , we get

[275.25 / 270.25] = [e^(r x 3) / e^(r x 1)]

r = 0.009167 or 0.9167% per month or 10.99 % per annum

S = F1 / e^(0.009167 x 1) = $ 267.78

Let us assume that observed risk free rate per month is 1% or 0.01 in decimal terms. A riskless arbitrage can be executed using the March futures and the underlying asset as described below:

- Short sell one unit of the asset. Payment received = $ 267.78 (the assets per unit spot price)

- Lend this payment at the monthly risk free rate of 1% with continuous compounding for one month and simultaneously buy a forward contract for March.

- Proceeds received from lending after one month = 267.78 x e^(0.01 x 1) = $ 270.47

- Forward contract is executed to purchase the shorted underlying asset at the March Forward price of $ 270.25

- Amount Left = Lending proceeds received - forward price = 270.47 - 270.25 = $ 0.22

- This amount left is the riskless profit or arbitrage profite generated owing to a mismatch in calculated (what it should be) interest rates and actual ( what it is) interest rates.

- Similar arbitrage can be executed using the May forward contract.