Arbitrage - futures contracts part (B) Flashcards
Valuing Forward and Futures Contracts
It is well known in practice that
It is well known in practice that, if interest rates are constant, a futures contract has the same value as an otherwise identical forward contract. That is, although a futures contract has a complicated cash flow pattern (via the marking to market feature), it can be valued as though it were a forward contract.
a forward contract has only a single cash flow, it is easy to value. Consequently, it is industry practice to value futures contracts as though they were forward contracts.
Forward and futures contracts can be valued by recognizing the two following ideas:
- Their payoff can be replicated by taking positions in the spot market; and,
- If there are two ways to get the same outcome, they must have the same price (i.e. valuation is through the concept of arbitrage).
Imagine that you are the maker of fine quality wool fabrics and have determined that you are going to need to buy 5,000 kg of wool exactly three months from today. However, you are worried about what might happen to the spot price of wool between now and that time.
two ways that you can “lock in” a guaranteed price for this wool.
Either way, you will
you will end up having the required wool three months from now, and this wool will have a spot market value of ST at this time. As the outcome will be the same regardless of whether you choose the first or second option, according to the laws of arbitrage, F must be equal to S0(1+rf+q)T. This gives us the valuation equation for forwards and futures contracts:
F = S0(1 + rf + q)T
when
F > S0(1 + rf + q)T
what happens?
as arbitrageurs would spot this difference and would take advantage of it immediately as follows:
–Borrow and buy the wool at time 0, also entering into a forward contract to sell it for F in 3 months;
–Hold the wool for 3 months, incurring the cost of borrowing (rf% p.a.) and storage (q% p.a.);
–In 3 months time, sell the wool, which they have paid S0(1+rf+q)T to buy and deliver today for F; and,
–As F exceeds S0(1+rf+q)T, make a riskless profit equal to F - S0(1+rf+q)T.
what is a riskless profit
The profit is riskless as, regardless of what happens in the market, the arbitrageur will make the same profit (and therefore there is no uncertainty).
The arbitrageur’s strategy can be summarised as follows:
when more and more arbitrageurs pot the mispricing and adopt a strategy identical, what will happen?
This will push up the wool’s spot price at time 0 as well as drive down the forward price to the point where F is equal to S0(1+rf+q)T. At this point, no more such opportunities exist.
In general, whenever F is not equal to S0(1+rf+q)T, it is because:
- The asset underlying the contract is incorrectly valued in the spot market
–The futures / forward contract is incorrectly valued.
elaborate
If F > S0(1+rf+q)T then either the forward / futures contract is overvalued and / or the asset underlying the contract is undervalued in the spot market
If F 0(1+rf+q)T then either the forward / futures contract is overvalued and / or the asset underlying the contract is undervalued in the spot market
arbitrageurs’ trading strategies will be designed to
take account of the potential mispricing in both markets. Moreover, they will always buy the relatively underpriced asset and simultaneously sell the relatively overpriced asset (i.e. they will buy low and sell high):
arbitrageurs’ trading strategies will be designed to take account of the potential mispricing in both markets
If F 0(1+rf+q)T then
they will sell the right hand side and buy the left hand side, which entails:
–Borrowing the asset at time 0, selling it for S0 in the spot market and going long in the derivative market to buy the asset for F at time T;
–Investing S0 and accruing interest (rf% p.a.), saving the cost of carry (q% p.a.) between 0 and time T;
–Buying the asset at time T in exchange for F and returning it to the person you borrowed it from; and,
–Making a positive profit equal to S0(1+rf+q)T – F.
short selling?
process of borrowing an asset from its owner at time 0, selling it and repurchasing it at a later date to return to the owner
Short-selling is
Short-selling is something that is not possible for all assets. However, this does not mean that the cost-of-carry relationship will not hold. This is because, if there are a large number of people who hold the underlying asset for investment purposes, a low forward price will make it attractive for them to sell the asset now and take a long position in the forward contract.
why do some assets have a negative cost of carry or storage cost?
A good example is a share that pays dividends at a constant rate or, alternatively, a stock index. The cost of carry is negative for these assets as they have a benefit associated with holding them, namely the receipt of dividends.
Also, in these cases, we use d rather than q, to represent holding benefits in the cost of carry model, or:
• F = S0(1 + rf – d)T
Suppose the spot price of beef is 350 cents per kilogram, the six-month futures price is 380, the riskless rate of interest is 5% p.a., and the cost of storing beef is 6% p.a. Is there an arbitrage opportunity present in this market?
Example: Share Contract Valuation
Suppose we observe that shares in BHP Billiton are currently trading at $26.00 each. We also observe that the futures contract on these shares with six months to expiration is trading at $25.90. If the prevailing Bank Bill rate is 7% p.a. and the dividend rate d = 5% p.a., does this represent an arbitrage opportunity?
