Valuing forward/futures using the no-arbitrage principle

We have discussed general theories of pricing of derivative products i.e. law of one price (LOP) and about the assumption that markets are frictionless. We studied about fundamental analysis and technical analysis. We will study in this article about how the pricing of forward and futures contracts are made. The focus of this section is to describe some important no-arbitrage relations for derivative contract prices in relation to these products.

If the investor does not book a forward or futures contract, the alternative form him is to buy at the spot market and hold the underly. In such a contingency he would incur the spot price + the cost of carry.

"The cost of carry refers to the difference between the costs and the benefits that accrue while holding an asset. Suppose a breakfast cereal producer needs 5000 bushels of wheat for processing in two months. To lock in the price of the wheat today, he can buy it and carry it for two months. One cost of this strategy is the opportunity cost of funds. To come up with the purchase price, he must either borrow money or reduce his earning assets by that amount. Beyond interest cost, however, carry costs vary depending upon the nature of the asset. For a physical asset such as wheat, he incurs storage costs (e.g., rent and insurance). At the same time, by storing wheat, he avoids the costs of possibly running out of his regular inventory before two months are up and having to pay extra for emergency deliveries. This benefit is called convenience yield. Thus, the cost of carry for a physical asset equals interest cost plus storage costs less convenience yield, that is,

Carry costs = Cost of funds + storage cost - convenience yield. (1a)

For a financial asset such as a stock or a bond, storage costs are negligible. Moreover, income (yield) accrues in the form of quarterly cash dividends or semi-annual coupon payments. The cost of carry for a financial asset is-

Carry costs = Cost of funds - income. (1b)

Carry costs and benefits are modeled either as continuous rates or as discrete flows. Some costs/benefits such as the cost of funds (i.e., the risk-free interest rate) are best modeled continuously. The dividend yield on a broadly-based stock portfolio and the interest income on a foreign currency deposit also fall into this category. Other costs/benefits like quarterly cash dividends on individual common stocks, semi-annual coupons on bonds, and warehouse rent payments for holding an inventory of grain are best modeled as discrete cash flows. In the interest of brevity, only continuous costs are considered here.

Dividend income from holding a broadly-based stock index portfolio or interest income from holding a foreign currency is typically modeled as a constant, continuous rate . The income, as it accrues, is re-invested in more units of the asset. In this way, buying exp[-iT ] units of a stock index portfolio today grows to exactly one unit at time T, and produces a net terminal value of _ST - S exp[(r - i) T]. The cost of carry rate equals the difference between the risk-free rate of interest r and the dividend yield rate i for a stock index portfolio investment, and equals the difference between the domestic interest rate r and the foreign interest rate i for a foreign currency investment. The total cost of carry paid at time T is

Carry costs = S (exp[(r - i) T] - 1) . (2)

The value of a forward contract is inextricably linked to the cost of carry of the underlying asset. Since a forward contract requires its buyer to accept delivery of the underlying asset at time T, buying a forward contract today is a perfect substitute for buying the asset today and carrying it until time T. The present value of the payment obligation under the forward contract strategy is f exp[-rT ], and the present value of the latter strategy is S exp[-iT ]. Since both strategies provide exactly one unit of the asset at time T, (i.e., _ST ), their costs must be identical,

f exp[-rT] = S exp[-iT ]. (3a)

If the relation (3a) does not hold, costless arbitrage profits would be possible by selling the over-priced instrument and simultaneously buying the under-priced one. The relation (3a) is the present value version of the cost of carry relation. A more familiar version is the future value form,

f = S exp[(r - i) T]. (3b)

When the prices of the forward and the asset are such that Equation (3a) and/or Equation (3b) hold exactly, the forward market is said to be at full carry. Unless costless arbitrage is somehow impeded, the forward market will always be at full carry. The difference between the forward (or futures) price and the asset price is frequently referred to as the basis

Futures contracts are like forward contracts, except that price movements are marked-to-market each day rather than receiving a single, once-and-for-all settlement on the contract’s expiration day . Obviously, the sum of the daily mark-to-market price moves over the life of the futures equals the overall price movement of a forward with the same maturity. With the futures position, however, the mark-to-market profits (losses) are invested (carried) at the risk-free interest rate until the futures expires. The value of the futures position at time T, therefore, may be greater or less than the terminal value of the forward position, depending on the path that futures price follows over the life of the contract.

Theories about pricing/valuation of Option contracts are discussed in the next module

The fair price of a derivative is the price at which profitable arbitrage is infeasible. In this sense, arbitrage (and arbitrage alone) determines the fair price of a derivative: this is the price at which there are no profitable arbitrage opportunities.

The pricing of index futures depends upon the spot index, the cost of carry, and expected dividends. For simplicity, suppose no dividends are expected, suppose the spot Nifty is at 1000 and suppose the one–month interest rate is 1.5%. Then the fair price of an index futures contract that expires in a month is 1015.

The difference between the spot and the futures price is called the basis. When a Nifty futures trades at 1015 and the spot Nifty is at 1000, “the basis” is said to be Rs.15 or 1.5%.

Basis risk is the risk that users of the futures market suffer, owing to unwanted fluctuations of the basis. In the ideal futures market, the basis should reflect interest rates, and interest rates alone. In reality, the basis fluctuates within a band. These fluctuations reduce the usefulness of the futures market for hedgers and speculators.

This is an error in the futures price of Rs.10. An arbitrageur can, in principle, capture the mispricing of Rs.10 using a series of transactions. He would (a) buy the spot Nifty, (b) sell the futures, and (c) hold till expiration. This strategy is equivalent to risklessly lending money to the market at 2.5% per month. As long as a person can borrow at 1.5%/month, he would be turning a profit of1% per month by doing this arbitrage, without bearing any risk.

This is an error in the futures price of Rs.10. An arbitrageur can, in principle, capture the mispricing of Rs.10 using a series of transactions. He would (a) sell the spot Nifty, (b) buy the futures, and (c) hold till expiration. This is equivalent to borrowing money from the market, using (Nifty) shares as collateral, at 0.5% per month. As long as a person can lend at 1.5%/month, he would be turning a profit of 1% per month by doing this arbitrage, without bearing any risk.

In practice, arbitrageurs will suffer transactions costs in doing Nifty program trades. The arbitrageur suffers one market impact cost in entering into a position on the Nifty spot, and another market impact cost when exiting. As a thumb rule, transactions of a million rupees suffer a one–way market impact cost of 0.1%, so the arbitrageur suffers a cost of 0.2% or so on the roundtrip. Hence, the actual return is lower than the apparent return by a factor of 0.2 percentage points or so.

The international experience is that in the first six months of a new index futures market, there are greater arbitrage opportunities that lie unexploited for relatively longer. After that, the increasing size and sophistication of the arbitrageurs ensures that arbitrage opportunities vanish very quickly. However, the international experience is that the glaring arbitrage opportunities only go away when extremely large amounts of capital are deployed into index arbitrage

What kinds of interest–rates are likely to show up on the index futures market – will they be like badla financing rates?

Arbitrage in the index futures market involves having the clearing corporation (NSCC) as the legal counterparty on both legs of the transaction. Hence the credit risk involved here will be equal to the credit risk of NSCC. This is in contrast with the risks of badla financing