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Futures and Options Prices and Call/Put Parity

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Forward Pricing Contracts (commonly “forwards”) may well be the world’s oldest derivative instruments.

In use for at least 4000 years, forwards allow producers of assets to agree with purchasers of the same asset to make an exchange at a certain time and for a certain price.

In its most rudimentary form, forwards allowed the producer (mostly farmers) of an agricultural commodity which would not be ready for market until some point in the future to lock in a price in the future with a buyer of the asset (distributors, merchants). This insulated both parties from the uncertainty of price fluctuations due to weather, crop yield, supply and demand issues, etc.

It’s a win-win as both the buyer and seller can plan their respective business more effectively with the knowledge of what the transaction price in the future would be.

In 1865, forwards took a big step well…forward…when the World’s largest agricultural exchange, the Chicago Board of Trade, standardized the terms of the forward contracts traded. The liquid market they created allowed either party to make a trade without negotiating all of the terms of the deal on every transaction.  It also allowed them to trade their position with a third party rather than always executing delivery or acceptance of the underlying commodity at the expiration of the contract.  Now referred to as “Futures” contracts, these instruments are functionally identical to forwards. The concept of standardization attracted speculators and hedgers from around the world and traders could readily find a counterparty for the trades they desired. The institution of centralized clearing also took the wildcard of counter-party credit worthiness out of the equation.

In the past 40 years, futures have been listed with not only physical commodities as the underlying asset, but also currencies, debt instruments, equity index futures, single stocks and myriad other assets that traders and investors have a stake in the future price of.

Future Prices

Contrary to popular belief, the price of a futures contract is not an indication of the market’s belief in whether the underlying asset is likely to appreciate or depreciate over the life of the contract.  Rather, the futures price is a component of the current price of the underlying (the “spot” price) and the cost of carrying the asset during the term of the contract.

Arbitrage (the simultaneous purchase and sale of two or more assets to lock in a price discrepancy) ensures fair prices in the Futures markets.

Example: If the spot price for a bushel of corn is $3.50, the interest cost of borrowing $3.50 for 6 months is $0.10, and the cost of transporting, storing and insuring a bushel of corn for 6 months is $0.20, then the price of the 6-month future must be very close to $3.80/bushel.  If it were higher, say $4.50, a savvy trader could borrow money and buy corn in the spot market for $3.50/bushel while selling the futures contract at $4.50/bushel.  After paying the interest and the other costs of carrying the physical asset, he could then let the contract expire and deliver the corn for $4.50/bushel, pocketing $0.70/bushel.

Since the CBOT Corn contract represents a lot of 5,000 bushels, the trader’s risk-free profit would be $3,500/contract, a tidy sum. Market forces ensure that risk-free profit opportunities are rare and short-lived by pushing prices back to their arbitrage-free equilibrium.

(Note: the above example ignores commissions and fees and differences in interest rates and carrying costs paid by various market participants, so the actual futures price doesn’t have to be exactly $3.80 to preclude successful arbitrage, but it should generally be very close.)

Futures on Stocks

Let’s apply the same concept to futures on individual stocks. Keep in mind that unlike commodities, stocks don’t have to be transported, stored or insured, so the cost of carry is made up of just the interest to finance the position minus any dividend or other income the stock pays the shareholder during the holding period.

Stock price: $100

Risk Free Interest Rate: 3%

Stock dividend annual yield: 1%

Futures term: 6 months to expiration.

Assuming simple interest and equal distribution of dividend payments, the futures contract above will be worth $101 – or the cost of the stock plus the cost of carrying the stock until the expiration of the futures contract. If the futures price were higher, an arbitrageur could borrow $100, buy the stock, pay $1.50 in interest, collect $0.50 in dividends and hold the stock for six months for a total cost of $101.  If he sold the futures contract for a higher price, say $102, at expiration he would be required to sell the stock for $102, keeping $1 in risk free profit, since he had spent a total of $101 to hold it.

In this example if the contract were trading lower than $101, the arbitrageur could make the exact opposite trades – sell the stock short, invest the proceeds of that sale for $1.50 in interest, and pay the $0.50 in dividends for total proceeds of $101, then buy the stock back at expiration for something lower than $101.

Turning Options into Futures

Futures on Single Stocks do exist in the U.S, though volumes are usually quite low.  However, ordinary call and put options can be combined in a spread that exactly mimics the behavior of a futures contract.

If we buy a call on the above stock with 6 months until expiration, if the price of the stock is above 100 when it expires, we will exercise the call and buy 100 shares of stock for $100/share. If the price of the stock is below $100 at expiration, we will not exercise the call and it will expire worthless.

If we sell a put with a strike price of 100, if the stock price is below $100 at expiration, the holder of the put will exercise it and we will buy 100 shares at $100/share.  If the price is above $100, the put will expire worthless.

If we combine these trades, buying a 100 strike call and selling a 100 strike put, we have a position that exactly mimics a futures contract – that is, our position will require us to purchase 100 shares of stock at expiration for $100/share, regardless of whether the then-current market price of the stock is above or below $100.  Add in whatever we paid for the net options trade and that’s the price at which we bought the future.  If we paid $5 for the call and sold the put for $4, we would have been long the futures contract for $101 ($100 for the stock and $1 for the options) – exactly what we determined the fair value to be earlier.

Just as was the case with the price of futures contract, the prospect of arbitrage keeps call and put prices in line.  In fact, arbitrage is much simpler in the equity and option markets because it’s much easier to make a series of electronic transactions than it is to arrange for physical delivery of a train car full of corn, so even small price discrepancies tend to go away very quickly.

The Simple Math

We can express the relationship between calls and puts mathematically with a simple equation known as “Call-Put Parity”

Call – Put + Strike = Future

The price of the call minus the price of the put, added to the strike price of the options must equal the value of a futures contract on the stock. 

Why Does it Matter?

It’s probably not practical for individual investors to engage in options arbitrage.  In liquid, large-cap names where a virtually unlimited number of shares can be bought or sold short, the opportunities to execute these trades profitably will occur so seldomly as to be ignorable.

But…it’s important to understand the concept because in less liquid stocks, especially those that are difficult to borrow in order to sell short, apparent violations of call-put parity can provide insight into the positions of big traders and give us an edge in determining how the stock price might behave in various scenarios.

More on that in Know Your Options next week…




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