In modern financial markets, instruments known as "derivatives" play an important role. The most important derivatives are futures and options.

A futures contract is a promise to deliver a well-specified commodity at some future date. A classic user of a futures contract would be a farmer trying to lock in the price of corn that is not yet ready for market. Suppose that today is the first of June, and the farmer sees that in the futures market the price of corn for delivery in August is $4 a bushel. The farmer, whose corn will be ready for delivery in August, could wait until then to sell the corn. However, the price of corn could go up or down between now and August.

If the farmer likes the price of $4 a bushel, the farmer can sell an August futures contract to deliver, say 10,000 bushels of corn. Such contracts are traded on the Chicago Mercantile Exchange, which is a bit like the New York Stock Exchange.

In theory, the farmer has promised to deliver corn in August to whoever purchased the futures contract. However, when August rolls around, the farmer is not going to come waltzing onto the floor of the Merc with 10,000 bushels of corn. In August, the farmer buys back the futures contract at the price that prevails in August. If the price of corn has fallen to, say, $3 a bushel, the farmer makes a profit of $10,000 on the futures market transactions. Conversely, if the price of corn goes up to $5 a bushel in August, the farmer loses $10,000 on the futures contract transactions.

Also in August, the farmer sells the 10,000 bushels of corn in a local market at the prevailing price. If the price is $3 a bushel, the farmer gets $30,000. If the price is $5 a bushel, the farmer gets $50,000.

Notice that no matter what happens to the price of corn, when the farmer sells a futures contract the farmer is bound to receive $40,000 one way or the other. The value of the corn that is sold in August plus (or minus) the profit (or loss) on the futures contract will equal $40,000. The farmer's transactions in the futures market are known as hedging. A hedge is a way of reducing your losses if prices move against you while limiting your gains if prices move favorably.

Today's price of corn is known as the spot price. The relationship between futures prices and spot prices is governed by the cost of storage. If the cost of storing corn for two months, which includes the interest cost of borrowing money to buy and hold corn for two months as well as the cost of renting a silo, is one percent of the spot price of corn, then the futures price should be one percent over the spot price. If the futures price were higher than that, then speculators would sell corn futures contracts, buy corn in the spot market and store it, making a sure profit. Conversely, if the futures price were too low, speculators could buy in the futures market and sell in the spot market and make a sure profit. These sorts of transactions that force prices at different times and places to stay in line with one another are called arbitrage.

Arbitrage between cash and futures prices ensures that something that affects the price of a commodity at any point in time will affect it at all points in time. For example, if information that arrives in April suggests that the demand for gasoline will be particularly high in June, this will cause the June futures price to rise. Speculators will see that it is profitable to buy gasoline now and store it until June, so that they will bid up the price of gasoline now. Thus, the spot price of gasoline will rise right away.

An option is the right to buy something at a particular price, known as the strike price. For example, suppose that Freddie Mac stock is trading for $60 a share in July, and I pay $50 for an August call option to buy 100 shares at a strike price of $62 a share. Between now and August, if the price of Freddie Mac stock goes above $62 a share, I may choose to exercise my option. If the price reaches $63 a share, then I can exercise my option to buy 100 shares at $62 a share and then immediately sell those 100 shares for $63 a share, for a profit of $100 (minus the $50 that I paid for the option).

On the other hand, if the price of the stock goes below the strike price, I do not have to exercise my option to buy it at the strike price. Instead, I leave the option unexercised.

Options only last a finite amount of time. The last day that I can exercise my option is called the expiration date. On the expiration date, if the price of the stock is below the strike price, then my call option expires worthless. Otherwise, the value of the option on the expiration date will be proportional to the difference between the price of the stock and the strike price of the option.

The cost of an option is fixed at the time that I buy it. I cannot lose more than what I pay for the option. If the price of the stock falls, I am not obligated to exercise the option. Thus, unlike a futures contract, my potential loss is limited.

Thus, the payoff from an option is asymmetric. For a call option, a big rise in the stock price can generate a big profit. However, an equally large drop in the stock price only means that the option expires worthless and I am out the cost of the option.

Because of the asymmetric payoff, options are more valuable when the underlying securities have prices that are highly volatile. The sharper the up-and-down swings in a stock price, the more valuable are options on that stock.

Other the factors that make an option valuable are:

1. Time to expiration. The more time that remains until an option expires, the more valuable is the option.

2. Strike price. Suppose that I have a call option on Freddie Mac stock with a strike price of 61. If the price of the stock is already at 62, then my option is "in the money." It is particularly valuable. If the stock price is at 61, then my option is "at the money." If the stock price is at 60, then the option is "out of the money." It still has value, but not as much as if it were at the money or in the money.

The option to sell a security at a strike price is called a put option. If I have a put option on Freddie Mac stock with a strike price of 61, and the price of the stock falls to 60, then I can exercise my option. I buy the stock for 60, then use the option to sell it for 61.

Selling an option also is known as writing an option or being short an option. Buying an at-the-money call option on a stock and writing an at-the-money put option on the same stock is like owning the stock. If the stock price goes down, then my call option will expire worthless and my short put will be costly. If the stock price goes up, then my short put will be unexercised and my call option will be profitable. Overall, the equivalence between owning a stock and being long an at-the-money call and short an at-the-money put is known as put-call parity.

There is a formula for determining the fair price of an option, taking into account all of the factors. The formula was developed by Myron Scholes and Fischer Black (Scholes won the Nobel Prize, but Black died several years before the award was given). The Black-Scholes formula is too complex to go into in this course.

The significance of market-traded options is that they tell us how investors estimate volatility. Using the Black-Scholes formula and observing the price of an option, you can determine the estimate of volatility that makes the price of the option fair. That is the market's estimate of volatility. When the at-the-money options on a particular stock are expensive, that suggests that the market views the stock as volatile, meaning that its price could go up or down a great deal. If the at-the-money options are less expensive, this means that the market expects the stock's price to be fairly stable.

Certain trading strategies also can be interpreted in terms of options, based on how they will be affected by volatility. For example, there is an aphorism that goes "ride the winners, sell the losers." This means that if a stock that you own goes up, you buy more of it, and if a stock that you own goes down, you sell it. Following this strategy is like owning a portfolio of call options--you are "long" volatility. Higher volatility will give you more profits.

Conversely, suppose you use price limits to determine when to buy and sell stocks. You have a low price that is a "buy" trigger and a high price that is a "sell" trigger. With this approach, it is as if you are writing options on your portfolio. Writing options on stocks that you own is known as writing covered options. You are "short" volatility, and you stand to do well in a market that does not fluctuate too widely.

In an efficient market, it does not pay to be either long volatility or short volatility. No trading strategy is superior to just buying and holding the market portfolio. Being short volatility (or writing covered options) is just another way of putting a larger share of your wealth in the risk-free asset. Being long volatility is just another way of putting a larger share of your wealth in the risky portfolio that offers higher expected returns.