Derivatives are financial instruments whose promised payoffs are derived from the value of something else, generally called the underlying. The underlying is often a financial asset or rate, but it does not have to be. For instance, derivatives exist with payments based on the level of the S&P 500, the temperature at Kennedy Airport, or the number of bankruptcies among a group of selected companies.

Derivatives come in different flavors: plain vanilla and exotic. Plain vanilla includes contracts to buy or sell for future delivery, called forward and futures contracts; contracts that involve an option to buy or sell at a fixed price sometime in the future, called options; and combinations of forward, futures, and option contracts. Exotic derivatives are everything else in the derivatives world.

**Forward contracts**

A forward contract obligates one party to buy the underlying at a fixed price at a certain time in the future, called ¡§maturity¡¨, from a counterparty who is obligated to sell the underlying at that fixed price. Consider a U.S. exporter who expects to receive €100 million in six months. Suppose that the price of the euro is $1.20 now. If the price of the euro falls by 10 percent over the next six months, the exporter loses $12 million. By selling euros forward, the exporter locks in the current forward rate (if the forward rate is $1.18, the exporter receives $118 million at maturity). Financial hedging involves taking a financial position to reduce one's exposure or sensitivity to a risk; hedging is perfect when all exposure to the risk is eliminated through the financial position. In our example, the financial position is a forward contract, the risk is the euro, the exposure is €100 million in six months, and the exposure to the euro is perfectly hedged with the forward contract. Since no money changes hands when the exporter buys euros forward, the market value of the forward contract must be zero when initiated since otherwise the exporter would get something for nothing.

**Options**

Although options can be written on any underlying, let's use options on common stock as an example. A call (put) option on a stock gives its holder the right to buy (sell) a fixed number of shares at a given price at some future date.1 The specified price is called the exercise price. Whoever sells an option at inception of the contract is called the option writer. When the holder of an option takes advantage of her right, she is said to ¡§exercise it.¡¨ An option holder who cannot gain from exercising an option before expiration does not exercise it. The purchase price of an option is called the option premium. Options enable their holders to lever their resources while at the same time limiting their risk. Suppose an investor, Smith, believes that the current price of $50 for Upside Inc. common stock is too low. Let's assume that the premium on a call option that gives the right to buy 100 shares at $50 per share is $10 per share. By investing $1,000, Smith can buy call options to purchase 100 shares. If she does so, she will gain from stock price increases as if she had invested in 100 shares, even though she invests only an amount equal to the value of 20 shares. With only $1,000 to invest, Smith could borrow $4,000 to buy 100 shares. At maturity, she would then have to repay the loan. The gain made upon exercising the option is therefore similar to the gain from a levered position in the stock ¡V a position consisting of purchasing shares with one's own money and some borrowed money. However, if Smith borrows, she could lose up to $5,000 plus interest due on the loan if the stock price falls to zero. With the call option, she can lose at most $1,000, which she does if the stock price does not rise above $50.

**Exotics**

An exotic derivative is a derivative that cannot be created by putting together option and forward contracts. Instead, the payoff of exotics is a complicated function of one or many underlyings. When Procter and Gamble lost $160 million on derivatives in 1994, the main culprit was an exotic swap. The amounts P&G had to pay on the swap depended on the five-year Treasury note yield and the price of the 30-year Treasury bond. Another example of an exotic derivative is a binary option, which pays a fixed amount if some condition is met. For instance, a binary option might pay $10 million if before a given future date one of the three largest banks has defaulted on its debt.

The Pricing of Derivatives

Derivatives are typically priced assuming that there are no frictions in financial markets. With that assumption (and, often, some other more technical assumptions), one can find a portfolio strategy that does not use the derivative and only requires an initial investment such that the portfolio pays the same as the derivative at maturity. The portfolio is called a replicating portfolio. The derivative must be worth the same as the replicating portfolio if financial markets are frictionless, since otherwise there is an opportunity to make a risk-free profit (the term of art is an arbitrage opportunity). Consider the euro forward contract described earlier. At maturity, the exporter has to pay €100 million and receives $118 million. A replicating portfolio can be constructed as follows: borrow the present value of €100 million and invest the present value of the payment to be made on the forward contract, $118 million, in Treasury bills that mature when the forward contract matures. At maturity, you have $118 million dollars and have to pay back the borrowed euros plus interest, which amounts to €100 million. The forward contract must be priced so that the exporter is indifferent between using the forward contract or the replicating portfolio. If one approach is cheaper than the other, any investor could make money for sure by buying dollars against euros using the cheap approach and selling dollars against euros using the expensive approach. The value of a forward contract changes over time. If the euro appreciates unexpectedly by 10 percent, the replicating portfolio makes a loss: the present value of the euro debt of the portfolio increases unexpectedly, but the value of the Treasury bills does not increase commensurately. Since the replicating portfolio has the same payoff as the forward contract, the loss means that the value of the forward contract has become negative. The replicating portfolio strategy is trickier to devise and implement for options. Consider the option to buy 100 shares of Upside at $50 per share. If the price of the stock exceeds $50 at maturity, the value of the option is 100 times the difference between the value of the underlying and $50; otherwise, the option is worth nothing. The replicating portfolio must therefore be worth zero at maturity if the stock price is less than $50 and 100 times the difference between the stock price and $50 otherwise. At inception of the replicating strategy, one must purchase shares partly with borrowed money. However, the number of shares held has to increase as the stock price increases because it becomes more likely that the option holder will exercise, and it has to fall as the stock price falls since the option becomes more likely to expire worthless. In the former case, the shares acquired are financed with more borrowing; in the latter case, the proceeds from selling shares are used to pay back money borrowed. Fischer Black and Myron Scholes, in their path-breaking work provide a mathematical solution for the option price, the Black-Scholes (1973) formula, and for the number of shares held in the replicating portfolio at any time during the life of the option.

Derivatives come in different flavors: plain vanilla and exotic. Plain vanilla includes contracts to buy or sell for future delivery, called forward and futures contracts; contracts that involve an option to buy or sell at a fixed price sometime in the future, called options; and combinations of forward, futures, and option contracts. Exotic derivatives are everything else in the derivatives world.

The benefits of derivatives

We saw that derivatives are priced by constructing a replicating portfolio. This suggests that derivatives are redundant assets. So, why the fuss? If it is true that everybody can replicate derivatives perfectly, banning derivatives would change nothing ¡V as long as the disclosure requirements for replicating portfolios are the same as for derivatives. Instead of buying a call option on Upside Inc., Smith would simply manufacture the option on her own. The assumption of pricing models that financial markets are frictionless is a powerful simplifying assumption. The reason it makes sense to typically use that assumption is that firms with large trading operations can trade often enough and cheaply enough to manage replicating portfolios well enough for derivatives on underlyings trading in very liquid markets. These firms can also make markets in derivatives so that they match buyers and sellers. When they do so, they do not have to hedge. For individuals and non-financial firms, derivatives are almost never redundant assets. There are three reasons for this. First, individuals and non-financial firms face much higher trading costs than the most efficient financial institutions. In general, for most investors and firms, replicating closely the payoff of a derivative such as a call option will be prohibitively expensive. Second, for derivatives that include option features, the replicating portfolio strategy typically requires trades to be made whenever the price of the underlying changes. Otherwise, the replicating portfolio works only approximately. Third, identifying the correct replicating strategy is often a problem. For instance, the strategy may differ depending on the dynamics of the price of the underlying, so that estimation error may affect the performance of the replicating strategy and make that approach risky. Because of these issues, individuals and firms will be willing to pay financial institutions to provide them with a derivative instead of attempting to manufacture it on their own. The main gain from derivatives is therefore to permit individuals and firms to achieve payoffs that they would not be able to achieve without derivatives or could only achieve at much greater cost. Because individuals and firms could not obtain the payoffs of derivatives efficiently by manufacturing them on their own, derivatives make markets more complete ¡V i.e., they make it possible to hedge risks that otherwise would be un-hedgeable ¡V for these individuals and firms. When individuals and firms can manage risk better, risks are born by those who are in the best position to bear them and firms and individuals can take on riskier but more profitable projects by hedging those risks that can be hedged. As a result, the economy is more productive and welfare is higher. A second important benefit from derivatives is that they can make underlying markets more efficient. First, derivatives markets produce information. For instance, in a number of countries, the only reliable information about long-term interest rates is obtained from swaps because the swap market is more liquid and more active than the bond market. Second, derivatives enable investors to trade on information that otherwise might be prohibitively expensive to trade on. For instance, short sales of stocks are often difficult to implement. This slows down the speed with which adverse information is incorporated in stock prices and makes markets less efficient. With puts, investors can more easily take advantage of adverse information about stock prices. Though derivatives can make underlying markets more efficient, observers have long been concerned that they can also disrupt markets because they make it easier to build speculative positions. Overall, there does not seem compelling evidence at this time that the introduction of derivatives trading on an underlying increases permanently the volatility of the return of the underlying. For instance, Conrad (1998) finds that the introduction of option trading on a stock trading reduce s the volatility of the underlying stock and Bollen (1998) finds no effect.

(*René M. Stulz is the Reese Professor of Banking and Monetary Economics, The Ohio State University, Columbus, Ohio, and Research Associate, National Bureau of Economic Research, Cambridge, Massachusetts.)

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