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ELEMENTARY PRINCIPLES OF DERIVATIVE PRICING

July 7th, 2009

In this post, we take a preliminary glance at how derivative contracts are priced. First, we introduce the concept of arbitrage. Arbitrage occurs when equivalent assets or combinations of assets sell for two different prices. This situation creates an opportunity to profit at no risk with no commitment of money. Let us start with the simplest (and least likely) opportunity for arbitrage: the case of a stock selling for more than one price at a given time. Assume that a stock is trading in two markets simultaneously. Suppose the stock is trading at $100 in one market and $98 in the other market. We simply buy a share for $98 in one market and immediately sell it for $100 in the other. We have no net position in the stock, so it does not matter what price the stock moves to. We make an easy $2 at no risk and we did not have to put up any funds of our own. The sale of the stock at $100 was more ! than adequate to finance the purchase of the stock at $98. Naturally, many market participants would do this, which would create downward pressure on the price of the stock in the market where it trades for $100 and upward pressure on the price of the stock in the market where it trades for $98. Eventually the two prices must come together so that there is but a single price for the stock. Accordingly, the principle that no arbitrage opportunities should be available is often referred to as the law of one price. Recall that we mentioned in earlier posts that an asset can potentially trade in different geographic markets and, therefore, have several spot prices. This potential would appear to violate the law of one price, but in reality, the law is still upheld. A given asset selling in two different locations is not necessarily the same asset. If a buyer in one location dis- covered that it is possible to buy the asset more cheaply in another location, the buyer would still have to incur the cost of moving the asset to the buyer’s location. Transportation costs could offset any such price differences.
We have just had a taste of not only the powerful forces of arbitrage but also a pricing model for one derivative, the forward contract. In this simple example, according to the pricing model, the forward price should be the spot price increased by the interest rate. Although there is a lot more to derivative pricing than shown here, the basic principle remains the same regardless of the type of instrument or the complexity of the setting: Prices are set to eliminate the opportunity to profit at no risk with no commitment of one’s own funds.
There are no opportunities for arbitrage profits.
Lest we be too naive, however, we must acknowledge that there is a large industry of arbitrageurs. So how can such an industry exist if there are no opportunities for riskless profit? One explanation is that most of the arbitrage transactions are more complex than this simple example and involve estimating information, which can result in differing opinions. Arbitrage involving options, for example, usually requires estimates of a stock’s volatility. Different participants have different opinions about this volatility. It is quite possible that two counterparties trading with each other can believe that each is arbitraging against the other.
But more importantly, the absence of arbitrage opportunities is upheld, ironically, only if participants believe that arbitrage opportunities do exist. If market traders believe that no opportunities exist to earn arbitrage profits, then they will not follow market prices and compare these prices with what they ought to be, as in the forward contract example given above. Without participants watching closely, prices would surely get out of line and offer arbitrage opportunities. Thus, eliminating arbitrage opportunities requires that participants be vigilant to arbitrage opportunities. In other words, strange as it may sound, disbelief and skepticism concerning the absence of arbitrage opportunities are required in order that it hold as a legitimate principle.
Markets in which arbitrage opportunities are either nonexistent or are quickly eliminated are relatively efficient markets. Recall from your study of portfolio theory and investment analysis that efficient markets are those in which it is not possible, except by chance, to earn returns in excess of those that would be fair compensation for the risk assumed. Although abnormal returns can be earned in a variety of ways, arbitrage profits are definitely examples of abnormal returns, relatively obvious to identify and easy to capture. Thus, they are the most egregious violations of the principle of market efficiency. A market in which arbitrage profits do not exist is one in which the most obvious violations of market efficiency have been eliminated.
Throughout this blog, we shall study derivatives by using the principle of arbitrage as a guide. We will assume that arbitrage opportunities cannot exist for any significant length of time. Thus, prices must conform to models that assume no arbitrage. On the other hand, we do not want to take the absence of arbitrage opportunities so seriously that we give up and believe that arbitrage opportunities never exist. Otherwise, they will arise, and someone else will take them from us.

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CRITICISMS OF DERIVATIVE MARKETS

June 22nd, 2009

Derivatives have been highly controversial for a number of reasons. For one, they are very complex. Much of the criticism has stemmed from a failure to understand derivatives. When derivatives fail to do their job, it is often the derivatives themselves, rather than the users of derivatives, that take the blame. Yet, in many cases, the critics of derivatives simply do not understand them well enough. As described in former posts, when homeowners take out mortgages, they usually receive a valuable option: the right to prepay their mortgages. When interest rates fall, homeowners often pay off their mortgages, refinancing them at lower rates. The holders of these mortgages usually sell them to other parties, which can include small organizations and individuals. Thus, we often find unsophisticated investors holding securities based on the payments from mortgages. When homeowners refinance, they capture huge interest savings. Where does this money come from? It comes from the pockets of the holders of mortgage securities. When these unsophisticated investors lose a lot of money, derivatives usually get the blame. Yet these losses went into the pockets of homeowners in the form of interest savings. Who is to blame? Probably the brokers, who sold the securities to investors who did not know what they were buying-which leads us to the next common criticism of derivatives.
The complexity of derivatives means that sometimes the parties that use them do not understand them well. As a result, they are often used improperly, leading to potentially large losses. Such an argument can, however, be used to describe fire, electricity, and chemicals. Used improperly, perhaps in the hands of a child or someone who does not know how to use them, all of these can be extremely dangerous. Yet, we know that sufficient knowledge of fire, electricity, and chemicals to use them properly is not very difficult to obtain. The same is true for derivatives; treat them with respect and healthy doses of knowledge.
Derivatives are also mistakenly characterized as a form of legalized gambling. Although gambling is certainly legal in many parts of the world, derivatives are often viewed as a government’s sanction of gambling via the financial markets. But there is an important distinction between gambling and derivatives: The benefits of derivatives extend much further across society. By providing a means of managing risk along with the other benefits discussed above, derivatives make financial markets work better. The organized gambling industry affects the participants, the owners of casinos, and perhaps some citizens who benefit from state lotteries. Organized gambling does not, however, make society function better, and it arguably incurs social costs.
We have taken a look at what derivatives are, where they come from, where they are now, why we have them, and what people think of them. Understanding derivatives, however, requires a basic understanding of the market forces that govern derivative prices. Although we shall cover derivative pricing in more detail in later posts, here we take a brief look at the process of pricing derivatives by examining some important fundamental principles.

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THE PURPOSES OF DERIVATIVE MARKETS

June 7th, 2009

Derivative markets serve a variety of purposes in global social and economic systems. One of the primary functions of futures markets is price discovery. Futures markets provide valuable information about the prices of the underlying assets on which futures contracts are based. They provide this information in two ways. First, many of these assets are traded in geographically dispersed markets. Recall that the current price of the underlying asset is called the spot price. With geographically dispersed markets, many different spot prices could exist. In the futures markets, the price of the contract with the shortest time to expiration often serves as a proxy for the price of the underlying asset. Second, the prices of all futures contracts serve as prices that can be accepted by those who trade contracts in lieu of facing the risk of uncertain future prices. For example, a company that mines gold can hedge by selling a futures contract on gold expiring in two months, which locks in the price of gold two months later. In this manner, the two-month futures price substitutes for the uncertainty of the price of gold over the next two months.
Futures contracts are not, however, the only derivatives that serve this purpose. In fact, forward contracts and swaps allow users to substitute a single locked-in price for the uncertainty of future spot prices and thereby permit the same form of price discovery as do futures.
Options work in a slightly different manner. They are used in a different form of hedging, one that permits the holder to protect against loss while allowing participation in gains if prices move favorably. Options do not so much reveal prices as they reveal volatility. As we shall see soon, the volatility of the underlying asset is a critical factor in the pricing of options. It is possible, therefore, to infer what investors feel about volatility from the prices of options.
Perhaps the most important purpose of derivative markets is risk management. We define risk management as the process of identifying the desired level of risk, identifying the actual level of risk, and altering the latter to equal the former. Often this process is described as hedging, which generally refers to the reduction, and in some cases the elimination, of risk. On the other side is the process called speculation. Traditional discussions of derivatives refer to hedging and speculation as complementary activities. In general, hedgers seek to eliminate risk and need speculators to assume risk, but such is not always the case. Hedgers often trade with other hedgers, and speculators often trade with other speculators. All one needs to hedge or speculate is a party with opposite beliefs or opposite risk exposure. For example, a corporation that mines gold could hedge the future sale of gold by entering into a derivative transaction with a company that manufactures jewelry. Both of these companies are hedgers, seeking to avoid the uncertainty of future gold prices by locking in a price for a future transaction. The mining corporation has concerns about a price decrease, and the jewelry manufacturer is womed about a price increase.
An unfortunate consequence of the use of the terms “hedging” and “speculating” is that hedgers are somehow seen as on the high moral ground and speculators are sometimes seen as evil-a distortion of the role of speculators. In fact, there need be very little difference between hedgers and speculators. To restate an example we used when discussing swaps, consider a corporation that currently borrows at a floating rate. A common response to a fear of rising interest rates is for the corporation to use an interest rate swap in which it will make payments at a fixed rate and receive payments at a floating rate. The floating-rate payments it receives from the swap offset the floating-rate payments on the loan, thereby effectively converting the loan to a fixed-rate loan. The company is now borrowing at a fixed rate and, in the eyes of many, hedging.
But is the company really hedging? Or is it simply making a bet that interest rates will increase? If interest rates decrease, the company will be losing money in the sense of the lost opportunity to borrow at a lower rate. From a budgeting and cash flow standpoint, however, its fixed interest payments are set in stone. Moreover, the market value of a fixedrate loan is considerably more volatile than that of a floating-rate loan. Thus, our “hedging” corporation can be viewed as taking more risk than it originally had.
The more modem view of the reason for using derivatives does not refer to hedging or speculation. Although we shall sometimes use those terms, we shall use them carefully and make our intentions clear. In the grander scheme of things, derivatives are tools that enable companies to more easily practice risk management. In the context of our corporation borrowing at the floating rate, it made a conscious decision to borrow at a fixed rate. Engaging in the swap is simply an activity designed to align its risk with the risk it wants, given its outlook for interest rates. Whether one calls this activity hedging or speculation is not even very important. The company is simply managing risk.
Derivative markets serve several other useful purposes. As we show later when exploring the pricing of derivative contracts, they improve market efficiency for the underlying assets. Efficient markets are fair and competitive and do not allow one party to easily take money from another. As a simple example, we shall learn in near future buying a stock index fund can be replicated by buying a futures on the fund and investing in risk-free bonds the money that otherwise would have been spent on the fund. In other words, the fund and the combination of the futures and risk-free bond will have the same performance. But if the fund costs more than the combination of the futures and risk-free bond, investors have the opportunity to avoid the overpriced fund and take the combination.18 This decreased demand for the fund will lower its price. The benefits to investors who do not even use derivatives should be clear: They can now invest in the fund at a more attractive price, because the derivatives market forced the price back to its appropriate level.
Derivative markets are also characterized by relatively low transaction costs. For example, the cost of investing in a stock index portfolio is as much as 20 times the cost of buying a futures contract on the index and a risk-free bond as described above. One might reasonably ask why derivatives are so much less expensive in terms of transaction costs. The answer is that derivatives are designed to provide a means of managing risk. As we have previously described, they serve as a form of insurance. Insurance cannot be a viable product if its cost is too high relative to the value of the insured asset. In other words, derivatives must have low transaction costs; otherwise, they would not exist. It would be remiss to overlook the fact that derivative markets have been subject to many criticisms. We next present some of these complaints and the reasons behind them.

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Personal and Intangible Factors

May 18th, 2009

Several communities may remain in the running at the end of the plant location evaluation process. It is at this point that the personal and intangible attributes of communities under consideration come into prominence. These attributes can be described best in terms of community leadership and attitudes, housing, schools, recreation, shopping, and overall community image. Most large companies are also concerned about their corporate image, and they want to be a good corporate neighbor. Their impressions of a potential location can be greatly enhanced if community leaders create an image of acceptance, cooperation, and fairness. This is a much easier task if community leaders can exhibit a history of creating a favorable environment for existing industrial plants. This is one factor that a community or area has a great deal of control. Personal factors have become increasingly important in recent years with the shift in industrial organization away from owner-manager firms and toward the corporate structure. In modern corporations, management and ownership are separated. Corporation owners (stockholders) do not make location decisions, managers do. Managers live with the plants, owners do not. Clearly, corporation management must select plant locations that will be profitable and earn sufficient net revenues for long term growth of the firm and to yield stockholders a satisfactory and competitive return to their investments. Beyond this constraint of a satisfactory profit, corporate managers may tend to emphasize personal factors rather than maximizing profits. The modern decision-making framework tends to increase the influence of desirable characteristics of plant location as a place to live and work more than would be expected in the owner-manager framework of the past. Worker productivity is always affected by these personal factors and can lead to attracting quality labor and management to a particular geographic area. Management is aware of these factors when considering plant location.
Making a community more attractive to industry also creates a better place to live for existing residents. Thus, even if a new industry does not come, the community reaps the benefit of its efforts.

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