Essays in Derivatives

Risk-Transfer Tools and Topics Made Easy

Don M. Chance

• Publisher: Wiley
• Year: 2011
• ISBN:
  9781118160640
• Edition: 2
• Topic: Betriebswirtschaft
• Subtopic: Investitionen & Wertpapiere

In the updated second edition of Don Chance's well-received Essays in Derivatives, the author once again keeps derivatives simple enough for the beginner, but offers enough in-depth information to satisfy even the most experienced investor. This book provides up-to-date and detailed coverage of various financial products related to derivatives and contains completely new chapters covering subjects that include why derivatives are used, forward and futures pricing, operational risk, and best practices.

SECTION Three

Derivative Pricing

The day is finally here. This is what I told you will challenge you. Understanding how derivatives are priced is probably the most difficult part of learning about derivatives. To help you get started, let me try to teach you something right here in this introduction.

Suppose you find that unleaded 89 octane gasoline sells for $3.0199 at one station, while across the street, it sells for $3.0399. Both are convenience stores operated by major oil companies. They offer virtually identical products and services. Their gas prices are posted and easily visible from all angles. Assume that both stations are equally accessible and the traffic flow from both directions is approximately equal. Thus, a customer would have no difficulty choosing one station over another. It should be apparent that the station with the cheaper gasoline will get most if not virtually all of the business. Hence, it is quite rare to see a widely used and nearly homogeneous product like 89 octane gasoline selling for different prices. One store would drive the other out of business, but of course, competition forces the other store to adjust its price. Now, we are not arguing that the same product will always cost the same. Wal-Mart, Target, and Macy’s may have different prices for the same product, but customers are typically not able to switch easily and costlessly from one seller to another. In the financial world, nearly costless switching is quite easy. It is as simple to buy a bond from Merrill Lynch as it is to buy it from Goldman Sachs.

Suppose you find that a stock is selling for $50 on one exchange and for $53 on another exchange. It is possible to buy the stock on the cheaper exchange and sell it on the more expensive exchange and net a profit, provided your costs are not more than $3. This situation is called an arbitrage opportunity and the transaction is referred to as arbitrage. In an arbitrage, someone is able to earn a profit by taking no risk and investing no money. It is essentially manufacturing money out of nothing. Financial markets work very well, and you are very unlikely to see a stock selling for two prices on two different markets.

So how does this work for derivatives? Remember that a derivative “derives” its value from the underlying. As such, the derivative behaves much like the underlying. It is possible to use the derivative to replicate the underlying and vice versa. Likewise, it is possible to combine the derivative and the underlying to eliminate all of the risk in which case, the position should return the risk-free rate. On that basis, it is possible to determine the appropriate price of the derivative, which will be the price that guarantees that the derivative replicates the underlying and that no one can earn more than the risk-free rate on a risk-free transaction. In other words, if the derivative is correctly priced, arbitrage should not be possible.

Does it always work this way for derivatives? Well, not exactly. There are very few instances in which a clean arbitrage profit is even theoretically possible, what more in reality. Some of the models require information that is not easy to get and figures on which people would disagree. Thus, a lot of what goes on in the market and may be called arbitrage is really not arbitrage. But those are technical issues you can think about at a more advanced stage. For now, I suggest you take the rule of no arbitrage as the truth as it will serve you quite well in helping you to understand derivative pricing.

This section consists of 17 essays covering a variety of topics. I will not list them all here, but suffice it to say that we will cover how to price options, forwards, futures, and swaps. We will also learn something about how the risks from these instruments is hedged, which plays an important role in how these instruments themselves serve as tools for hedging.

ESSAY 21

Forward and Futures Pricing

In Essay 4 we introduced forward and futures contracts. Forward contracts are the simplest of all derivatives. Recall that they represent an agreement between two parties in which one party commits to purchase an underlying asset at a fixed price at a later date. The other party commits to sell the underlying asset at the fixed price at the later date. A forward contract is a customized, over-the-counter agreement and is subject to the possibility of default by the party that ends up owing the other the greater amount. A futures contract is essentially a standardized version of a forward contract in which a futures exchange has assigned a fixed set of terms and conditions concerning the identity of the underlying, the number of units of the underlying, the expiration date, and delivery procedure. The futures exchange provides a market for trading of the contract and also a clearinghouse for transferring all payments and providing a credit guarantee that assures each party that it will incur no credit loss. To manage the credit risk, futures exchanges provide for a daily settling of gains and losses that requires that parties with accumulated losses during the trading day pay parties with accumulated gains during that day. Minimum margin requirements assure that adequate funds are on hand. In this essay we look at how these contracts are priced.

The first principle of forward and futures pricing is that the contract price at expiration is the price of the underlying. Consider a forward or futures contract that expires in one month. Now consider that contract when it has one week to go. Then consider that contract with one day to go. Continue to shorten the life of the contract until its expiration is instantaneous. At that point the contract is essentially an ordinary transaction to purchase the asset. Thus, the forward or futures price of a contract at expiration is the price of the underlying. That result is, hopefully, obvious. The less obvious task is to price the forward or futures contract prior to expiration. Let us start with forward contracts for they are simpler.

Suppose an investor holds one unit of an asset. This asset is traded in a market and has an observable price S, which is commonly known as the spot price or cash price.1 We do not question whether S represents the true value of the asset. We simply take S as given.2 Now suppose the investor enters into a forward contract to sell the asset at a fixed price at a specific later date. This investor is taking the short position, which is balancing or hedging his long position in the underlying. The counterparty is taking the long position in the forward contract, thereby agreeing to buy the asset at the fixed price at the later date. All terms and conditions are agreed upon by the two parties. The fixed price, called the forward price, is denoted as F.

It should be easy to see that the owner of the asset is now engaged in a riskless transaction. He owns an asset worth S and has contracted to sell it at a later date at a price F. So his return is guaranteed. In well-functioning markets, that return has to be the risk-free rate. Thus, F has to be the value of S compounded at the risk-free rate. When S is converted to F by compounding S at the risk-free rate, the forward price accounts for the cost of financing S. That is, the investor who owns the asset has money tied up in the asset and, therefore, incurs an opportunity cost.3 If the investor sells the asset at F and F reflects the growth of S at the risk-free rate, then the opportunity cost of the investor’s money has been properly accounted for in the model.

If the forward price in the market is not equal to S compounded at the risk-free rate, an opportunity for arbitrage is possible. If the forward price in the market is too high, an arbitrageur can buy the asset and sell the forward contract, which would lead to a risk-free position that earns more than the risk-free rate. If the forward price in the market is too low, an arbitrageur can sell the asset and buy the forward contract. This transaction is the opposite of the other one and creates an initial cash inflow that the arbitrageur must pay back later with interest, making it an implicit loan. If the forward price is too low, however, the implied interest rate is lower than the risk-free rate. One could, therefore, invest the proceeds from the sale of the asset at the risk-free rate, which would more than cover the cost of paying back the implicit loan. To engage in this transaction, however, requires that the arbitrageur either already own the asset or be able to borrow it so as to engage in a short sale.

So we have seen that the forward price equals the spot price compounded at the risk-free rate. In most cases, however, some other adjustments are needed to correctly arrive at the value of F.

Suppose the asset is one that incurs costs to hold. These are called carrying costs. Heavy physical assets such as oil and gold incur extensive storage costs. Assets that can be destroyed, such as bushels of soybean that could be burned, must be insured. These costs in total are the carrying costs, sometimes referred to as the cost of carry. The investor engaged in this risk-free transaction will not earn a net return of the risk-free rate if the carrying costs are not considered. Thus, F must not only reflect the value of S compounded at the risk-free rate but also any costs incurred to hold the assets. These costs should be adjusted for the time value of money so that when factored in to determine F, they reflect the costs plus the accrual of any interest on these costs. If the forward price reflects these values, it will be higher than without. Because the forward price would be high enough to compensate for the costs incurred, it would a return of the risk-free rate.

Some assets generate income while being held. The classic example is stock and bonds. Some stocks pay dividends and some bonds pay interest. These dividend and interest amounts are usually known at the time of the transaction. The investor who holds the asset would receive these payments and would therefore earn more than the risk-free rate. Thus, the forward price should be adjusted downward to reflect these cash payments. The investor holding the asset would receive a return less than the risk-free rate from selling the asset at the forward price, but these cash payments would augment the return to bring it up to the risk-free rate.

Some forward contracts, however, do not trade at values that imply a risk-free rate. This puzzling phenomenon has been formalized into a theory called the convenience yield. It is believed that certain assets offer nonpecuniary benefits to their holders. The convenience yield captures this notion. It is said to arise from the advantage of havi...

Table of contents

• Cover
• Contents
• Title
• Copyright
• Preface to the New Edition
• Preface to the First Edition
• Section One: Derivatives and Their Markets
• Section Two: The Basic Instruments
• Section Three: Derivative Pricing
• Section Four: Derivative Strategies
• Section Five: Exotic Instruments
• Section Six: Fixed Income Securities and Derivatives
• Section Seven: Other Topics and Issues
• Recommended Reading
• Answers to End-of-Essay Questions