Chapter 3: Derivatives Overview of Derivatives A derivative is a financial claim whose value is derived from the value of the underlying contract. A gold futures contract is a derivative that depends on the underlying value of gold while an option on a Eurodollar Futures contract depends on the value of the underlying Eurodollar Futures contract. Derivatives are also referred to as contingent claims . Derivatives range from a simple forward contract to highly complex options that depend on the price history of the underlying asset. Derivatives can be purchased on exchanges or custom-tailored to the parties needs as OTC transactions. Derivatives exist to enable market participants to transfer risk, to take a position or to exploit mis-pricings in a market. Market participants that wish to transfer risk are called hedgers . Those participants taking a view in the market are called speculators . A third category, arbitrageurs , exploits profit opportunities that may arise from a variety of reasons. Derivatives play a vital and crucial role in capital markets. In the following, we will define and value derivatives including forwards, futures, swaps and options. As long as you can find a willing counterparty, it seems that you can get a derivative on almost any underlying contract. As of 2000, the list of futures and options authorized by the Commodity Futures Trading Commission (CFTC) included a bankruptcy index; catastrophe, health and homeowners insurance derivatives; wood and fertilizer futures. There are futures on degree-days in Atlanta as well as on gas and electricity. Credit derivatives are becoming increasingly important, accounting for almost $700 billion by mid-2001, according to the Bank for International Settlements (BIS). Although modeling and pricing derivatives is important, it is equally important to understand and be able to manage the inherent risk. Sometimes, the unexpected does happen and catastrophic losses can occur. Forward Contracts Overview The forward contract is the most basic form of derivative. It is an agreement between two parties (the counterparties ) to make a trade at some future date. One party is S, the seller who agrees to deliver the underlying. The other party is B, the buyer, who agrees to take delivery of the underlying. The seller is also called “short” the contract and the buyer is “long” the contract. The agreement is binding on both parties since a contract is drawn up and executed upon entering the contract. Forward contracts are traded over the counter. The counterparties determine the nature and quality of the asset to be delivered, when delivery will take place ( the expiration date of the forward ), where delivery will take place, how pricing will be carried out and so forth. The OTC contract is custom-tailored to meet the needs of the counterparties. Forward contracts enable market participants such as farmers to hedge their price risk. If a farmer has corn in the ground in May that he anticipates selling next July, he may worry that the price will fall before he can get the corn to the market. To hedge away this undesirable price risk, he can arrange to lock in a forward price today. The forward price he locks in may or may not be the price that actually prevails in July, but he has locked in a guaranteed price. On the other hand, the counterparty is speculating that the ultimate price for corn in July will be higher than the contracted price. The counterparty does not own the asset and so is not subject to the same price risk as the farmer. The forward contract establishes a shifting of price risk from the farmer, who does not wish to bear it, to the speculator, who does wish to take this risk on. Entering a forward contract can help the farmer eliminate the unknown but will not guarantee a profit. Since a forward contract is essentially a private transaction between two individuals, the creditworthiness of each party is a big factor. A forward transaction is a zero-sum game: one party’s loss is the other’s gain, so if the losing party decides to default on the contract, the other party is subject to loss. Usually, the contracts are non-transferable which greatly reduces liquidity. No money changes hands at contract initiation. The payoff occurs only at expiration. An example of such a contract is when I sell an item on Ebay.com. If someone bids on the item, assuming they have met all of the conditions I have set (such as my reserve price and so on), I am bound to deliver the item at the end of the auction and they are bound to pay for it at the ending price. The “contract” specifies what will be delivered, where it will be delivered and how delivery is to take place in addition to Page 1 how payment should be made. In a way, we are agreeing on a forward price for my item. If the buyer defaults, I have some recourse through ebay and if I default by refusing to deliver the item, the buyer can get recourse through Ebay.com. The items are unique so that if I change my mind, the buyer can’t go to another seller to get the identical item at the identical price, and if the buyer changes his mind, I can’t easily transfer his contract to another. (I could contact the next bidder on the list, but probably wouldn’t get as high a price.) Additionally, an analogy of short selling is if I decide to sell books on Amazon.com. Suppose that I notice that a particular Adobe Photoshop book is selling on the site for $31. I decide to offer one for sale at $29 even though I don’t own the book, because coincidentally, I saw street vendors selling the books for $20 in Chinatown recently. If I sell the book, I plan to get one from the vendor to deliver. I think that this is a smart strategy since the vendor has to hold the inventory and I get cash first to purchase the asset. This is a great idea in theory, but in practice, what if the vendors are not there that day, raise their prices or are out of the book? I still have to deliver and will have to go to a bookstore to purchase it, possibly just breaking even or even suffering a loss. Forwards are traded on a diverse array of underlying assets including: financial assets such as stocks, stock indices, interest rates and currencies; on grains and oils including coconut oil, corn and soybeans ; on commodities such as live hogs, sugar, coffee and orange juice; natural resources such as heating oil, gasoline, coal and crude oil; and metals including gold, platinum, copper and US silver coins. We will find that futures have much in common with forwards, but they also have very important differences. Valuing Forward Contracts Suppose the spot price of corn today is $1.82 per bushel and a farmer can lock in a July price of $2.05 per bushel. If he enters into the contract he is guaranteed a price of $2.05 per bushel when he harvests his corn in July, no matter what. The contract size for corn is 5,000 bushels. If the July price turns out to be $2.50 per bushel, the farmer will be disappointed because he could have received $2.50 per bushel but he has agreed to accept $2.05. Locking into this forward price causes him to lose out on the potential excess profit over the forward price of $(2.50 – 2.05)*5,000 = $2,250. But he will have the $2.05*5,000 = $10,250 realized from the sale of his corn. (This is what is meant by the statement that forward contracts don’t guarantee a profit, they just guarantee a price.) On the other hand, what if corn prices fall to $1.70/bushel by July? The farmer gets a bad price for his corn, $1.70*5,000 = $8,500 but has a gain from his forward position of $(2.05-1.7)*5,000 =1,750. Total value of his portfolio of corn and forward contract is $10,250. Either way, he has received a net per-bushel price of $2.05, as guaranteed by the forward contract. Is there a linkage between the forward price to the spot price? The market participant will want to understand this relationship this so he can decide whether to enter into a forward contract. Because the forward contract is derived from an underlying asset, it can be priced by appealing to the no- arbitrage agreement. We classify our underlying assets as those that can be stored, those that cannot be stored and those that make cash payments such as coupons and dividend payments. Define spot price as the current price of the underlying. What is the relationship of the forward price to the underlying spot price? Defining t = time 0 ≤ t ≤ T T = expiration or maturity date of contract St = spot price of underlying at time t Ft,T = forward price at time t for maturity T r = risk-free interest rate We can use no-arbitrage arguments to form a relationship between S t and F t,T . Suppose that an arbitrageur (Gustavo) can take either side of the contract. He can borrow and lend money at the risk-free rate r and has no transactions costs (he is a large investment bank, for example.) The spot price of the asset today is S t. His counterparty (a grain elevator, say) quotes a forward price F t,T .