Derivative Securities Pricing

Chapter 1 - Introduction

A derivative can be defined as a financial instrument whose value depends on (or derives from) the values of other, more basic, underlying variables. Very often the variables underlying derivatives are the prices of traded assets.

1.1 Exchange-traded markets

Markets where individuals trade standardized contracts that have been defined by the exchange. Once two traders have agreed on a trade, it is handled by the exchange clearing house. This stands between the two traders and manages the risks. The advantage of this arrangement is that traders do not have to worry about the creditworthiness of the people they are trading with. The clearing house takes care of credit risk by requiring each of the two traders to deposit funds (margin) in order to live up to their obligations.

1.2 Over-the-counter markets

Not all derivatives trading is on exchanges. Many trades take place in the over-the-counter (OTC) market. Banks, other large financial institutions, fund managers, and corporations are the main participants in OTC derivatives markets. Once an OTC trade has been agreed, the two parties can either present it to a central counterparty (CCP) or clear the trade bilaterally. A CCP is like an exchange clearing house. Banks often act as market makers for the more commonly traded instruments. This means that they are always prepared to quote a bid price (at which they are prepared to take one side of a derivatives transaction) and an offer price (at which they are prepared to take the other side).

Prior to the credit crisis in 2007, OTC derivatives markets were largely unregulated. Following this, there has been development in the regulations, in order to improve transparency and market efficiency and to reduce systemic risk in OTC markets.

Markets Size: the number of derivatives transactions per year in OTC markets is smaller than in exchange-traded markets, but the average size of the transactions is much greater (also the OTC market is much larger than the exchange-traded market).

1.3 Forward contracts

It is an agreement to buy or sell an asset at a certain future time for a certain price. It can be contrasted with a spot contract, which is an agreement to buy or sell an asset almost immediately. A forward contract is traded in the OTC market—usually between two financial institutions or between a financial institution and one of its clients. The party (side) that agrees to buy assumes a long position, while the other party a short one.

Payoffs from Forwards: the payoff from a long position in a forward is S_T-K, where K is the delivery price and S_T is the spot price of the asset at maturity of contract. For the short, it is K-S_T.

1.4 Future contracts

Like a forward contract, a futures contract is an agreement between two parties to buy or sell an asset at a certain time in the future for a certain price. Unlike forward contracts, futures contracts are normally traded on an exchange. To make trading possible, the exchange specifies certain standardized features of the contract.

1.5 Options

Options are traded both on exchanges and in the over-the-counter market. There are two types of option. A call option gives the holder the right to buy the underlying asset by a certain date for a certain price. A put option gives the holder the right to sell the underlying asset by a certain date for a certain price. The price in the contract is known as the exercise price or strike price; the date in the contract is known as the expiration date or maturity.

American options can be exercised at any time up to the expiration date. European options can be exercised only on the expiration date itself. Most of the options that are traded on exchanges are American. European options are generally easier to analyze than American options, and some of the properties of an American option are frequently deduced from those of its European counterpart.

The holder does not have to exercise the right. This is what distinguishes options from forwards and futures, where the holder is obligated to buy or sell the underlying asset. Whereas it costs nothing to enter into a forward or futures contract, there is a cost to acquiring an option. In option markets there are 4 types of participants: buyers of calls, sellers of calls, buyers of puts, seller of puts. Buyers have long positions, sellers short. Selling an option is also known as writing the option.

1.6 Types of traders

Three types: hedgers (use derivatives to reduce the risk that they face from potential future movements in a market variable), speculators (use them to bet on the future direction of a market variable), arbitrageurs (take offsetting positions in two or more instruments to lock in a profit).

1.7 Hedgers

Hedging using forwards

Consider a US company, which we will refer to as ExportCo, that is exporting goods to the United Kingdom and, on May 6, 2013, knows that it will receive £30 million 3 months later. ExportCo can hedge its foreign exchange risk by selling £30 million in the 3-month forward market at an exchange rate of 1.5533. This would have the effect of locking in the US dollars to be realized for the sterling at $46,599,000. (They are sure that after 3 months they will have to pay that amount, nothing more, nothing less).

The purpose of hedging is to reduce risk. There is no guarantee that the outcome with hedging will be better than the outcome without hedging. If the exchange rate in August proves to be less than 1.5533, the company will wish that it had hedged; if the rate is greater than 1.5533, it will be pleased that it has not done so. It is all about locking the price.

Hedging using options

Consider an investor who in May of a particular year owns 1,000 shares of a particular company. The share price is $28 per share. The investor is concerned about a possible share price decline in the next 2 months and wants protection. The investor could buy ten July put option contracts on the company’s stock with a strike price of $27.50. This would give the investor the right to sell a total of 1,000 shares for a price of $27.50. If the quoted option price is $1, the total cost of the hedging strategy would be $1,000.

The strategy costs $1,000 but guarantees that the shares can be sold for at least $27.50 per share. If the market price of the stock falls below $27.50, the options will be exercised, so that $27,500 is realized for the entire holding. When the cost of the options is taken into account, the amount realized is $26,500. If the market price stays above $27.50, the options are not exercised and expire worthless. However, in this case the value of the holding is always above $26,500.

A comparison

There is a fundamental difference between the use of forward contracts and options for hedging. Forward contracts are designed to neutralize risk by fixing the price that the hedger will pay or receive for the underlying asset. Option contracts, by contrast, provide insurance. They offer a way for investors to protect themselves against adverse price movements in the future while still allowing them to benefit from favorable price movements. Unlike forwards, options involve the payment of an up-front fee.

1.8 Speculators

Whereas hedgers want to avoid exposure to adverse movements in the price of an asset, speculators wish to take a position in the market. Either they are betting that the price of the asset will go up or they are betting that it will go down.

Speculation using futures

• Purchase £250,000 in the spot market in the hope that the sterling can be sold later at a higher price
• Take a long position in four April futures contracts on sterling. (Each futures contract is for the purchase of £62,500 in April; they will be sold in April) on the assumption that the current exchange rate is 1.4540 dollars per pound; if the exchange rate turns out to be 1.5000 dollars per pound in April, the futures contract alternative enables the speculator to realize a profit of (1:5000-1:4543)*250,000=$11,425. The spot market alternative leads to 250,000 units of an asset being purchased for $1.4540 in February and sold for $1.5000 in April, so that a profit of (1:5000-1:4540)* 250,000 = $11,500 is made.

What then is the difference between the two alternatives? The first alternative of buying sterling requires an up-front investment of 250,000* 1:4540 = $363; 500. In contrast, the second alternative requires only a small amount of cash to be deposited by the speculator in what is termed a 'margin account' (the operation of margin accounts is explained in Chapter 2). The futures market allows the speculator to obtain leverage. With a relatively small initial outlay, a large speculative position can be taken.

Speculation using options

Suppose that it is October and a speculator considers that a stock is likely to increase in value over the next 2 months. The stock price is currently $20, and a 2-month call option with a $22.50 strike price is currently selling for $1. The speculator is willing to invest $2,000. One alternative is to purchase 100 shares; the other involves the purchase of 2,000 call options (i.e., 20 call option contracts). Suppose the price of the stock rises to $27 by December. The first alternative of buying the stock yields a profit of 100*(27-20)=700.

However, the second alternative is far more profitable. The total payoff from the 2,000 options that are purchased under the second alternative is 2000*4.5=9000; we subtract the initial investment of 2000 and obtain 7000. Options like futures provide a form of leverage. Good outcomes become very good, while bad outcomes result in the whole initial investment being lost (if the price falls below the level, the option is not exercised and this leads to the loss of the initial investment, in this case of 2000).

A comparison

Futures and options are similar instruments for speculators in that they both provide a way in which a type of leverage can be obtained. However, when a speculator uses futures, the potential loss as well as the potential gain is very large. When options are used, no matter how bad things get, the speculator’s loss is limited to the amount paid for the options.

1.9 Arbitrageurs

Suppose that the stock price is $140 in New York and £100 in London at a time when the exchange rate is $1.4300 per pound. An arbitrageur could simultaneously buy 100 shares of the stock in New York and sell them in London to obtain a risk-free profit of 100*[(1.43*100)-140]=300 in the absence of transactions costs. Transactions costs would probably eliminate the profit for a small trader. However, a large investment bank faces very low transactions costs in both the stock market and the foreign exchange market. It would find the arbitrage opportunity very attractive and would try to take as much advantage of it as possible.

Arbitrage opportunities such as the one just described cannot last for long. As arbitrageurs buy the stock in New York, the forces of supply and demand will cause the dollar price to rise. Similarly, as they sell the stock in London, the sterling price will be driven down. (In this book most of the arguments concerning futures, forward and options will be based on the assumption that no arbitrage opportunities exist.)

1.10 Dangers

Derivatives are very versatile instruments. They can be used for hedging, for speculation, and for arbitrage. It is this very versatility that can cause problems. Sometimes traders who have a mandate to hedge risks or follow an arbitrage strategy become (consciously or unconsciously) speculators. The results can be disastrous.

To avoid the sort of problems SocGen encountered, it is very important for both financial and nonfinancial corporations to set up controls to ensure that derivatives are being used for their intended purpose. Risk limits should be set and the activities of traders should be monitored daily to ensure that these risk limits are adhered to.

SocGen: a junior trader (whose job was to look up for arbitrage opportunities) used his knowledge of the bank’s procedures to speculate while giving the appearance of arbitraging. He took big positions in equity indices and created fictitious trades to make it appear that he was hedged. In reality, he had large bets on the direction in which the indices would move. The size of his unhedged position grew over time to tens of billions of euros.

Chapter 2 - Future markets and central counterparties

The vast majority of futures contracts do not lead to delivery. The reason is that most traders choose to close out their positions prior to the delivery period specified in the contract. A trader who shorted a contract can close out the position by buying one contract later. The trader’s total gain or loss is determined by the change in the futures price between the first date and the day when the contract is closed out. Delivery is so unusual that traders sometimes forget how the delivery process works.

2.2 Specification of a futures contract

The asset: non-financial assets can be ambiguous (f.e. when trading orange juice futures specifications on the quality needs to be clear). Financial assets are defined (f.e. Japanese yen).

The contract size: specifies the amount of the asset that has to be delivered under one contract. The correct size for a contract clearly depends on the likely user. Whereas the value of what is delivered under a futures contract on an agricultural product might be $10,000 to $20,000, it is much higher for some financial futures.

2.3 Convergence of futures price to spot price

As the delivery period for a futures contract is approached, the futures price converges to the spot price of the underlying asset. This happens because if the futures price is above the spot price, traders have a clear arbitrage opportunity: sell a future, buy the asset, make delivery. As this opportunity is exploited, the futures price will fall. The opposite happens if we have the spot price above the future one. The result of these movements is that future and spot prices tend to converge.

2.4 The operation of margin accounts

They come useful in order to organize trading so that contract defaults are avoided (one of the parties backs out or simply hasn’t got the financial resources to honor the agreement).

Daily settlement

We consider a trader who contacts his broker to buy two December gold futures contracts. We suppose that the current futures price is $1,250 per ounce. Because the contract size is 100 ounces, the trader has contracted to buy a total of 200 ounces at this price. The broker will require the trader to deposit funds in a margin account. The amount that must be deposited at the time the contract is entered into is known as the initial margin. We suppose this is $6,000 per contract, $12,000 in total.

At the end of each trading day, the margin account is adjusted to reflect the trader’s gain or loss. This practice is referred to as daily settlement or marking to market. Suppose, for example, that by the end of the first day the futures price has dropped by $9 from $1,250 to $1,241. The trader has a loss of $1,800 (200*9). The balance in the margin account would therefore be reduced by $1,800 to $10,200. Similarly, if the price of December gold rose to $1,259 by the end of the first day, the balance in the margin account would be increased by $1,800 to $13,800.

A trade is settled at the close of trading on each day. To ensure that the balance in the margin account never becomes negative, a maintenance margin is set. If the balance in the margin account falls below the maintenance margin, the trader receives a margin call and is expected to top up the margin account to the initial margin level by the end of the next day. The extra funds deposited are known as a variation margin. If the trader does not provide the variation margin, the broker closes out the position. In the case of the trader considered earlier, closing out the position would involve neutralizing the existing contract by selling 200 ounces of gold for delivery in December.

Further details

Whereas a forward contract is settled at the end of its life, a futures contract is, as we have seen, settled daily. At the end of each day, the trader’s gain (loss) is added to (subtracted from) the margin account, bringing the value of the contract back to zero. A futures contract is in effect closed out and rewritten at a new price each day. Minimum levels for the initial and maintenance margin are set by the exchange clearing house. Individual brokers may require greater margins from their clients. Minimum margin levels are determined by the variability of the price of the underlying asset and are revised when necessary. The higher the variability, the higher the margin levels. The maintenance margin is usually about 75% of the initial margin.

The clearing house and its members

A clearing house acts as an intermediary in futures transactions. It guarantees the performance of the parties to each transaction. The clearing house has a number of members. Brokers who are not members themselves must channel their business through a member and post margin with the member. At the end of each day, the transactions being handled by the clearing house member are settled through the clearing house. If in total the transactions have lost money, the member is required to provide variation margin to the exchange clearing house (usually by the beginning of the next day); if there has been a gain on the transactions, the member receives variation margin from the clearing house.

Clearing house members are required to contribute to a guaranty fund. This may be used by the clearing house in the event that a member defaults and the member’s margin proves insufficient to cover losses.

2.5 OTC markets

OTC markets are markets where companies agree to derivatives transactions without involving an exchange. Credit risk has traditionally been a feature of OTC derivatives markets. Consider two companies, A and B, that have entered into a number of derivatives transactions. If A defaults when the net value of the outstanding transactions to B is positive, a loss is likely to be taken by B. In an attempt to reduce credit risk, the OTC market has borrowed some ideas from exchange-traded markets. We now discuss this.

Central counterparties

Members of the CCP, similarly to members of an exchange clearing house, have to provide both initial margin and daily variation margin. Like members, they...