## Info

On the same stock with the same strike price and exercise date Se i(T ') _x < C - P < S- Xe-'V-') where S is the stock price, r is the risk-free interest rate, and r > 0. (Hint To obtain the first half of the inequality, consider possible values of Portfolio A A European call option plus an amount X invested at the risk-free rate Portfolio B An American put option plus of stock with dividends being To obtain the second half of the inequality consider possible values of Portfolio C An...

## The Operation Of Margins

If two investors get in touch with each other directly and agree to trade an asset in the future for a certain price, there are obvious risks. One of the investors may regret the deal and try to back out. Alternatively, the investor simply may not have the financial resources to honor the agreement. One of the key roles of the exchange is to organize trading so that contract defaults are minimized. This is where margins come in. To illustrate how margins work consider an investor who contacts...

## Questions And Problems

A company caps 3-month LIBOR at 10 per annum. The principal amount is 20 million. On a reset date, 3-month LIBOR is 12 per annum. What payment would this lead to under the cap When would the payment be made 15.2. Explain what mortgage-backed securities are. Explain why mortgage-backed securities are more risky than regular fixed income instruments such as government bonds. 15.3. Explain why a swaption can be regarded as a type of bond option. 15.4. Use the Black-Scholes model to value a...

## Dividends

Up to now we have assumed that the stock upon which the option is written pays no dividends. In practice, this is not usually true. In this section we assume that the dividends that will be paid during the life of an option can be predicted with certainty. As traded options typically last for less than 9 months, this is not an unreasonable assumption. A dividend-paying stock can reasonably be expected to follow the stochastic process developed in Chapter 9 except when the stock goes...

## Factors Affecting Option Prices

There are six factors affecting the price of a stock option 4. The volatility of the si price S 5. The risk-free interest rale 6. The dividends expected during the life of jhe-option In this section, we consider what happens to option prices when one of these factors changes with all the others remaining fixed. The results are summarized in Table 7.1. If it is exercised at some time in the future, the payoff from a call option will be the amount by which the stock price exceeds the strike...

## The Lognormal Property Of Stock Prices

A variable has a lognormal distribution if the natural logarithm of the variable is normally distributed. It has just been shown that the model of stock price behavior developed in Chapter 9 implies that In ST - In S 0 x - T - t , a T - t 10.6 where St is the stock price at a future time T S is the stock price at the current time, f, and lt p m, s denotes a normal distribution with mean m and standard deviation i. From the properties of the normal distribution, it follows from Equation 10.6...

## Mechanics Of Interest Rate Swaps

The most common type of swap is a plain vanilla interest rate swap. In this, one party, B, agrees to pay to the other party, A, cash flows equal to interest at a predetermined fixed rate on a notional principal for a number of years. At the same time, party A agrees to pay party B cash flows equal to interest at a floating rate on the same notional principal for the same period of time. The currencies of the two sets of interest cash flows are the same. The life of the swap can range from 2...

## Forward Prices Versus Futures Prices

Appendix 3A provides an arbitrage argument to show that, when the risk-free interest rate is constant and the same for all maturities, the forward price for a contract with a certain delivery date is the same as the futures price for a contract with the same delivery date. The argument in Appendix 3A can be extended to cover situations where the interest rate is a known function of time. When interest rates vary unpredictably as they do in the real world , forward and futures prices are in...

## Specification Of Stock Options

In the rest of this chapter, we will focus on exchange-traded stock options. The contract specifications and trading of index options, currency options, and futures options are discussed further in Chapter 11. As already mentioned, a stock option contract is an American-style option contract to buy or sell 100 shares of the stock. Details of the contract, such as the expiration date, the strike price, what happens when dividends are declared, how large a position investors can hold, and so on,...

## Early Exercise Calls On A Nondividendpaying Stock

In this section, we show that it is never optimal to exercise an American call option on a non-dividend-paying stock early. To illustrate the general nature of the argument, consider an American call option on a non-dividend-paying stock with one month to expiration when the stock price is 50 and the strike price is 40. The option is deep in the money and the investor who owns the option might well be tempted to exercise it immediately. However, if the investor plans to hold the stock for more...

## Forward Contracts On A Security That Provides A Known Dividend Yield

As will be explained in later sections, both currencies and stock indices can be regarded as securities that provide known dividend yields. In this section, we provide a general analysis of forward contracts on such securities. A known dividend yield means that the income when expressed as a percentage of the security price is known. We will assume that the dividend yield is paid continuously at an annual rate q. To illustrate what this means, suppose that q 0.05 so that the dividend yield is 5...

## Financial Institution Has Agreed To Pay 10 Per Annum And To Receive Three-month Libor

Company X requires a floating rate loan company Y requires a fixed rate loan. Design a swap that will net a bank, acting as intermediary, 0.2 per annum and which will appear equally attractive to X and Y. 5.9. Company A, a British manufacturer, wishes to borrow U.S. dollars at a fixed rate of interest. Company B, a U.S. multinational, wishes to borrow sterling at a fixed rate of interest. They have been quoted the following rates per annum Design a swap that will net a bank, acting as...

## Futures Prices And The Expected Future Spot Price

One question that is often raised is whether the futures price of an asset is equal to its expected future spot price. If you had to guess what the price of an asset will be in 3 months, is the futures price an unbiased estimate John Maynard Keynes and John Hicks in the 1930s argued that, if hedgers tend to hold short positions and speculators tend to hold long positions, the futures price will be below the expected future spot price. This is because speculators require compensation for the...

## Credit Risk

Contracts such as swaps that are private arrangements between two companies entail credit risks. Consider a financial institution that has entered into offsetting contracts with two companies, A and B. See Figure 5.3 or Figure 5.6. If neither party defaults, the financial institution remains fully hedged. A decline in the value of one contract will always be offset by an increase in the value of the other contract. However, there is a chance that one party will get into financial difficulties...

## Suppose That A Bond Portfolio With A Duration Of 12 Years Is Hedged Using A Futures

The cheapest-to-deliver bond in a September 1992 Treasury bond futures contract is a 13 percent coupon bond, and delivery is expected to be made on September 30, 1992. Coupon payments on the bond are made on February 4 and August 4 each year. The term structure is flat and the rate of interest with semiannual compounding is 12 per annum The conversion factor for the bond is 1.5. The current quoted bond price is 110. Calculate the quoted futures price for the contract....

## Forward Contracts On A Security That Provides A Known Cash Income

In this section we consider a forward contract on a security that will provide a perfectly predictable cash income to the holder. Examples are stocks paying known dividends and coupon-bearing bonds. Define as the present value, using the risk-free discount rate, of income to be received during the life of the forward contract. For there to be no arbitrage, the relationship between F and S must be To show this, suppose first that F gt S I er T 'K An arbitrageur can borrow money, buy the asset,...

## Hedging Using Futures

A company that knows it is due to sell an asset at a particular time in the future can hedge by taking a short futures position. This is known as a short hedge. If the price of the asset goes down, the company does not fare well on the sale of the asset, but makes a gain on the short futures position. If the price of the asset goes up, the company gains from the sale of the asset, but makes a loss on the futures position. Similarly, a company that knows it is due to buy an asset in the f uture...

## Ft Si Fi

In our example, gt i 0.30 and 2 0.10. Consider first the situation of a hedger who knows that the asset will be sold at time t2 and takes a short futures position at time t . The price realized for the asset is S2 and the profit on the futures position is F F2. The effective price that is obtained for the asset with hedging is therefore In our example, this is 2.30. The value of F is known at time t . If b2 were also known at this time, a perfect hedge that is, a hedge eliminating all...

## The Specification Of The Futures Contract

When developing a new contract, an exchange must specify in some detail the cxact nature of the agreement between the two parties. In particular, it must specify the asset, the contract size i.e., exactly how much of the asset will be delivered under one contract , how prices will be quoted, where delivery will be made, when delivery will be made, and how the price paid will be determined. Sometimes alternatives are specified for the asset that will be delivered and for the delivery...

## General Approach To Pricing Derivative Securities

12.1 A Single Underlying Variable 274 12.3 Securities Dependent on Several State Variables 279 12.4 Derivative Securities Dependent on Commodity Prices 282 12.5 Cross-Currency Futures and Options 284 Suggestions for Further Reading 288 Questions and Problems 288 Appendix 12A A Generalization of Ito's Lemma 290 Appendix 12B Derivation of the General Differential Equation Satisfied by Derivative Securities 291