## 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 < p (fi - )(T - t), ojT - t (10.6) where St is the stock price at a future time T S is the stock price at the current time, and < j> (m, s) denotes a normal distribution with mean m and standard deviation s. From the properties of the normal distribution, it follows from...

## Questions And Problems

Explain the difference between a forward start option and an as you like it option. 16.2. Describe the payoff from a combination of a lookback call and a lookback put. 16.3. Explain the purpose of the call feature in convertible bonds. 16.4. Consider an as you like it option where the holder has the right to choose between a European call and a European put at any time during a 2-year period. The maturity dates and strike prices for the calls and puts are the same regardless of when the...

## Monte Carlo Simulation

American-style derivative security to refer to a derivative security where there are early exercise or other decisions that may have to be made. Consider a European-style derivative security that pays off fr at time T. From Equation (12.15), its value at time zero is given by where E denotes expectations in a risk-neutral world and r is the average instantaneous risk-free interest rate between time zero and time T. If the risk-free interest rate is assumed to be known with certainty, Equation...

## 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...

## The Heath Jarrow And Morton Approach

The Heath, Jarrow, and Morton HJM approach is to specify the volatilities of all instantaneous forward rates at all future times. This is sometimes referred to as the volatility structure. Equation 15.27 is then used to calculate the drift of each instantaneous forward rate and a binomial tree describing the evolution of the term structure of forward rates is constructed. Unfortunately, the HJM tree is, in general, nonrecombining in the sense that an up movement followed by a down movement does...

## 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...

## Suggestions For Further Reading

Emanuel, Discretely Adjusted Option Hedges, Journal of Financial Economics, 8 1980 , 259-82. Dillman, S., and J. Harding, Life after Delta the Gamma Factor, Euromoney, Supplement February 1985 , pp. 14-17. Figlewski, S., Options Arbitrage in Imperfect Markets, Journal of Finance 44 December 1989 , 1289-1311. Galai, D., The Components of the Return from Hedging Options Against Stocks, Journal of Business, 56 January 1983 , 45-54. Hull, J., and A. White, Hedging the Risks...

## C

Calendar spread, 181 Callable bonds, 149, 371 Call option, 5 definition, 5 early exercise, 158, 234 Cancellable forward, 415 Capital allocation, 466 Capital Asset pricing model CAPM , 61, 280 Chicago Board of Trade, 4, 18, 20, 22, 27, 42, 88, 106, 137, 259 Chicago Board Options Exchange, 137, 149, 249 Chicago Mercantile Exchange, 4, 20, 22, 58, 259 Clearinghouse, 26 Clearing margin, 26 Closing out position, 19 Commodity price process risk-neutral world , 282 Collars, 378 Combinations, 183-86...

## F

Financial News Composite Index, 249 Finite difference methods, 352 application of, 361-62 backward difference approximation, 354 explicit, 356, 358 forward difference approximation, 354 implicit, 354 relation to tree approaches, 359-61 Flexible forward contract, 11 Floating interest rates, 112 Floor-ceiling agreement, 378 Floor broker, 144 Floors, 378 Foreign currency options, 137, 255 Foreign exchange quotes, 64 Forward bond yield, 382 Forward contract, 2 comparison with futures, 4-5...

## Upper And Lower Bounds For Option Prices

In this section, we derive upper and lower bounds for option prices. These do not depend on any particular assumptions about the factors mentioned in the previous section except r gt 0 . If the option price is above the upper bound or below the lower bound, there are profitable opportunities for arbitrageurs. An American or European call option gives the holder the right to buy one share of a stock for a certain price. No matter what happens, the option can never be worth more than the stock....

## 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...

## Model of the Behavior of Stock Prices

Any variable whose value changes over time in an uncertain way is said to follow a stochastic process. Stochastic processes can be classified as discrete time or continuous time. A discrete-time stochastic process is one where the value of the variable can only change at certain fixed points in time, whereas a continuous-time stochastic process is one where changes can take place at any time. Stochastic processes can also be classified as continuous variable or discrete variable. In a...

## 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...

## The Process For Stock Prices

In this section we discuss the stochastic process followed by the price of a non-dividend-paying stock. The effects of dividends on the process will be discussed in Chapter 10. It is tempting to suggest that a stock price follows a generalized Wiener process that is, that it has a constant expected drift rate and a constant variance rate. However, this model fails to capture a key aspect of stock prices. This is that the expected percentage return required by investors from a stock is...

## Futures On Commodities

We now move on to consider commodity futures contracts. Here it will prove to be important to distinguish between commodities that are held solely for investment by a significant number of investors e.g., gold and silver and those that are held primarily for consumption. Arbitrage arguments can be used to obtain exact futures prices in the case of investment commodities. However, it turns out that they can only be used to give an upper bound to the futures price in the case of consumption...

## 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...

## The Markov Property

A Markov process is a particular type of stochastic process where only the present value of a variable is relevant for predicting the future. The past history of the variable and the way in which the present has emerged from the past are irrelevant. Stock prices are usually assumed to follow a Markov process. Suppose that the price of IBM stock is 100 now. If the stock price follows a Markov process, our predictions for the future should be unaffected by the price 1 week ago, 1 month ago, or 1...

## Durationbased Hedging Strategies

We assume that the change in the yield, Ay, is the same for all maturities which means that only parallel shifts in the yield curve can occur. From Equation 4.8 it is approximately true that To a reasonable approximation, it is also true that The number of contracts required to hedge against an uncertain Ay is therefore This is the duration-based hedge ratio. It is sometimes also called the price sensitivity hedge ratio. Using it has the effect of making the duration of the whole position zero....

## 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...

## Company X Is Based In The United Kingdom And Would Like To Borrow 50 Million At A Fixed Rate Of Interest For Five Years

Assume that A wants to borrow dollars at a floating rate of interest and B wants to borrow marks at a fixed rate of interest. A financial institution is planning to arrange a swap and requires a 50 basis point spread. If the swap is to appear equally attractive to A and B, what rates of interest will A and B end up paying 5.15. Company X is based in the United Kingdom and would like to borrow U.S. 50 million at a fixed rate of interest for 5 years in U.S. funds. As the company is not well known...

## 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...

## Fixed Rate Bond Underlying Swap

LIBOR 0.1 x - LIBOR 0.0986 per annum Consider the swap between company B and the financial institution in Figure 5.3. Although the principal is not exchanged, we can assume without changing the value of the swap that, at the end of its life, A pays B the notional principal of 10 million and B pays A the same notional principal. The swap is then the same as an arrangement in which 1. Company B has lent the financial institution 10 million at the 6-month 2. The financial institution has lent...

## 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....

## 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...

## Convergence Of Futures Price To Spot Price

As the delivery month of a futures contract is approached, the futures price converges to the spot price of the underlying asset. When the delivery period is reached, the futures price equals or is very close to the spot price. To show why this is so, suppose first that the futures price is above the spot price during the delivery period. This gives rise to a clear arbitrage opportunity for traders This is certain to lead to a profit equal to the amount by which the futures price exceeds the...

## Optimal Hedge Ratio

The hedge ratio is the ratio of the size-of the position taken in futures contracts to the size of the exposure. Up to now we have always assumed a hedge ratio of 1.0. We now show that, if the objective of the hedger is to minimize risk, a hedge ratio of 1.0 is not necessarily optimal. AS Change in spot price, S, during a period of time equal to the life of the hedge AF Change in futures price, F, during a period of time equal to the life of the hedge ers Standard deviation of A 5 lt tf''...

## 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