金融工程教材英文版课后习题

END-OF-CHAPTER QUESTIONS AND PROBLEMS

Chapter 1

1. Business risk is the risk associated with a particular line of business, whereas financial risk is the risk associated with stock prices, exchange rates, interest rates and commodity prices. An example would be that a firm might be in the business of manufacturing furniture. Business risk would reflect the uncertainty of the furniture market, whereas financial risk would reflect the risk associated with the interest rates that would impact their borrowing costs. In addition a business might also be affected by exchange rates. Some of the financial risks can also be business risks. The business risk of a bank, for example, is highly affected by interest rate and exchange rate risk.

2. Real assets consist of the tangible assets of the economy; however, for our purposes we also define them to include such intangible assets as management talent, ideas, brand names, etc. They are distinguished from financial assets, which are securities. These securities represent claims on business firms, which own the real assets, or on governments.

3. An investor who is risk averse does not like risk and will not take on additional risk without the expectation of higher return. Such an individual will try to get the highest return for a given amount of risk or the lowest risk for a given amount of return. An individual who is risk neutral will simply seek the highest return without regard to risk.

4. Financial markets distinguish the qualities of stocks and bonds by their prices. The markets set the price so that the return expected by investors is appropriate for the level of risk. In competitive and efficient markets, the expected return will vary directly with the level of risk. If one wishes to earn a higher return, it is necessary to assume more risk. This is the risk-return trade-off. Investors trade off risk against return.

The risk-return trade-off will also hold in derivative markets. The risk-return trade-off is a fundamental result of the nature of competition in the marketplace. Attractive, low-risk investments will have their prices driven up and this will lower the expected returns. High-risk investments will have their prices driven down and this will result in higher expected returns.

5. The expected return consists of the risk-free rate and a risk premium. The risk-free rate is the return one expects from investing money today and, thereby, foregoing the consumption that could be obtained. A risk premium is the additional return that one expects to receive by virtue of assuming risk.

6. An efficient market is one in which prices reflect the true economic values of the assets trading therein. Inefficient markets, no one can earn returns that are more than commensurate with the level of risk. Efficient markets are characterized by low transaction costs and by the rapid rate at which new information is incorporated into prices.

7. Arbitrage is a type of investment transaction that seeks to profit when identical goods are priced differently. Buying an item at one price and immediately selling it at another is a type of arbitrage. Because of the combined activities of arbitrageurs, identical goods, primarily financial assets, cannot sell for different prices for long. This is the law of one price. Arbitrage helps make our markets efficient by assuring that prices are in line with what they are supposed to be. In short, we cannot get something for nothing. A situation involving two identical goods or portfolios that are not priced equivalently would be exploited by arbitrageurs until their prices were equal. The "one price" that an asset must be is called the "theoretical fair value."

11. Derivative markets provide a means of adjusting the risk of spot market investments to a more acceptable level and identifying the consensus market beliefs. They make trading easier and less costly and spot markets more efficient. These markets also provide a means of speculating.

Chapter 3

3. European call: We know that its price cannot exceed S0 but must exceed Max(0, S0 - X(1+r)-T). With an infinite time to expiration, the present value of X is zero so the lower bound is S0 and, since the upper bound is S0, the call price must be S0. American call: We know that its price cannot exceed S0 but it must be at least as valuable as a European call. Thus its value must also be S0. Note that if exercised early it would be worth only S0 - X so it will never be exercised early.

4. Ordinarily the option with the longer time to expiration would sell for more. If both options were deep out of-the-money, however, they could both sell for essentially nothing. The market would be expecting that both the shorter- and longer-lived options will expire out-of-the-money.

5. If both options were deep out-of-the-money, they might have prices of zero. As in the previous question, the two options are expected to expire out-of-the-money.

6. The call is underpriced, so buy the call, sell short the stock, and buy risk-free bonds with face value of X. The cash received from the stock is greater than the cost of the call and bonds. Thus, there is a positive cash flow up front. The payoffs from the portfolio at expiration are as follows:

If S T < X,

- S T (from the stock)

X (from the bonds)

the total, X - S T, is positive

3-2

If S T ≥ X

S T - X (from the call)

- S T (from the stock)

X (from the bonds)

the total is zero.

The portfolio generates a positive cash flow up front and there is no cash outflow at expiration.

7. Time value is a measure of the amount of uncertainty in an option. Uncertainty relates to whether the option will expire in- or out-of-the-money. When the option is deep in-the-money, there is little uncertainty about the fact that the option will expire in-the-money. The option will then begin to behave about like the stock. Then the option is deep-out-the-money, there is also little uncertainty since it is likely to finish out-of-the money. When the option is at-the-money there is considerable uncertainty about how it will finish.

8. Assuming the stock pays no dividends, there is no reason to exercise a call early (this obviously presumes the call is American). The tendency to believe that exercising an option because the stock can go up no further ignores the fact that an option can generally be sold. Exercising an option throws away any chance that the stock can go up further. If the stock falls, the option holder would be hurt, but if the option holder exercised and became a stock holder, he would also be hurt by a falling stock price. There is simply no reason to give away the time value that arises because of the possibility that the stock can always go further upward. In simple, mathematical terms, exercising captures only the intrinsic value S0 - X. The call can always be sold for at least S0 - X (1 + r) -T.

9. The paradox is resolved by recalling that if the option expires out-of-the-money, it does not matter how far out-of-the-money it is. The loss to the option holder is limited to the premium paid. For example, suppose the stock price is $24, the exercise price is $20, and the call price is $6. Higher volatility increases the chance of greater gains to the holder of the call. It also increases the chance of a larger stock price decrease. If, however, the stock price does end up below $20, the investor's loss is the same regardless of whether the stock price at expiration is $19 or $1. If the stock were purchased instead of the call, the loss would obviously be greater if the stock price went to $1 than if it went to $19. For this reason, holders of stocks dislike volatility, while holders of calls like volatility. A similar argument applies to puts.

10. The minimum value of an American put is Max(0, X - S0). This is always higher than the lower bound of a European put, X (1 + r) -T - S0, except at expiration when the two are equal.

11. When buying a call option, one hopes to exercise it at a later date. Thus, the exercise price will be paid out later. If interest rates are higher, additional interest can be earned on the money that will eventually be paid out as the exercise price. When buying a put option, one hopes to exercise it later, thus receiving the exercise price. If interest rates are higher, the put is less valuable because the holder is foregoing interest by having to wait to exercise the put. Higher interest rates make the present value of the exercise price be lower. In the case of the call, this is good because the call holder anticipates having to pay out the exercise price. For the put holder this is bad because the put holder anticipates receiving the exercise price.

12. If the put price is higher than predicted by the model, the put is overpriced. Then the put should be sold. The funds should be used to construct a portfolio consisting of a long call and risk-free bonds with face value of X, and a short position in the stock. You can (and should) verify that this portfolio has a payoff at expiration that is the same as that of the put. Thus, if you sell the put for more than it costs to construct this portfolio, you can create a portfolio that offsets your put and have some money left over.

13. An American call is exercised early only to capture a dividend. When a stock goes ex-dividend, the call will lose value as the stock drops. This will cause a loss in value to the holder of the call. The call holder knows this loss will be incurred as soon as the stock goes ex-dividend. If the call were exercised just before the stock goes ex-dividend, however, the call holder would capture the stock and the dividend, which might be enough to offset the otherwise loss in the value of the call. For a put, however, dividends are not necessary to make the argument that it might be optimal to exercise early. The holder of an American put faces a situation in which the gains are limited to the exercise price. Since the stock price can go down only to zero, early exercise of a put on a bankrupt firm would obviously be advisable. But the firm does not have to go bankrupt. If the stock price is low enough, the gains from waiting for it to go lower are not worth the wait. If dividends were added to the picture, however, they would discourage early exercise. The more dividends paid, the lower the stock price is driven and the more valuable it is to hold on to the put.

Chapter 4

1. When we price an option according to its boundary conditions, we do not find an exact price for the option. We provide only limits on the maximum and minimum price of an option or group of options. In the case of options differing by exercise prices or in the case of put-call parity, we can price only the relationship or difference between the option prices. We cannot price each option individually without an option pricing odel; however, an option pricing model must provide prices that conform to the boundary conditions. Because boundary condition rules require fewer assumptions, we can say that they are more generally applicable and are more likely to hold in practice. They are incomplete, however, in the sense that they do not tell us exactly what the option price should be, which is what an option pricing model does tell us.

2. A binomial option pricing model enables us to see the relationship between the stock price and the call price. The model shows, in a simple framework, how to construct a riskless portfolio by appropriately weighting the stock against the option. By noting that the riskless portfolio should return the risk-free rate, we can see what the call price must be. We can also understand the forces that bring the call option price in line if it is not priced according to the model. In addition, the model illustrates the importance of revising the hedge ratio. Finally, the model is probably the best way to handle the problem of pricing an American option.

3. For a call, a hedge portfolio will consist of n shares of stock and one short call. The shares of stock, n, will be a long position that will reflect the next two possible values of the call and stock. The call is short because the call and stock move opposite each other, so a short position in the call is needed to offset a long position in the stock. If the option is a put, a long position in n shares will be hedged by a long position in one put. The number of shares, n, is also a reflection of the next two possible option and stock values. The put is long because it already moves opposite to the stock.

4. This means that the option is trading in the market for a price that is lower than its theoretical fair value. Consequently, the option is underpriced. An underpriced option should be bought. To hedge the position, one should sell n shares of stock, where n is the hedge ratio as defined in the chapter.

5. At each point in the binomial tree, the option price is computed based on the next two possible prices, weighted by the appropriate probability values and discounted back one period. If the option can be exercised early, we determine its intrinsic value, i.e., the value it would have if it were exercised at that point. If the intrinsic value is greater, it replaces the computed value. This is done at all nodes. Note that it is quite possible that the option will be exercised at many different nodes.

6. A path independent problem is one in which the only values that matter in determining the current price are the values at expiration. A particular expiration value is the same regardless of the sequence of up and down moves the stock took to get to that point. A path dependent model is one in which the sequence of up and down moves does matter. For example, suppose the stock goes down, up and then up. Contrast that with up, up and down. If the option is a European put, it does not matter which sequence the stock followed. Where it is at expiration is the only thing that matters. If the option is, however, an American put, the down, up, up sequence could lead to early exercise when it makes that first move down, whereas the up, up, down sequence is unlikely to lead to early exercise.

Chapter 5

1. In a discrete time model, the stock price can make a jump to only one of two possible values. The length of time over which the move can be made is finite. In a continuous time model, the stock price can jump to an infinite number of possibilities. The length of time over which the move can be made is infinitesimal (very, very small). The difference between the two models is perhaps best described as in the text as the difference between still photos and a movie.

2. The familiar normal or bell-shaped distribution is a symmetric probability distribution that depends only on the mean and variance. A lognormal distribution is skewed, having more extreme right values. A lognormal distribution, however, is normal in the logarithm. Thus, if x is lognormally distributed, its logarithm, lnx, is normally distributed. With respect to stock prices, the logarithm is of the rate of return. That is, let (S1 –S0)/S0 be defined as the return over period 0 to period 1. Then if it is lognormally distributed, ln(S1/S0) is normally distributed. Note what appears to be slight inconsistency: (S1 – S0)/S0 is a percentage return whereas S1/S0 is 1.0 plus the percentage return. That does not matter as we can always shift a normal distribution by adding a constant such as 1.0 and it does not affect the fact that it is normally distributed. It just slides it over.

3. The Black-Scholes model assumes no dividends, but actually this was just for convenience to allow us to start at the simplest level. If the dividends are appropriately modeled, the Black-Scholes model handles stocks with dividends with only the minor adjustment that we must remove the present value of the dividends from the stock price before using it in the model. We can do this by subtracting the present value of the stream of discrete dividends over the life of the model or by discounting the stock price by the dividend yield rate over the life of the model.

4. The variables in the binomial model are S0 (the stock price), X (the exercise price), r (the discrete risk-free rate), u (one plus the return on the stock if it goes up), d (one plus the return on the stock if it goes down) and n, the number of time periods. The variables in the Black-Scholes model are S0 (the stock price), X (the exercise price), r c (the continuously compounded risk-free rate), ó (the standard deviation of the continuously compounded return on the stock) and T (the time to expiration). The variables S0 and X are the same in both models. In the binomial model, r is the discrete interest rate per period. In the Black-Scholes model, the interest rate must be expressed in continuously compounded form. The annual discrete interest rate (r) is related to the annual continuously compounded rate (r c) by the formula, r c = ln(1 + r). Then the binomial rate per period is found as (1 + r)T/n - 1. The up and down factors in the binomial model are directly related to ó in the Black-Scholes model by the formula u = T/n ó e – 1 and d = (1/(1+u)) – 1. The time to expiration (T) in the Black-Scholes model is related to the length of each binomial period by the relationship T/n where n is the number of periods. Thus, all of the Black-Scholes variables are either equal to or directly convertible to binomial variables.

Chapter6

Chapter7

Chapter 8