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Solutions to end of Chapter Problems Part Paper

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Fin 4910/6990 Further Questions Problem 7. 19 (a) Company A has been offered the rates shown in Table 73. It can borrow for three years at 6. 45%. What floating rate can it swap this fixed rate into? (b) Company B has been offered the rates shown in Table It can borrow for S years at LABOR plus 75 basis points. What fixed rate can it swap this floating rate into? (a) Company A can pay LABOR and receive 6. 21% for three years. It can therefore exchange a loan at 6. 45% into a loan at LABOR plus 0. 24% or LABOR plus 24 basis points (b) Company B can receive LABOR and pay 6. 51% for five years.

It can Hereford exchange a loan at LABOR plus 0. 75% for a loan at 7. 26%. Problem 7. 21. The one-year LABOR rate is With annual compounding. A bank trades swaps where a fixed rate of interest is exchanged for 12-month LABOR with payments being exchanged annually. Two. And three-year swap rates (expressed with annual compounding) are 11% and 12% per annum. Estimate the two- and three year LABOR zero rates. The u-year swap rate implies that a two-year LABOR bond with a coupon of 11% sells for par. If is the two-year zero rate so that The three-year swap rate implies that a three-year LABOR bond with a coupon of 12% sells for par.

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If is the three-year zero rate so that The and three-year rates are therefore 11. 05% and 12. 17% with annual compounding. Problem 7. 22_ Company A, a British manufacturer, wishes to borrow U. S. Dollars at a fixed rate of interest. Company B, a US multinational, Wishes to borrow sterling at a fixed rate of interest. They have been quoted the following rates per annum (adjusted for differential tax effects): Sterling LIST Dollars Company A 11. 0% Company B 10. 6% 6. 2% Design a swap that will net a bank, acting as intermediary, 10 basis points per annum and that Will produce a gain Of IS basis points per annum for each Of the no companies.

The spread between the interest rates offered to A and B is 0. 4% (or 40 basis points) on sterling loans and 0. 8% (or 80 basis points) on U. S. Dollar loans. The total benefit to all parties from the swap is therefore It is therefore possible to design a swap which will earn 10 basis points for the bank while making each of A and B 15 basis points better off than they would be by going directly to financial markets. One possible swap is shown in Figure SO. 5. Company A borrows at an detective rate of 6. 85% per annum in U. S. Dollars.

Company B borrows at an effective rate of 10. 45% per annum in sterling. The ann. earns a ID-basis-point spread. The way in which currency swaps such as this operate is as follows. Principal amounts in dollars and sterling that are roughly equivalent are chosen. These principal amounts flow in the opposite direction to the arrows at the time the swap is initiated. Interest payments then flow in the same direction as the arrows during the life of the swap and the principal amounts flow in the same direction as the arrows at the end Of the life of the swap.

Note that the bank is exposed to some exchange rate risk in the swap. It earns 65 basis points in US. Dollars and pays AS basis points in sterling. This exchange rate risk could be hedged using forward contracts. Figure 57. 5 One Possible Swap for Problem 7. 22 Problem 7. 23. Lender the terms of an interest rate swap, a financial institution has agreed to pay 10% per annum and receive three-month LABOR in return on a notional principal to $100 million with payments being exchanged every three months.

The swap has a remaining life of 14 months. The average of the bid and offer fixed rates currently being swapped tort three-month LABOR is 12% per annum for all maturities. The three-month LABOR rate one month ago was II per annum. All rates are compounded quarterly. What is the value of the swap? The swap can be regarded as a long position in a floating-rate bond combined with a short position in a fixed-rate bond. The correct discount rate is 12% per annum with quarterly compounding or 1 182% per annum with continuous compounding.

Immediately after the next payment the floating-rate bond will be worth $100 million. The next floating payment ($ million) is The value of the floating-rate bond is therefore The value of the fixed-rate bond is The value Of the swap is therefore As an alternative approach we can value the swap as a series of forward rate agreements. The calculated value is which is in agreement with the answer obtained using the first approach. Problem 7. 24. Suppose that the term structure of interest rates is flat in the United States and Australia.

The USED interest rate is 7% per annum and the LAID rate is 9% per annum_ The current value of the SAID is C,62 SAID, In a swap agreement, a financial institution pays 8% per annum in LAID and receives per annum in LESS. The principals in the two currencies are $12 million SAID and 20 million ADD. Payments are exchanged every year, with one exchange having just taken place. The Swap will last two more years. What is the value of the Swap to the financial institution? Assume all interest rates are continuously compounded. The financial institution is long a dollar bond and short a LESS bond.

The value of the dollar bond (in millions Of dollars) is The value of the ADD bond (in millions of LAID) is The value of the swap (in millions of dollars) is therefore or -$795,000. As an alternative we can value the swap as a series of forward foreign exchange contracts. The one-year forward exchange rate is . The non-year tankard exchange rate is . The value of the swap in millions of dollars is therefore which is in agreement with the first calculation, problem 7,25_ Company X is based in the United Kingdom and would like to borrow $50 million at a fixed rate of interest for five years in funds.

Because the company is not well known in the United States, this has proved to be impossible. However, the company has been quoted per annum on fixed-rate five-year sterling funds, Company Y is based in the United States and would like to borrow the equivalent of $50 million in sterling funds for five years at a fixed rate of interest It has been unable to get a quote but has been offered US. Dollar funds at 10. % per annum. Vive-year government bonds currently yield 9. 5% per annum in the Lignite States and 10. 5% in the united Kingdom.

Suggest an appropriate currency swap that Will net the financial intermediary 0. 5% per annum. There is a 1% differential bet. ‘en the yield on sterling and dollar 5-year bonds. The financial intermediary could use this differential when designing a swap. For example, it could (a) allow company X to borrow dollars at 1% per annum less than the rate offered on sterling funds, that is, at 11% per annum and (b) allow company Y to borrow sterling at 1% per annum more than the rate offered on alular funds, that is, at 11% per annum. However, as shown in Figure SO. , the financial intermediary would not then earn a positive spread. Figure 57. 6 First attempt at designing swap for Problem 7. 25

To make C. S% per annum, the financial intermediary could add 0 25% per annum, to the rates paid by each of X and Y. This means that X pays 11. 25% per annum, for dollars and Y pays I I _ per annum, for sterling and leads to the swap shown in Figure SO. 7. The financial intermediary would be exposed to some foreign exchange risk in this swap. This could be hedged using forward contracts. Figure SO. Final swap for Problem 7. 5 Problem 9. 23. The price of a stock is $40.

The price of a one-year European put option on the stock with a strike price to $30 is quoted as $7 and the price to a one-year European call option on the stock with a strike price of is quoted as Suppose that an investor buys 100 shares, shorts 100 call options, and buys 100 put options, Draw a diagram illustrating how the investor’s profit or loss varies with the stock price over the next year.

How does your answer change if the investor buys 100 shares, shorts 200 call options, and buys 200 put options? Figure SO. Wows the way in which the investor’s profit varies with the stock price in the first case. For stock prices less than $30 there is a loss of $1,200. As the stock price increases from $30 to $50 the profit increases from -51,200 to $800. Above $50 the profit is $800. Students may express surprise that a call which is $10 out Of the money is less expensive than a put Which is SIS out Of the money.

This could be because of dividends or the acrophobia phenomenon discussed in Chapter leg. Figure So. 8 shows the way in Which the profit varies with stock price in the second case.

In this case the profit pattern has a zigzag heap. The problem illustrates how many different patterns can be obtained by including calls, puts, and the underlying asset in a portfolio. Figure SO. 7 Figure 59. 8 Profit in first case considered Problem g_23 Profit for the second case considered Problem 9. 23 Problem 9. 25. Use Derivable to calculate the value of an American put option on a undivided paying stock when the stock price is $30, the strike price is $32, the risk-free rate is 5%, the volatility is 30%, and the time to maturity is IS years. (Choose Binomial American for the “option type” and 50 time steps. A.

What is he option’s intrinsic value? B. What is the option’s time value? C. What would a time value Of zero indicate? What is the value Of an option With zero time value? D. Using a trial and error approach calculate how low the stock price would have to be for the time value Of the option to be zero. Derivable shows that the value of the option is 4. 57. The option’s intrinsic value is . The option’s time value is therefore . A time value of zero would indicate that it is optimal to exercise the option immediately. In this case the value of the option would equal its intrinsic value.

When the stock price is 20, Derivable ivies the value of the option as 12, which is its intrinsic value. When the stock price is 25, Derivable gives the value of the options as 7. 54, indicating that the time value is still positive Keeping the number to time steps equal to 50, trial and error indicates the time value disappears when the stock price is reduced to 21. 6 or lower. (With 500 time steps this estimate of how low the stock price must become is reduced to 21. 3. ) Problem 9,26_ On July 20, 2004 Microsoft surprised the market by announcing a $3 dividend.

The ex-dividend date was November 1 7, 2004 and the payment date was December 2, 2004. Its stock price at the time was about $28. It also changed the terms Of its employee stock options so that each exercise price was adjusted downward to Pre-dividend Exercise Price The number Of shares covered by each stock option outstanding was adjusted upward to Number of Shares Pre-dividend “Closing Price” means the Official NASDAQ closing price Of a share Of Microsoft common stock on the last trading day before the ex-civilized date. Evaluate this adjustment.

Compare it with the system used by exchanges to adjust for extraordinary dividends (see Business Snapshot 9. 1). Suppose that the closing stock price is SIB and an employee has 1000 options with a strike price of $24. Microsoft’s adjustment involves changing the strike price to and changing the number of options to The system used by exchanges would involve keeping the number of options the same and reducing the strike price why 53 to $21, The Microsoft adjustment is more complicated than that used by the exchange because it requires a knowledge to the Microsoft stock price immediately before the stock goes ex-dividend.

However, arguably it is a better adjustment than the one used by the exchange. Before the adjustment he employee has the right to pay 524,000 for Microsoft stock that is worth $28,000. After the adjustment the employee also has the option to pay $24,000 for Microsoft stock worth 528,000 Under the adjustment rule used by exchanges the employee would have the right to buy stock worth $25,000 for $21 ,OHO. If the volatility of Microsoft remains the same this is a less valuable option. One complication here is that Microsoft volatility does not remain the same.

It can be expected to go up because some cash (a zero risk asset) has been transferred to shareholders. The employees therefore have the same basic option as before UT the volatility Of Microsoft can be expected to increase. The employees are slightly better off because the value of an option increases with volatility. Problem 10. 22. A European call option and put option on a stock both have a strike price of $20 and an expiration date in three months. Both sell for $3. The risk-free interest rate is 10% per annum, the current stock price is $19, and a $1 dividend is expected in one month.

Identify the arbitrage opportunity open to a trader. Fifth call is worth $3, put-call parity shows that the put should be worth This is greater than $3. The put is therefore undervalued relative to the call. The correct arbitrage strategy is to buy the put, buy the stock, and short the call. This costs $ leg, If the stock price in three months is greater than $20, the call is exercised. If it is less than $20, the put is exercised. In either case the arbitrageur sells the stock for 520 and collects the $1 dividend in one month.

The present value of the gain to the arbitrageur is Problem 10. 6. Consider an option on a stock when the stock price is $41, the strike price is $40, the risk-free rate is 6%, the volatility is 35%, and the time to maturity is I year. Assume that a dividend of $0. 50 is expected after six months. . Use Derivable to value the option assuming it is a European call. B. Use Derivable to value the option assuming it is a European put. C. Verify that put-call parity holds. D. Explore using Derivable what happens to the price of the options as the time to maturity becomes very large. For this purpose assume there are no dividends. Explain the results you get.

Derivable shows that the price of the call option is 6. 9686 and the price of the put option is 41244 In this case Also As the time to maturity becomes very large and there are no dividends, the price of the call option approaches the stock price of 41. For example, when T = 100 it is 40,94_) This is because the call option can be regarded as a position in the stock where the price does not have to be paid for a very long time.

The present value of what has to be paid is close to Zero As the time to maturity becomes very large the price of the European put option becomes close to zero. For example, when T 100 it is 014_) This is because the present value of the expected payoff from the put option tends to zero as the time to maturity increases. Problem 10. 27 Consider a put option on a Nan-dividend-paying stock When the stock price is 40, the strike price is $42, the risk-free rate of interest is 2%, the volatility is 25% per inurn, and the time to maturity is 3 months. Use Drainage to determine: a. The price of the option if it is European (use Black-Schools: European) b. The price Of the option if it is American (Else Binomial: American With 100 tree steps) c.

Point B in Figure 10. 7 (a) 9. 06 (b) $3. 08 (c) 535. 4 (using trial and error to determine when the European option price equals its intrinsic value). Problem 12. 16. A stock price is currently It is known that at the end of six months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is per annum_ Calculate the value of a six-month European call option on the stock with an exercise price of $48. Verity that no-arbitrage arguments and risk- neutral valuation arguments give the same answers.

At the end Of six months the value Of the option Will be either $12 (if the stock price is $60) or SO (if the stock price is $42). Consider a portfolio consisting of: The value of the portfolio is either or in six months. If the value of the portfolio is certain to be 28. For this value of the portfolio is therefore reckless. The current value of the portfolio is: here is the value of the option. Since the portfolio must earn the risk-free rate of interest The value of the option is therefore $6. 96, This can also be calculated using risk-neutral valuation.

Suppose that is the probability of an upward stock price movement in a risk-neutral world. We must have The expected value of the option in a risk-neutral world is: This has a present value Of Hence the above answer is consistent With risk-neutral valuation. Problem 12. 17. A stock price is currently $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. The risk

This is greater than the value Of the European option because it is optimal to exercise early at node C. Figure SIS. S Tree to evaluate European and American put options in Problem 12. 17. At each node, upper number is the stock price, the next number is the European put price, and the final number is the American put price Problem 12. 19. A stock price is currently $30. During each two-month period for the next four months it is expected to increase by 8% or reduce by 10%. The risk-tree interest ate is use a two-step tree to calculate the value of a derivative that pays off where is the stock price in four months?

If the derivative is American-style, should it be exercised early? This type of option is known as a power option. A tree describing the behavior of the stock price is shown in Figure SIS_6. The risk-neutral probability of an up move, , is given by The value Of the European option is 5. 394. This can also be calculated by working back through the tree as shown in Figure SIS. 6. The second number at each node is the value Of the European option. Early exercise at node C would give 9. 0 which is less than 13. 2449. The option should therefore not be exercised early if it is American.

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