Chapter 14, Advanced Derivacives and Strategies Video Solutions, An Introduction to Derivatives and Risk Management: With Stock-Trak Coupon

Problem 1

Explain the advantages and disadvantages of implementing portfolio insurance using stock and puts in comparison to using stock and futures in a dymamic hedge strategy.

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Problem 2

Explain how a portfolio manager might justify the purchase of an inverse floatingrate note.

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Demonstrate that the payoffs of a chooser option with an exercise price of $\mathrm{X}$ and a time to expiration of $\mathrm{T}$ that permits the user to designate it as a call or a put at $\mathrm{t}$, can be replicated with two transactions. Specifically, by (1) buying a call with an exercise price of $X$ and time to expiration of $T$ and (2) buying a put with an exercise price equal to $X(1+r)^{-(T-t)}$ and time to expiration of $t$. This proof will require that you consider two possible outcomes at $t$ (user designates it as a call or user designates it as a put according to the rule given in this chapter). For each outcome at $\mathrm{t}$, there are two possible outcomes at $\mathrm{T}, \mathrm{S}_{\mathrm{T}} \geq \mathrm{X}$ or $\mathrm{S}_{\mathrm{T}}<\mathrm{X}$. Explain why a chooser option is less expensive than a straddle.

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01:07

Problem 4

Explain why an interest-only (IO) mortgage strip has a value that is extremely volatile with respect to interest rates. What two factors determine its value?

Nick Johnson

Numerade Educator

00:14

Problem 5

Explain the difference between path-dependent options and path-independent options and give examples of each.

Becky Rahm

Numerade Educator

00:58

Problem 6

Give an example of a situation in which someone might wish to use a barrier option.

Bailey Latka

Numerade Educator

02:38

Problem 7

Explain how weather derivatives could be used by an electric utility to manage the risk associated with power consumption as affected by the weather.

Karan Soni

Numerade Educator

Problem 8

In modern financial derivatives markets, there are many exotic options. Briefly explain compound options, multi-asset options, shout options, and forward start options.

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05:04

Problem 9

On July 5 a market index is at 492.54 . You hold a portfolio that duplicates the index and is worth 20,500 times the index. You wish to insure the portfolio at a particular value over the period until September 20 . You can buy risk-free debt maturing on September 20 with a face value of $$\$ 100$$ for $$\$ 98.78$$.
a. You plan to use puts, which are selling for $$\$ 23.72$$ and have an exercise price of 510. Determine the appropriate number of puts and shares to hold. What is the insured value of the portfolio?
b. Determine the value of the portolio if the index on September 20 is at 507.35.
c. Determine the value of the portfolio if the index on September 20 is at 515.75. Compute the upside capture and the cost of the insurance.

James Kiss

Numerade Educator

Problem 10

Use the information in problem 9 to set up a dynamic hedge using stock index futures. Assume a multiplier of 500. The futures price is 496.29 . The volatility is 17.5 percent. The continuously compounded risk-free rate is 3.6 percent, and the call delta is 0.3826 . Let the stock price increase by $$\$ 1$$, and show that the change in the portfolio value is almost the same as it would have been had a put been used.

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Problem 11

Determine the price of an average price Asian call option. Use an exercise price of 95. Count the current price in determining the average. Comment on whether you would expect a standard European call to have a lower or higher price.

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Problem 12

Determine the prices of lookback and modified lookback calls and puts. For the modified lookbacks, use an exercise price of 95.

Victor Salazar

Numerade Educator

Problem 13

Determine the prices of the following barrier options.
a. A down-and-out call with the barrier at 90 and the exercise price at 95
b. An up-and-out put with the barrier at 110 and the exercise price at 105
c. Select any other barrier option but base your selection on the following instructions: Calculate the value of your selected barrier option and use it with the results you obtained in part $a$ or $b$ to determine the price of a standard European call or put. Then calculate the actual value of the European call or put and compare that answer with your answer obtained from the barrier options. Explain why this result is obtained.

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01:17

Problem 14

A portfolio manager is interested in purchasing an instrument with a call optionlike payoff but does not want to have to pay money up front. The manager learns from a banker that one can do this by entering into a break forward contract. The manager wants to learn if the banker is quoting a fair price. The stock price is 437.55. The contract expires in 270 days. The volatility is 18 percent and the continuously compounded risk-free rate is 3.75 percent. The exercise price will be set at the forward price of the stock.
a. Determine the exercise price.
b. The loan implicit in the break forward contract will have a face value of 40.19 . Determine if this is a fair amount by using your answer in $a$ and computing the value of $K$
c. Regardless of whether the break forward is found to be fairly priced, determine the value of the position if the stock price ends up at 465 and at 425 .

Breanna Ollech

Numerade Educator

Problem 15

Consider a stock priced at 100 with a volatility of 25 percent. The continuousiy compounded risk-free rate is 5 percent. Answer the following questions about various options, all of which have an original maturity of one year.
a. Find the premium on an at-the-money pay-later call option. Then determine the market value of the option nine months later if the stock is at 110.
b. Find the value of $F$ and $K$ on a break forward contract. Then determine the market value of the break forward nine months later if the stock is at 110 .
c. Find the premium on an at-the-money contingent-pay call option. Then determine the market value of the option nine months later if the stock is at 110.

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Problem 16

A stock is priced at 125.37 , the continuously compounded risk-free rate is 4.4 percent, and the volatility is 21 percent. There are no dividends. Answer the following questions.
a. Determine a fair price for a two-year asset-or-nothing option with exercise price of 120.
b. Assuming you purchased the asset-or-nothing option at the price you determined in $a$, calculate your profir if the asset price at expiration is (1) 138 and (2) 114.
c. Determine a fair price for a two-year cash-or-nothing option with exercise price of 120 that pays 120 if it expires in-the-moncy.
d. Assuming you purchased the cash-or-nothing option at the price you determined in c, calculate your profit if the asset price at expiration is (1) 138 or (2) 114.

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03:53

Problem 17

Consider a 10-year, fixed-rate mortgage of $$\$ 500,000$$ that has an interest rate of 12 percent. For simplification assume that payments are made annually.
a. Determine the amortization schedule.
b. Using your answer in $a$, determine the value of both IO and PO strips with a discount rate of 10 percent under the assumption that the mortgage will not be prepaid.
c. Now recompute the values of the IO and $\mathrm{PO}$ under the assumption that interest rates immediately fall to 8 percent and the mortgage is prepaid in year 6 . d. Explain the risk characteristics of $\mathrm{IO}$ and $\mathrm{PO}$ strips.

Aman Gupta

Numerade Educator

Problem 18

An investment manager expects a stock to be quite volatile and is considering the purchase of either a straddle or a chooser option. The stock is priced at 44, the exercise price is 40 , the continuously compounded risk-free rate is 5.2 percent, and the volatility is 51 percent. The options expire in 194 days. The chooser option must be declared a call or a put exactly 90 days before expiration.
a. Determine the prices of the straddle and the chooser.
b. Suppose at 90 days before expiration, the stock is at 28 . Find the value of the chooser option at expiration if the stock price ends up at 50 and at 30 .
c. Suppose at 90 days before expiration, the stock is at 60 . Find the value of the chooser option at expiration if the stock price ends up at 50 and at 30 .
d. Compare your answers in $c$ and $d$ to the performance of the straddle.

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Problem 19

Suppose FRM, Inc. issued a zero-coupon, equity index-linked note with a five-year maturity. The par value is $$\$ 1,000$$ and the coupon payment is stated as $75 \%$ of the equity index return or as zero. Calculate the cash flow at maturity assuming the equity index appreciates by $30 \%$ over this five-year period.

Rashmi Sinha

Numerade Educator

Problem 20

(Concept Problem) Suppose you are asked to assist in the design of an equitylinked security. The instrument is a five-year zero coupon bond with a guaranteed return of 1 percent, compounded annually. At the end of five years the bond will pay an additional return based on any appreciation of the Nikkei 300 stock index, a measure of the performance of 300 Japanese stocks. The risk-free rate is 5.5 percent, compounded annually, and the volatility of the index is 15 percent. In addition the index pays a dividend of 1.7 percent continuously compounded. Presently the index is at 315.55 and the additional return is based on appreciation above the current level of the index. You expect to sell these bonds in minimum increments of $$\$ 100$$. Overall you expect to sell $$\$ 10$$ million of these securities. Your firm has determined that it needs a margin of $$\$ 175,000$$ in cash today to cover costs and earn a reasonable profit. Determine the percentage of the Nikkei return that your firm should offer to cover its costs. Your firm would then set the percentage offered at less than this. If your firm sells this security, comment on the risk it creates for itself and suggest how it might deal with that risk.

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Problem 21

(Concept Problem) A convertible bond is a bond that permits the holder to turn in the bond and convert it into a certain number of shares of stock. Conversion would, thus, occur only when the stock does well. As a result of the option to convert the bond to stock, the coupon rate on the bond is lower than it otherwise would be. A new type of financial instrument, the reverse convertible, pays a higher-than-normal coupon, but the principal payoff can be reduced if the stock falls. Let us specify that the principal payoff of the reverse convertible is $\mathrm{FV}$, the face value, if $S_{\mathrm{T}}>\mathrm{S}_0$ where $\mathrm{S}_0$ is the stock price when the bond is issued. If $\mathrm{S}_{\mathrm{T}} \leq \mathrm{S}_0$, the principal payoff is $\mathrm{FV}\left(\mathrm{S}_{\mathrm{T}} / \mathrm{S}_0\right)$. Thus, for example, if the stock falls by 10 percent, $\mathrm{S}_{\mathrm{T}} / \mathrm{S}_0$, the principal payoff, is $0.9 \mathrm{FV}$. Show that this payoff ( $\mathrm{FV}$ if $\mathrm{S}_T>\mathrm{S}_0$, and $\mathrm{FV}\left(\mathrm{S}_{\mathrm{T}} / \mathrm{S}_0\right)$ if $\left.\mathrm{S}_{\mathrm{T}} \leq \mathrm{S}_0\right)$ is equivalent to a combination of an ordinary bond and a certain number of European puts with an exercise price of $S_0$. Determine how many puts you would need.

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