Repeat the previous problem calculating prices for American options instead of European. What happens? Previous Problem For a stock index, S = $100, σ = 30%, r = 5%, δ = 3%, and T = 3. Let n = 3. a. What is the price of a European call option with a strike of $95? b. What is the price of a European put option with a strike of $95? c. Now let S = $95, K = $100, σ = 30%, r = 3%, and δ = 5%. (You have exchanged values for the stock price and strike price and for the interest rate and dividend yield.) Value both options again. What do you notice?
> Verify that equation (21.12) satisfies the Black-Scholes equation. What is the boundary condition for which this is a solution? v't, T) = e=r(T-1) (21.12)
> A $50 stock pays a $1 dividend every 3 months, with the first dividend coming 3 months from today. The continuously compounded risk-free rate is 6%. a. What is the price of a prepaid forward contract that expires 1 year from today, immediately after the
> Consider a perpetual call option with S = $50, K = $60, r = 0.06, σ = 0.40, and δ = 0.03. a. What is the price of the option and at what stock price should it be exercised? b. Suppose δ = 0.04 with all other inputs the same. What happens to the price and
> Consider AAAPI, the Nikkei ADR in disguise. To answer this question, use the information in Table 23.4. a. What is the volatility of Y, the price of AAAPI? b. What is the covariance between Y and x, the dollar-yen exchange rate? c. What is the correlati
> Suppose the 1-year copper forward price were $0.80 instead of $1. If XYZ were to sell forward its expected copper production, what is its estimated profit 1 year from now? Should XYZ produce copper? What if the forward copper price is $0.45?
> Consider the same 3-year swap. Suppose you are a dealer who is paying the fixed oil price and receiving the floating price. Suppose that you enter into the swap and immediately thereafter all interest rates rise 50 basis points (oil forward prices are un
> Use a spreadsheet to verify the option prices in Examples 12.1 and 12.2. Example 12.1 Let S = $41, K = $40, o = 0.3, r=8%, T = 0.25 (3 months), and 8 = 0. Computing the Black-Scholes call price, we obtain' In() + (0.08 – 0 + 0) x 0.25 0.3/0.25 $41 x
> Suppose you observe the following par coupon bond yields: 0.03000 (1-year), 0.03491 (2-year), 0.03974 (3-year), 0.04629 (4-year), 0.05174 (5-year). For each maturity year compute the zero-coupon bond prices, effective annual and continuously compounded z
> Let S = $100, K = $120, σ = 30%, r = 0.08, and δ = 0. a. Compute the Black-Scholes call price for 1 year to maturity and for a variety of very long times to maturity. What happens to the option price as T →∞? b. Set δ = 0.001. Repeat (a). Now what happen
> Suppose call and put prices are given by What no-arbitrage property is violated? What spread position would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage. Strike 50 55 Call premium 9. 10 Put premium 6
> Let S = $40, K = $45, σ = 0.30, r = 0.08, T = 1, and δ = 0. a. What is the price of a standard call? b. What is the price of a knock-in call with a barrier of $44? Why? c. What is the price of a knock-out call with a barrier of $44? Why?
> Suppose XYZ is a non-dividend-paying stock. Suppose S = $100, σ = 40%, δ = 0, and r = 0.06. a. What is the price of a 105-strike call option with 1 year to expiration? b. What is the 1-year forward price for the stock? c. What is the price of a 1-year 10
> Suppose that S = $100, K = $100, r = 0.08, σ = 0.30, δ = 0, and T = 1. Construct a standard two-period binomial stock price tree using the method in Chapter 10. a. Consider stock price averages computed by averaging the 6-month and 1-year prices. What ar
> Repeat Problem 11.4, only set r = 0 and δ = 0.08. What is the lowest strike price (if there is one) at which early exercise will occur? If early exercise never occurs, explain why not. For the following problems, note that the BinomCall and BinomPut func
> Consider a one-period binomial model with h = 1, where S = $100, r = 0, σ = 30%, and δ = 0.08. Compute American call option prices for K = $70, $80, $90, and $100. a. At which strike(s) does early exercise occur? b. Use put-call parity to explain why ear
> A lender plans to invest $100m for 150 days, 60 days from today. (That is, if today is day 0, the loan will be initiated on day 60 and will mature on day 210.) The implied forward rate over 150 days, and hence the rate on a 150-day FRA, is 2.5%. The actu
> Let S = $100, K = $95, σ = 30%, r = 8%, T = 1, and δ = 0. Let u = 1.3, d = 0.8, and n = 2. Construct the binomial tree for a call option. At each node provide the premium, ∆, and B.
> Suppose you observe the following zero-coupon bond prices per $1 of maturity payment: 0.96154 (1-year), 0.91573 (2-year), 0.87630 (3-year), 0.82270 (4-year), 0.77611 (5-year). For each maturity year compute the zero-coupon bond yields (effective annual a
> Suppose a security has a bid price of $100 and an ask price of $100.12. At what price can the market-maker purchase a security? At what price can a market-maker sell a security? What is the spread in dollar terms when 100 shares are traded?
> Let S = $40, σ = 0.30, r = 0.08, T = 1, and δ = 0. Also let Q = $40, σQ = 0.30, δQ = 0, and ρ = 1. Consider an exchange call with S as the price of the underlying asset and Q as the price of the strike asset. a. What is the price of an exchange call with
> Suppose the yield curve is flat at 6%. Consider a 4-year 5%-coupon bond and an 8-year 7%-coupon bond. All coupons are annual. a. What are the prices and durations of both bonds? b. Consider buying one 4-year bond and duration-hedging by selling an approp
> a. What are some possible explanations for the shape of this forward curve? b. What annualized rate of return do you earn on a cash-and-carry entered into in December of Year 0 and closed in March of Year 1? Is your answer sensible? c. What annualized ra
> Suppose the spot $/¥ exchange rate is 0.008, the 1-year continuously compounded dollar-denominated rate is 5% and the 1-year continuously compounded yen-denominated rate is 1%. Suppose the 1-year forward exchange rate is 0.0084. Explain precisely the tra
> Suppose that firms face a 40% income tax rate on positive profits and that net losses receive no credit. (Thus, if profits are positive, after-tax income is (1− 0.4) × profit, while if there is a loss, after-tax income is the amount lost.) Firms A and B
> A default-free zero-coupon bond costs $91 and will pay $100 at maturity in 1 year. What is the effective annual interest rate? What is the payoff diagram for the bond? The profit diagram?
> Suppose a 10-year zero-coupon bond with a face value of $100 trades at $69.20205. a. What is the yield to maturity and modified duration of the zero-coupon bond? b. Calculate the approximate bond price change for a 50-basis-point increase in the yield, b
> Suppose S = $100, K = $95, r = 8% (continuously compounded), t = 1, σ = 30%, and δ = 5%. Explicitly construct an eight-period binomial tree using the lognormal expressions for u and d: Compute the prices of European and American calls and puts.
> Compute profit diagrams for the following ratio spreads: a. Buy 950-strike call, sell two 1050-strike calls. b. Buy two 950-strike calls, sell three 1050-strike calls. c. Consider buying n 950-strike calls and selling m 1050-strike calls so that the prem
> Using the information in Table 4.9 about Scenario C: a. What is the expected quantity of production? b. Suppose you short the expected quantity of corn. What is the standard deviation of hedged revenue? TABLE 4.9 Three scenarios illustrating differ
> Using the information in Table 4.9 about Scenario C: a. Using your answer to the previous question, use equation (4.7) to compute the variance-minimizing hedge ratio. b. Run a regression of revenue on price to compute the variance-minimizing hedge ratio
> Four years after the option grant, the stock price for Analog Devices was about $40. Using the same input as in the previous problem, compute the market value of the options granted in 2000, assuming that they were issued at strikes of $44.50 and $63.25.
> Construct a four-period, three-step (eight terminal node) binomial interest rate tree where the initial interest rate is 10% and rates can move up or down by 2%; model your tree after that in Figure 25.3. Compute prices and yields for 1-, 2-, 3-, and 4-y
> Value the M&I stock purchase contract assuming that the 3-year interest rate is 3% and the M&I volatility is 15%. How does your answer change if volatility is 35%?
> Suppose you have a project that will produce a single widget. Widgets today cost $1 and the project costs $0.90. The risk-free rate is 5%. Under what circumstances would you invest immediately in the project? What conditions would lead you to delay the p
> Construct a spreadsheet for which you can input up to five strike prices and quantities of put and call options bought or sold at those strikes, and which will automatically construct the total expiration payoff diagram for that position. Modify the spre
> a. What is the 1-year bond forward price in year 1? b. What is the price of a call option that expires in 1 year, giving you the right to pay $0.9009 to buy a bond expiring in 1 year? c. What is the price of an otherwise identical put? d. What is the pri
> You have written a 35–40–45 butterfly spread with 91 days to expiration. Compute and graph the 1-day holding period profit if you delta- and gamma-hedge this position using the stock and a 40-strike call with 180 days to expiration.
> Suppose that u < e(r−δ)h. Show that there is an arbitrage opportunity. Now suppose that d >e(r−δ)h. Show again that there is an arbitrage opportunity.
> A 6-year bond with a 4% coupon sells for $102.46 with a 3.5384% yield. The conversion factor for the bond is 0.90046. An 8-year bond with 5.5% coupons sells for $113.564 with a conversion factor of 0.9686. (All coupon payments are semiannual.) Which bond
> Using the information in Table 4.9 about Scenario C: a. Compute total revenue when correlation between price and quantity is positive. b. What is the correlation between price and revenue? TABLE 4.9 Three scenarios illustrating different correlatio
> The strike price of a compensation option is generally set on the day the option is issued. On November 10, 2000, the CEO of Analog Devices, Jerald Fishman, received 600,000 options. The stock price was $44.50. Four days later, the price rose to $63.25 a
> Suppose that in order to hedge interest rate risk on your borrowing, you enter into an FRA that will guarantee a 6%effective annual interest rate for 1 year on $500,000.00. On the date you borrow the $500,000.00, the actual interest rate is 5%. Determine
> What are 95% and 99% 1-, 10-, and 20-dayVaRs for a portfolio that has $4m invested in stock A, $3.5m in stock B, and $2.5m in stock C?
> Suppose S (0) = $100, r = 0.06, σS= 0.4, and δ = 0. Use equation (20.32) to compute prices for claims that pay the following: a. S2 b. √S c. S−2 Compare your answers to the answers you obtained
> A stock purchase contract with a zero initial premium calls for you to pay for one share of stock in 3 years. The stock price is $100 and the 3-year interest rate is 3%. a. If you expect the stock to have a zero dividend yield, what price in 3 years woul
> Use the same inputs as in the previous problem, except that K = $1.00. a. What is the price of a 9-month European put? b. What is the price of a 9-month American put?
> Using weekly price data (constructed Wednesday to Wednesday), compute historical annual volatilities for IBM, Xerox, and the S&P 500 index for 1991 through 2004. Annualize your answer by multiplying by √52. Also compute volatility for each for the entire
> Suppose you are a market-maker in S&R index forward contracts. The S&R index spot price is 1100, the risk-free rate is 5%, and the dividend yield on the index is 0. a. What is the no-arbitrage forward price for delivery in 9 months? b. Suppose a customer
> You wish to insure a portfolio for 1 year. Suppose that S = $100, σ = 30%, r = 8%, and δ = 0. You are considering two strategies. The simple insurance strategy entails buying one put option with a 1-year maturity at a strike price that is 95% of the stoc
> An 8-year bond with 6% annual coupons and a 5.004% yield sells for $106.44 with a Macaulay duration of 6.631864. A 9-year bond has 7% annual coupons with a 5.252% yield and sells for $112.29 with a Macaulay duration of 7.098302. You wish 228 Chapter 7. I
> Compute estimated profit in 1 year if XYZ buys collars with the following strikes: a. $0.95 for the put and $1.00 for the call. b. $0.975 for the put and $1.025 for the call. c. $1.05 for the put and $1.05 for the call. Draw a graph of profit in each cas
> Using the information in Table 4.11, verify that a regression of revenue on price gives a regression slope coefficient of about 100,000. Table 4.11: TABLE 4.11 Results in Scenario B (negative correlation between the price of com and the quantity of
> Repeat Problem 17.18 assuming that the volatility of gold is 20% and that once opened, the mine can be costlessly shut down once, and then costlessly reopened once. What is the value of the mine? What are the prices at which the mine will be shut down an
> Firm A has a stock price of $40, and has made an offer for firm B where A promises to pay 1.5 shares for each share of B, as long as A’s stock price remains between $35 and $45. If the price of A is below $35, A will pay $52.50/share, and if the price of
> Use the Black-Scholes equation to verify the solution in Chapter 20, given by Proposition 20.3, for the value of a claim paying Sa. Z(T) – Z(0) = vT Y (ih) (20.3) i=l
> The S&R index spot price is 1100, the risk-free rate is 5%, and the continuous dividend yield on the index is 2%. a. Suppose you observe a 6-month forward price of 1120. What arbitrage would you undertake? b. Suppose you observe a 6-month forward price o
> Suppose that S = $50, K = $45, σ = 0.30, r = 0.08, and t = 1. The stock will pay a $4 dividend in exactly 3 months. Compute the price of European and American call options using a four-step binomial tree.
> A DECS contract pays two shares if ST < 27.875, 1.667 shares if the price is above ST > 33.45, and $27.875 and $55.75 otherwise. The quarterly dividend is $0.87. Value this DECS assuming that S = $26.70, σ = 35%, r = 9%, and T = 3.3 and that the underlyi
> Suppose that S = $100, σ = 30%, r = 8%, and δ = 0. Today you buy a contract which, 6 months from today, will give you one 3-month to expiration at-the-money call option. (This is called a forward start option.) Assume that r, σ, and δ are certain not to
> Consider a 5-year equity-linked note that pays one share of XYZ at maturity. The price of XYZ today is $100, and XYZ is expected to pay its annual dividend of $1 at the end of this year, increasing by $0.50 each year. The fifth dividend will be paid the
> Repeat the previous problem, but this time for perpetual options. What do you notice about the prices? What do you notice about the exercise barriers? Previous Problem Let S = $100, K = $90, σ = 30%, r = 8%, δ = 5%, and T = 1. a. What is the Black-Schole
> Suppose the yield curve is flat at 8%. Consider 3- and 6-year zero-coupon bonds. You buy one 3-year bond and sell an appropriate quantity of the 6-year bond to duration hedge the position. Any additional investment is in short-term (zero-duration) bonds.
> In this problem we consider whether parity is violated by any of the option prices in Table 9.1. Suppose that you buy at the ask and sell at the bid, and that your continuously compounded lending rate is 0.3% and your borrowing rate is 0.4%. Ignore tr
> For a stock index, S = $100, σ = 30%, r = 5%, δ = 3%, and T = 3. Let n = 3. a. What is the price of a European call option with a strike of $95? b. What is the price of a European put option with a strike of $95? c. Now let S = $95, K = $100, σ = 30%, r
> Consider the following two bonds which make semiannual coupon payments: a 20- year bond with a 6% coupon and 20% yield, and a 30-year bond with a 6% coupon and a 20% yield. a. For each bond, compute the price value of a basis point. b. For each bond, com
> Here is a quote from an investment website about an investment strategy using options: One strategy investors are applying to the XYZ options is using “synthetic stock.”Asynthetic stock is created when an investor simultaneously purchases a call option a
> Let S = $40, σ = 0.30, r = 0.08, T = 1, and δ = 0. Also let Q = $60, σQ= 0.50,δQ= 0, and ρ = 0.5. In this problem we will compute prices of exchange calls with S as the price of the underlying asset and Q as the price of the strike asset. a. Vary δ from
> A collect-on-delivery call (COD) costs zero initially, with the payoff at expiration being 0 if S
> Use the same assumptions as in the preceding problem, without the bid-ask spread. Suppose that we want to construct a paylater strategy using a ratio spread. Instead of buying a 440-strike call, Auric will sell one 440-strike call and use the premium to
> Using Monte Carlo, simulate the process dr = a(b − r)dt + σ, assuming that r = 6%, a = 0.2, b = 0.08, φ = 0 and σ = 0.02. Compute the prices of 1-, 2-, and 3-year zero coupon bonds, and verify that your answers match those of the Cox- Ingersoll-Ross for
> Compute estimated profit in 1 year if Telco buys a call option with a strike of $0.95, $1.00, or $1.05. Draw a graph of profit in each case.
> Using the CEV option pricing model, set β = 3 and generate option prices for strikes from 60 to 140, in increments of 5, for times to maturity of 0.25, 0.5, 1.0, and 2.0. Plot the resulting implied volatilities.
> Consider Example 6.1. Suppose the February forward price had been $2.80. What would the arbitrage be? Suppose it had been $2.65. What would the arbitrage be? In each case, specify the transactions and resulting cash flows in both November and February. W
> Repeat Problem 17.18 assuming that the volatility of gold is 20% and that once opened, the mine can be costlessly shut down forever. What is the value of the mine? What is the price at which the mine will be shut down? Repeat Problem 17.18 A mine costin
> Firm A has a stock price of $40 and has made an offer for firm B where A promises to pay $60/share for B, as long as A’s stock price remains between $35 and $45. If the price of A is below $35, A will pay 1.714 shares, and if the price of A is above $45,
> Consider again the Netscape PEPS discussed in this chapter and assume the following: the price of Netscape is $39.25, Netscape is not expected to pay dividends, the interest rate is 7%, and the 5-year volatility of Netscape is 40%. What is the theoretica
> A chooser option (also known as an as-you-like-it option) becomes a put or call at the discretion of the owner. For example, consider a chooser on the S&R index for which both the call, with value C (St , K, T − t), and the put, with value P(St , K, T −
> Repeat the previous problem, except that instead of hedging volatility risk, you wish to hedge interest rate risk, i.e., to rho-hedge. In addition to delta-, gamma-, and rho-hedging, can you delta-gamma-rho-vega hedge? Data from Previous Problem You hav
> You have been asked to construct an oil contract that has the following characteristics: The initial cost is zero. Then in each period, the buyer pays S −, with a cap of $21.90 − and a floor of $19.90 −. Assume oil volatility is 15%. What is?
> Use the same inputs as in the previous problem. Suppose that you observe a bid option price of $50 and an ask price of $50.10. a. Explain why you cannot compute an implied volatility for the bid price. b. Compute an implied volatility for the ask price,
> Using the information in Table 8.9, what are the euro-denominated fixed rates for 4- and 8-quarter swaps? TABLE 8.9 Quarter 1 2 3 4 5 6 7 8. Oil forward price 21 21.1 20.8 20.5 20.2 20 19.9 19.8 Gas swap price Zero-coupon bond price 0.9852 0.9701 0.
> Estimate a GARCH (1,1) for the S&P 500 index, using data from January 1999 to December 2003.
> Verify that the butterfly spread in Figure 3.14 can be duplicated by the following transactions (use the option prices in Table 3.4): a. Buy 35 call, sell two 40 calls, buy 45 call. b. Buy 35 put, sell two 40 puts, buy 45 put. c. Buy stock, buy 35 put,
> Let S = $100, K = $90, σ = 30%, r = 8%, δ = 5%, and T = 1. a. What is the Black-Scholes call price? b. Now price a put where S = $90, K = $100, σ = 30%, r = 5%, δ = 8%, and T = 1. c. What is the link between your answers to (a) and (b)? Why?
> Suppose XYZ stock pays no dividends and has a current price of $50. The forward price for delivery in one year is $53. If there is no advantage to buying either the stock or the forward contract, what is the 1-year effective interest rate?
> Suppose you sell a 45-strike call with 91 days to expiration. What is delta? If the option is on 100 shares, what investment is required for a delta-hedged portfolio? What is your overnight profit if the stock tomorrow is $39? What if the stock price is
> Suppose the S&P 500 futures price is 1000, σ = 30%, r = 5%, δ = 5%, T = 1, and n = 3. a. What are the prices of European calls and puts for K = $1000? Why do you find the prices to be equal? b. What are the prices of American calls and puts for K = $1000
> Compute Macaulay and modified durations for the following bonds: a. A 5-year bond paying annual coupons of 4.432% and selling at par. b. An 8-year bond paying semiannual coupons with a coupon rate of 8% and a yield of 7%. c. A10-year bond paying annual c
> Suppose we wish to borrow $10 million for 91 days beginning next June, and that the quoted Eurodollar futures price is 93.23. a. What 3-month LIBOR rate is implied by this price? b. How much will be needed to repay the loan?
> Let c be consumption. Under what conditions on the parameters λ0 and λ1 could the following functions serve as utility functions for a risk-averse investor? (Remember that marginal utility must be positive and the function must be concave.) a. U(c) = λ0
> Suppose that LMN Investment Bank wishes to sell Auric a zero-cost collar of width 30 without explicit premium (i.e., there will be no cash payment from Auric to LMN). Also suppose that on every option the bid price is $0.25 below the Black- Scholes price
> In this problem you will price various options with payoffs based on the Eurostoxx index and the dollar/euro exchange rate. Assume that Q= 2750 (the index), x = 1.25 ($/=C), s = 0.08 (the exchange rate volatility), σ = 0.2 (index volatility), r = 0.01(th
> Using the CEV option pricing model, set β = 1and generate option prices for strikes from 60 to 140, in increments of 5, for times to maturity of 0.25, 0.5, 1.0, and 2.0. Plot the resulting implied volatilities. (This should reproduce Figure 2
> Suppose that S = $100, σ = 30%, r = 6%, t = 1, and δ = 0. XYZ writes a European put option on one share with strike price K = $90. a. Construct a two-period binomial tree for the stock and price the put. Compute the replicating portfolio at each node. b.
> XYZ wants to hedge against depreciations of the euro and is also concerned about the price of oil, which is a significant component of XYZ’s costs. However, there is a positive correlation between the euro and the price of oil: The euro appreciates when
> You have purchased a 40-strike call with 91 days to expiration. You wish to delta-hedge, but you are also concerned about changes in volatility; thus, you want to vega-hedge your position as well. a. Compute and graph the 1-day holding period profit if y
> Use Itˆo’s Lemma to evaluate d(√S). For the following four problems, use Itˆo’s Lemma to determine the process followed by the specified equation, assuming that S(t) follows (a)
> Construct an asymmetric butterfly using the 950-, 1020-, and 1050-strike options. How many of each option do you hold? Draw a profit diagram for the position.
> Suppose that the exchange rate is 1 dollar for 120 yen. The dollar interest rate is 5% (continuously compounded) and the yen rate is 1% (continuously compounded). Consider an at-the-money American dollar call that is yen-denominated (i.e., the call permi
> Consider the June 165, 170, and 175 call option prices in Table 9.1. a. Does convexity hold if you buy a butterfly spread, buying at the ask price and selling at the bid? b. Does convexity hold if you sell a butterfly spread, buying at the ask price and
