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 are the possible arithmetic and geometric averages after 1 year? b. Construct a binomial tree for the average. How many nodes does it have after 1 year? (Hint: While the moves ud and du give the same year-1 price, they do not give the same average in year 1.) c. What is the price of an Asian arithmetic average price call? d. What is the price of an Asian geometric average price call?
> Repeat the previous problem, but set φ = 0.05. Be sure that you simulate the risk neutral process, obtained by including the risk premium in the interest rate process. Repeat the previous problem Using Monte Carlo, simulate the process dr = a(b − r)dt +
> Suppose the firm issues a single zero-coupon bond with maturity value $100. a. Compute the yield, probability of default, and expected loss given default for times to maturity of 1, 2, 3, 4, 5, 10, and 20 years. b. For each time to maturity compute the a
> Suppose you short the S&R index for $1000 and buy a 1050-strike call. Construct payoff and profit diagrams for this position. Verify that you obtain the same payoff and profit diagram by borrowing $1029.41 and buying a 1050-strike put.
> a. What is the 2-year forward price for a 1-year bond? b. What is the price of a call option that expires in 2 years, giving you the right to pay $0.90 to buy a bond expiring in 1 year? c. What is the price of an otherwise identical put? d. What is the p
> Suppose that S1 and S2 follow geometric Brownian motion and pay continuous proportional dividends at the rates δ1 and δ2. Use the martingale argument to show that the value of a claim paying S1(T ) if S1(T) > KS2(T ) is where and δ1 and δ2 are the div
> Compute daily volatilities for 1991 through 2004 for IBM, Xerox, and the S&P 500 index. Annualize by multiplying by √252. How do your answers compare to those in Problem 24.1? Answer Problem 24.1 Here are the results for all 14 sample years. SP500 IB
> A barrier COD option is like a COD except that payment for the option occurs whenever a barrier is struck. Price a barrier COD put for the same values as in the previous problem, with a barrier of $95 and a strike of $90. Compute the delta and gamma for
> Use a change of numeraire and measure to verify that the value of a claim paying K if ST
> Verify that you earn the same profit and payoff by (a) shorting the S&R index for $1000 and (b) selling a 1050-strike S&R call, buying a 1050-strike put, and borrowing $1029.41.
> Verify that ASaeγt satisfies the Black-Scholes PDE for
> Verify that you earn the same profit and payoff by (a) buying the S&R index for $1000 and (b) buying a 950-strike S&R call, selling a 950-strike S&R put, and lending $931.37.
> Use Itˆo’s Lemma to evaluate dS2. For the following four problems, use Itˆo’s Lemma to determine the process followed by the specified equation, assuming that S(t) follows (a) arithmetic Brownian
> Suppose you invest in the S&R index for $1000, buy a 950-strike put, and sell a 1050- strike call. Draw a profit diagram for this position. What is the net option premium? If you wanted to construct a zero-cost collar keeping the put strike equal to $950
> Assume r = 8%, σ = 30%, δ = 0. Using 1-year-to-expiration European options, construct a position where you sell two 80-strike puts, buy one 95-strike put, buy one 105-strike call, and sell two 120-strike calls. For a range of stock prices from $60 to $14
> Construct payoff and profit diagrams for the purchase of a 950-strike S&R call and sale of a 1000-strike S&R call. Verify that you obtain exactly the same profit diagram for the purchase of a 950-strike S&R put and sale of a 1000-strike S&R put. What is
> A mine costing $1000 will produce 1 ounce of gold per year forever at a marginal extraction cost of $250, with production commencing 1 year after the mine opens. Gold volatility is zero. What is the value of the mine?
> Let ui∼ U (0, 1). Compute − 6, 1000 times. (This will use 12,000 random numbers.) Construct a histogram and compare it to a theoretical standard normal density. What are the mean and standard deviation? (This is a way to compute a random approximately
> You draw these five numbers from a standard normal distribution: {−1.7, 0.55, −0.3, −0.02, .85}. What are the equivalent draws from a normal distribution with mean 0.8 and variance 25?
> Assuming that the stock price satisfies equation (20.20), verify that Ke−r(T−t) +S(t)e−δ(T−t) satisfies the Black-Scholes equation, where K is a constant. What is the boundary condition for which this is a solution?
> The premium of a 100-strike yen-denominated put on the euro is ¥8.763. The current exchange rate is 95 ¥/=C. What is the strike of the corresponding euro-denominated yen call, and what is its premium?
> The price of a 6-month dollar-denominated call option on the euro with a $0.90 strike is $0.0404. The price of an otherwise equivalent put option is $0.0141. The annual continuously compounded dollar interest rate is 5%. a. What is the 6-month dollar-eur
> Let S = $100,K = $95, r = 8%, T = 0.5, and δ = 0. Let u = 1.3, d = 0.8, and n = 1. a. Verify that the price of a European put is $7.471. b. Suppose you observe a put price of $8. What is the arbitrage? c. Suppose you observe a put price of $6. What is th
> You have a project costing $1.50 that will produce two widgets, one each the first and second years after project completion. Widgets today cost $0.80 each, with the price growing at 2% per year. The effective annual interest rate is 5%. When will you in
> Suppose the gold spot price is $300/oz, the 1-year forward price is 310.686, and the continuously compounded risk-free rate is 5%. a. What is the lease rate? b. What is the return on a cash-and-carry in which gold is not loaned? c. What is the return on
> Suppose that copper costs $3.00 today and the continuously compounded lease rate for copper is 5%. The continuously compounded interest rate is 10%. The copper price in 1 year is uncertain and copper can be stored costlessly. a. If you short-sell a poun
> There is a single debt issue. Compute the yield on this debt assuming that it matures in 1 year and has a maturity value of $127.42, 2 years with a maturity value of $135.30, 5 years with a maturity value of $161.98, or 10 years with a maturity value of
> Suppose the effective semiannual interest rate is 3%. a. What is the price of a bond that pays one unit of the S&P index in 3 years? b. What semiannual dollar coupon is required if the bond is to sell at par? c. What semiannual payment of fractional unit
> Suppose you observe the prices {5, 4, 5, 6, 5}. What are the arithmetic and geometric averages? Now you observe {3, 4, 5, 6, 7}. What are the two averages? What happens to the difference between the two measures of the average as the standard deviation o
> Suppose you sell a 40-strike put 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 price tomorrow is $39? What if it is $40.50?
> Suppose that top executives of XYZ are told they will receive at-the-money call options on 10,000 shares each year for the next 3 years. When granted, the options have 5 years to maturity. XYZ’s stock price is $100, volatility is 30%, a
> a. Suppose the March Year 1 forward price were $3.10. Describe two different transactions you could use to undertake arbitrage. b. Suppose the September Year 1 forward price fell to $2.70 and subsequent forward prices fell in such a way that there is no
> Let S = $100,K = $95, r = 8%, T = 0.5, and δ = 0. Let u = 1.3, d = 0.8, and n = 1. a. Verify that the price of a European call is $16.196. b. Suppose you observe a call price of $17. What is the arbitrage? c. Suppose you observe a call price of $15.50. W
> Supposing the effective quarterly interest rate is 1.5%, what are the per-barrel swap prices for 4-quarter and 8-quarter oil swaps? (Use oil forward prices in Table 8.9.) What is the total cost of prepaid 4- and 8-quarter swaps? TABLE 8.9 Quarter 3 4
> Using Table 6.6, what is your best guess about the current price of gold per ounce? TABLE 6.6 Gold forward and prepaid forward prices on 1 day for gold delivered at 1-year intervals, out to 6 years. The continuously compounded interest rate is 6% an
> Suppose that oil forward prices for 1 year, 2 years, and 3 years are $20, $21, and $22. The 1-year effective annual interest rate is 6.0%, the 2-year interest rate is 6.5%, and the 3-year interest rate is 7.0%. a. What is the 3-year swap price? b. What i
> Suppose you short the S&R index for $1000 and buy a 950-strike call. Construct payoff and profit diagrams for this position. Verify that you obtain the same payoff and profit diagram by borrowing $931.37 and buying a 950-strike put.
> Using the information in the previous problem, find the price of a 5-year coupon bond that has a par payment of $1,000.00 and annual coupon payments of $60.00.
> 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
> 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
> 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 stri
> 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
