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, in what direction would you have to change the call strike?
> Given a continuously compounded risk-free rate of 3% annually, at what lease rate will forward prices equal the current commodity price? (Recall the copper example in Section 6.3.) If the lease rate were 3.5%, would there be contango or backwardation?
> Suppose you buy theS&Rindex for $1000 and buy a 950-strike put. Construct payoff and profit diagrams for this position. Verify that you obtain the same payoff and profit diagram by investing $931.37 in zero-coupon bonds and buying a 950-strike call.
> The current price of oil is $32.00 per barrel. Forward prices for 3, 6, 9, and 12 months are $31.37, $30.75, $30.14, and $29.54. Assuming a 2% continuously compounded annual risk-free rate, what is the annualized lease rate for each maturity? Is this an
> Suppose that you short the S&R index for $1000 and sell a 1000-strike put. Construct a table mimicking Table 3.1 that summarizes the payoff and profit of this position. Verify that your table matches Figure 3.5. For the following problems assume th
> Repeat the previous problem, only use Monte Carlo simulation. Repeat the previous problem Using the delta-approximation method and assuming a $10m investment in stock A, compute the 95% and 99% 1-, 10-, and 20-day VaRs for a position consisting of stock
> Use Itˆo’s Lemma to evaluate d[ln(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) arithmetic Brow
> Suppose that you buy the S&R index for $1000, buy a 1000-strike put, and borrow $980.39. Perform a payoff and profit calculation mimicking Table 3.1. Graph the resulting payoff and profit diagrams for the combined position. TABLE 3.1 Payoff and
> Suppose the firm issues a single zero-coupon bond. a. Suppose the maturity value of the bond is $80. Compute the yield and default probability for times to maturity of 1, 2, 3, 4, 5, 10, and 20 years. b. Repeat part (a), only supposing the maturity value
> Using the same information as the previous question, draw payoff and profit diagrams for a short position in the stock. Verify that profit is 0 at a price in 1 year of $55.
> If XYZ does nothing to manage copper price risk, what is its profit 1 year from now, per pound of copper? If on the other hand XYZ sells forward its expected copper production, what is its estimated profit 1 year from now? Construct graphs illustrating b
> Suppose you buy a 40–45 bull spread with 91 days to expiration. If you delta-hedge this position, what investment is required? What is your overnight profit if the stock tomorrow is $39? What if the stock is $40.50?
> Replicate the GARCH(1,1) estimation in Example 24.2, using daily returns from on IBM from January 1999 to December 2003. Compare your estimates with and without the four largest returns. Example 24.2 Estimating a GARCH(1,1) model for IBM using daily
> Consider production ratios of 2:1:1, 3:2:1, and 5:3:2 for oil, gasoline, and heating oil. Assume that other costs are the same per gallon of processed oil. a. Which ratio maximizes the per-gallon profit if oil costs $80/barrel, gasoline is $2/gallon, and
> Using Monte Carlo, simulate the process dr = a(b − r)dt + σdZ, 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 Vasicek formula.
> Assume that S = $45, K = $40, r = 0.05, δ = 0.02, and σ = 0.30. Using the up rebate formula (equation (23.21)), find the value of H that maximizes (H − K) × UR(S, σ, r , T , δ), for T = 1, 10, 100, 1000, and 10,000. Compare both H and (H − K) × UR to the
> Under the same assumptions as the previous problem, show that the value of a claim paying is where σ2, δ1, and δ2 are defined as in the previous problem. In the next set of problems you will use Monte Carlo valuation. Assume that S0 = $41, K = $40, P0
> Verify that S(t)e−δ(T−t)N(d1) satisfies the Black-Scholes equation.
> Heating degree-day and cooling degree-day futures contracts make payments based on whether the temperature is abnormally hot or cold. Explain why the following businesses might be interested in such a contract: a. Soft-drink manufacturers. b. Ski-resort
> Swaps often contain caps or floors. In this problem, you are to construct an oil contract that has the following characteristics: The initial cost is zero. Then in each period, the buyer pays the market price of oil if it is between K1 and K2; otherwise,
> Again consider the widget investment problem in Section 17.1. Verify that with S = $50, K = $30, r = 0.04879, σ = 0, and δ = 0.009569, the perpetual call price is $30.597 and exercise optimally occurs when the present value of cash flows is $152.957. Wha
> If XYZ does nothing to manage copper price risk, what is its profit 1 year from now, per pound of copper? If on the other hand XYZ sells forward its expected copper production, what is its estimated profit 1 year from now? Construct graphs illustrating b
> Suppose you invest in the S&R index for $1000, buy a 950-strike put, and sell a 1107- strike call. Draw a profit diagram for this position. How close is this to a zero-cost collar?
> Repeat the previous problem for debt instead of equity. Previous Problem Suppose there is a single 5-year zero-coupon debt issue with a maturity value of $120. The expected return on assets is 12%. What is the expected return on equity? The volatility of
> Consider the example of Auric. a. Suppose that Auric insures against a price increase by purchasing a 440-strike call. Verify by drawing a profit diagram that simultaneously selling a 400- strike put will generate a collar. What is the cost of this colla
> Suppose you enter into a put ratio spread where you buy a 45-strike put and sell two 40-strike puts, both 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-s
> The exchange rate is ¥95/=C, the yen-denominated interest rate is 1.5%, the euro-denominated interest rate is 3.5%, and the exchange rate volatility is 10%. a. What is the price of a 90-strike yen-denominated euro put with 6 months to expiration? b. What
> A $50 stock pays an 8% continuous dividend. The continuously compounded risk free rate is 6%. a. What is the price of a prepaid forward contract that expires 1 year from today? b. What is the price of a forward contract that expires at the same time?
> Compute estimated profit in 1 year if XYZ buys a put option with a strike of $0.95, $1.00, or $1.05. Draw a graph of profit in each case.
> Suppose you desire to short-sell 400 shares of JKI stock, which has a bid price of $25.12 and an ask price of $25.31. You cover the short position 180 days later when the bid price is $22.87 and the ask price is $23.06. a. Taking into account only the bi
> What position is the opposite of a purchased call? The opposite of a purchased put?
> a. Suppose you enter into a short 6-month forward position at a forward price of $50. What is the payoff in 6 months for prices of $40, $45, $50, $55, and $60? b. Suppose you buy a 6-month put option with a strike price of $50. What is the payoff in 6 mo
> Obtain at least 5 years of daily data for at least three stocks and, if you can, one currency. Estimate annual volatility for each year for each asset in your data. What do you observe about the pattern of historical volatility over time? Does historical
> Using the base case parameters, plot the implied volatility curve you obtain for the base case against that for the case where there is a jump to zero, with the same λ.
> Consider the bonds in Example 7.8. What hedge ratio would have exactly hedged the portfolio if interest rates had decreased by 25 basis points? Increased by 25 basis points? Repeat assuming a 50-basis-point change. Example 7.8 Suppose we own a 7-yea
> Repeat the previous problem supposing that the brokerage fee is quoted as 0.3% of the bid or ask price. Previous Problem ABC stock has a bid price of $40.95 and an ask price of $41.05. Assume there is a $20 brokerage commission. a. What amount will you
> Construct Table 5.1 from the perspective of a seller, providing a descriptive name for each of the transactions. TABLE 5.1 Four different ways to buy a share of stock that has price So at time 0. At time 0 you agree to a price, which is paid either
> Suppose the businesses in the previous problem use futures contracts to hedge their temperature-related risk. Who do you think might accept the opposite risk?
> 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
> 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
> 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.
