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Derivatives Markets 4th edition Rober L. Macdonald - Solutions
A stock currently sells for $32.00. A 6-month call option with a strike of $35.00 has a premium of $2.27. Assuming a 4% continuously compounded risk-free rate and a 6% continuous dividend yield, what is the price of the associated put option? Discuss.
Suppose call and put prices are given byFind the convexity violations. What spread would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage.
Suppose call and put prices are given byFind the convexity violations. What spread would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage.
In each case identify the arbitrage and demonstrate how you would make money by creating a table showing your payoff.a. Consider two European options on the same stock with the same time to expiration. The 90-strike call costs $10 and the 95-strike call costs $4.b. Now suppose these options have 2
Suppose the interest rate is 0% and the stock of XYZ has a positive dividend yield. Is there any circumstance in which you would early-exercise an American XYZ call? Is there any circumstance in which you would early-exercise an American XYZ put? Explain. Discuss.
In the following, suppose that neither stock pays a dividend.a. Suppose you have a call option that permits you to receive one share of Apple by giving up one share of AOL. In what circumstance might you early exercise this call?b. Suppose you have a put option that permits you to give up one share
The price of a non-dividend-paying stock is $100 and the continuously compounded risk free rate is 5%. A 1-year European call option with a strike price of $100 × e0.05 × 1= $105.127 has a premium of $11.924. A 1 1/2 year European call option with a strike price of $100 × e0.05×1.5 = $107.788
Suppose that to buy either a call or a put option you pay the quoted ask price, denoted Ca(K, T) and Pa(K, T), and to sell an option you receive the bid, Cb(K, T) and Pb(K, T). Similarly, the ask and bid prices for the stock are Sa and Sb. Finally, suppose you can borrow at the rate rH and lend at
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 transaction costs on the stock, for which the
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 selling at the bid?c. Does convexity hold if
A stock currently sells for $32.00. A 6-month call option with a strike of $30.00 has a premium of $4.29, and a 6-month put with the same strike has a premium of $2.64. Assume a 4% continuously compounded risk-free rate. What is the present value of dividends payable over the next 6 months?
Suppose the S&R index is 800, the continuously compounded risk-free rate is 5%, and the dividend yield is 0%. A 1-year 815-strike European call costs $75 and a 1 year 815-strike European put costs $45. Consider the strategy of buying the stock, selling the 815-strike call, and buying the
Suppose the exchange rate is 0.95 $/=C, the euro-denominated continuously compounded interest rate is 4%, the dollar-denominated continuously compounded interest rate is 6%, and the price of a 1-year 0.93-strike European call on the euro is $0.0571. What is the price of a 0.93-strike European put?
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? Discuss.
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-euro forward price?b. If the euro-denominated
Suppose the dollar-denominated interest rate is 5%, the yen-denominated interest rate is 1% (both rates are continuously compounded), the spot exchange rate is 0.009 $/¥, and the price of a dollar-denominated European call to buy one yen with 1 year to expiration and a strike price of $0.009 is
Suppose call and put prices are given byWhat no-arbitrage property is violated? What spread positionwould you use to effect arbitrage? Demonstrate that the spread position is an arbitrage.
Suppose call and put prices are given byWhat no-arbitrage property is violated? What spread positionwould you use to effect arbitrage? Demonstrate that the spread position is an arbitrage.
Let S = $100, K = $105, r = 8%, T = 0.5, and δ = 0. Let u = 1.3, d = 0.8, and n = 1.a. What are the premium, Δ, and B for a European call?b. What are the premium, Δ, and B for a European put?
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 an American put option. At each node provide the premium, Δ, and B.
Suppose S0 = $100, K = $50, r = 7.696% (continuously compounded), δ = 0, and T = 1.a. Suppose that for h = 1, we have u = 1.2 and d = 1.05. What is the binomial option price for a call option that lives one period? Is there any problem with having d > 1?b. Suppose now that u = 1.4 and d = 0.6.
Let S = $100, K = $95, r = 8% (continuously compounded), σ = 30%, δ = 0, T = 1 year, and n = 3.a. Verify that the binomial option price for an American call option is $18.283. Verify that there is never early exercise; hence, a European call would have the same price.b. Show that the binomial
Repeat the previous problem assuming that the stock pays a continuous dividend of 8% per year (continuously compounded). Calculate the prices of the American and European puts and calls. Which options are early-exercised?In previous problemLet S = $100, K = $95, r = 8% (continuously compounded), σ
Let S = $40, K = $40, r = 8% (continuously compounded), σ = 30%, δ = 0, T = 0.5 year, and n = 2.a. Construct the binomial tree for the stock. What are u and d?b. Show that the call price is $4.110.c. Compute the prices of American and European puts.
Use the same data as in the previous problem only suppose that the call price is $5 instead of $4.110.In previous problem Let S = $40, K = $40, r = 8% (continuously compounded), σ = 30%, δ = 0, T = 0.5 year, and n = 2.a. At time 0, assume you write the option and form the replicating portfolio to
Suppose that the exchange rate is $0.92/ = C. Let r$ = 4%, and r = C = 3%, u = 1.2, d = 0.9, T = 0.75, n = 3, and K = $0.85.a. What is the price of a 9-month European call?b. What is the price of a 9-month American call?
Use the same inputs as in the previous problem, except that K = $1.00.In previous problem Suppose that the exchange rate is $0.92/ = C. Let r$ = 4%, and r = C = 3%, u = 1.2, d = 0.9, T = 0.75, n = 3, and K = $0.85.a. What is the price of a 9-month European put?b. What is the price of a 9-month
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 permits you to buy 1 dollar for 120 yen). The
An option has a gold futures contract as the underlying asset. The current 1-year gold futures price is $300/oz, the strike price is $290, the risk-free rate is 6%, volatility is 10%, and time to expiration is 1 year. Suppose n = 1. What is the price of a call option on gold? What is the
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. What is the arbitrage?
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?c. What are the time-0 replicating
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
Repeat the previous problem calculating prices for American options instead of European. What happens?In previous problem 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
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.
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 the arbitrage?
Obtain at least 5 years' worth of daily or weekly stock price data for a stock of your choice.1. Compute annual volatility using all the data.2. Compute annual volatility for each calendar year in your data. How does volatility vary over time?3. Compute annual volatility for the first and second
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 volatility move in tandem for different
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, Δ , and B.
Repeat the option price calculation in the previous question for stock prices of $80, $90, $110, $120, and $130, keeping everything else fixed. What happens to the initial option Δ as the stock price increases?In previous questionsLet S = $100, K = $95, σ = 30%, r = 8%, T = 1, and δ = 0. Let u =
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 European put option. At each node provide the premium, Δ, and B.
Repeat the option price calculation in the previous question for stock prices of $80, $90, $110, $120, and $130, keeping everything else fixed. What happens to the initial put Δ as the stock price increases?In previous question Let S = $100, K = $95, σ = 30%, r = 8%, T = 1, and δ = 0. Let u =
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 early exercise does not occur at the higher
Let S = $100, σ = 30%, r = 0.08, t = 1, and δ = 0. Suppose the true expected return on the stock is 15%. Set n = 10. Compute European put prices, Δ, and B for strikes of $70, $80, $90, $100, $110, $120, and $130. For each strike, compute the expected return on the option. What effect does the
Repeat the previous problem, except that for each strike price, compute the expected return on the option for times to expiration of 3 months, 6 months, 1 year, and 2 years. What effect does time to maturity have on the option's expected return?In previous problem Let S = $100, σ = 30%, r =
Let S = $100, σ = 0.30, r = 0.08, t = 1, and δ = 0. Using equation (11.12) to compute the probability of reaching a terminal node and Suidnˆ’i to compute the price at that node, plot the risk-neutral distribution of year-1 stock prices as in Figures 11.7 and 11.8 for n = 3 and n = 10.
Repeat the previous problem for n = 50. What is the risk-neutral probability that S1 < $80? S1 > $120?In previous problemLet S = $100, σ = 0.30, r = 0.08, t = 1, and δ = 0. Using equation (11.12) to compute the probability of reaching a terminal node and Suidn−i to compute the price at
We saw in Section 10.1 that the undiscounted risk-neutral expected stock price equals the forward price. We will verify this using the binomial tree in Figure 11.4.a. Using S = $100, r = 0.08, and δ = 0, what are the 4-month, 8-month, and 1-year forward prices?b. Verify your answers in (a) by
Compute the 1-year forward price using the 50-step binomial tree in Problem 11.13.
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 Cox-Ross Rubinstein expressions for u and d:Compute the prices of European and American calls and puts.
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.
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.
Repeat Problem 11.1, only assume that r = 0.08. What is the greatest strike price at which early exercise will occur? What condition related to put-call parity is satisfied at this strike price?
Repeat Problem 11.1, only assume that r = 0.08 and δ = 0.Will early exercise ever occur? Why?
Consider a one-period binomial model with h = 1, where S = $100, r = 0.08, σ = 30%, and δ = 0. Compute American put option prices for K = $100, $110, $120, and $130.a. At which strike(s) does early exercise occur?b. Use put-call parity to explain why early exercise does not occur at the other
Repeat Problem 11.4, only set δ = 0.08. What is the lowest strike price at which early exercise will occur? What condition related to put-call parity is satisfied at this strike price? Discuss.
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 functions are array functions that return the
Let S = $100, K = $100, σ = 30%, r = 0.08, t = 1, and δ = 0. Let n = 10. Suppose the stock has an expected return of 15%.a. What is the expected return on a European call option? A European put option?b. What happens to the expected return if you increase the volatility to 50%? Discuss.
Let S = $100, σ = 30%, r = 0.08, t = 1, and δ = 0. Suppose the true expected return on the stock is 15%. Set n = 10. Compute European call prices, Δ, and B for strikes of $70, $80, $90, $100, $110, $120, and $130. For each strike, compute the expected return on the option. What effect does the
Repeat the previous problem, except that for each strike price, compute the expected return on the option for times to expiration of 3 months, 6 months, 1 year, and 2 years. What effect does time to maturity have on the option's expected return?In previous problem Let S = $100, σ = 30%, r =
Use a spreadsheet to verify the option prices in Examples 12.1 and 12.2.
"Time decay is greatest for an option close to expiration." Use the spreadsheet functions to evaluate this statement. Consider both the dollar change in the option value and the percentage change in the option value, and examine both in-the-money and out-of-the-money options.
In the absence of an explicit formula, we can estimate the change in the option price due to a change in an input-such as σ-by computing the following for a small value of ϵ:Vega = BSCall(S, K, σ + ϵ, r, t, δ) − BSCall(S, K, σ − ϵ, r, t, δ)/2ϵa. What is the logic behind this
Suppose S = $100, K = $95, σ = 30%, r = 0.08, δ = 0.03, and T = 0.75. Using the technique in the previous problem, compute the Greek measure corresponding to a change in the dividend yield. What is the predicted effect of a change of 1 percentage point in the dividend yield?
Consider a bull spread where you buy a 40-strike call and sell a 45-strike call. Suppose S = $40, σ = 0.30, r = 0.08, δ = 0, and T = 0.5. Draw a graph with stock prices ranging from $20 to $60 depicting the profit on the bull spread after 1 day, 3 months, and 6 months. Discuss.
Consider a bull spread where you buy a 40-strike call and sell a 45-strike call. Suppose σ = 0.30, r = 0.08, δ = 0, and T = 0.5. a. Suppose S = $40. What are delta, gamma, vega, theta, and rho? b. Suppose S = $45. What are delta, gamma, vega, theta, and rho? c. Are any of your answers to (a) and
Consider a bull spread where you buy a 40-strike put and sell a 45-strike put. Suppose σ = 0.30, r = 0.08, δ = 0, and T = 0.5.a. Suppose S = $40. What are delta, gamma, vega, theta, and rho?b. Suppose S = $45. What are delta, gamma, vega, theta, and rho?c. Are any of your answers to (a) and (b)
Assume r = 8%, σ = 30%, δ = 0. In doing the following calculations, use a stock price range of $60-$140, stock price increments of $5, and two different times to expiration: 1 year and 1 day. Consider purchasing a 100-strike straddle, i.e., buying one 100-strike put and one 100-strike call. a.
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 $140, compute delta, vega, theta, and rho of
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 exercise barrier? Why?c. Suppose r = 0.07
Consider a perpetual put 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 exercise barrier? Why?c. Suppose r = 0.07
Using the BinomCall and BinomPut functions, compute the binomial approximations for the options in Examples 12.1 and 12.2. Be sure to compute prices for n = 8, 9, 10, 11, and 12. What do you observe about the behavior of the binomial approximation?
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?
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?In previous problem 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 =
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 happens to the option price? What accounts for
Let S = $120, K = $100, σ = 30%, r = 0, and δ = 0.08.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 price as T → ∞?b. Set r = 0.001. Repeat (a). Now what happens? What accounts for the difference?
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 is the price of a 1/90-strike
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 105-strike option, where the underlying asset is
Suppose S = $100, K = $95, σ = 30%, r = 0.08, δ = 0.03, and T = 0.75.a. Compute the Black-Scholes price of a call.b. Compute the Black-Scholes price of a call for which S = $100 × e−0.03×0.75, K = $95 × e−0.08×0.75, σ = 0.3, T = 0.75, δ = 0, r = 0. How does your answer compare to that
Make the same assumptions as in the previous problem.a. What is the 9-month forward price for the stock?b. Compute the price of a 95-strike 9-month call option on a futures contract.c. What is the relationship between your answer to (b) and the price you computed in the previous question? Why?
Assume K = $40, σ = 30%, r = 0.08, T = 0.5, and the stock is to pay a single dividend of $2 tomorrow, with no dividends thereafter.a. Suppose S = $50. What is the price of a European call option? Consider an otherwise identical American call. What is its price?b. Repeat, only suppose S = $60.c.
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 $40.50? Discuss.
Consider a 40-strike call with 365 days to expiration. Graph the results from the following calculations. a. Compute the actual price with 360 days to expiration at $1 intervals from $30 to $50. b. Compute the estimated price with 360 days to expiration using a delta approximation. c. Compute the
Using the parameters in Table 13.1, verify that equation (13.9) is zero.
Consider a put for which T = 0.5 and K = $45. Compute the Greeks and verify that equation (13.9) is zero.
You own one 45-strike call with 180 days to expiration. Compute and graph the 1-day holding period profit if you delta-and gamma-hedge this position using a 40-strike call with 180 days to expiration.
You have sold one 45-strike put with 180 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.
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 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-strike call with 180 days to expiration.
You have purchased a 40-strike call with 91 days to expiration. You wish to deltahedge, 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 you delta-and vegahedge this position using the
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?
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 rhohedging, can you delta-gamma-rho-vega hedge? a. Compute and graph the 1-day holding period profit if you delta-and vegahedge
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? Discuss.
Suppose you enter into a put ratio spread where you buy a 45-strike put and sell two 40-strike puts. 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?
Reproduce the analysis in Table 13.2, assuming that instead of selling a call you sell a 40-strike put.
Reproduce the analysis in Table 13.3, assuming that instead of selling a call you sell a 40-strike put.
Consider a 40-strike 180-day call with S = $40. Compute a delta-gamma-theta approximation for the value of the call after 1, 5, and 25 days. For each day, consider stock prices of $36 to $44.00 in $0.25 increments and compare the actual option premium at each stock price with the predicted premium.
Repeat the previous problem for a 40-strike 180-day put. Previous problem Consider a 40-strike 180-day call with S = $40. Compute a delta-gamma-theta approximation for the value of the call after 1, 5, and 25 days. For each day, consider stock prices of $36 to $44.00 in $0.25 increments and compare
Consider a 40-strike call with 91 days to expiration. Graph the results from the following calculations. a. Compute the actual price with 90 days to expiration at $1 intervals from $30 to $50. b. Compute the estimated price with 90 days to expiration using a delta approximation. c. Compute the
Examine the prices of up-and-out puts with strikes of $0.9 and $1.0 in Table 14.3. With barriers of $1 and $1.05, the 0.90-strike up-and-outs appear to have the same premium as the ordinary put. However, with a strike of 1.0 and the same barriers, the up-and-outs have lower premiums than the
Suppose S = $40, K = $40, σ = 0.30, r = 0.08, and δ = 0.a. What is the price of a standard European call with 2 years to expiration?b. Suppose you have a compound call giving you the right to pay $2 1 year from today to buy the option in part (a). For what stock prices in 1 year will you exercise
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