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Questions and Answers of
Corporate Finance
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
Use the same data as in the previous problem, only suppose that the call price is $5 instead of $4.110.a. At time 0, assume you write the option and form the replicating portfolio to offset the
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
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?
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
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
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
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.
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
Repeat the previous problem calculating prices for American options instead of European. What happens?For a stock index, S = $100, σ = 30%, r = 5%, δ = 3%, and T = 3. Let n = 3.a. What is the price
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. Many (but not all) of these questions can
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
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
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
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.
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 _ as the stock price
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, ∆ 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 _ as the stock price
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
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, ∆ and B for strikes of $70, $80, $90, $100,
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
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 Suidn−i to compute the price at that node, plot the
Repeat the previous problem for n = 50. What is the risk-neutral probability that S1< $80? S1> $120? We sawin Section 10.1 that the undiscounted risk-neutral expected stock price equals the
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
Compute the 1-year forward price using the 50-step binomial tree in Problem 11.13. show your calculations.
Suppose S = $100, K = $95, r = 8% (continuously compounded), t = 1, Ï = 30%, and δ = 5%. Explicitly construct an eight-period binomial tree using the Cox-Ross Rubinstein
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. Calculate the price of European and American call options using a four-step
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?In
Repeat Problem 11.1, only assume that r = 0.08 and δ = 0.Will early exercise ever occur? Why? In Problem 11.1 Consider a one-period binomial model with h = 1, where S = $100, r = 0, σ = 30%, and δ
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
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?
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
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
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, ∆ and B for strikes of $70, $80, $90, $100,
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
Use a spreadsheet to verify the option prices in Examples 12.1 and 12.2. Show calculations
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
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 :a. What is the logic
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
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
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.
Consider a bull spread where you buy a 40-strike put and sell a 45-strike put. Supposeσ = 0.30, r = 0.08, δ = 0, and T = 0.5.a. Suppose S = $40. What are delta, gamma, vega, theta, and rho?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
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
Consider a perpetual call option with S = $50, K = $60, r = 0.06, σ = 0.40, andδ = 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
Consider a perpetual put option with S = $50, K = $60, r = 0.06, σ = 0.40, andδ = 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
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 forn = 8, 9, 10, 11, and 12. What do you observe
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%, andT = 1.c. What is the link
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?
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
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
The exchange rate is ¥95/=C, the yen-denominated interest rate is 1.5%, the eurodenominated interest rate is 3.5%, and the exchange rate volatility is 10%.a. What is the price of a 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
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
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
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
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
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
Repeat Problem 13.9 for a 91-day 40-strike put.
Repeat Problem 13.10 for a 365-day 40-strike put.
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
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 by the stock and a 40-strike call with 180
You have written a 35-40-45 butterfly spread with 91 days to expiration. Calculate and graph the 1-day holding period profit if delta- and gamma-hedge this position using the stock and a 40 strike
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-
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.
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
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
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
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
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
Repeat the previous problem for a 40-strike 180-day put.
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.
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.
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
Make the same assumptions as in the previous problem.a. What is the price of a standard European put with 2 years to expiration?b. Suppose you have a compound call giving you the right to
Consider the hedging example using gap options, in particular the assumptions and prices in Table 14.4.a. Implement the gap pricing formula. Reproduce the numbers in Table 14.4.b. Consider the option
Problem 12.11 showed how to compute approximate Greek measures for an option. Use this technique to compute delta for the gap option in Figure 14.3, for stock prices ranging from $90 to $110 and for
Consider the gap put in Figure 14.4. Using the technique in Problem 12.11, compute vega for this option at stock prices of $90, $95, $99, $101, $105, and $110, and for times to expiration of 1 week,
Let S = $40, σ = 0.30, r = 0.08, T = 1, and δ = 0. Also let Q = $60, σQ= 0.50, δQ = 0.04, and ρ = 0.5. What is the price of a standard 40-strike call with S as the underlying asset? What is the
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 withS as the price of the underlying
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
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
Suppose you observe the prices {5, 4, 5, 6, 5}. What are the arithmetic and geometric averages? Nowyou observe {3, 4, 5, 6, 7}. What are the two averages? What happens to the difference between the
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
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
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
Using the information in the previous problem, compute the prices ofa. An Asian arithmetic average strike call.b. An Asian geometric average strike call.
Repeat Problem 14.3, except construct a three-period binomial tree. Assume that Asian options are based on averaging the prices every 4 months.a. What are the possible geometric and arithmetic
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
Let S = $40, K = $45, σ = 0.30, r = 0.08, δ = 0, and T = {0.25, 0.5, 1, 2, 3, 4, 5, 100}.a. Compute the prices of knock-out calls with a barrier of $38.b. Compute the ratio of the
Repeat the previous problem for up-and-out puts assuming a barrier of $44.
Let S = $40, K = $45, σ = 0.30, r = 0.08, and δ = 0. Compute the value of knockout calls with a barrier of $60 and times to expiration of 1 month, 2 months, and so on, up to 1 year. As you increase
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
Compute λ if the dividend on the CD is 0 and the payoff is $1300 - max (0, 1300 − S5.5) + λ × max(0, S5.5 − 2600) and the initial price is to be $1300.
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