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Derivatives Markets 4th edition Rober L. Macdonald - Solutions

An options trader purchases 1000 1-year at-the-money calls on a non-dividend paying stock with S0 = $100, α = 0.20, and σ = 0.25. Assume the options are priced according to the Black-Scholes formula and r = 0.05. a. Use Monte Carlo (with 1000 simulations) to estimate the expected return, standard
Refer to Table 19.1.a. Verify the regression coefficients in equation (19.12).b. Perform the analysis for t = 1, verifying that exercise is optimal on paths 4, 6, 7, and 8, and not on path 1.
Refer to Figure 19.2.a. Verify that the price of a European put option is $0.0564.b. Verify that the price of an American put option is $0.1144. Be sure to allow for the possibility of exercise at time 0.
Assume S0 = $50, r = 0.05, σ = 0.50, and δ = 0. The Black-Scholes price for a 2-year at-the-money put is $10.906. Suppose that the stock price is lognormal but can also jump, with the number of jumps Poisson-distributed. Assume α = 0.05 (the expected return to the stock is equal to the
Let ui ∼ U(0, 1). Compute Σ12i=1 ui − 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 normally distributed variable.)
The Black-Scholes price for a European put option with S = $40, K = $40, σ = 0.30, r = 0.08, δ = 0, and t = 0.25 is $1.99. Use Monte Carlo to compute this price. Compute the standard deviation of your estimates. How many trials do you need to achieve a standard deviation of $0.01 for your
Let r = 0.08, S = $100, δ = 0, and σ = 0.30. Using the risk-neutral distribution, simulate 1/S1. What is E(1/S1)? What is the forward price for a contract paying 1/S1?
Suppose S0 = 100, r = 0.06, σS = 0.4 and δ = 0. Use Monte Carlo to compute prices for claims that pay the following:a. S21b. √S1c. S1−2
Assume that the market index is 100. Show that if the expected return on the market is 15%, the dividend yield is zero, and volatility is 20%, then the probability of the index falling below 95 over a 1-day horizon is approximately 0.0000004. Discuss.
Suppose that on any given day the annualized continuously compounded stock return has a volatility of either 15%, with a probability of 80%, or 30%, with a probability of 20%. This is a mixture of normals model. Simulate the daily stock return and construct a histogram and normal plot. What happens
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 Brownian motion, equation (20.8); (b) A mean reverting process, equation (20.9); and (c) Geometric
The formula for an infinitely lived call is given in equation (12.18). Suppose that S follows equation (20.20), with Î± replaced by r, and that Eˆ—(dV ) = rV dt. Use ItË†o€™s Lemma to verify that the value of the call, V (S), satisfies this equation:
Suppose that the processes for S1 and S2 are given by these two equations:dS1= α1S1dt + σ1S1dZ1dS2 = α2S2dt + σ2S2dZ2The diffusions dZ1 and dZ2 are different. In this problem we want to find the expected return on Q, αQ, where Q follows the processdQ = αQQdt + Qη1dZ1+ η2dZ2Show that, to
Suppose that S1 follows equation (20.26) with δ = 0. Consider an asset that follows the process dS2 = α2S2 dt − σ2S2 dZShow that (α1 − r)/σ1=−(α2 − r)/σ2. (Hint: Find a zero-investment position in S1 and S2 that eliminates risk.)
Suppose that S and Q follow equations (20.36) and (20.37). Derive the value of a claim paying S(T)a Q(T )b by each of the following methods:a. Compute the expected value of the claim and discounting at an appropriate rate. b. Compute the lease rate and substituting this into the formula for the
Assume that one stock follows the process dS/S = αdt + σdZ (20.44) Another stock follows the process dQ/Q = αQdt + σdZ + dq1+ dq2 (20.45) (The σdZ terms for S and Q are identical.) Neither stock pays dividends. dq1 and dq2 are both Poisson jump processes with Poisson parameters λ1 and
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 motion, equation (20.8); (b) A mean reverting process, equation (20.9); and (c) Geometric Brownian
Use Itˆo’s Lemma to evaluate dS−1.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 motion, equation (20.8); (b) A mean reverting process, equation (20.9); and (c) Geometric
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) Arithmetic Brownian motion, equation (20.8); (b) A mean reverting process, equation (20.9); and (c) Geometric
Suppose that S follows equation (20.36) and Q follows equation (20.37). Use Itˆo’s Lemma to find the process followed by S2Q0.5. Discuss.
Suppose that S follows equation (20.36) and Q follows equation (20.37). Use Itˆo’s Lemma to find the process followed by ln(SQ).
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. S2b. √Sc. S−2
Suppose that ln(S) and ln(Q) have correlation ρ =−0.3 and that S(0) = $100, Q(0) = $100, r = 0.06, σS = 0.4, and σQ = 0.2. Neither stock pays dividends. Use equation (20.38) to find the price today of claims that paya. SQb. S/Qc. √SQd. 1/(SQ)e. S2Q
Suppose that X(t) follows equation (20.9). Use It^o's Lemma to verify that a solution to this differential equation is Discuss.
Verify that equation (21.12) satisfies the Black-Scholes equation. What is the boundary condition for which this is a solution? Discuss.
Suppose that a derivative claim makes continuous payments at the rate. Show that the Black-Scholes equation becomes
What is the value of a claim paying Q(T)2S(T)? Check your answer using Proposition 20.4.
What is the value of a claim paying Q(T)−1S(T)? Check your answer using Proposition 20.4.
You are offered the opportunity to receive for free the payoff[Q(T ) − F0,T (Q)]× max[0, S(T ) − K]
An agricultural producer wishes to insure the value of a crop. Let Q represent the quantity of production in bushels and S the price of a bushel. The insurance payoff is therefore Q(T) × V [S(T), T], where V is the price of a put with K = $50. What is the cost of insurance?
Verify that ASaeÎ³t satisfies the Black-Scholes PDE for
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.
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? Discuss.
Verify that S(t)e−δ(T−t)N(d1) satisfies the Black-Scholes equation. Discuss.
Verify that e−r(T−t) N(d2) satisfies the Black-Scholes equation.
Use the answers to the previous two problems to verify that the Black-Scholes formula, equation (12.1), satisfies the Black-Scholes equation. Verify that the boundary condition V [S(T ), T ]= max[0, S(T ) − K] is satisfied.
Consider Joe and Sarah’s bet in Examples 21.2 and 21.3.a. In this bet, note that $106.184 is the forward price. A bet paying $1 if the share price is above the forward price is worth less than a bet paying $1 if the share price is below the forward price. Why?b. Suppose the bet were to be
Consider again the bet in Example 21.3. Suppose the bet is S − $106.184 if the price is above $106.184, and $106.184 − S if the price is below $106.184. What is the value of this bet to each party? Why? Discuss.
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 exp(λ1c)b. U(c) = λ0cλ1c. U(c) = λ0c +
Repeat the previous problem assuming that δ1= 0.05 and δ2 = 0.12. Verify that both procedures give a price of approximately $15.850.Previous problem Suppose that S1 and S2 are correlated, non-dividend-paying assets that follow geometric Brownian motion. Specifically, let S1(0) = S2(0) = $100, r =
Suppose there are 1-, 2-, and 3-year zero-coupon bonds, with prices given by P1, P2, and P3. The implied forward interest rate from year 1 to 2 is r0(1, 2) = P1/P2 − 1, and from year 2 to 3 is r0(2, 3) = P2/P3 − 1. Denote the rates as r(1) and r(2). Suppose that you select the 3-year bond as
Assume the same bonds and numeraire as in the previous question. Suppose that P1/P3 is a martingale following a geometric Brownian process with annual standard deviation σ1= 0.10, and that P2/P3 is a martingale following a geometric Brownian process with annual standard deviation σ2 = 0.05. The
The box on page 282 discusses the following result: If the strike price of a European put is set to equal the forward price for the stock, the put premium increases with maturity. a. How is this result related to Warren Buffett's critique of put-pricing, discussed in Section 22.6? b. In Chapter 9
Under the social security system in the United States, workers pay taxes and receive a monthly annuity after retirement. Some have argued that the United States should invest the social security tax proceeds in stocks. The rationale is that, over time, there is a decreasing probability that stocks
Warren Buffett stated in the 2009 Letter to Shareholders: "Our derivatives dealings require our counterparties to make payments to us when contracts are initiated. Berkshire therefore always holds the money, which leaves us assuming no meaningful counterparty risk." Buffett also was not required to
Use a change of numeraire and measure to verify that the value of a claim paying K if ST < K is Ke−rT N(−d2)
Use a change of numeraire and measure to verify that the value of a claim paying ST if ST
Suppose that S1 and S2 follow geometric Brownian motion and pay continuous proportional dividends at the rates Î´1 and Î´2. Use the martingale argument to show that the value of a claim paying S1(T) if S1(T) > KS2(T) iswhere Ïƒ2 = Ïƒ21 + Ïƒ22 ˆ’ 2Ï1, 2Ïƒ1Ïƒ2 and Î´1 and
Under the same assumptions as the previous problem, show that the value of a claim paying S2(T) if S1(T) > KS2(T) iswhere Ïƒ2, Î´1, and Î´2 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 = 0.9802,
In this problem we will use Monte Carlo to simulate the behavior of the martingale St/Pt, with Pt as numeraire. Let x0 = S0/P0(0, T). Simulate the processxt+h= (1+ σ√hZt+h)xtLet h be approximately 1 day.a. Evaluate P0E [ST /PT (T, T) > K].b. Compute the mean and standard deviation of the
We now use Monte Carlo to simulate the behavior of the martingale Pt/St, with St as numeraire. Let x0 = P0(0, T)/S0. Simulate the process xt+h= (1+ σ√hZt+h)xtLet h be approximately 1 day.a. Evaluate S0EPT (T, T)/ST < 1/K.b. Compute the mean and standard deviation of the difference xT− x0.
Suppose that the stock price follows a jump-diffusion process as outlined in Section 20.7. Let the jump intensity be λ = 0.75, the expected jump exp(αJ), with αJ = −0.15, and let the jump volatility be σJ = 0.25. You can simulate the behavior of the martingale St/Pt asxt+h = [1− λkh + σ
Suppose that S1 and S2 are correlated, non-dividend-paying assets that follow geometric Brownian motion. Specifically, let S1(0) = S2(0) = $100, r = 0.06, σ1 = 0.35, σ2 = 0.25, ρ = 0.40 and T = 1. Verify that the following two procedures for valuing an outperformance option give a price of
A collect-on-delivery call (COD) costs zero initially, with the payoff at expiration being 0 if S
Suppose an option knocks in at H1 > S, and knocks out at H2 > H1. Suppose that K <H2 and the option expires at T. Call this a “knock-in, knock-out” option. Here is an equation summarizing the payoff to this option (note that because H2 > H1, it is not possible to hit H2 without
Suppose the stock price is $50, but that we plan to buy 100 shares if and when the stock reaches $45. Suppose further that σ = 0.3, r = 0.08, T − t = 1, and δ = 0. This is a noncancellable limit order. a. What transaction could you undertake to offset the risk of this obligation? b. You can
Covered call writers often plan to buy back the written call if the stock price drops sufficiently. The logic is that the written call at that point has little “upside,” and, if the stock recovers, the position could sustain a loss from the written call. a. Explain in general how this buy-back
For the lookback call:a. What is the value of a lookback call as St approaches zero? Verify that the formula gives you the same answer.b. Verify that at maturity the value of the call is ST ˆ’ ST.A European lookback call at maturity pays ST ˆ’ ST. A European lookback put at maturity pays ST
For the lookback put:a. What is the value of a lookback put if St = 0? Verify that the formula gives you the same answer.b. Verify that at maturity the value of the put is ST ˆ’ ST.A European lookback call at maturity pays ST ˆ’ ST. A European lookback put at maturity pays ST ˆ’ ST.
A European shout option is an option for which the payoff at expiration is max(0, S − K, G − K), where G is the price at which you shouted. (Suppose you have an XYZ shout call with a strike price of $100. Today XYZ is $130. If you shout at $130, you are guaranteed a payoff of max($30, ST −
Consider the Level 3 outperformance option with a multiplier, discussed in Section 16.2. This can be valued binomially using the single state variable SLevel3/SS&P, and multiplying the resulting value by SS&P.a. Compute the value of this option if it were European, assuming the Level 3
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 correlation between Y and x, the dollar-yen exchange
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 the paylater put. Compare the behavior of
Verify that equation (23.7) satisfies the appropriate boundary conditions for Pr(ST ≤ H and ST >K). Discuss.
Verify that equation (23.14) (for both cases K > H and K < H) solves the boundary conditions for an up-and-in cash put.
Assume that S = $45, K = $40, r = 0.05, δ = 0.02, and σ = 0.30. Using the up rebate formula (equation (23.21)), find the value of H that maximizes (H − K) × UR(S, σ, r, T, δ), for T = 1, 10, 100, 1000, and 10,000. Compare both H and (H − K) × UR to the perpetual option solution. Explain
Verify in Example 23.12 that you obtain the same answer if you use x0Q0 as the stock price, δQ + ρsσQ + r − rf as the dividend yield, r as the interest rate, and σQ as the volatility.
In this problem you will price various options with payoffs based on the Eurostoxx index and the dollar/euro exchange rate. Assume that Q= 2750 (the index), x = 1.25 ($/=C), s = 0.08 (the exchange rate volatility), σ = 0.2 (index volatility), r = 0.01(the U.S. risk-free rate), rf = 0.03 (the
The quanto forward price can be computed using the risk-neutral distribution as E(Yx−1). Use Proposition 20.4 to derive the quanto forward price given by equation (23.30). Discuss.
In this problem we use the lognormal approximation (see equation (11.14)) to draw one-step binomial trees from the perspective of a yen-based investor. Use the information in Table 23.4.a. Construct a one-step tree for the Nikkei index.b. Construct a one-step tree for the exchange rate
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 period.
Compute January 12 2004 bid and ask volatilities (using the Black-Scholes implied volatility function) for IBM options expiring February 21.a. Do you observe a volatility smile?b. For which options are you unable to compute a plausible implied volatility? Why? Discuss.
Compute January 12 2004 implied volatilities using the average of the bid and ask prices for IBM options expiring February 21 (use the Black-Scholes implied volatility function). Compare your answers to those in the previous problem. Why might someone prefer to use implied volatilities based on the
In this problem you will compute January 12 2004 bid and ask volatilities (using the Black-Scholes implied volatility function) for 1-year IBM options expiring the following January. Note that IBM pays a dividend in March, June, September, and December.a. Compute implied volatilities ignoring the
For this problem, use the implied volatilities for the options expiring in January 2005, computed in the preceding problem. Compare the implied volatilities for calls and puts. Where is the difference largest? Why does this occur? Discuss.
Suppose S = $100, r = 8%, σ = 30%, T = 1, and δ = 0. Use the Black-Scholes formula to generate call and put prices with the strikes ranging from $40 to $250, with increments of $5. Compute the implied volatility from these prices by using the formula for the VIX (equation (24.29)). What happens
Explain why the VIX formula in equation (24.29) overestimates implied volatility if options are American.
Using the Merton jump formula, generate an implied volatility plot for K = 50, 55, . . . 150.a. How is the implied volatility plot affected by changing αJ to−0.40 or−0.10?b. How is the implied volatility plot affected by changing λ to 0.01 or 0.05?c. How is the implied volatility plot
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.As a base case, assume S = $100, r = 8%, σ = 30%, T = 1, and δ = 0. Also assume that λ = 0.02, αJ =−0.20, and σJ = 0.30.
Repeat Problem 24.16, except let αJ = 0.20, and in part (b) consider expected alternate jump magnitudes of 0.10 and 0.50.As a base case, assume S = $100, r = 8%, σ = 30%, T = 1, and δ = 0. Also assume that λ = 0.02, αJ =−0.20, and σJ = 0.30.
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.
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?
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.
For the period 1999–2004, using daily data, compute the following: a. An EWMA estimate, with b = 0.95, of IBM’s volatility using all data.b. An EWMA estimate, with b = 0.95, of IBM’s volatility, at each date using only the previous 60 days of data.Plot both estimates. How different are they?
Estimate a GARCH (1,1) for the S&P 500 index, using data from January 1999 to December 2003.
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.
Use the following inputs to compute the price of a European call option: S = $100, K = $50, r = 0.06, σ = 0.30, T = 0.01, δ = 0.a. Verify that the Black-Scholes price is $50.0299.b. Verify that the vega for this option is almost zero. Why is this so?c. Verify that if you compute the option price
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, but be sure to set the initial volatility at
Use the following inputs to compute the price of a European call option: S = $50, K = $100, r = 0.06, σ = 0.30, T = 0.01, δ = 0. a. Verify that the Black-Scholes price is zero. b. Verify that the vega for this option is zero. Why is this so? c. Suppose you observe a bid price of zero and an ask
Compute January 12 2004 bid and ask volatilities (using the Black-Scholes implied volatility function) for IBM options expiring January 17. For which options are you unable to compute a plausible implied volatility? Why? Discuss.
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 price of an interest rate caplet that provides an
Verify that the price of the 12% interest rate cap in Figure 25.6 is $3.909.In Figure 25.6
Verify that the 1-year forward rate 3 years hence in Figure 25.5 is 14.0134%.In Figure 25.5
What are the 1-, 2-, 3-, 4-, and 5-year zero-coupon bond prices implied by the two trees?For the next four problems, here are two BDT interest rate trees with effective annual interest rates at each node.
What volatilities were used to construct each tree? (You computed zero-coupon bond prices in the previous problem; now you have to compute the year-1 yield volatility for 1-, 2-, 3-, and 4-year bonds.) Can you unambiguously say that rates in one tree are more volatile than the other?For the next
For years 2€“5, compute the following:a. The forward interest rate, rf, for a forward rate agreement that settles at the time borrowing is repaid. That is, if you borrow at t ˆ’ 1 at the 1-year rate Ëœr, and repay the loan at t, the contract payoff in year t is(Ëœr ˆ’ rf)b. The
You are going to borrow $250m at a floating rate for 5 years. You wish to protect yourself against borrowing rates greater than 10.5%. Using each tree, what is the price of a 5-year interest rate cap? (Assume that the cap settles each year at the time you repay the borrowing.)For the next four
Suppose that the yield curve is given by y(t) = 0.10 − 0.07e −0.12t , and that the short-term interest rate process is dr(t) = (θ(t) − 0.15r(t)) + 0.01dZ. Compute the calibrated Hull-White tree for 5 years, with time steps of h = 1.a. What is the probability transition matrix Q?b. What is
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.
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.In previous problem Using Monte Carlo, simulate the process dr = a(b − r)dt + σdZ, assuming that r = 6%, a = 0.2, b = 0.08, φ =

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