Question: Suppose that a derivative claim makes continuous
Suppose that a derivative claim makes continuous payments at the rate. Show that the Black-Scholes equation becomes For the following four problems, assume that S follows equation (21.5) and Q follows equation (21.35). Suppose S0 = $50, Q0 = $90, T = 2, r = 0.06, δ = 0.02, δQ = 0.01, σ = 0.3, σQ = 0.5, and Ï =−0.2. Use Proposition 21.1 to find solutions to the problems. Optional: For each problem, verify the solution using Monte Carlo. 
Transcribed Image Text:
V,+o'sVs+(r-8)Sv, +T-rV =0 %3D ds = (a – 8)dt + odZ (21.5)
> 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.
> 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 s
> What is the value of a claim paying Q(T )−1S(T )? Check your answer using Proposition 20.4. (20.4)
> Suppose that S1 follows equation (20.26) with δ = 0. Consider an asset that follows the process dS2 = α2S2 dt − σ2S2 dZ Show that (α1 − r)/σ1=âˆ
> Let h = 1/52. Simulate both the continuously compounded actual return and the actual stock price, St+h. What are the mean, standard deviation, skewness, and kurtosis of both the continuously compounded return on the stock and the stock price? Use the sam
> Let KT= S0erT. Compute Pr(ST
> 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. Compu
> A project costing $100 will produce perpetual net cash flows that have an annual volatility of 35% with no expected growth. If the project existed, net cash flows today would be $8. The project beta is 0.5, the effective annual risk-free rate is 5%, and
> Suppose a firm has 20 shares of equity, a 10-year zero-coupon debt with a maturity value of $200, and warrants for 8 shares with a strike price of $25. What is the value of the debt, the share price, and the price of the warrant?
> 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 pay $2 1 year from today to buy the option in (a). For what stoc
> 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 risk-neutral distribution of year-1 stock prices as in Figures 11.7
> 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
> 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
> Using the zero-coupon bond prices and natural gas swap prices in Table 8.9, what is the implicit loan amount in each quarter in an 8-quarter natural gas swap? TABLE 8.9 Quarter 2 3 4 5 7 8 20.5 Oil forward price Gas swap price Zero-coupon bond price
> Suppose you are the counterparty for a lender who enters into an FRA to hedge the lending rate on $10m for a 90-day loan commencing on day 270. What positions in zero-coupon bonds would you use to hedge the risk on the FRA?
> Suppose the S&P 500 index is currently 950 and the initial margin is 10%. You wish to enter into 10 S&P 500 futures contracts. a. What is the notional value of your position? What is the margin? b. Suppose you earn a continuously compounded rate of 6% on
> Suppose that Wirco does nothing to manage the risk of copper price changes. What is its profit 1 year from now, per pound of copper? Suppose that Wirco buys copper forward at $1. What is its profit 1 year from now?
> Consider the 3-year swap in the previous example. Suppose you are the fixed-rate payer in the swap. How much have you overpaid relative to the forward price after the first swap settlement? What is the cumulative overpayment after the second swap settlem
> For each entry in Table 2.5, explain the circumstances in which the maximum gain or loss occurs. TABLE 2.5 Maximum possible profit and loss at maturity for long and short forwards and purchased and written calls and puts. FV(premium) denotes the fut
> Suppose that you go to a bank and borrow $100. You promise to repay the loan in 90 days for $102. Explain this transaction using the terminology of short-sales.
> Suppose that in Figure 27.6 the tranches have promised payments of $160 (senior), $50 (mezzanine), and $90 (subordinated). Reproduce the table for this case, assuming zero default correlation. dA - (α- δ)dt+ σdZ Α (27.6)
> Suppose you write a 1-year cash-or-nothing put with a strike of $50 and a 1-year cash-or-nothing call with a strike of $215, both on stock A. a. What is the 1-year 99% VaR for each option separately? b. What is the 1-year 99% VaR for the two written opti
> Verify that the 1-year forward rate 3 years hence in Figure 25.5 is 14.0134%. For the next four problems, here are two BDT interest rate trees with effective annual interest rates at each node. Tree #1 0.08000 0.07676 0.08170 0.07943 0.07552 0.10362
> 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 th
> 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). Supp
> What is the value of a claim paying ? Check your answer using Proposition 20.4. (20.4)
> Suppose that the processes for S1 and S2 are given by these two equations: Note that the diffusions dZ1 and dZ2 are different. In this problem we want to find the expected return on Q, αQ, where Q follows the process Show that, to avoid arbitrage,
> Assume S0 = $100, r = 0.05, σ = 0.25, δ = 0, and T = 1. Use Monte Carlo valuation to compute the price of a claim that pays $1 if ST > $100, and 0 otherwise. (This is called a cash-or-nothing call and will be further discussed in Chapter 23. The actual
> Compute estimated profit in 1 year if XYZ sells a call option with a strike of $0.95, $1.00, or $1.05. Draw a graph of profit in each case.
> Let t = 1. What is E (St |St < $98)? What is E (St |St < $120)? How do both expectations change when you vary t from 0.05 to 5? Let σ = 0.1. Does either answer change? How?
> Verify the binomial calculations in Figure 17.3. FIGURE 17.3 YEAR O YEAR 1 YEAR 2 Value of the investment $307.74 option for the project in Figure 17.2. $147.31 $55.80 $50 $15.64
> A firm has outstanding a bond with a 5-year maturity and maturity value of $50, convertible into 10 shares. There are also 20 shares outstanding. What is the price of the warrant? The share price? Suppose you were to compute the value of the convertible
> 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
> 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: a. What is the log
> 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? Previou
> 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?
> Suppose call and put prices are given by Find the convexity violations. What spread would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage. Strike 80 100 105 Call premium Put premium 22 4 21 24.80
> Using the zero-coupon bond prices and natural gas swap prices in Table 8.9, what are gas forward prices for each of the 8 quarters? TABLE 8.9 Quarter 2 3 4 5 7 8 20.5 Oil forward price Gas swap price Zero-coupon bond price 0.9852 0.9701 0.9546 0.938
> What is the rate on a synthetic FRA for a 180-day loan commencing on day 180? Suppose you are the counterparty for a borrower who uses the FRA to hedge the interest rate on a $10m loan. What positions in zero-coupon bonds would you use to hedge the risk
> a. Suppose you enter into a long 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 call option with a strike price of $50. What is the payoff in 6 mo
> Suppose the S&P 500 index futures price is currently 1200. You wish to purchase four futures contracts on margin. a. What is the notional value of your position? b. Assuming a 10% initial margin, what is the value of the initial margin?
> Compute estimated profit in 1 year if Telco buys paylater calls as follows (the net premium may not be exactly zero): a. Sell one 0.975-strike call and buy two 1.034-strike calls. b. Sell two 1.00-strike calls and buy three 1.034-strike calls. Draw a gra
> For Figure 2.8, verify the following: a. The S&R index price at which the put option diagram intersects the x-axis is $924.32. b. The S&R index price at which the put option and forward contract have the same profit is $1095.68. FIGURE 2.8
> Short interest is a measure of the aggregate short positions on a stock. Check an online brokerage or other financial service for the short interest on several stocks of your choice. Can you guess which stocks have high short interest and which have low?
> Consider two firms, one with an FF rating and one with an FFF rating. What is the probability that after 4 years each will have retained its rating? What is the probability that each will have moved to one of the other two ratings? For the next two probl
> Compute the 95% 10-day tail VaR for the position in Problem 26.8. Problem 26.8. Compute the 95% 10-day VaR for a written strangle (sell an out-of-the-money call and an out-of-the-money put) on 100,000 shares of stock A. Assume the options have strikes o
> Verify that the price of the 12% interest rate cap in Figure 25.6 is $3.909. FIGURE 25.6 Year 0 Year 1 Year 2 Year 3 $6.689 Tree showing the payoff to a 12% interest rate cap on a $100 3-year loan, assuming that interest rates evolve $6.799 accordin
> Suppose an option knocks in at H1> S, and knocks out at H2 >H1. Suppose that K H1, it is not possible to hit H2 without hitting H1): What is the value of this option? if Hj not hit Payoff = max (0, ST – K) if Hj hit and H2 not hit if Hj hit and
> 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
> 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 pay to buy 100 shares? b. What amount will you receive for selling 100 shares? c. Suppose you buy 100 shares, then immedia
> The formula for an infinitely lived call is given in equation (12.18). Suppose that S follows equation (20.20), with α replaced by r, and that E∗(dV ) = rV dt. Use Itˆo’s Lemma to verify that t
> Consider a project that in one year pays $50 if the economy performs well (the stock market goes up) and that pays $100 if the economy performs badly (the stock market goes down). The probability of the economy performing well is 60%, the effective annua
> 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
> “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
> 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, $110, $120, and $130. For each strike, compute the expected ret
> 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, ∆, and B.
> Suppose call and put prices are given by Find the convexity violations. What spread would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage. Strike 50 55 60 Call premium 18 14 9.50 Put premium 7 10.75 14.45
> What is the rate on a synthetic FRA for a 90-day loan commencing on day 90? A 180-day loan commencing on day 90? A 270-day loan commencing on day 90?
> The S&R index spot price is 1100 and the continuously compounded risk-free rate is 5%. You observe a 9-month forward price of 1129.257. a. What dividend yield is implied by this forward price? b. Suppose you believe the dividend yield over the next 9 mon
> Compute estimated profit in 1 year if Telco sells 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. $0.95 for the put and $0.95 for the call. Draw a graph of profit in each
> Suppose the firm issues a single zero-coupon bond with time to maturity 3 years and maturity value $110. a. Compute the price, yield to maturity, default probability, and expected recovery (E[BT |Default]). b. Verify that equation (27.5) holds. 1 P-r
> For Figure 2.6, verify the following: a. The S&R index price at which the call option diagram intersects the x-axis is $1095.68. b. The S&R index price at which the call option and forward contract have the same profit is $924.32. FIGURE 2.
> Suppose a stock pays a quarterly dividend of $3. You plan to hold a short position in the stock across the dividend ex-date. What is your obligation on that date? If you are a taxable investor, what would you guess is the tax consequence of the payment?
> Consider a firm with an F rating. a. What is the probability that after 4 years it will still have an F rating? b. What is the probability that after 4 years it will have an FF or FFF rating? c. From examining the transition matrix, are firms tending ove
> Using Monte Carlo, compute the 95% and 99% 1-, 10-, and 20-day tail VaRs for the position in Problem 26.2. Problem 26.2. Assuming a $10m investment in one stock, compute the 95% and 99% VaR for stocks A and B over 1-day, 10-day, and 20-day horizons.
> 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?
> 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 o
> 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
> 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? Example 21
> 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 constr
> What is E(St|St > $105) for t = 1? How does this expectation change when you change t, σ, and r?
> Assuming a $10m investment that is 40% stock A and 60% stock B, compute the 95% and 99% VaR for the position over 1-day, 10-day, and 20-day horizons.
> To answer this question, use the assumptions of Example 17.1 and the risk-neutral valuation method (and risk-neutral probability) described in Example 17.2. a. Compute the value of a claim that pays the square root of the cash flow in period 1. b. Comput
> Now suppose the firm finances the project by issuing debt that has higher priority than existing debt. How much must a $10 or $25 project be worth if the shareholders are willing to fund it?
> 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 time to expiration, what happens to the price of the kno
> 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 it
> 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? Previou
> 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 increases?
> 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 Put premium 16 10 7 14
> Given an 8-quarter oil swap price of $20.43, construct the implicit loan balance for each quarter over the life of the swap.
> Using the same information as the previous problem, suppose the interest rate on the borrowing date is 7.5%. Determine the dollar settlement of the FRA assuming a. Settlement occurs on the date the loan is initiated. b. Settlement occurs on the date the
> Suppose that 10 years from now it becomes possible for money managers to engage in time travel. In particular, suppose that a money manager could travel to January 1981, when the 1-year Treasury bill rate was 12.5%. a. If time travel were costless, what
> 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. Pl
> Compute estimated profit in 1 year if Telco sells a put option with a strike of $0.95, $1.00, or $1.05. Draw a graph of profit in each case.
> An off-market forward contract is a forward where either you have to pay a premium or you receive a premium for entering into the contract. (With a standard forward contract, the premium is zero.) Suppose the effective annual interest rate is 10% and the
> When you open a brokerage account, you typically sign an agreement giving the broker the right to lend your shares without notifying or compensating you. Why do brokers want you to sign this agreement?
> Repeat the previous problem, except that the time to maturity can be 1, 2, 3, 4, 5, 10, or 20 years. How does the bond yield change with time to maturity?
> Compute the 95% 10-day VaR for a written strangle (sell an out-of-the-money call and an out-of-the-money put) on 100,000 shares of stock A. Assume the options have strikes of $90 and $110 and have 1 year to expiration. Use the delta-approximation method
> Verify that the 4-year zero-coupon bond price generated by the tree in Figure 25.5 is $0.6243. Figure 25.5 FIGURE 25.5 Year O Year 1 Year 2 Year 3 20.03% Black-Derman-Toy interest rate tree constructed using the data in Table 25.1. Each rate is an e
> 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 y
> 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). Fo.T(Q) = Qoe"s-80-)T (23.30) %3D
> 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 marti
> 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 pric
> Verify that equation (23.7) satisfies the appropriate boundary conditions for Pr(ST≤ H and ST >K). Pr(S, <H and Sy > K) = N (d,) – N(d6) + S (4)* 25-1 N (dg) (23.7)
> 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 pay a. SQ b. S/Q c. d. 1/(SQ) e. S2Q Com
> 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.
> What is Pr(St > $105) for t = 1? How does this probability change when you change t? How does it change when you change σ?
> Consider the widget investment problem of Section 17.1 with the following modification. The expected growth rate of the widget price is zero. (This means there is no reason to consider project delay.) Each period, the widget price will be $0.25 with prob
> Now suppose the firm finances the project by issuing debt that has lower priority than existing debt. How much must a $1, $10, or $25 project be worth if the shareholders are willing to fund it?
> Consider the equity-linked CD in Section 15.3. Assuming that profit for the issuing bank is zero, draw a graph showing how the participation rate, γ , varies with the coupon, c. Repeat assuming the issuing bank earns profit of 5%.
