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Principles Of Financial Engineering 3rd Edition Robert Kosowski, Salih N. Neftci - Solutions
9. Suppose you issue a CLN. How would you hedge your position? Mention at least two ways of doing this. By the way, why do you need to hedge your position? Be specific.
8. What is a credit-linked note (CLN)? Why would investors buy credit-linked notes instead of, say, corporate bonds?Analyze the risks and the cash flows generated by these two instruments to see in what sense CLNs are preferable.
7. Consider the following reading, which deals with collateralized debt obligations (CDOs).
6. (Reduced form approach to CoCos valuation). The text mentions that CoCos can be valued from three different perspectives. One of the approaches is based on the reduced form or default intensity approach outlined in Chapter 18. Assume that a CoCo has a 5-year maturity.The underlying share price
5. Consider the following quote:Until last year, this correlation pricing of single-tranche CDOs and first-to-default baskets was dependent on each bank or hedge fund’s assessment of correlation. However, in 2003 the banks behind iBoxx and Trac-x started trading tranched versions of the
4. Consider the following quote:It is only when portfolios are tranched that the relative value of default correlation becomes meaningful.So, for subordinate tranches, the risk and spreads decrease as correlation between defaults increases, while for senior tranches the risk and spreads increase as
3. Consider the following news from Reuters:1008 GMT [Dow Jones] LONDON—SG recommends selling 7-year 03% tranche protection versus buying 5-year and 10-year 03% protection. 7-year equity correlation tightened versus 5-year and 10-year last year. SG’s barbell plays a steepening of the 7-year
2. What is the effect of default probabilities on CDO tranches? What is the effect of default correlations on CDO tranches? Explain.
1. What is the difference between an ABS and a CDO?
5. Show how you would engineer the following CMS spread note.Issuer: ABC Notional: $10mio Tenor: 10 years Principal: Guaranteed at maturity Coupon:Yr 1: 11.50%Yr 210: 16 3 (CMS30 2 CMS10), max of 30%, min 0%Call: Callable on each coupon date by the issuera. What is the view of the investor?b. What
4. Show how you would engineer the following Snowball Note.Issuer: ABC bank Notional: $10 mio Tenor: 10 years; Principal: Guaranteed at maturity Coupon: Yr 1; Q1: 9.00%Q2: Previous Coupon 1 CMS10 4.65%Q3: Previous Coupon 1 CMS10 4.85%Q4: Previous Coupon 1 CMS10 5.25%Yr 2 Q1: Previous Coupon 1 CMS10
3. What follows is the description of a rather complex swap structured by a bank. The structure is sold for the purpose of liability management and involves an exotic option (digital cap)and a CMS component.
2. Consider the swap and LIBOR curves available in Reuters or Bloomberg.a. Obtain the 3-month discount and forward curvesb. Obtain the 2-year forward curvec. Find the components for the following note: maturity: 3 years Callable: each coupon payment date?
1. Case Study: Reverse-Convertibles and Volatility Trading This case study shows another example of volatility trading and reverse convertibles.Read the case study below and answer the following questions.a. What is a reverse-convertible bond? How would you decompose this instrument? How would a
5. Consider two convertible bonds X and Y, which for simplicity are assumed to be riskless.The following table provides information about the two bonds. Calculate the conversion value. What is the yield to maturity of the convertible bonds based on the actual bond price?Convertible Bond X
4. What variables and real-world complications are important in practice but ignored by the basic Merton model.
3. Consider company B which issued equity and zero-coupon bonds with a maturity of 1 year.Assume that the value of the firm is $100 and the value of the equity is $50 million. The risk-free rate is 2%. The equity volatility is 30%.a. What is the market value of debt and the implied credit spread?b.
2. Assume that company A has an asset volatility of 20%. The current value of its assets is$100 million and the face value of its 1-year maturity zero-coupon debt is $50 million. The risk-free rate of interest is 2%. Use the Merton (1974) model to calculate the value of the firm’s equity. What is
1. Explain why debt in the Merton (1974) model is viewed as an option.
8. Explain the logic behind the two following strategies using cash flow diagrams.a. WestLB mortgage Pfandbriefe trade too tight. Sell the WestLB 3% 2009s at 5.4 bp under swaps and buy the zero risk weighted Land Berlin 2.75% 2010s at 2.7 bp under. (TMA)b. The following quote deals with implied
7. Consider the following news from Reuters:HVB Suggests Covered Bond Switches 0843 GMT [Dow Jones] LONDON—Sell DG Hyp 4.25% 2008s at 6.5 bp under swaps and buy Landesbank Baden-Wuerttemberg(LBBW) 3.5% 2009s at swaps-4.2 bp, HVB says. The LBBW deal is grandfathered and will continue to enjoy
6. Consider the following news from Reuters:1407 GMT [Dow Jones] LONDON—According to a large investment bank investors can boost yields using the following strategies:a. In the strategy, sell 5-yr CDS on basket of Greece (9 bp), Italy (8.5 bp), Japan (4 bp), Poland (12 bp), and Hungary (16 bp),
5.a. Consider the following quote from Reuters:The poor correlation between CDS and cash in Swedish utility Attentat (VTT.XE) is an anomaly and investors can benefit by setting up negative basis trades, says ING. 5-yr CDS for instance has tightened by approx. 5 bp since mid-May while the Attentat
4. In the CDS pricing example in the text we assumed a hazard rate to derive the CDS spread and to price the contract. Now assume that the hazard rate is unknown, but assume that you can observe a CDS spread of 200 bp in the market for this credit. The recovery rate is still 40%, the maturity is 5
3. Consider the CDS pricing example in Section 18.7. Assume that hazard rate is 3% instead of 5% but all other input parameters remain the same. Calculate the value of the CDS by finding the CDS spread cds that sets the expected present value of the protection leg payments equal to the expected
2. You are given two risky bonds with the following specifications:Bond Aa. Par: 100b. Currency: USDc. Coupon: 10d. Maturity: 4 yearse. Callable after 3 yearsf. Credit: AABond Ba. Par: 100b. Currency: EURc. Coupon: LIBOR 1 78 bpd. Maturity: 5 yearse. Credit: AAA
1. This exercise deals with value-at-risk calculations for credit portfolios. Using the data on a corporate financial statement, answer the following questions:a. How would you calculate the default probabilities?b. How can one obtain the migration matrix for a credit?c. How can one obtain the
5. (Interest Rate Floor Pricing). Write a VBA program to determine the price of interest rate floor which makes the payment if the floating USD LIBOR rate is below the fixed level of 4.60%. Use the data given in the “Floor Pricing Input” worksheet on the chapter webpage for the calculation.
4. (Interest Rate Cap Pricing). Write a VBA program to determine the price of an interest rate cap which makes payments if the floating USD LIBOR rate is above the fixed level of 5.00%. Use the data given in the “Cap Pricing Input” worksheet on the chapter webpage for the calculation.
2. The reading below deals with some typical swaption strategies and the factors that originate them. First, read it carefully.Lehman Brothers and Credit Suisse First Boston are recommending clients to buy long-dated swaption vol ahead of the upcoming US Federal Open Markets Committee meeting. In
1. Consider the following statement:One prop trader noted that cap/floor volatility should be slightly higher than swaptions.Corporates buy caps and investors sell swaptions through callable bonds, said one Londonbased prop trader. The market is structurally short caps and long swaptions.a.
7. (VIX calculation) Using the S&P 500 option data on the chapter website (collected on July 4, 2013 for options expiring on July 25, 2013), calculate the VIX index value following the procedure discussed in the book. Take the risk-free interest rate value to be 5% for the purpose of calculation.
6. (Stock price crashes and jumps)Write a MATLAB program to observe the implied volatility smile for the option based on the stock price having jumps (particularly crashes). Use the following data for the calculation.Sð0Þ 5 100;K 5 100; T 5 0:5; r 5 8%; σ 5 30%; λ 5 0:3; αJ 5 22%; σJ 5 5%Plot
5. (Nongeometric Brownian motion) Write a MATLAB program to observe the implied volatility smile for the option based on the stock price following the dynamics of nongeometric Brownian motion. Use the following data for the calculation.Sð0Þ 5 100;K 5 100; T 5 0:5; r 5 8%; σ 5 30%; α 5 0:8 Plot
4. (Stochastic volatility) Write a MATLAB program to observe the implied volatility smile for the option based on the stock price having stochastic volatility. Use the following data for the calculation.Sð0Þ 5 100;K 5 100; T 5 0:5; r 5 8%; σ 5 30%; σσ 5 10%;rσ 5 10%Plot the graph of the
3. Write a MATLAB program to observe the volatility payoff of stop loss hedged long call position and measure the performance of this strategy of replicating long call position when the frequency of adjustment is increased during the time interval until the call expires. Use the following data for
2. (Stop-loss hedge). Write a VBA program to show the stop-loss hedged portfolio adjustments and cash flows for 100 long calls from the dealers’ perspective with the following data:Sð0Þ 5 100;K 5 100; T 5 1; r 5 8%; σ 5 30%; & realized volatility 5 50%
1. Consider the following table displaying the bidask prices for all options on the OEX index passed on January 10, 2002, at 9:46. These options have February 22, 2002, expiry and at the time of data collection, the underlying was at 589.14.
4. (Volatility swap price). Write a MATLAB program to show the invariance of the volatility payoff of a delta hedged long call for the various frequency of delta adjustment until the time of expiry of the call. Use the following data:• S(0) 5 100; K 5 105; T 5 1; r 5 8%; σ 5 30% and realized
3. (Variance swap price). Write a MATLAB program to illustrate the variation of the price of an (equity) variance swap with respect to the change in the mean return of the underlying stock and also show this on a graph. Use the following parameters for the specification of the stochastic process
2. The following reading deals with another example of how spread positions on volatility can be taken. Yet, of interest here are further aspects of volatility positions. In fact, the episode is an example of the use of knock-in and knock-out options in volatility positions.a. Suppose the investor
1. Read the quote carefully and describe how you would take this position using volatility swaps. Be precise about the parameters of these swaps.a. How would you price this position? What does pricing mean in this context anyway?Which price are we trying to determine and write in the contract?b. In
3. Going back to the data given in Exercise 2, calculate the following:a. The bidask on a forward swap that starts in 2 years with maturity in 3 years. The swap is against 12-month LIBOR.b. The forward price of a coupon bond that will be delivered at time 2. The bond pays coupon 7% and matures in
2. You are given the following quotes for liquid swap rates. Assume that all time intervals are measured in years.
1. You are given the following quotes for liquid FRAs paid in arrears. Assume that all time intervals are measured in months of 30 days.Term Bid/Ask 3 3 6 4.54.6 6 3 9 4.74.8 9 3 12 5.05.1 12 3 15 5.55.7 15 3 18 6.16.3 You also know that the current 3-month LIBOR rate is 4%.a. Calculate the
11. (BDT Model Calibration.) Write a MATLAB program to calibrate the BDT model based on the following data on bond prices and implied volatilities• B(t0, t1) 5 0.95; B(t0, t2) 5 0.93; B(t0, t3) 5 0.91; B(t0, t4) 5 0.89• σ(0,1) 5 20%; σ(0,2) 5 25%; σ(0,3) 5 20%; σ(0,4) 5 18%Draw the LIBOR
9. (Barrier Option.) Write a MATLAB program to document the efficiency of a Monte Carlo approach to the estimation of the price of Down-and-Out and Down-and-In Call options based on the following data:• S(0) 5 100; K 5 110; T 5 1; r 5 8%; σ 5 30%; H 5 90• Plot a graph of estimated prices as a
8. (European Options.) Write a MATLAB program to document the efficiency of a Monte Carlo approach to the estimation of European Call and Put option prices based on the following data:• S(0) 5 100; K 5 105; T 5 1; r 5 8%; σ 5 30%Plot a graph of estimated prices as a function of the number of
7. (Barrier Option.) Write a VBA program to simulate M 5 100 stock prices using a Monte Carlo technique to calculate the price of Barrier down-and-out and down-and-in call options based on the following data: S(0) 5 100; K 5 110; T 5 1; r 5 8%; σ 5 50%; H 5 90 Gradually increase the value of M and
6. (Digital Currency Options.) Write a VBA program to simulate M 5 100 stock prices using a Monte Carlo technique to calculate the prices of digital call and put options FX options as discussed in the text. Use the following parameters:S(0) 5 $1.54; K 5 $1.58; T 5 1; r 5 8%; rf 5 6%; σ 5 30%;
5. (European Option.)Write a VBA program to simulate M 5 100 stock prices using a Monte Carlo technique to calculate the prices of European Call and Put options based on the following data:
4. (European FX option.) Suppose you know that the current value of the pesodollar exchange rate is 3.75 pesos per dollar. The yearly volatility of the Mexican peso is 20%.The Mexican interest rate is 8%, whereas the US rate is 3%. You will price a dollar option written on the Mexican peso. The
3. (European option.) Consider again the data given in the previous question.a. Use Δ 5 1 year to discretize the system.b. Generate five sets of standard normal random numbers with five random numbers in each set. How do you know that these five trajectories are arbitrage-free?c. Calculate the
2. (Exchange rates and LIBOR rates.) You know that the euro/dollar exchange rate et follows the real-world dynamics:det 5 μdt 1 0:15etdWt (13.152)The current value of the exchange rate is eo 5 1.1015. You also know that the price of a 1-year USD discount bond is given by Bðt; t11ÞUS 5 98:93
1. (BlackDermanToy model.) You observe the following default-free discount bond prices B(t, Ti), where time is measured in years:Bð0; 1Þ 5 95; Bð0; 2Þ 5 93; Bð0; 3Þ 5 91; Bð0; 4Þ 5 89 (13.150)These prices are assumed to be arbitrage-free. In addition, you are given the following cap-floor
14. FX European Option (Nonrecombining B-Tree)Write a VBA program to determine the initial price of European-style options on the British pound in a nonrecombining binomial tree model based on the following data:S(0) 5 $1.54; K 5 $1.54; T 5 1; r 5 8%; rf 5 6%; σ 5 30%; M 5 10 Create a function to
13. FX European Option Write a VBA program to determine the initial price of European-style options on the British pound in the binomial model based on the following data:S(0) 5 $1.54; K 5 $1.54; T 5 1; r 5 8%; rf 5 6%; σ 5 30%; M 5 10 Create a function to calculate the option price using the
12. (FX American Option)Write a VBA program to determine the initial price of an American-style options on the British pound in a binomial model based on the following data:S(0) 5 $1.54; K 5 $1.54; T 5 1; r 5 8%; rf 5 6%; σ 5 30%; M 5 10 Gradually increase the value of M and report the subsequent
11. European option on dividend paying index (nonrecombining binomial tree)Write a VBA program to determine the initial price of a European Call and European Put option in nonrecombining binomial model based on the following data:S(0) 5 100; K 5 105; T 5 1; r 5 8%; σ 5 30%; div 5 8%; M 5 10 Create
10. European option on dividend paying index Write a VBA program to determine the initial price of a European Call and European Put option in a binomial model based on the following data:S(0) 5 100; K 5 105; T 5 1; r 5 8%; σ 5 30%; div 5 8%; M 5 10 Create function to calculate the option price
9. (American Option)Write a VBA program to determine the initial price of an American Call and American Put option in a binomial model based on the following data:• S(0) 5 100; K 5 105; T 5 1; r 5 8%; σ 5 30%; div 5 8%; M 5 10 Gradually increase the value of M and report the subsequent option
8. (European Option Nonrecombining Binomial Tree)Write a VBA program to determine the initial price of a European Call and European Put option in a nonrecombining binomial tree model based on the following data:• S(0) 5 100; K 5 105; T 5 1; r 5 8%; σ 5 30%; M 5 10
7. (European Option)Write a VBA program to determine the initial price of a European Call and European Put option in a binomial model based on the following data:• S(0) 5 100; K 5 105; T 5 1; r 5 8%; σ 5 30%; M 5 10 Create a function to calculate the option price using the BlackScholes formula
6. Barrier options belong to one of four main categories. They can be up-and-out, down-andout, up-and-in, or down-and-in. In each case, there is a specified “barrier,” and when the underlying asset price down- or up-crosses this barrier, the option either expires automatically (the “out”
5. We use binomial trees to value American-style options on the British pound. Assume that the British pound is currently worth $1.40. Volatility is 20%. The current British risk-free rate is 6% and the US risk-free rate is 3%. The put option has a strike price of $1.50. It expires in 200 days.a.
4. Suppose the stock discussed in the previous exercise pays dividends. Assume all parameters are the same. Consider three forms of dividends paid by the firm:a. The stock pays a continuous, known stream of dividends at a rate of 4% per time.b. The stock pays 5% of the value of the stock at the
3. Suppose you are given the following data. The risk-free interest rate is 4%. The stock price follows:dSt 5 μSt 1 σStdWt (12.124)The percentage annual volatility is 18% a year. The stock pays no dividends and the current stock price is 100.Using these data, you are asked to calculate the
2. The current time is t 5 1 and our framework is the LIBOR model. We consider a situation with four states of the world ωi at time t 5 3.
1. The following prices of the four different stocks are reported from an arbitrage-free market at time t0 S1 t0 5 12:66; S2 t0 5 12:24; S3 t0 5 20:66; S4 t0 5 18:25 Find out the price of the 5th asset given the following possible values of these assets at time T.14 3 25 4 11 10 9 2 9 23 24 22 8 25
10. Write a MATLAB program to plot the following spreads including the time value of the spread as well as its payoff at the expiration.a. Bull spread
9. Write a VBA program to show the fabrication of the butterfly spread composed of a strangle and straddle. Assume the following parameters:S(0) 5 100; T 5 1; r 5 8%; σ 5 30%; Spread 5 5%Plot the straddle and strangle payoff along with the payoff of the butterfly spread.Repeat the above
8. Write a VBA program to show the construction of a bull and bear spread, first using calls only and then using puts only. Also plot both the call spread and put spread in both bull and bear phases. Explain your observation for the difference in the initial cost of call and put spreads.Assume the
7. The next question deals with a different type range option, called a range accrual option. Range accrual options can be used to take a view on volatility directly. When a trader is short volatility, the trader expects the actual volatility to be less than the implied volatility. Yet, within the
6. Double no-touch options is another name for range binaries. Read the following carefully, and then answer the questions at the end.Fluctuating U.S. dollar/yen volatility is prompting option traders managing their books to capture high volatilities through range binary structures while hedging
5. The following questions deal with range binaries. These are another example of exotic options. Read the following carefully and then answer the questions at the end.Investors are looking to purchase range options. The product is like a straightforward range binary in that the holder pays an
4. Consider this reading carefully and then answer the questions that follow.A bank suggested risk reversals to investors that want to hedge their Danish krone assets, before Denmark’s Sept. 28 referendum on whether to join the Economic and Monetary Union.A currency options trader in New York
3. Consider a bear spread. An investor takes a short position in a futures denoted by xt. But he or she thinks that xt will not fall below a level xmin.a. How would you create a position that trades off gains beyond a certain level against large losses if xt increases above what is expected?b. How
2. Assume that a trader believes that during the vacation periods actual realized volatility is lower than the implied volatility. To exploit this opportunity a trader takes a short position in a strangle to cover the cost of the long straddle position. If the actual volatility is 30% less than the
1. Construct a payoff and profit diagram for the purchase of a 105-strike call and sale of a 95-strike call. Verify that you obtain exactly the same profit diagram for the purchase of a 105-strike put and a sale of 95-strike put. Explain the difference in the initial cost of these positions. Assume
7. Consider the “CIR (Cox-Ingersoll-Ross)” model dr 5 αðμ 2 rÞdt 1 σOrdW t Plot the term structure (i.e., plot yield versus time) for the following parameter sets [α,μ,σ,r(0)]:½0:02; 0:7; 0:02; 0:1½0:7; 0:1; 0:3; 0:2½0:06; 0:09; 0:5; 0:02Now for the third parameter set, plot the
6. Consider the Vasicek model dr 5 αðμ 2 rÞdt 1 σ dW t Plot the term structure (i.e., plot yield versus time) for the following parameter sets [α,μ,σ,r(0)]:½5:9; 0:3; 0:2; 0:1½3:9; 0:1; 0:3; 0:2½0:1; 0:4; 0:11; 0:1
5. Answer the following questions related to the case study “Convexity of long bonds, convexity and arbitrage.”a. First the preliminaries. Explain what is meant by convexity of long-dated bonds.b. What is meant by the convexity of long-dated interest rate swaps?c. Explain the notion of
4. Assuming that the yield curve is flat and has only parallel shifts, determine the spread between the paid-in-arrear FRAs and market-traded linear FRAs if the FRA rates are expected to oscillate as follows around an initial rate:f10:02; 0:02; 1 0:02; 0:02; 1 0:02; 0:02g (10.99)
3. Consider the data given in the previous question.a. Suppose an anticipated movement as in the previous question. Market participants suddenly move to an anticipated trajectory such as f0:08; 0:02; 0:08; 0:02; 0:08; 0:02g (10.98)Assuming that this was the only exogenous change in the market, what
2. You are given a 30-year bond with yield y. The yield curve is flat and will have only parallel shifts. You have a liquid 3-month Eurodollar contract at your disposition. You can also borrow and lend at a rate of 5% initially.
1. You are given the following default-free long bond:Face value: 100 Issuing price: 100 Currency: USD Maturity: 30 years Coupon: 6%No implicit calls or puts.Further, in this market there are no bidask spreads and no trading commissions. Finally, the yield curve is flat and moves only parallel to
11. MATLAB exercise on a Delta-Hedged Portfolio Write a MATLAB program to delta hedge the portfolio consisting only of the stock and the risk-free asset to cover the long call option. Observe the number of shorted shares and the cash flow during the adjustment of the portfolio to make it delta
10. (MATLAB exercise on Barrier Option)Write a MATLAB program to determine the initial price of Barrier Down and In Call option and Barrier Down and Out Call option using the BSM formula with the following data:S(0) 5 100; K 5 105; T 5 1; r 5 8%; σ 5 30%; H 5 95 Now plot the initial price of both
9. (MATLAB exercise on Chooser Option)Write a MATLAB program to determine the initial price of chooser option using the BSM formula with the following data:S(0) 5 100; K 5 105; T0 5 0.5; T 5 1; r 5 8%; σ 5 30%Now plot the initial price of option by varying the following parameters at a time while
8. (MATLAB exercise on European Option)Write a MATLAB program to determine the initial price of European Call and European Put option using the BSM formula with the following data:S(0) 5 100; K 5 105; T 5 1; r 5 8%; σ 5 30%Now plot the initial price of both call and put options by varying the
7. Answer the questions related to the case study Swiss Central Bank, 2012:a. What is the rationale for the proposed futures strategyb. What options strategy could you consider that would also benefit from the peg?
6. Search the Internet for the following questions.a. Which sensitivities do the Greeks, volga and Vanna represent?b. Why are they relevant for vega hedging?
5. You are given the following table concerning the price of a put option satisfying all BlackScholes assumptions. The strike is 20 and the volatility is 30%. The risk-free rate is 2.5%.Option Price Underlying Asset Price 10 10 5 15 1.3 20 0.25 25 0.14 30 The option expires in 100 days. Assume
4. Consider the following episode:EUR/USD one-month implied volatility sank by 2.7% to 10% Wednesday as traders hedged this euro exposure against the greenback, as the euro plunged to historic lows on the spot market. After the European Central Bank raised interest rates by 25 basis points, the
3. Consider the following quote:Implied U.S. dollar/New Zealand dollar volatility fell to 10.1%/11.1% on Tuesday. Traders bought at-the-money options at the beginning of the week, ahead of the Federal Reserve interest-rate cut. They anticipated a rate cut which would increase short-term volatility.
2. (Delta Hedge Portfolio)Write a VBA program to show the delta-hedged portfolio adjustments and cash flows for a long call from the dealers’ perspective with the following data:Total number of long calls N 5 100 Number of adjustments M 5 15 S(0) 5 100; K 5 100; T 5 1; r 5 8%;σ 5 30%; and
1. Consider the following comment dealing with options written on the eurodollar exchange rate:Some traders, thinking that implied volatility was too high entered new trades. One example was to sell one-year in-the-money euro Puts with strikes around USD1.10 and buy one-year at-the-money euro
7. Consider the replication of a European call option. Write a VBA program to show the dynamic hedging strategy using only stocks and a saving account to replicate a short European option.Calculate the position in both the stock and the savings account for all the intermediate time points and all
6. How could you determine whether the trees in Figure 8.7 are arbitrage-free or not?
5. Consider the reading that follows which deals with the effects of straightforward delta hedging.Read the events described and then answer the questions that follow.
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