Show that the spread for a new plain vanilla CDS should be times the spread for a similar new binary CDS, where R is the recovery rate.
> Suppose that in the risk-neutral Vasicek process a=0:15, b=0:025, and =0.012. The market price of interest rate risk is -0:2. What are the risk-neutral and real-world processes for (a) the short rate and (b) a zero-coupon bond with a current maturity
> Explain the difference between a regular credit default swap and a binary credit default swap.
> An Excel spreadsheet containing over 900 days of daily data on a number of different exchange rates and stock indices can be downloaded from the author’s website: www-2.rotman.utoronto.ca/hull/data. Choose one exchange rate and one stock index. Estimate
> Explain the difference between an unconditional default probability density and a hazard rate.
> How are recovery rates usually defined?
> Assume that S&P 500 at close of trading yesterday was 1,040 and the daily volatility of the index was estimated as 1% per day at that time. The parameters in a GARCH(1,1) model are , , . If the level of the index at close of trading today is 1,060, wh
> The most recent estimate of the daily volatility of the U.S. dollar/sterling exchange rate is 0.6% and the exchange rate at 4 p.m. yesterday was 1.5000. The parameter in the EWMA model is 0.9. Suppose that the exchange rate at 4 p.m. today proves to be
> Should researchers use real-world or risk-neutral default probabilities for (a) calculating credit value at risk and (b) adjusting the price of a derivative for defaults?
> The volatility of a certain market variable is 30% per annum. Calculate a 99% confidence interval for the size of the percentage daily change in the variable.
> A company uses an EWMA model for forecasting volatility. It decides to change the parameter from 0.95 to 0.85. Explain the likely impact on the forecasts.
> The most recent estimate of the daily volatility of an asset is 1.5% and the price of the asset at the close of trading yesterday was $30.00. The parameter in the EWMA model is 0.94. Suppose that the price of the asset at the close of trading today is $
> Suppose that in Problem 24.1 the spread between the yield on a 5-year bond issued by the same company and the yield on a similar risk-free bond is 60 basis points. Assume the same recovery rate of 30%. Estimate the average hazard rate per year over the 5
> The spread between the yield on a 3-year corporate bond and the yield on a similar risk-free bond is 50 basis points. The recovery rate is 30%. Estimate the average hazard rate per year over the 3-year period.
> Suppose that in Problem 23.17 the price of silver at the close of trading yesterday was $16, its volatility was estimated as 1.5% per day, and its correlation with gold was estimated as 0.8. The price of silver at the close of trading today is unchanged
> Suppose that the market price of risk for gold is zero. If the storage costs are 1% per annum and the risk-free rate of interest is 6% per annum, what is the expected growth rate in the price of gold? Assume that gold provides no income.
> ‘‘If X is the expected value of a variable, X follows a martingale.’’ Explain this statement.
> Explain the difference between the way a forward interest rate is defined and the way the forward values of other variables such as stock prices, commodity prices, and exchange rates are defined.
> The variable S is an investment asset providing income at rate q measured in currency A. It follows the process in the real world. Defining new variables as necessary, give the process followed by S, and the corresponding market price of risk, in: (a
> ‘‘The IVF model correctly values any derivative whose payoff depends on the value of the underlying asset at only one time.’’ Explain why.
> At time 0 the price of a non-dividend-paying stock is S0. Suppose that the time interval between 0 and T is divided into two subintervals of length t1 and t2. During the first subinterval, the risk-free interest rate and volatility are r1 and , respecti
> Consider the case of Merton’s jump–diffusion model where jumps always reduce the asset price to zero. Assume that the average number of jumps per year is . Show that the price of a European call option is the same as in a world with no jumps except that
> Confirm that Merton’s jump–diffusion model satisfies put–call parity when the jump size is lognormal.
> What is Merton’s mixed jump–diffusion model price for a European call option when r =5%, q =0, =0:3, k = 50%, =25%, S0= 30, K= 30, s = 50%, and T = 1. Use DerivaGem to check your price.
> Consider an 18-month zero-coupon bond with a face value of $100 that can be converted into five shares of the company’s stock at any time during its life. Suppose that the current share price is $20, no dividends are paid on the stock, the risk-free rate
> Consider a variable that is not an interest rate: (a) In what world is the futures price of the variable a martingale? (b) In what world is the forward price of the variable a martingale? (c) Defining variables as necessary, derive an expression for the
> When there are two barriers how can a tree be designed so that nodes lie on both barriers?
> Consider a European put option on a non-dividend paying stock when the stock price is $100, the strike price is $110, the risk-free rate is 5% per annum, and the time to maturity is one year. Suppose that the average variance rate during the life of an o
> Examine the early exercise policy for the eight paths considered in the example in Section 27.8. What is the difference between the early exercise policy given by the least squares approach and the exercise boundary parameterization approach? Which gives
> Verify that the 6.492 number in Figure 27.3 is correct. Figure 27.3 Part of tree for valuing option on the arithmetic average. S=54.68 Average S Option price 47.99 7.575 51.12 8.101 54,26 Y 57.39 8.635 9.178 S- 50.00 Average S Option price 46.65 5.6
> Can the approach for valuing path-dependent options in Section 27.5 be used for a 2-year American-style option that provides a payoff equal to , where is the average asset price over the three months preceding exercise? Explain your answer.
> Use a three-time-step tree to value an American put option on the geometric average of the price of a non-dividend-paying stock when the stock price is $40, the strike price is $40, the risk-free interest rate is 10% per annum, the volatility is 35% per
> What happens to the variance-gamma model as the parameter v tends to zero?
> Use a three-time-step tree to value an American floating lookback call option on a currency when the initial exchange rate is 1.6, the domestic risk-free rate is 5% per annum, the foreign risk-free interest rate is 8% per annum, the exchange rate volatil
> How can the value of a forward start put option on a non-dividend-paying stock be calculated if it is agreed that the strike price will be 10% greater than the stock price at the time the option starts?
> Suppose that the strike price of an American call option on a non-dividend-paying stock grows at rate g. Show that if g is less than the risk-free rate, r, it is never optimal to exercise the call early.
> Consider a chooser option where the holder has the right to choose between a European call and a European put at any time during a 2-year period. The maturity dates and strike prices for the calls and puts are the same regardless of when the choice is ma
> In a 3-month down-and-out call option on silver futures the strike price is $20 per ounce and the barrier is $18. The current futures price is $19, the risk-free interest rate is 5%, and the volatility of silver futures is 40% per annum. Explain how the
> Explain why a regular European call option is the sum of a down-and-out European call and a down-and-in European call. Is the same true for American call options?
> Does a down-and-out call become more valuable or less valuable as we increase the frequency with which we observe the asset price in determining whether the barrier has been crossed? What is the answer to the same question for a down-and-in call?
> Answer the following questions about compound options: (a) What put–call parity relationship exists between the price of a European call on a call and a European put on a call? Show that the formulas given in the text satisfy the relationship. (b) What p
> If a stock price follows geometric Brownian motion, what process does follow where is the arithmetic average stock price between time zero and time t?
> Suppose that the risk-free zero curve is flat at 7% per annum with continuous compounding and that defaults can occur halfway through each year in a new 5-year credit default swap. Suppose that the recovery rate is 30% and the hazard rate is 3%. Estimate
> Explain why a total return swap can be useful as a financing tool.
> Explain the difference between base correlation and compound correlation.
> Suppose that in a one-factor Gaussian copula model the 5-year probability of default for each of 125 names is 3% and the pairwise copula correlation is 0.2. Calculate, for factor values of -2, -1, 0, 1, and 2: (a) the default probability conditional on
> A security’s price is positively dependent on two variables: the price of copper and the yen/dollar exchange rate. Suppose that the market price of risk for these variables is 0.5 and 0.1, respectively. If the price of copper were held fixed, the volati
> ‘‘The position of a buyer of a credit default swap is similar to the position of someone who is long a risk-free bond and short a corporate bond.’’ Explain this statement.
> Explain how forward contracts and options on credit default swaps are structured.
> Verify that, if the CDS spread for the example in Tables 25.1 to 25.4 is 100 basis points, the hazard rate must be 1.63% per year. How does the hazard rate change when the recovery rate is 20% instead of 40%? Verify that your answer is consistent with th
> How does a 5-year nth-to-default credit default swap work? Consider a basket of 100 reference entities where each reference entity has a probability of defaulting in each year of 1%. As the default correlation between the reference entities increases wha
> What is the credit default swap spread in Problem 25.8 if it is a binary CDS? Data from Problem 25.8: Suppose that the risk-free zero curve is flat at 7% per annum with continuous compounding and that defaults can occur halfway through each year in a n
> Explain the difference between the Gaussian copula model for the time to default and CreditMetrics as far as the following are concerned: (a) the definition of a credit loss and (b) the way in which default correlation is modeled.
> Verify (a) that the numbers in the second column of Table 24.3 are consistent with the numbers in Table 24.1 and (b) that the numbers in the fourth column of Table 24.4 are consistent with the numbers in Table 24.3 and a recovery rate of 40%. Tabl
> ‘‘A long forward contract subject to credit risk is a combination of a short position in a no-default put and a long position in a call subject to credit risk.’’ Explain this statement.
> A company has issued 3- and 5-year bonds with a coupon of 4% per annum payable annually. The yields on the bonds (expressed with continuous compounding) are 4.5% and 4.75%, respectively. Risk-free rates are 3.5% with continuous compounding for all maturi
> In an annual-pay cap, the Black volatilities for at-the-money caplets which start in 1, 2, 3, and 5 years and end 1 year later are 18%, 20%, 22%, and 20%, respectively. Estimate the volatility of a 1-year forward rate in the LIBOR Market Model when the t
> Give an example of (a) right-way risk and (b) wrong-way risk.
> A 4-year corporate bond provides a coupon of 4% per year payable semiannually and has a yield of 5% expressed with continuous compounding. The risk-free yield curve is flat at 3% with continuous compounding. Assume that defaults can take place at the end
> Show that the value of a coupon-bearing corporate bond is the sum of the values of its constituent zero-coupon bonds when the amount claimed in the event of default is the no-default value of the bond, but that this is not so when the claim amount is the
> Suppose that in an asset swap B is the market price of the bond per dollar of principal, is the default-free value of the bond per dollar of principal, and V is the present value of the asset swap spread per dollar of principal. Show that .
> Does put–call parity hold when there is default risk? Explain your answer.
> ‘‘When a bank is negotiating currency swaps, it should try to ensure that it is receiving the lower interest rate currency from companies with low credit risk.’’ Explain why.
> Explain why the impact of credit risk on a matched pair of interest rate swaps tends to be less than that on a matched pair of currency swaps.
> Suppose that the LIBOR/swap curve is flat at 6% with continuous compounding and a 5-year bond with a coupon of 5% (paid semiannually) sells for 90.00. How would an asset swap on the bond be structured? What is the asset swap spread that would be calculat
> At the end of Section 23.8, the VaR and ES for the four-index example were calculated using the model-building approach. How do the VaR and ES estimates change if the investment is $2.5 million in each index? Carry out calculations when (a) volatilities
> Show that the GARCH (1,1) model 1in equation (23.9) is equivalent to the stochastic volatility model , where time is measured in days, V is the square of the volatility of the asset price, and What is the stochastic volatility model when time is measured
> The payoff from a derivative will occur in 8 years. It will equal the average of the 1-year risk-free interest rates observed at times 5, 6, 7, and 8 years applied to a principal of $1,000. The risk-free yield curve is flat at 6% with annual compounding
> Suppose that in Problem 23.12 the correlation between the S&P 500 Index (measured in dollars) and the FTSE 100 Index (measured in sterling) is 0.7, the correlation between the S&P 500 Index (measured in dollars) and the dollar/sterling exchange rate is 0
> Suppose that the daily volatility of the FTSE 100 stock index (measured in pounds sterling) is 1.8% and the daily volatility of the dollar/sterling exchange rate is 0.9%. Suppose further that the correlation between the FTSE 100 and the dollar/sterling e
> Suppose that the current daily volatilities of asset X and asset Y are 1.0% and 1.2%, respectively. The prices of the assets at close of trading yesterday were $30 and $50 and the estimate of the coefficient of correlation between the returns on the two
> The parameters of a GARCH(1,1) model are estimated as , , and. What is the long-run average volatility and what is the equation describing the way that the variance rate reverts to its long-run average? If the current volatility is 20% per year, what
> Suppose that the daily volatilities of asset A and asset B, calculated at the close of trading yesterday, are 1.6% and 2.5%, respectively. The prices of the assets at close of trading yesterday were $20 and $40 and the estimate of the coefficient of corr
> ‘‘DVA can improve the bottom line when a bank is experiencing financial difficulties.’’ Explain why this statement is true.
> Describe how netting works. A bank already has one transaction with a counterparty on its books. Explain why a new transaction by a bank with a counterparty can have the effect of increasing or reducing the bank’s credit exposure to the counterparty.
> A driver entering into a car lease agreement can obtain the right to buy the car in 4 years for $10,000. The current value of the car is $30,000. The value of the car, S, is expected to follow the process where , , and dz is a Wiener process. The ma
> Show that if y is a commodity’s convenience yield and u is its storage cost, the commodity’s growth rate in the traditional risk-neutral world is r − y + u, where r is the risk-free rate. Deduce the relationship between the market price of risk of the co
> The market price of risk for copper is 0.5, the volatility of copper prices is 20% per annum, the spot price is 80 cents per pound, and the 6-month futures price is 75 cents per pound. What is the expected percentage growth rate in copper prices over the
> Suppose that the payoff from a derivative will occur in 10 years and will equal the 3-year U.S. dollar swap rate for a semiannual-pay swap observed at that time applied to a certain principal. Assume that the swap yield curve is flat at 8% (semiannually
> Explain the difference between the net present value approach and the risk-neutral valuation approach for valuing a new capital investment opportunity. What are the advantages of the risk-neutral valuation approach for valuing real options?
> Suppose that you have 50 years of temperature data at your disposal. Explain carefully the analyses you would carry out to value a forward contract on the cumulative CDD for a particular month.
> Distinguish between the historical data and the risk-neutral approach to valuing a derivative. Under what circumstance do they give the same answer?
> Consider two bonds that have the same coupon, time to maturity, and price. One is a B-rated corporate bond. The other is a CAT bond. An analysis based on historical data shows that the expected losses on the two bonds as a function of time are the same.
> Explain how CAT bonds work.
> Explain how a option contract for May 2017 on electricity with daily exercise works. Explain how a option contract for May 2017 on electricity with monthly exercise works. Which is worth more?
> In the accrual swap discussed in the text, the fixed side accrues only when the floating reference rate lies below a certain level. Discuss how the analysis can be extended to cope with a situation where the fixed side accrues only when the floating refe
> Calculate the total convexity/timing adjustment in Example 34.3 of Section 34.4 if all cap volatilities are 18% instead of 20% and volatilities for all options on 5-year swaps are 13% instead of 15%. What should the 5-year swap rate in 3 yearsâ
> Calculate all the fixed cash flows and their exact timing for the swap in Business Snapshot 34.1. Assume that the day count conventions are applied using target payment dates rather than actual payment dates. Business Snapshot 34.1 Hypothetical Conf
> Explain why a sticky cap is more expensive than a similar ratchet cap.
> Suppose that the risk-free yield curve is flat at 8% (with continuous compounding). The payoff from a derivative occurs in 4 years. It is equal to the 5-year rate minus the 2-year rate at this time, applied to a principal of $100 with both rates being co
> Provide an intuitive explanation of why a ratchet cap increases in value as the number of factors increase.
> ‘‘When the forward rate volatility, , in HJM is , the Hull–White model results.’’ Verify that this is true by showing that HJM gives a process for bond prices that is consistent with the Hull–White model in Chapter 32.
> ‘‘When the forward rate volatility in HJM is constant, the Ho–Lee model results.’’ Verify that this is true by showing that HJM gives a process for bond prices that is consistent with the Ho–Lee model in Chapter 32.
> ‘‘An option adjusted spread is analogous to the yield on a bond.’’ Explain this statement.
> Suppose a = 0:05, , and the term structure is flat at 10%. Construct a trinomial tree for the Hull–White model where there are two-time steps, each 1 year in length.
> Suppose that a = 0:05 and in the Hull–White model with the initial term structure being flat at 6% with semiannual compounding. Calculate the price of a2.1-year European call option on a bond that will mature in 3 years. Suppose that the bond pays a cou
> Suppose that a =0:05, b =0:08, and in Vasicek’s model with the initial shortterm interest rate being 6%. Calculate the price of a 2.1-year European call option on a bond that will mature in 3 years. Suppose that the bond pays a coupon of 5% semiannually
> Repeat Problem 32.3 valuing a European put option with a strike of $87. What is the put–call parity relationship between the prices of European call and put options? Show that the put and call option prices satisfy put–
> Calculate the price of an 18-month zero-coupon bond from the tree in Figure 32.8 and verify that it agrees with the initial term structure. Figure 32.8 Tree for lognormal model. Node: A в с B E н D F H I -3.373 -2.875 -3.181 -3.487 -2.430 -2.736 -3.
> Calculate the price of a 2-year zero-coupon bond from the tree in Figure 32.7 and verify that it agrees with the initial term structure. Figure 32.7 Tree for R in Hull-White model (the second stage). H A Node: A B D E F G H I R (%) 3.824 6.937 5.205
> Consider an instrument that will pay off S dollars in 2 years, where S is the value of the Nikkei index. The index is currently 20,000. The yen/dollar exchange rate is 100 (yen per dollar). The correlation between the exchange rate and the index is 0.3 a
