Why is there a potential asymmetric information problem in credit default swaps?
> Suppose that a =0:1 and b=0:1 and in both the Vasicek and the Cox, Ingersoll, Ross model. In both models, the initial short rate is 10% and the initial standard deviation of the short-rate change in a short time is . Compare the prices given by the mo
> Suppose that the risk-free zero curve is flat at 6% per annum with continuous compounding and that defaults can occur at times 0.25 years, 0.75 years, 1.25 years, and 1.75 years in a 2-year plain vanilla credit default swap with semiannual payments. Supp
> Observations spaced at intervals t are taken on the short rate. The ith observation is ri (1 ≤ i ≤ m). Show that the maximum-likelihood estimates of a, b*, and in Vasicek’s model are given by maximizing
> a) What is the second partial derivative of P(t,T) with respect to r in the Vasicek and CIR models? (b) In Section 31.2, is presented as an alternative to the usual duration measure, D. What is a similar alternative, , to the convexity measure in Secti
> Explain whether any convexity or timing adjustments are necessary when: (a) We wish to value a spread option that pays off every quarter the excess (if any) of the 5-year swap rate over the 3-month LIBOR rate applied to a principal of $100. The payoff oc
> If the yield volatility for a 5-year put option on a bond maturing in 10 years’ time is specified as 22%, how should the option be valued? Assume that, based on today’s interest rates the modified duration of the bond at the maturity of the option will
> Calculate the price of an option that caps the 3-month rate, starting in 15 months’ time, at 13% (quoted with quarterly compounding) on a principal amount of $1,000. The forward interest rate for the period in question is 12% per annum (quoted with quart
> Use the Black’s model to value a 1-year European put option on a 10-year bond. Assume that the current cash price of the bond is $125, the strike price is $110, the 1-year risk-free interest rate is 10% per annum, the bond’s forward price volatility is 8
> Suppose that the yield R on a zero-coupon bond follows the process where and are functions of R and t, and dz is a Wiener process. Use Itoˆ ’s lemma to show that the volatility of the zero-coupon bond price declines to zero as it approaches maturity.
> When a bond’s price is lognormal can the bond’s yield be negative? Explain your answer.
> A company caps 3-month LIBOR at 2% per annum. The principal amount is $20 million. On a reset date, 3-month LIBOR is 4% per annum. What payment would this lead to under the cap? When would the payment be made?
> Suppose that an interest rate x follows the process where a, x0, and c are positive constants. Suppose further that the market price of risk for x is . What is the process for x in the traditional risk-neutral world?
> A new European-style floating lookback call option on a stock index has a maturity of 9 months. The current level of the index is 400, the risk-free rate is 6% per annum, the dividend yield on the index is 4% per annum, and the volatility of the index is
> Consider two securities both of which are dependent on the same market variable. The expected returns from the securities are 8% and 12%. The volatility of the first security is 15%. The instantaneous risk-free rate is 4%. What is the volatility of the s
> Show that when w = h/g and h and g are each dependent on n Wiener processes, the ith component of the volatility of w is the ith component of the volatility of h minus the ith component of the volatility of g. Use this to prove the result that if is t
> Prove the result in Section 28.5 that when and with the dzi uncorrelated, f/g is a martingale for (Hint: Start by using equation (14A.11) to get the processes for ln f and ln g.)
> How is the market price of risk defined for a variable that is not the price of an investment asset?
> ‘‘The IVF model does not necessarily get the evolution of the volatility surface correct.’’ Explain this statement.
> Write down the equations for simulating the path followed by the asset price in the stochastic volatility model in equations (27.2) and (27.3). ds/s = (r - q) dt + Vī dzs dV = a(V1 – V)dt +§Vª dzv (27.2) (27.3) %3D
> Suppose that the volatility of an asset will be 20% from month 0 to month 6, 22% from month 6 to month 12, and 24% from month 12 to month 24. What volatility should be used in Black–Scholes–Merton to value a 2-year option?
> Confirm that the CEV model formulas satisfy put–call parity.
> Explain why a down-and-out put is worth zero when the barrier is greater than the strike price.
> Section 26.9 gives two formulas for a down-and-out call. The first applies to the situation where the barrier, H, is less than or equal to the strike price, K. The second applies to the situation where . Show that the two formulas are the same when H= K
> Estimate parameters for EWMA and GARCH(1, 1) from data on the euro–USD exchange rate between July 27, 2005, and July 27, 2010. This data can be found on the author’s website: www-2.rotman.utoronto.ca/hull/data.
> The text derives a decomposition of a particular type of chooser option into a call maturing at time T2 and a put maturing at time T1. Derive an alternative decomposition into a call maturing at time T1 and a put maturing at time T2
> Suppose that c1 and p1 are the prices of a European average price call and a European average price put with strike price K and maturity T, c2 and p2 are the prices of a European average strike call and European average strike put with maturity T, and c3
> Verify that the results in Section 26.2 for the value of a derivative that pays Q when S = H are consistent with those in Section 15.6.
> Value the variance swap in Example 26.4 of Section 26.16 assuming that the implied volatilities for options with strike prices 800, 850, 900, 950, 1,000, 1,050, 1,100, 1,150, 1,200 are 20%, 20.5%, 21%, 21.5%, 22%, 22.5%, 23%, 23.5%, 24%, respectively. /
> Explain adjustments that have to be made when r = q for (a) the valuation formulas for floating lookback call options in Section 26.11 and (b) the formulas for M1 and M2 in Section 26.13.
> What is the relationship between a regular call option, a binary call option, and a gap call option?
> Carry out the analysis in Example 26.4 of Section 26.16 to value the variance swap on the assumption that the life of the swap is 1 month rather than 3 months. //
> The 1-, 2-, 3-, 4-, and 5-year CDS spreads are 100, 120, 135, 145, and 152 basis points, respectively. The risk-free rate is 3% for all maturities, the recovery rate is 35%, and payments are quarterly. Use DerivaGem to calculate the hazard rate each year
> Calculate DVA in Example 24.6. Assume that default can happen in the middle of each month. The default probability of the bank is 0.001 per month for the two years and the recovery rate in the event of a bank default is 40%. Example 24.6 A bank has
> Extend Example 24.6 to calculate CVA when default can happen in the middle of each month. Assume that the default probability per month during the first year is 0.001667 and the default probability per month during the second year is 0.0025. //
> The calculations for the four-index example at the end of Section 23.8 assume that the investments in the DJIA, FTSE 100, CAC 40, and Nikkei 225 are $4 million, $3 million, $1 million, and $2 million, respectively. How do the VaR and ES estimates change
> Suppose that a bank has a total of $10 million of exposures of a certain type. The 1-year probability of default averages 1% and the recovery rate averages 40%. The copula correlation parameter is 0.2. Estimate the 99.5% 1-year credit VaR.
> In Example 25.2, what is the tranche spread for the 6% to 9% tranche assuming a tranche correlation of 0.15? Example 25.2 Consider the mezzanine tranche of iTraxx Europe (5-year maturity) when the copula correlation is 0.15 and the recovery rate is
> Calculate the price of a 1-year European option to give up 100 ounces of silver in exchange for 1 ounce of gold. The current prices of gold and silver are $1,520 and $16, respectively; the risk-free interest rate is 10% per annum; the volatility of each
> Explain why delta hedging is easier for Asian options than for regular options.
> Show that, if there is no recovery from the bond in the event of default, a convertible bond can be valued by assuming that (a) both the expected return and discount rate are r+and (b) there is no chance of default.
> Explain the difference between risk-neutral and real-world default probabilities. Which should be used for valuing CDSs?
> The credit spreads for 1-, 2-, 3-, 4-, and 5-year zero-coupon bonds are 50, 60, 70, 80, and 87 basis points, respectively. The recovery rate is 35%. Estimate the average hazard rate each year.
> In Example 25.2, what is the tranche spread for the 9% to 12% tranche assuming a tranche correlation of 0.15? //
> Calculate the price of a cap on the 90-day LIBOR rate in 9 months’ time when the principal amount is $1,000. Use Black’s model with LIBOR discounting and the following information: (a) The quoted 9-month Eurodollar futures price =92. (Ignore differences
> Modify Sample Application G in the DerivaGem Application Builder software to test the convergence of the price of the trinomial tree when it is used to price a 2-year call option on a 5-year bond with a face value of 100. Suppose that the strike price (q
> Suppose that the parameters in a GARCH (1,1) model are , , and . (a) What is the long-run average volatility? (b) If the current volatility is 1.5% per day, what is your estimate of the volatility in 20, 40, and 60 days? (c) What volatility should be
> What is the difference between the exponentially weighted moving average model and the GARCH(1,1) model for updating volatilities?
> Technical Note 13 at www-2.rotman.utoronto.ca/hull/Technical Notes provides a different approach to valuing lookbacks. Value the lookback in Problem 27.19 using this approach. Show that it gives the same answer as the approach in Section 27.5.
> Does valuing a CDS using real-world default probabilities rather than risk-neutral default probabilities overstate or understate its value? Explain your answer.
> Suppose that the price of gold at close of trading yesterday was $600 and its volatility was estimated as 1.3% per day. The price at the close of trading today is $596. Update the volatility estimate using (a) The EWMA model with (b) The GARCH(1,1) mod
> What is the effect of changing from 0.94 to 0.97 in the EWMA calculations in the four index example at the end of Section 23.8. Use the spreadsheets on the author’s website.
> In the two-factor extension of Vasicek given in Section 31.5, derive the differential equations which must be satified by a bond price, P(t,T). Use this to derive differential equations that must be satisfied by A(t,T),, B(t,T),, and C(t,T) in P(t,T)=A(t
> Suppose that the market price of risk of the short rate is . Show that if the real-world process for the short rate is the one assumed by CIR, the risk-neutral process has the same functional form. Derive the relationship between (a) the real-world rev
> Suppose that in a risk-neutral world the CIR parameters are a =0:15, b = 0:025, and =0.075. What is the price of a 5-year zero-coupon bond with a principal of $1 when the short rate is 2.5%?
> 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
