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Corporate Finance
The spot price of copper is $0.60 per pound. Suppose that the futures prices (dollars per pound) are as follows:3 months...............................0.596 months...............................0.579
Calculate the price of a three-month American put option on a non-dividend-paying stock when the stock price is $60, the strike price is $60, the risk-free interest rate is 10% per annum, and the
Use the binomial tree in Problem 21.19 to value a security that pays off x2 in one year where x is the price of copper.
When do the boundary conditions for S = 0 and S→∞ affect the estimates of derivative prices in the explicit finite difference method?
How would you use the antithetic variable method to improve the estimate of the European option in Business Snapshot 21.2 and Table 21.2?
A company has issued a three-year convertible bond that has a face value of $25 and can be exchanged for two of the company's shares at any time. The company can call the issue, forcing conversion,
Provide formulas that can be used for obtaining three random samples from standard normal distributions when the correlation between sample i and sample j is ρij.
The current value of the British pound is $1.60 and the volatility of the pound-dollar exchange rate is 15% per annum. An American call option has an exercise price of $1.62 and a time to maturity
Answer the following questions concerned with the alternative procedures for constructing trees in Section 21.4.(a) Show that the binomial model in Section 21.4 is exactly consistent with the mean
Explain how the control variate technique is implemented when a tree is used to value American options.
How much is gained from exercising early at the lowest node at the nine-month point in Example 21.4?
Calculate the price of a nine-month American call option on corn futures when the current futures price is 198 cents, the strike price is 200 cents, the risk-free interest rate is 8% per annum, and
"For a dividend-paying stock, the tree for the stock price does not recombine; but the tree for the stock price less the present value of future dividends does recombine." Explain this statement.
Show that the probabilities in a Cox, Ross, and Rubinstein binomial tree are negative when the condition in footnote 8 holds.
Use stratified sampling with 100 trials to improve the estimate of π in Business Snapshot 21.1 and Table 21.1.
Explain why the Monte Carlo simulation approach cannot easily be used for American-style derivatives.
The text calculates a VaR estimate for the example in Table 22.9 assuming two factors. How does the estimate change if you assume (a) one factor and (b) three factors.
A bank has a portfolio of options on an asset. The delta of the options is -30 and the gamma is -5. Explain how these numbers can be interpreted. The asset price is 20 and its volatility per day is
Suppose that in Problem 22.12 the vega of the portfolio is -2 per 1% change in the annual volatility. Derive a model relating the change in the portfolio value in one day to delta, gamma, and vega.
The one-day 99% VaR is calculated for the four-index example in Section 22.2 as $253,385. Look at the underlying spreadsheets on the author's website and calculate the a) the one-day 95% VaR, b) the
Describe three ways of handling interest-rate-dependent instruments when the model building approach is used to calculate VaR. How would you handle interest-rate-dependent instruments when historical
Use equation (22.1) to show that when the loss distribution is normal, VaR with 99% confidence is almost exactly the same as ES with 97.5% confidence.
A financial institution owns a portfolio of options on the U.S. dollar-sterling exchange rate. The delta of the portfolio is 56.0. The current exchange rate is 1.5000. Derive an approximate linear
Suppose you know that the gamma of the portfolio in the previous question is 16.2. How does this change your estimate of the relationship between the change in the portfolio value and the percentage
Suppose that the daily change in the value of a portfolio is, to a good approximation, linearly dependent on two factors, calculated from a principal components analysis. The delta of a portfolio
Suppose a company has a portfolio consisting of positions in stocks and bonds Assume there are no derivatives. Explain the assumptions underlying (a) the linear model and (b) the historical
Explain how a forward contract to sell foreign currency is mapped into a portfolio of zero-coupon bonds with standard maturities for the purposes of a VaR calculation.
Explain the exponentially weighted moving average (EWMA) model for estimating volatility from historical data.
The parameters of a GARCH(1,1) model are estimated as ω = 0.000004, α = 0.05, and β = 0.92. What is the long-run average volatility and what is the equation describing the way that the variance
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
Show that the GARCH (1,1) model in equation (23.9) is equivalent to the stochastic volatility model where time is measured in days and is the square of the volatility of the asset
What is the difference between the exponentially weighted moving average model and the GARCH(1,1) model for updating volatilities?
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 λ in the EWMA model is 0.94. Suppose that
A company uses the GARCH(1,1) model for updating volatility. The three parameters are ω, α, and β. Describe the impact of making a small increase in each of the parameters while keeping the others
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 ω = 0.000002,
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
The spread between the yield on a three-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
Suppose that the LIBOR/swap curve is flat at 6% with continuous compounding and a five-year bond with a coupon of 5% (paid semiannually) sells for 90.00. How would an asset swap on the bond be
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
A four-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
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
Suppose that a financial institution has entered into a swap dependent on the sterling interest rate with counterparty X and an exactly offsetting swap with counterparty Y. Which of the following
"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.
Why does the credit exposure on a matched pair of forward contracts resembles a straddle?
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.
"When a bank is negotiating currency swaps, it should try to ensure that it is receiving the lower interest rate currency from a company with a low credit risk." Explain.
Does put-call parity hold when there is default risk? Explain your answer.
Suppose that in Problem 24.1 the spread between the yield on a five-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
Suppose that in an asset swap B is the market price of the bond per dollar of principal, B* is the default-free value of the bond per dollar of principal, and V is the present value of the asset swap
Show that under Merton’s model in Section 24.6 the credit spread on a T-year zero-coupon bond is –ln[N(d2) + N(–d1)/L] / T where L = De–rT / V0.
Suppose that the spread between the yield on a 3-year zero-coupon riskless bond and a 3-year zero-coupon bond issued by a corporation is 1%. By how much does Black-Scholes-Merton overstate the value
Give an example of a) right-way risk and b) wrong-way risk.
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.
The LIBOR/swap curve is flat at 3% with continuous compounding and a 4-year with a coupon of 4% per annum (paid semiannually) sells for 101. How would an asset swap on the bond be structured? What is
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?
Explain the difference between an unconditional default probability density and a hazard rate.
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
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
"DVA can improve the bottom line when a bank is experiencing financial difficulty." Explain why this statement is true.
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 lossb) The way in which default
Explain the difference between a regular credit default swap and a binary credit default swap.
A company enters into a total return swap where it receives the return on a corporate bond paying a coupon of 5% and pays LIBOR. Explain the difference between this and a regular swap where 5% is
Why is there a potential asymmetric information problem in credit default swaps?
Does valuing a CDS using real-world default probabilities rather than risk-neutral default probabilities overstate or understate its value? Explain your answer.
A five-year credit default swap requires quarterly payments at the rate of 60 basis points per year. The principal is $300 million and the credit default swap is settled in cash. A default occurs
What is the difference between a total return swap and an asset swap?
Suppose that in a one-factor Gaussian copula model the five-year probability of default for each of 125 names is 3% and the pair wise copula correlation is 0.2. Calculate, for factor values of -2,
Explain the difference between base correlation and compound correlation
Assume that the hazard rate for a company is λ and the recovery rate is R. The risk-free interest rate is 5% per annum. Default always occurs half way through a year. The spread for a five-year
In Example 25.3, what is the spread for a) a first-to-default CDS and b) a second-to-default CDS?
Explain the difference between a cash CDO and a synthetic CDO.
Explain the term "single tranche trading."
What is a first-to-default credit default swap? Does its value increase or decrease as the default correlation between the companies in the basket increases? Explain your answer..
Explain the difference between risk-neutral and real-world default probabilities. Which should be used for valuing CDSs.
Explain why a total return swap can be useful as a financing tool.
Explain the difference between a forward start option and a chooser option.
If a stock price follows geometric Brownian motion, what process does A(t) follow where A(t) is the arithmetic average stock price between time zero and time t?
Explain why delta hedging is easier for Asian options than for regular options.
Calculate the price of a one-year European option to give up 100 ounces of silver in exchange for one ounce of gold. The current prices of gold and silver are $1520 and $16, respectively; the
Is a European down-and-out option on an asset worth the same as a European down-and-out option on the asset's futures price for a futures contract maturing at the same time as the option?
Does a floating lookback call become more valuable or less valuable as we increase the frequency with which we observe the asset price in calculating the minimum?
In a three-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
A new European-style floating lookback call option on a stock index has a maturity of nine months. The current level of the index is 400, the risk-free rate is 6% per annum, the dividend yield on the
Estimate the value of a new six-month European-style average price call option on a non-dividend-paying stock. The initial stock price is $30, the strike price is $30, the risk-free interest rate is
Explain adjustments that have to be made when r = q for a) the valuation formulas for lookback call options in Section 26.11 and b) the formulas for M1 and M2 in Section 26.13.
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%,
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.
Sample Application F in the DerivaGem Application Builder Software considers the static options replication example in Section 26.17. It shows the way a hedge can be constructed using four options
Consider a down-and-out call option on a foreign currency. The initial exchange rate is 0.90, the time to maturity is two years, the strike price is 1.00, the barrier is 0.80, the domestic
Suppose that a stock index is currently 900. The dividend yield is 2%, the risk-free rate is 5%, and the volatility is 40%. Use the results in the Technical Note 27 to calculate the value of a
Use the DerivaGem Application Builder software to compare the effectiveness of daily delta hedging for (a) the option considered in Tables 19.2 and 19.3 and (b) an average price call with the same
In the DerivaGem Application Builder Software modify Sample Application D to test the effectiveness of delta and gamma hedging for a call on call compound option on a 100,000 units of a foreign
Out performance certificates (also called "sprint certificates", "accelerator certificates", or "speeders") are offered to investors by many European banks as a way of investing in a company's stock.
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.
Produce a formula for valuing a cliquet option where an amount Q is invested to produce a payoff at the end of n periods. The return each period is the greater of the return on an index (excluding
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
Explain why a down-and-out put is worth zero when the barrier is greater than the strike price.
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
Confirm that the CEV model formulas satisfy put-call parity.
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
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