Question: Use stratified sampling with 100 trials to
Use stratified sampling with 100 trials to improve the estimate of in Business Snapshot 21.1 and Table 21.1.
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Business Snapshot 21.1 Calculating Pi with Monte Carlo Simulation Suppose the sides of the square in Figure 21.13 are one unit in length. Imagine that you fire darts randomly at the square and calculate the percentage that lie in the circle. What should you find? The square has an area of 1.0 and the circle has a radius of 0.5 The area of the circle is a times the radius squared or 1/4. It follows that the proportion of darts that lie in the circle should be 1/4. We can estimate a by multiplying the proportion that lie in the circle by 4. We can use an Excel spreadsheet to simulate the dart throwing as illustrated in Table 21.1. We define both cell Al and cell B1 as =RAND(). Al and B1 are random numbers between 0 and 1 and define how far to the right and how high up the dart lands in the square in Figure 21.13. We then define cell C1 as =IF((Al-0.5) 2+(B1-0.5) 2<0.5*2,4,0) This has the effect of setting Cl equal to 4 if the dart lies in the circle and 0 otherwise. Define the next 99 rows of the spreadsheet similarly to the first one. (This is a "select and drag" operation in Excel.) Define C102 as =AVERAGE(Cl:C100) and C103 as =STDEV(CI:C100). C102 (which is 3.04 in Table 21.1) is an estimate of a calculated from 100 random trials. C103 is the Standard Deviation of our results and as we will see in Example 21.7 can be used to assess the accuracy of the estimate. Increasing the number of trials improves accuracy-but convergence to the correct value of 3.14159 is slow. Table 21.1 Sample spreadsheet calculations in Business Snapshot 21.1. A B C 1 0.207 0.690 4 2 0.271 0.520 4 3 0.007 0.221 100 0.198 0.403 101 102 Mean: 3.04 103 SD: 1.69
> The 6-month and 1-year zero rates are both 5% per annum. For a bond that has a life of 18 months and pays a coupon of 4% per annum (with semiannual payments and one having just been made), the yield is 5.2% per annum. What is the bond’s price? What is th
> ‘‘An interest rate swap where 6-month LIBOR is exchanged for a fixed rate of 5% on a principal of $100 million for 5 years involves a known cash flow and a portfolio of nine FRAs.’’ Explain this statement.
> Explain how LIBOR is determined.
> Use the risk-free rates in Problem 4.14 to value an FRA where you will pay 5% (annually compounded) and receive LIBOR for the third year on $1 million. The forward LIBOR rate (annually compounded) for the third year is 5.5%. Risk-free rates in Problem 4.
> Suppose that risk-free zero interest rates with continuous compounding are as follows: Calculate forward interest rates for the second, third, fourth, and fifth years. Maturity (vears) Rate (% per annum) 1 2.0 2 3.0 3 3.7 4 4.2 5 4.5
> Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are, respectively, 4%, 4.2%, 4.4%, 4.6%, and 4.8% per annum, with continuous compounding. Estimate the cash price of a bond with a face value of 100 that will mature in 30 months
> A deposit account pays 4% per annum with continuous compounding, but interest is actually paid quarterly. How much interest will be paid each quarter on a $10,000 deposit?
> ‘‘Buying a put option on a stock when the stock is owned is a form of insurance.’’ Explain this statement.
> A bank quotes an interest rate of 7% per annum with quarterly compounding. What is the equivalent rate with (a) continuous compounding and (b) annual compounding?
> Prove the result in equation (11.7). (Hint: For the first part of the relationship, consider (a) a portfolio consisting of a European call plus an amount of cash equal to K, and (b) a portfolio consisting of an American put option plus one share.)
> Does a perfect hedge always succeed in locking in the current spot price of an asset for a future transaction? Explain your answer.
> In the corn futures contract traded on an exchange, the following delivery months are available: March, May, July, September, and December. Which of the available contracts should be used for hedging when the expiration of the hedge is in (a) June, (b) J
> The expected return on the S&P 500 is 12% and the risk-free rate is 5%. What is the expected return on an investment with a beta of (a) 0.2, (b) 0.5, and (c) 1.4?
> Suppose that, on October 24, 2018, a company sells one April 2019 live cattle futures contract. It closes out its position on January 21, 2019. The futures price (per pound) is 121.20 cents when it enters into the contract, 118.30 cents when it closes ou
> The standard deviation of monthly changes in the spot price of live cattle is (in cents per pound) 1.2. The standard deviation of monthly changes in the futures price of live cattle for the closest contract is 1.4. The correlation between the futures pri
> ‘‘Options and futures are zero-sum games.’’ What do you think is meant by this?
> The CME Group offers a futures contract on long-term Treasury bonds. Characterize the traders likely to use this contract.
> ‘‘When the futures price of an asset is less than the spot price, long hedges are likely to be particularly attractive.’’ Explain this statement.
> On July 1, an investor holds 50,000 shares of a certain stock. The market price is $30 per share. The investor is interested in hedging against movements in the market over the next month and decides to use an index futures contract. The index futures pr
> A company knows that it is due to receive a certain amount of a foreign currency in 4 months. What type of option contract is appropriate for hedging?
> An American put option on a non-dividend-paying stock has 4 months to maturity. The exercise price is $21, the stock price is $20, the risk-free rate of interest is 10% perannum, and the volatility is 30% per annum. Use the explicit version of the finite
> At the end of one day a clearing house member is long 100 contracts, and the settlement price is $50,000 per contract. The original margin is $2,000 per contract. On the following day the member becomes responsible for clearing an additional 20 long cont
> Explain what a stop–limit order to sell at 20.30 with a limit of 20.10 means.
> ‘‘If the minimum variance hedge ratio is calculated as 1.0, the hedge must be perfect.’’ Is this statement true? Explain your answer.
> Explain what is meant by basis risk when futures contracts are used for hedging.
> When first issued, a stock provides funds for a company. Is the same true of a stock option? Discuss.
> Explain why a short hedger’s position improves when the basis strengthens unexpectedly and worsens when the basis weakens unexpectedly.
> What are the most important aspects of the design of a new futures contract?
> The party with a short position in a futures contract sometimes has options as to the precise asset that will be delivered, where delivery will take place, when delivery will take place, and so on. Do these options increase or decrease the futures price?
> A company has a $20 million portfolio with a beta of 1.2. It would like to use futures contracts on a stock index to hedge its risk. The index futures price is currently standing at 1080, and each contract is for delivery of $250 times the index. What is
> Show by substituting for the various terms in equation (19.4) that the equation is true for: (a) A single European call option on a non-dividend-paying stock (b) A single European put option on a non-dividend-paying stock (c) Any portfolio of European pu
> Suppose that the standard deviation of quarterly changes in the prices of a commodity is $0.65, the standard deviation of quarterly changes in a futures price on the commodity is $0.81, and the coefficient of correlation between the two changes is 0.8. W
> An investor enters into a short forward contract to sell 100,000 British pounds for U.S. dollars at an exchange rate of 1.5000 USD per pound. How much does the investor gain or lose if the exchange rate at the end of the contract is (a) 1.4900 and (b) 1.
> Under what circumstances does a minimum variance hedge portfolio lead to no hedging at all?
> What is the difference between entering into a long forward contract when the forward price is $50 and taking a long position in a call option with a strike price of $50?
> What is the difference between a local and a futures commission merchant?
> Distinguish between the terms open interest and trading volume.
> Explain why the linear model can provide only approximate estimates of VaR for a portfolio containing options.
> Suppose that a company has a portfolio consisting of positions in stocks and bonds. Assume that there are no derivatives. Explain the assumptions underlying (a) the linear model and (b) the historical simulation model for calculating VaR.
> 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 relationship between the change in the portfolio value an
> Describe three ways of handling instruments that are dependent on interest rates when the model-building approach is used to calculate VaR. How would you handle these instruments when historical simulation is used to calculate VaR?
> Stock A, whose price is $30, has an expected return of 11% and a volatility of 25%. Stock B, whose price is $40, has an expected return of 15% and a volatility of 30%. The processes driving the returns are correlated with correlation parameter. In Excel,
> Use the spreadsheets on the author’s website to calculate the one-day 99% VaR and ES, employing the basic methodology in Section 22.2, if the four-index portfolio considered in Section 22.2 is equally divided between the four indices.
> 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 1 day to delta, gamma, and vega. Explain without doing detailed calculations how you would u
> 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 is 1% per day. Adapt Sample Application E in the DerivaGem Applicati
> 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. Table 22.9 Change in portfolio value for a l-basis-point rate move (S milli ons).
> Some time ago a company entered into a forward contract to buy £1 million for $1.5 million. The contract now has 6 months to maturity. The daily volatility of a 6-month zero-coupon sterling bond (when its price is translated to dollars) is 0.06% and the
> Consider a position consisting of a $100,000 investment in asset A and a $100,000 investment in asset B. Assume that the daily volatilities of both assets are 1% and that the coefficient of correlation between their returns is 0.3. Estimate the 5-day 99%
> Explain why the Monte Carlo simulation approach cannot easily be used for American-style derivatives
> Show that the probabilities in a Cox, Ross, and Rubinstein binomial tree are negative when the condition in footnote 8 holds.
> ‘‘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.
> Explain carefully the arbitrage opportunities in Problem 11.16 if the American put price is greater than the calculated upper bound. Data from Problem 11.16: The price of an American call on a non-dividend-paying stock is $4. The stock price is $31, the
> Suppose that you enter into a short futures contract to sell July silver for $17.20 per ounce. The size of the contract is 5,000 ounces. The initial margin is $4,000, and the maintenance margin is $3,000. What change in the futures price will lead to a m
> Consider an option that pays off the amount by which the final stock price exceeds the average stock price achieved during the life of the option. Can this be valued using the binomial tree approach? Explain your answer.
> Calculate the price of a 9-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 the volatility is 30% per annum. Use a binomial tree with a
> Explain how the control variate technique is implemented when a tree is used to value American options.
> 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 .
> 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? Business Snapshot 21.2 Checking Black-Scholes-Merton in Excel The Black-Scholes-Merton formula for a European ca
> When do the boundary conditions for and S→∞ affect the estimates of derivative prices in the explicit finite difference method?
> Use the binomial tree in Problem 21.19 to value a security that pays off in 1 year where x is the price of copper. Binomial tree in Problem 21.19: 1.335 0.735 1.093 0.493 0.895 0.895 0.295 0.295 0.733 0.133 0.733 0.133 0.600 0.062 0.600 0.042 0.600 0
> Calculate the price of a 3-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% perannum, and the volatility is 45% per annum. Use a binomial tree with a time in
> The spot price of copper is $0.60 per pound. Suppose that the futures prices (dollars per pound) are as follows: 3 months……………….. 0.59 6 months………………… 0.57 9 months ………………..0.54 12 months ………………0.50 The volatility of the price of copper is 40% per annum
> Suppose that Monte Carlo simulation is being used to evaluate a European call option on a non-dividend-paying stock when the volatility is stochastic. How could the control variate and antithetic variable technique be used to improve numerical efficiency
> How do equations (21.27) to (21.30) change when the implicit finite difference method is being used to evaluate an American call option on a currency?
> How can the control variate approach improve the estimate of the delta of an American option when the tree approach is used?
> A 2-month American put option on a stock index has an exercise price of 480. The current level of the index is 484, the risk-free interest rate is 10% per annum, the dividend yield on the index is 3% per annum, and the volatility of the index is 25% per
> A 1-year American put option on a non-dividend-paying stock has an exercise price of $18. The current stock price is $20, the risk-free interest rate is 15% per annum, and the volatility of the stock price is 40% per annum. Use the DerivaGem software wit
> A 3-month American call option on a stock has a strike price of $20. The stock price is $20, the risk-free rate is 3% per annum, and the volatility is 25% per annum. A dividend of $2 is expected in 1.5 months. Use a three-step binomial tree to calculate
> Use a three-time-step binomial tree to value a 9-month American call option on wheat futures. The current futures price is 400 cents, the strike price is 420 cents, the risk-free rate is 6%, and the volatility is 35% per annum. Estimate the delta of the
> A 9-month American put option on a non-dividend-paying stock has a strike price of $49. The stock price is $50, the risk-free rate is 5% per annum, and the volatility is 30% per annum. Use a three-step binomial tree to calculate the option price.
> Which of the following can be estimated for an American option by constructing a single binomial tree: delta, gamma, vega, theta, rho?
> Explain what is meant by ‘‘crashophobia.’’
> The market price of a European call is $3.00 and its price given by Black–Scholes– Merton model with a volatility of 30% is $3.50. The price given by this Black–Scholes–Merton model for a European put option with the same strike price and time to maturit
> ‘‘Resecuritization was a badly flawed idea. AAA tranches created from the mezzanine tranches of ABSs are bound to have a higher probability of default than the AAA-rated tranches of ABSs.’’ Discuss this point of view.
> Using Table 20.2, calculate the implied volatility a trader would use for an 8-month option with K=S0 ¼ 1:04. Table 20.2 Volatility surface. K/So 0.90 0.95 1.00 1.05 1.10 1 month 14.2 13.0 12.0 13.1 14.5 3 month 14.0 13.0 12.0 13.1 14.2
> ‘‘The Black–Scholes–Merton model is used by traders as an interpolation tool.’’ Discuss this view.
> Suppose that the result of a major lawsuit affecting a company is due to be announced tomorrow. The company’s stock price is currently $60. If the ruling is favorable to the company, the stock price is expected to jump to $75. If it is unfavorable, the s
> A European call option on a certain stock has a strike price of $30, a time to maturity of 1 year, and an implied volatility of 30%. A European put option on the same stock has a strike price of $30, a time to maturity of 1 year, and an implied volatilit
> Option traders sometimes refer to deep-out-of-the-money options as being options on volatility. Why do you think they do this?
> Explain the problems in testing a stock option pricing model empirically.
> Suppose that a stock price is currently $20 and that a call option with an exercise price of $25 is created synthetically using a continually changing position in the stock. Consider the following two scenarios: (a) Stock price increases steadily from $
> The Black–Scholes–Merton price of an out-of-the-money call option with an exercise price of $40 is $4. A trader who has written the option plans to use a stop-loss strategy. The trader’s plan is to buy at $40.10 and to sell at $39.90. Estimate the expect
> Why did portfolio insurance not work well on October 19, 1987?
> ‘‘The procedure for creating an option position synthetically is the reverse of the procedure for hedging the option position.’’ Explain this statement.
> A stock price is $40. A 6-month European call option on the stock with a strike price of $30 has an implied volatility of 35%. A 6-month European call option on the stock with a strike price of $50 has an implied volatility of 28%. The 6-month risk-free
> What is meant by the gamma of an option position? What are the risks in the situation where the gamma of a position is highly negative and the delta is zero?
> What does it mean to assert that the theta of an option position is 0:1 when time is measured in years? If a trader feels that neither a stock price nor its implied volatility will change, what type of option position is appropriate?
> A bank’s position in options on the dollar/euro exchange rate has a delta of 30,000 and a gamma of . Explain how these numbers can be interpreted. The exchange rate (dollars per euro) is 0.90. What position would you take to make the position delta neut
> Suppose that $70 billion of equity assets are the subject of portfolio insurance schemes. Assume that the schemes are designed to provide insurance against the value of the assets declining by more than 5% within 1 year. Making whatever estimates you fin
> What does it mean to assert that the delta of a call option is 0.7? How can a short position in 1,000 options be made delta neutral when the delta of each option is 0.7?
> Repeat Problem 19.16 on the assumption that the portfolio has a beta of 1.5. Assume that the dividend yield on the portfolio is 4% per annum. Data from Problem 19.16: A fund manager has a well-diversified portfolio that mirrors the performance of the S&P
> A fund manager has a well-diversified portfolio that mirrors the performance of the S&P 500 and is worth $360 million. The value of the S&P 500 is 1,200, and the portfolio manager would like to buy insurance against a reduction of more than 5% in the val
> Under what circumstances is it possible to make a European option on a stock index both gamma neutral and vega neutral by adding a position in one other European option?
> A financial institution has just sold 1,000 7-month European call options on the Japanese yen. Suppose that the spot exchange rate is 0.80 cent per yen, the exercise price is 0.81 cent per yen, the risk-free interest rate in the United States is 8% per a
> In Problem 19.10, what initial position in 9-month silver futures is necessary for delta hedging? If silver itself is used, what is the initial position? If 1-year silver futures are used, what is the initial position? Assume no storage costs for silver.
> The price of an American call on a non-dividend-paying stock is $4. The stock price is $31, the strike price is $30, and the expiration date is in 3 months. The risk-free interest rate is 8%. Derive upper and lower bounds for the price of an American put
> What is the delta of a short position in 1,000 European call options on silver futures? The options mature in 8 months, and the futures contract underlying the option matures in 9 months. The current 9-month futures price is $8 per ounce, the exercise pr
> Explain how a stop-loss trading rule can be implemented for the writer of an out-of-themoney call option. Why does it provide a relatively poor hedge?
> Suppose you sell a call option contract on April live cattle futures with a strike price of 130 cents per pound. Each contract is for the delivery of 40,000 pounds. What happens if the contract is exercised when the futures price is 135 cents?
> Calculate the value of a five-month European futures put option when the futures price is $19, the strike price is $20, the risk-free interest rate is 12% per annum, and the volatility of the futures price is 20% per annum.
