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Stan Loper is unfamiliar with the Black-Scholes-Merton (BSM) option pricing model and plans to use a two-period binomial model to value some call options. The stock of Arbor Industries pays no dividends and currently trades for $45. The up-move factor for the stock is 1.15, and the risk-free rate is 4%. He is considering buying two-period European style options on Arbor Industries with a strike price of S40. The delta of these options over the first period is 0.83.
Loper is curious about the effect of time on the value of the calls in the binomial model, so he also calculates the value of a one-period European style call option with a strike price of 40.
Loper is also interested in using the BSM model to price European and American call and put options. He is concerned, however, whether the assumptions necessary to derive the model are realistic. The assumptions he is particularly concerned about are:
* The volatility of the option value is known and constant.
* Stock prices are lognormally distributed.
* The continuous risk-free rate is known and constant.
Loper would also like to value options on Rapid Repair, Inc., common stock, but Rapid pays dividends, so Loper is uncertain what the effect will be on the value of the options. Loper uses the two-period model to value long positions in the Rapid Repair call and put options without accounting for the fact that Rapid Repair pays common dividends.
The value of a two-period 40 call on Arbor Industries stock is closest to:
Answer : B
Stan Loper is unfamiliar with the Black-Scholes-Merton (BSM) option pricing model and plans to use a two-period binomial model to value some call options. The stock of Arbor Industries pays no dividends and currently trades for $45. The up-move factor for the stock is 1.15, and the risk-free rate is 4%. He is considering buying two-period European style options on Arbor Industries with a strike price of S40. The delta of these options over the first period is 0.83.
Loper is curious about the effect of time on the value of the calls in the binomial model, so he also calculates the value of a one-period European style call option with a strike price of 40.
Loper is also interested in using the BSM model to price European and American call and put options. He is concerned, however, whether the assumptions necessary to derive the model are realistic. The assumptions he is particularly concerned about are:
* The volatility of the option value is known and constant.
* Stock prices are lognormally distributed.
* The continuous risk-free rate is known and constant.
Loper would also like to value options on Rapid Repair, Inc., common stock, but Rapid pays dividends, so Loper is uncertain what the effect will be on the value of the options. Loper uses the two-period model to value long positions in the Rapid Repair call and put options without accounting for the fact that Rapid Repair pays common dividends.
The position in calls necessary to hedge a long position in 1,000 shares of stock over the first period is closest to:
Answer : C
Stan Loper is unfamiliar with the Black-Scholes-Merton (BSM) option pricing model and plans to use a two-period binomial model to value some call options. The stock of Arbor Industries pays no dividends and currently trades for $45. The up-move factor for the stock is 1.15, and the risk-free rate is 4%. He is considering buying two-period European style options on Arbor Industries with a strike price of S40. The delta of these options over the first period is 0.83.
Loper is curious about the effect of time on the value of the calls in the binomial model, so he also calculates the value of a one-period European style call option with a strike price of 40.
Loper is also interested in using the BSM model to price European and American call and put options. He is concerned, however, whether the assumptions necessary to derive the model are realistic. The assumptions he is particularly concerned about are:
* The volatility of the option value is known and constant.
* Stock prices are lognormally distributed.
* The continuous risk-free rate is known and constant.
Loper would also like to value options on Rapid Repair, Inc., common stock, but Rapid pays dividends, so Loper is uncertain what the effect will be on the value of the options. Loper uses the two-period model to value long positions in the Rapid Repair call and put options without accounting for the fact that Rapid Repair pays common dividends.
The value of the one-period European style call option is closest to:
Answer : B
The payoff is zero for a down-move and 11.75 for an up-move. Since the probability of
an up-move is 0.607 the present value is
(Study Session 17, LOS60.b)
Charles Mabry manages a portfolio of equity investments heavily concentrated in the biotech industry. He just returned from an annual meeting among leading biotech analysts in San Francisco. Mabry and other industry experts agree that the latest industry volatility is a result of questionable product safety testing methodologies. While no firms in the industry have escaped the public attention brought on by the questionable safety testing, one company in particular is expected to receive further attention---Biological Instruments Corporation (BIC), one of several long biotech positions in Mabry's portfolio. Several regulatory agencies as well as public interest groups have heavily criticized the rigor of BIC's product safety testing.
In an effort to manage the risk associated with BIC, Mabry has decided to allocate a portion of his portfolio to options on BIC's common stock. After surveying the derivatives market, Mabry has identified the following European options on BIC common stock:
Mabry wants to hedge the large BIC equity position in his portfolio, which closed yesterday (June 1) at $42 per share. Since Mabry is relatively inexperienced with utilizing derivatives in his portfolios, Mabry enlists the help of an analyst from another firm, James Grimell.
Mabry and Grimell arrange a meeting in Boston where Mabry discusses his expectations regarding the future returns of BIC's equity. Mabry expects BIC equity to make a recovery from the intense market scrutiny but wants to provide his portfolio with a hedge in case BIC has a negative surprise. Grimell makes the following suggestion:
"If you want to avoid selling the BIC position and are willing to earn only the risk-free rate of return, you should sell calls and buy puts on BIC stock with the same market premium. Alternatively, you could buy put options to manage the risk of your portfolio. I recommend waiting until the vega on the options rises, making them less attractive and cheaper to purchase."
If the gamma of Put E is equal to 0.081, which of the following correctly interprets the option's gamma?
Answer : A
An option's gamma measures the change in the delta for a change in the price of the underlying asset. The gamma of an option is highest when an option is at-the-money since the probability of moving in or out of the money is high. Put E is close to being at-the-money and because it has a gamma of greater than zero, the sensitivity of Put Es price to changes in BlC's stock price (i.e., the delta) is likely to change. The higher the gamma, the greater the change in delta given a change in stock price. (Study Session 17, LOS 60. f)
Michelle Norris, CFA, manages assets for individual investors in the United States as well as in other countries. Norris limits the scope of her practice to equity securities traded on U .S . stock exchanges. Her partner, John Witkowski, handles any requests for international securities. Recently, one of Norris's wealthiest clients suffered a substantial decline in the value of his international portfolio. Worried that his U .S . allocation might suffer the same fate, he has asked Norris to implement a hedge on his portfolio. Norris has agreed to her client's request and is currently in the process of evaluating several futures contracts. Her primary interest is in a futures contract on a broad equity index that will expire 240 days from today. The closing price as of yesterday, January 17, for the equity index was 1,050. The expected dividends from the index yield 2% (continuously compounded annual rate). The effective annual risk-free rate is 4.0811%, and the term structure is flat. Norris decides that this equity index futures contract is the appropriate hedge for her client's portfolio and enters into the contract.
Upon entering into the contract, Norris makes the following comment to her client:
"You should note that since we have taken a short position in the futures contract, the price we will receive for selling the equity index in 240 days will be reduced by the convenience yield associated with having a long position in the underlying asset. If there were no cash flows associated with the underlying asset, the price would be higher. Additionally, you should note that if we had entered into a forward contract with the same terms, the contract price would most likely have been lower but we would have increased the credit risk exposure of the portfolio."
Sixty days after entering into the futures contract, the equity index reached a level of 1,015. The futures contract that Norris purchased is now trading on the Chicago Mercantile Exchange for a price of 1,035. Interest rates have not changed. After performing some calculations, Norris calls her client to let him know of an arbitrage opportunity related to his futures position. Over the phone, Norris makes the following comments to her client:
"We have an excellent opportunity to earn a riskless profit by engaging in arbitrage using the equity index, risk-free assets, and futures contracts. My recommended strategy is as follows: We should sell the equity index short, buy the futures contract, and pay any dividends occurring over the life of the contract. By pursuing this strategy, we can generate profits for your portfolio without incurring any risk."
Sixty days after the inception of the futures contract on the equity index, Norris has suggested an arbitrage strategy. Evaluate the appropriateness of the strategy. The strategy is:
Answer : B
First, calculate the continuously compounded risk-free rate as ln( 1.040811) = 4% and then calculate the theoretically correct futures price as follows:
Then, compare the theoretical price to the observed market price: 1.035 - 1,025 = 10. The futures contract is overpriced. To take advantage of the arbitrage opportunity, the investor should sell the (overpriced) futures contract and buy the underlying asset (the equity index) using borrowed funds. Norris has suggested the opposite. (Study Session 16, LOS 59.f)
