标题: Portfolio Management【 Reading 56】Sample [打印本页]
作者: kmf229 时间: 2012-4-3 14:23 标题: [2012 L2] Portfolio Management【Session 17- Reading 56】Sample
Referring to put-call parity, which one of the following alternatives would allow you to create a synthetic riskless pure-discount bond? A)
| Buy a European put option; sell the same stock; sell a European call option. |
|
B)
| Buy a European put option; buy the same stock; sell a European call option. |
|
C)
| Sell a European put option; sell the same stock; buy a European call option. |
|
According to put-call parity we can write a riskless pure-discount bond position as:
X/(1+Rf)T = P0 + S0 – C0.
We can then read off the right-hand side of the equation to create a synthetic position in the riskless pure-discount bond. We would need to buy the European put, buy the same underlying stock, and sell the European call.
作者: kmf229 时间: 2012-4-3 14:23
Referring to put-call parity, which one of the following alternatives would allow you to create a synthetic European call option? A)
| Buy the stock; sell a European put option on the same stock with the same exercise price and the same maturity; short an amount equal to the present value of the exercise price worth of a pure-discount riskless bond. |
|
B)
| Sell the stock; buy a European put option on the same stock with the same exercise price and the same maturity; invest an amount equal to the present value of the exercise price in a pure-discount riskless bond. |
|
C)
| Buy the stock; buy a European put option on the same stock with the same exercise price and the same maturity; short an amount equal to the present value of the exercise price worth of a pure-discount riskless bond. |
|
According to put-call parity we can write a European call as: C0 = P0 + S0 – X/(1+Rf)TWe can then read off the right-hand side of the equation to create a synthetic position in the call. We would need to buy the European put, buy the stock, and short or issue a riskless pure-discount bond equal in value to the present value of the exercise price.
作者: kmf229 时间: 2012-4-3 14:25
Referring to put-call parity, which one of the following alternatives would allow you to create a synthetic stock position? A)
| Sell a European call option; buy a European put option; short the present value of the exercise price worth of a riskless pure-discount bond. |
|
B)
| Buy a European call option; buy a European put option; invest the present value of the exercise price in a riskless pure-discount bond. |
|
C)
| Buy a European call option; short a European put option; invest the present value of the exercise price in a riskless pure-discount bond. |
|
According to put-call parity we can write a stock position as: S0 = C0 – P0 + X/(1+Rf)TWe can then read off the right-hand side of the equation to create a synthetic position in the stock. We would need to buy the European call, sell the European put, and invest the present value of the exercise price in a riskless pure-discount bond.
作者: kmf229 时间: 2012-4-3 14:26
A stock is priced at 38 and the periodic risk-free rate of interest is 6%. What is the value of a two-period European put option with a strike price of 35 on a share of stock using a binomial model with an up factor of 1.15 and a risk-neutral probability of 68%?
Given an up factor of 1.15, the down factor is simply the reciprocal of this number 1/1.15=0.87. Two down moves produce a stock price of 38 × 0.872 = 28.73 and a put value at the end of two periods of 6.27. An up and a down move, as well as two up moves leave the put option out of the money. You are directly given the probability of up = 0.68. The down probability = 0.32. The value of the put option is [0.322 × 6.27] / 1.062 = $0.57.
作者: kmf229 时间: 2012-4-3 14:26
A two-period interest rate tree has the following expected one-period rates:
t = 0
|
| t = 1 | | t = 2 |
|
| | | 7.12% |
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| 6.83% | |
|
6.00%
|
| | | 6.84% |
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| 6.17% | |
|
|
| | | 6.22% |
The price of a two-period European interest-rate call option on the one-period rate with a strike rate of 6.25% and a principal amount of $100,000 is closest to:
- Calculate the payoffs on the call in percent for I++ and I+− (= I−+):
I++ value = (0.0712 − 0.0625) / 1.0712 = 0.00812173.
I+− value = (0.0684 − 0.0625) / 1.0684 = 0.00552228.
Remember that the payoff on the call value is the present value of the interest rate difference based on the rate realized at t = 2 because the payment is received at t = 3. - Calculate the t = 1 values (the probabilities in an interest rate tree are 50%):
At t = 1 the values are I+ = [0.5(0.00812173) + 0.5 (0.00552228)] / 1.0683 = 0.00638585.
At t = 1 the values are I− = [0.5(0) + 0.5 (0.00552228)] / 1.0617 = 0.00260068. - Calculate the t = 0 value:
At t = 0 the option value is [0.5(0.00638585) + 0.5(0.00260068)] / 1.06 = 0.00423893 0.00423893 × 100,000 = $423.89.
作者: kmf229 时间: 2012-4-3 14:26
stock is priced at 40 and the periodic risk-free rate of interest is 8%. The value of a two-period European call option with a strike price of 37 on a share of stock using a binomial model with an up factor of 1.20 is closest to:
First, calculate the probability of an up move or a down move:
U = 1.20 so D = 0.833
Pu = (1 + 0.08 − 0.833) / (1.20 − 0.833) = 0.673
Pd = 1 − 0.673 = 0.327
Two up moves produce a stock price of 40 × 1.44 = 57.60 and a call value at the end of two periods of 20.60. An up and a down move leave the stock price unchanged at 40 and produce a call value of 3. Two down moves result in the option being out of the money. The value of the call option is discounted back one year and then discounted back again to today. The calculations are as follows:
C+ = [20.6(0.673) + 3(0.327)] / 1.08 = 13.745
C- = [3(0.673) + 0 (0.327)] / 1.08 = 1.869
Call value today = [13.745(0.673) + 1.869(0.327)] / 1.08 = 9.13
作者: kmf229 时间: 2012-4-3 14:27
Al Bingly, CFA, is a derivatives specialist who attempts to identify and make short-term gains from trading mispriced options. One of the strategies that Bingly uses is to look for arbitrage opportunities in the market for European options. This strategy involves creating a synthetic call from other instruments at a cost less than the market value of the call itself, and then selling the call. During the course of his research, he observes that Hilland Corporation’s stock is currently priced at $56, while a European-style put option with a strike price of $55 is trading at $0.40 and a European-style call option with the same strike price is trading at $2.50. Both options have 6 months remaining until expiration. The risk-free rate is currently 4 percent.
Bingly often uses the binomial model to estimate the fair price of an option. He then compares his estimated price to the market price. He observes that Dale Corporation’s stock has a current market price of $200, and he predicts that its price will either be $166.67 or $240 in one year. The risk-free rate is currently 4 percent. He also observes that the price of a one-year call with a $220 strike price is $11.11.
Bingly also uses the Black-Scholes-Merton model to price options. His stated rationale for using this model is that he believes the prices of the stocks he analyzes follow a lognormal distribution, and because the model allows for a varying risk-free rate over the life of the option. His plan is to use a statistical technique to estimate the volatility of a stock, enter it into the Black-Scholes-Merton model, and see if the associated price is higher or lower than the observed market price of the options on the stock.
Bingly wishes to apply the Black-Scholes-Merton model to both non-dividend paying and dividend paying stocks. He investigates how the presence of dividends will affect the estimated call and put price. In the case of the options on Hilland Corporation’s stock, if Bingly were to establish a long protective put position, he could: A)
| earn an arbitrage profit of $0.03 per share by selling the call and borrowing the remaining funds needed for the position at the risk-free rate. |
|
B)
| earn an arbitrage profit of $0.30 per share by selling the call and lending $57.20 at the risk-free rate. |
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C)
| not earn an arbitrage profit because he should short the protective put position. |
|
Under put-call parity, the value of the call = put + stock – PV(exercise price). Therefore, the equilibrium value of the call = $0.40 + $56 - $55/(1.040.5) = $2.47. Thus, the call is overpriced, and arbitrage is available. If Bingly sells the call for $2.50 and borrows $53.93= $55/(1.040.5), he will have $56.43 > $56.40 (= $56 + $0.40), which is the price he would pay for the protective put position. The arbitrage profit is the difference ($0.03 = $56.43 - $56.40).
The one-year call option on Dale Corporation: |
B)
| may be over or underpriced. The given information is not sufficient to give an answer. |
|
|
The up movement parameter U=1.20, and the down movement parameter D=0.833. We calculate the probability of an up move πU = (1 + 0.04 – 0.833)/(1.2 – 0.833) = 0.564. The call is out of the money in the event of a down movement, and has an intrinsic value of $20 in the event of an up movement. Therefore, the estimated value of the call is C = (0.564) × $20 / (1.04) = $10.85. Thus, the price of $11.11 is too high and the call is overpriced.
Bingly’s sentiments towards the Black-Scholes-Merton (BSM) model regarding a lognormal distribution of prices and a variable risk-free rate are: A)
| correct concerning the distribution of stocks but incorrect concerning the risk-free rate. |
|
B)
| incorrect for both reasons. |
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C)
| correct for both reasons. |
|
The model requires many assumptions, e.g., the distribution of stock prices is lognormal and the risk-free rate is known and constant. Other assumptions are frictionless markets, the options are European, and the volatility is known and constant.
Which of the following is least accurate regarding the limitations of the BSM model? A)
| The BSM is not useful in pricing options on bonds and interest rates. |
|
B)
| The BSM is designed to price American options but not European options. |
|
C)
| The BSM is not useful in situations where the volatility of the underlying asset changes over time. |
|
The following are limitations of the BSM:- The assumption of a known and constant risk free rate means the BSM is not useful for pricing options on bond prices and interest rates.
- The assumption of a known and constant asset return volatility makes the BSM not useful in situations where the volatility is not constant which occurs much of the time.
- The assumption of no taxes and transaction costs makes the BSM less useful.
- The BSM is designed to price European options and not American options.
If Bingly forecasts the volatility for a stock and find that it is significantly greater than that implied by the prices of the puts and calls of the stock, he would conclude that: A)
| puts and calls are underpriced. |
|
B)
| puts and calls are overpriced. |
|
C)
| the puts are overpriced and the calls are underpriced. |
|
There is a positive relationship between the volatility of the stock and the price of both puts and calls. A higher estimate of volatility implies that the prices of both puts and calls should be higher.
All else being equal, the greater the dividend paid by a stock the: A)
| higher the call price and the lower the put price. |
|
B)
| lower the call price and the lower the put price. |
|
C)
| lower the call price and the higher the put price. |
|
When dividend payments occur during the life of the option, the price of the underlying stock is reduced (on the ex-dividend date). All else being equal, the lower price reduces the value of call options and increases the value of put options.
作者: kmf229 时间: 2012-4-3 14:28
A bond analyst decides to use the BSM model to price options on bond prices. This model will most likely be inadequate because: A)
| the risk free rate must be constant and known. |
|
B)
| BSM cannot be modified to deal with cash flows like coupon payments. |
|
C)
| the price of the underlying asset follows a lognormal distribution. |
|
The BSM model is not useful for pricing options on bond prices and interest rates. In those cases, interest rate volatility is a key factor in determining the value of the option. BSM can be modified to deal with cash flows like coupon payments. The assumption that “the price of the underlying asset follows a lognormal distribution” is not applicable.
作者: kmf229 时间: 2012-4-3 14:28
Rachel Barlow is a recent graduate of Columbia University with a Bachelor’s degree in finance. She has accepted a position at a large investment bank, but first must complete an intensive training program to gain experience in several of the investment bank’s areas of operations. Currently, she is spending three months at her firm's Derivatives Trading desk. One of the traders, Jason Coleman, CFA, is acting as her mentor, and will be giving her various assignments over the three month period.
One of the first projects he asks her to do is to compare different option trading strategies. Coleman would like Barlow to pay particular attention to strategy costs and their potential payoffs. Barlow is not very comfortable with option models, and knows she needs to be able to fully understand the most basic concepts in order to move on. She decides that she must first investigate how to properly price European and American style equity options. Coleman has given her software that provides a variety of analytical information using three valuation approaches: the Black-Scholes model, the Binomial model, and Monte Carlo simulation. Barlow has decided to begin her analysis using a variety of different scenarios to evaluate option behavior. The data she will be using in her scenarios is provided in Exhibits 1 and 2. Note that all of the rates and yields are on a continuous compounding basis.
Exhibit 1| Stock Price (S) | $100 |
| Strike Price (X) | $100 |
| Interest Rate (r) | 7% |
| Dividend Yield (q) | 0% |
Time to Maturity
(years) | 0.5 |
| Volatility (Std. Dev.) | 20% |
Exhibit 2| Stock Price (S) | $110 |
| Strike Price (X) | $100 |
| Interest Rate (r) | 7% |
| Dividend Yield (q) | 0% |
Time to Maturity
(years) | 0.5 |
| Volatility (Std. Dev.) | 20% |
Value of European
Call | $14.8445 |
Barlow notices that the stock in Exhibit 1 does not pay dividends. If the stock begins to pay a dividend, how will the price of a call option on that stock be affected?
The call option value will decrease since the payment of dividends reduces the value of the underlying, and the value of a call is positively related to the value of the underlying. (Study Session 17, LOS 60.g)
Barlow determines the price of an American call option on the stock shown in Exhibit 2. Which of the following is the most accurate?
The value of the American-style call option is the same as the value of the equivalent European-style call option. Since the underlying stock does not pay a dividend, it is never optimal to exercise the American option early. Hence the early-exercise option imbedded in the American-style call has no value in this case. This makes the American option worth exactly the same as the European option. (Study Session 17, LOS 60.j)
Using the information in Exhibit 2, Barlow computes the value of a European put option. Which of the following is closest to Barlow's answer?
Using the information in Exhibit 2, this value can be determined from put-call parity as follows:
Put = Call + Xe−rt − S
So we have Put = $14.8445 + $100.00e(−7.00% × 0.5) − $110.00 = $1.4050
(Study Session 17, LOS 60.c)
Barlow notices that the stock in Exhibit 2 does not pay dividends. If the stock starts to pay a dividend, how will the price of a put option on that stock be affected?
The put option value will increase since the payment of dividends reduces the value of the underlying, and the value of a put is negatively related to the value of the underlying. (Study Session 17, LOS 60.g)
作者: HuskyGrad2010 时间: 2012-4-3 14:31
Which of the following is NOT one of the assumptions of the Black-Scholes-Merton option-pricing model? A)
| There are no taxes and transactions costs are zero for options and arbitrage portfolios. |
|
B)
| There are no cash flows over the term of the options. |
|
C)
| The yield curve for risk-free assets is fixed over the term of the option. |
|
The yield curve is assumed to be flat so that the risk-free rate of interest is known and constant over the term of the option. Having a fixed yield curve does not necessarily imply that the yield curve is flat.
作者: HuskyGrad2010 时间: 2012-4-3 14:32
Which of the following is least likely one of the assumptions of the Black-Scholes-Merton option pricing model? A)
| There are no cash flows on the underlying asset. |
|
B)
| The risk-free rate of interest is known and does not change over the term of the option. |
|
C)
| Changes in volatility are known and predictable. |
|
The BSM model assumes that volatility is known and constant. The term predictable would allow for non-constant changes in volatility.
作者: HuskyGrad2010 时间: 2012-4-3 14:32
Which of the following is NOT one of the assumptions of the Black-Scholes-Merton (BSM) option-pricing model? A)
| Any dividends are paid at a continuously compounded rate. |
|
|
C)
| Options valued are European style. |
|
The BSM model assumes there are no cash flows on the underlying asset.
作者: HuskyGrad2010 时间: 2012-4-3 14:32
The value of a put option is positively related to all of the following EXCEPT:
The value of a put option is negatively related to increases in the risk-free rate.
作者: HuskyGrad2010 时间: 2012-4-3 14:33
The value of a European call option on an asset with no cash flows is positively related to all of the following EXCEPT:
The value of a call option decreases as the exercise price increases.
作者: HuskyGrad2010 时间: 2012-4-3 14:33
The value of a European call option on an asset with no cash flows is positively related to all of the following EXCEPT:
The value of a call option decreases as the exercise price increases.
作者: HuskyGrad2010 时间: 2012-4-3 14:33
For a change in which of the following inputs into the Black-Scholes-Merton option pricing model will the direction of the change in a put’s value and the direction of the change in a call’s value be the same?
A decrease/increase in the volatility of the price of the underlying asset will decrease/increase both put values and call values. A change in the values of the other inputs will have opposite effects on the values of puts and calls.
作者: HuskyGrad2010 时间: 2012-4-3 14:35
John Fairfax is a recently retired executive from Reston Industries. Over the years he has accumulated $10 million worth of Reston stock and another $2 million in a cash savings account. He hires Richard Potter, CFA, a financial adviser from Stan Morgan, LLC, to help him develop investment strategies. Potter suggests a number of interesting investment strategies for Fairfax's portfolio. Many of the strategies include the use of various equity derivatives. Potter's first recommendation includes the use of a total return equity swap. Potter outlines the characteristics of the swap in Table 1. In addition to the equity swap, Potter explains to Fairfax that there are numerous options available for him to obtain almost any risk return profile he might need. Potter suggests that Fairfax consider options on both Reston stock and the S&P 500. Potter collects the information needed to evaluate options for each security. These results are presented in Table 2.
Table 1: Specification of Equity Swap Term | 3 years |
Notional principal | $10 million |
Settlement frequency | Annual, commencing at end of year 1 |
Fairfax pays to broker | Total return on Reston Industries stock |
Broker pays to Fairfax | Total return on S&P 500 Stock Index |
Table 2: Option Characteristics | Reston | S&P 500 |
Stock price | $50.00 | $1,400.00 |
Strike price | $50.00 | $1,400.00 |
Interest rate | 6.00% | 6.00% |
Dividend yield | 0.00% | 0.00% |
Time to expiration (years) | 0.5 | 0.5 |
Volatility | 40.00% | 17.00% |
Beta Coefficient | 1.23 | 1 |
Correlation | 0.4 | |
Potter presents Fairfax with the prices of various options as shown in Table 3. Table 3 details standard European calls and put options. Potter presents the option sensitivities in Tables 4 and 5.
Table 3: Regular and Options (Option Values) | Reston | S&P 500 |
European call | $6.31 | $6.31 |
European put | $4.83 | $4.83 |
American call | $6.28 | $6.28 |
American put | $4.96 | $4.96 |
Table 4: Reston Stock Option Sensitivities | Delta |
European call | 0.5977 |
European put | −0.4023 |
American call | 0.5973 |
American put | −0.4258 |
Table 5: S&P 500 Option Sensitivities | Delta |
European call | 0.622 |
European put | −0.378 |
American call | 0.621 |
American put | −0.441 |
Given the information regarding the various Reston stock options, which option will increase the most relative to an increase in the underlying Reston stock price?
Using its delta in Table 4, if the Reston stock increases by a dollar the European call on the stock will increase by 0.5977. (Study Session 17, LOS 56.a)
Fairfax is very interested in the total return swap and asks Potter how much it would cost to enter into this transaction. Which of the following is the cost of the swap at inception?
Swaps are always priced so that their value at inception is zero. (Study Session 17, LOS 57.a)
Fairfax would like to consider neutralizing his Reston equity position from changes in the stock price of Reston. Using the information in Table 4 how many standard Reston European options would have to be either bought or sold in order to create a delta neutral portfolio? A)
| Sell 334,616 put options. |
|
B)
| Sell 334,616 call options. |
|
C)
| Buy 300,703 put options. |
|
Number of call options = (Reston Portfolio Value / Stock PriceReston)(1 / Deltacall).
Number of call options = ($10,000,000 / $50.00/sh)(1 / 0.5977) = 334,616. (Study Session 17, LOS 56.e)
Fairfax remembers Potter explaining something about how options are not like futures and swaps because their risk-return profiles are non-linear. Which of the following option sensitivity measures does Fairfax need to consider to completely hedge his equity position in Reston from changes in the price of Reston stock?
Vega measures the sensitivity relative to changes in volatility. Theta measures sensitivity relative to changes in time to expiration. (Study Session 17, LOS 56.d)
Fairfax has heard people talking about "making a portfolio delta neutral." What does it mean to make a portfolio delta neutral? The portfolio: A)
| is insensitive to stock price changes. |
|
B)
| is insensitive to volatility changes in the returns on the underlying equity. |
|
C)
| is insensitive to interest rate changes. |
|
The delta of the option portfolio is the change in value of the portfolio if the stock price changes. A delta neutral option portfolio has a delta of zero. (Study Session 17, LOS 56.e)
After discussing the various equity swap options with Fairfax, Potter checks his e-mail and reads a message from Clark Ali, a client of Potter and the treasurer of a firm that issued floating rate debt denominated in euros at London Interbank Offered Rate (LIBOR) + 125 basis points. Now Ali is concerned that LIBOR will rise in the future and wants to convert this into synthetic fixed rate debt. Potter recommends that Ali: A)
| enter into a pay-fixed swap. |
|
B)
| take a short position in Eurodollar futures. |
|
C)
| enter into a receive-fixed swap. |
|
The floating-rate debt will be effectively converted into fixed rate debt if he entered into a pay-fixed swap. A short position in Eurodollar futures would create a hedge, but in the wrong currency. (Study Session 17, LOS 57.d, e)
作者: HuskyGrad2010 时间: 2012-4-3 14:35
As a portfolio manager for the Herron Investments, an analyst is interested in establishing a dynamic hedge for one of his clients, Lou Gier. Gier has 200,000 shares of a stock that he believes could take a dive in the near future. Suppose that a call option with an exercise price of $100 and a maturity of 90 days has a price of $7. Also assume that the current stock price is $95 and the risk free rate is 5%. Assuming that the delta value of call option is 0.70, how many call option contracts would be needed to create a delta neutral hedge?
Click for Answer and Explanation
The number of call options needed is 200,000 / 0.70 = 285,714 options or approximately 2,857 contracts. Since Gier is long the stock, he should short the calls.
When a delta neutral hedge has been established using call options, which of the following statements is most correct? As the price of the underlying stock: A)
| changes, no changes are needed in the number of call options purchased. |
|
B)
| increases, some option contracts would need to be repurchased in order to retain the delta neutral position. |
|
C)
| increases, some option contracts would need to be sold in order to retain the delta neutral position. |
|
As the stock price increases, the delta of the call option increases as well, requiring fewer option contracts to hedge against the underlying stock price movements. Therefore, some options contracts would need to be repurchased in order to maintain the hedge.
作者: HuskyGrad2010 时间: 2012-4-3 14:36
The price of a June call option with an exercise price of $50 falls by $0.50 when the underlying stock price falls by $2.00. The delta of a June put option with an exercise price of $50 is closest to:
The call option delta is:
The put option delta is 0.25 – 1 = –0.75.
作者: HuskyGrad2010 时间: 2012-4-3 14:37
Ronald Franklin, CFA, has recently been promoted to junior portfolio manager for a large equity portfolio at Davidson-Sherman (DS), a large multinational investment-banking firm. He is specifically responsible for the development of a new investment strategy that DS wants all equity portfolio managers to implement. Upper management at DS has instructed its portfolio managers to begin overlaying option strategies on all equity portfolios. The relatively poor performance of many of their equity portfolios has been the main factor behind this decision. Prior to this new mandate, DS portfolio managers had been allowed to use options at their own discretion, and the results were somewhat inconsistent. Some portfolio managers were not comfortable with the most basic concepts of option valuation and their expected return profiles, and simply did not utilize options at all. Upper management of DS wants Franklin to develop an option strategy that would be applicable to all DS portfolios regardless of their underlying investment composition. Management views this new implementation of option strategies as an opportunity to either add value or reduce the risk of the portfolio.
Franklin gained experience with basic options strategies at his previous job. As an exercise, he decides to review the fundamentals of option valuation using a simple example. Franklin recognizes that the behavior of an option's value is dependent on many variables and decides to spend some time closely analyzing this behavior. His analysis has resulted in the information shown in Exhibits 1 and 2 for European style options.Exhibit 1: Input for European Options |
Stock Price (S) | 100 |
Strike Price (X) | 100 |
Interest Rate (r) | 0.07 |
Dividend Yield (q) | 0.00 |
Time to Maturity (years) (t) | 1.00 |
Volatility (Std. Dev.)(Sigma) | 0.20 |
Black-Scholes Put Option Value | $4.7809 |
| Exhibit 2: European Option Sensitivities |
| Sensitivity | Call | Put |
| Delta | 0.6736 | -0.3264 |
| Gamma | 0.0180 | 0.0180 |
| Theta | -3.9797 | 2.5470 |
| Vega | 36.0527 | 36.0527 |
| Rho | 55.8230 | -37.4164 |
Using the information in Exhibit 1, Franklin wants to compute the value of the corresponding European call option. Which of the following is the closest to Franklin's answer?
This result can be obtained using put-call parity in the following way:Call Value = Put Value − Xe−rt + S = $4.78 − $100.00e(−0.07 × 1.0) + 100 = $11.54
The incorrect value of $4.78 does not discount the strike price in the put-call parity formula. (Study Session 17, LOS 56.i)
Franklin is interested in the sensitivity of the European call option to changes in the volatility of the underlying equity's returns. What happens to the value of the call option if the volatility of the underlying equity's returns decreases? The call option value: |
B)
| increases or decreases. |
|
|
Due to the limited potential downside loss, changes in volatility directly effect option value. Vega measures the option’s sensitivity relative to volatility changes. (Study Session 17, LOS 56.d)
Franklin is interested in the sensitivity of the European put option to changes in the volatility of the underlying equity's returns. What happens to the value of the put option if the volatility of the underlying equity's returns increases? The put option value: |
|
C)
| increases or decreases. |
|
Due to the limited potential downside loss, changes in volatility directly effect option value. Vega measures the option price sensitivity relative to the volatility of the underlying stock. (Study Session 17, LOS 56.d)
Franklin wants to know how the put option in Exhibit 1 behaves when all the parameters are held constant except the delta. Which of the following is the best estimate of the change in the put option's price when the underlying equity increases by $1?
The correct value is simply the delta of the put option in Exhibit 2.
The incorrect value −$3.61 represents the change due to the volatility divided by 10 multiplied by −1.
The incorrect value −$0.37 calculates the change by dividing the short-term interest rate divided by 100. (Study Session 17, LOS 56.e)
Franklin computes the rate of change in the European put option delta value, given a $1 increase in the underlying equity. Using the information in Exhibits 1 and 2, which of the following is the closest to Franklin's answer?
The correct value 0.0180 is referred to as the put option's Gamma.
The incorrect value −0.3264 is the delta of the put option.
The incorrect value 0.6736 is the call option's delta. (Study Session 17, LOS 56.e)
Franklin wants to know if the option sensitivities shown in Exhibit 2 have minimum or maximum bounds. Which of the following are the minimum and maximum bounds, respectively, for the put option delta? |
B)
| There are no minimum or maximum bounds. |
|
|
The lower bound is achieved when the put option is far in the money so that it moves exactly in the opposite direction as the stock price. When the put option is far out of the money, the option delta is zero. Thus, the option price does not move even if the stock price moves since there is almost no chance that the option is going to be worth something at expiration. (Study Session 17, LOS 56.e)
作者: HuskyGrad2010 时间: 2012-4-3 14:37
Mark Washington, CFA, is an analyst with BIC, a Bermuda-based investment company that does business primarily in the U.S. and Canada. BIC has approximately $200 million of assets under management, the bulk of which is invested in U.S. equities. BIC has outperformed its target benchmark for eight of the past ten years, and has consistently been in the top quartile of performance when compared with its peer investment companies. Washington is a part of the Liability Management group that is responsible for hedging the equity portfolios under management. The Liability Management group has been authorized to use calls or puts on the underlying equities in the portfolio when appropriate, in order to minimize their exposure to market volatility. They also may utilize an options strategy in order to generate additional returns. One year ago, BIC analysts predicted that the U.S. equity market would most likely experience a slight downturn due to inflationary pressures. The analysts forecast a decrease in equity values of between 3 to 5% over the upcoming year and one-half. Based upon that prediction, the Liability Management group was instructed to utilize calls and puts to construct a delta-neutral portfolio. Washington immediately established option positions that he believed would hedge the underlying portfolio against the impending market decline.
As predicted, the U.S. equity markets did indeed experience a downturn of approximately 4% over a twelve-month period. However, portfolio performance for BIC during those twelve months was disappointing. The performance of the BIC portfolio lagged that of its peer group by nearly 10%. Upper management believes that a major factor in the portfolio’s underperformance was the option strategy utilized by Washington and the Liability Management group. Management has decided that the Liability Management group did not properly execute a delta-neutral strategy. Washington and his group have been told to review their options strategy to determine why the hedged portfolio did not perform as expected. Washington has decided to undertake a review of the most basic option concepts, and explore such elementary topics as option valuation, an option’s delta, and the expected performance of options under varying scenarios. He is going to examine all facets of a delta-neutral portfolio: how to construct one, how to determine the expected results, and when to use one. Management has given Washington and his group one week to immerse themselves in options theory, review the basic concepts, and then to present their findings as to why the portfolio did not perform as expected. Which of the following best explains a delta-neutral portfolio? A delta-neutral portfolio is perfectly hedged against: A)
| small price decreases in the underlying asset. |
|
B)
| all price changes in the underlying asset. |
|
C)
| small price changes in the underlying asset. |
|
A delta-neutral portfolio is perfectly hedged against small price changes in the underlying asset. This is true both for price increases and decreases. That is, the portfolio value will not change significantly if the asset price changes by a small amount. However, large changes in the underlying will cause the hedge to become imperfect. This means that overall portfolio value can change by a significant amount if the price change in the underlying asset is large. (Study Session 17, LOS 56.e)
After discussing the concept of a delta-neutral portfolio, Washington determines that he needs to further explain the concept of delta. Washington draws the payoff diagram for an option as a function of the underlying stock price. Using this diagram, how is delta interpreted? Delta is the: A)
| slope in the option price diagram. |
|
B)
| curvature of the option price graph. |
|
C)
| level in the option price diagram. |
|
Delta is the change in the option price for a given instantaneous change in the stock price. The change is equal to the slope of the option price diagram. (Study Session 17, LOS 56.e)
Washington considers a put option that has a delta of −0.65. If the price of the underlying asset decreases by $6, then which of the following is the best estimate of the change in option price?
The estimated change in the price of the option is:
Change in asset price × delta = −$6 × (−0.65) = $3.90
(Study Session 17, LOS 56.e)
Washington is trying to determine the value of a call option. When the slope of the at expiration curve is close to zero, the call option is:
When a call option is deep out-of-the-money, the slope of the at expiration curve is close to zero, which means the delta will be close to zero. (Study Session 17, LOS 56.e)
BIC owns 51,750 shares of Smith & Oates. The shares are currently priced at $69. A call option on Smith & Oates with a strike price of $70 is selling at $3.50, and has a delta of 0.69 What is the number of call options necessary to create a delta-neutral hedge?
The number of call options necessary to delta hedge is = 51,750 / 0.69 = 75,000 options or 750 option contracts, each covering 100 shares. Since these are call options, the options should be sold short. (Study Session 17, LOS 56.e)
Which of the following statements regarding the goal of a delta-neutral portfolio is most accurate? One example of a delta-neutral portfolio is to combine a: A)
| long position in a stock with a short position in a call option so that the value of the portfolio changes with changes in the value of the stock. |
|
B)
| long position in a stock with a short position in call options so that the value of the portfolio does not change with changes in the value of the stock. |
|
C)
| long position in a stock with a long position in call options so that the value of the portfolio does not change with changes in the value of the stock. |
|
A delta-neutral portfolio can be created with any of the following combinations: long stock and short calls, long stock and long puts, short stock and long calls, and short stock and short puts. (Study Session 17, LOS 56.e)
作者: HuskyGrad2010 时间: 2012-4-3 14:38
In order to form a dynamic hedge using stock and calls with a delta of 0.2, an investor could buy 10,000 shares of stock and:
Each call will increase in price by $0.20 for each $1 increase in the stock price. The hedge ratio is –1/delta or –5. A short position of 50,000 calls will offset the risk of 10,000 shares of stock over the next instant.
作者: HuskyGrad2010 时间: 2012-4-3 14:38
The delta of an option is equal to the: A)
| dollar change in the stock price divided by the dollar change in the option price. |
|
B)
| dollar change in the option price divided by the dollar change in the stock price. |
|
C)
| percentage change in option price divided by the percentage change in the asset price. |
|
The delta of an option is the dollar change in option price per $1 change in the price of the underlying asset.
作者: HuskyGrad2010 时间: 2012-4-3 14:39
John Williamson is a recently retired executive from Reston Industries. Over the years he has accumulated $10 million worth of Reston stock and another $2 million in a cash savings account. He hires Frank Potter, CFA, a financial adviser from Star Financial, LLC, to help him with his investment strategies. Potter has a number of interesting investment strategies for Williamson's portfolio. Many of the strategies include the use of various equity derivatives.
Potter's first recommendation includes the use of a total return equity swap. Potter outlines the characteristics of the swap in Table 1. In addition to the equity swap, Potter explains to Williamson that there are numerous options available for him to obtain almost any risk return profile he might need. Potter suggest that Williamson consider options on both Reston stock and the S&P 500. Potter collects the information needed to evaluate options for each security. These results are presented in Table 2.
Table 1: Specification of Equity Swap| Term | 3 years |
| Notional principal | $10 million |
| Settlement frequency | Annual, commencing at end of year 1 |
| Fairfax pays to broker | Total return on Reston Industries stock |
| Broker pays to Fairfax | Total return on S&P 500 Stock Index |
Table 2: Option Characteristics | Reston | S&P 500 |
| Stock price | $50.00 | $1,400.00 |
| Strike price | $50.00 | $1,400.00 |
| Interest rate | 6.00% | 6.00% |
| Dividend yield | 0.00% | 0.00% |
| Time to expiration (years) | 0.5 | 0.5 |
| Volatility | 40.00% | 17.00% |
| Beta Coefficient | 1.23 | 1 |
| Correlation | 0.4 | |
Table 3: Regular and Exotic Options (Option Values)
| Reston | S&P 500 |
| European call | $6.31 | $6.31 |
| European put | $4.83 | $4.83 |
| American call | $6.28 | $6.28 |
| American put | $4.96 | $4.96 |
Table 4: Reston Stock Option Sensitivities | Delta |
| European call | 0.5977 |
| European put | -0.4023 |
| American call | 0.5973 |
| American put | -0.4258 |
Table 5: S&P 500 Option Sensitivities | Delta |
| European call | 0.622 |
| European put | -0.378 |
| American call | 0.621 |
| American put | -0.441 |
Williamson would like to consider neutralizing his Reston equity position from changes in the stock price of Reston. Using the information in Tables 3 and 4 how many standard Reston European options would have to be bought/sold in order to create a delta neutral portfolio? A)
| Sell 497,141 put options. |
|
B)
| Sell 370,300 call options. |
|
C)
| Buy 497,141 put options. |
|
Number of put options = (Reston Portfolio Value / Stock PriceReston) / −DeltaPut
Number of put options = ($10,000,000 / $50.00) / −0.4023 = −497,141 meaning buy 497,141 put options.Selling put options does not deliver any downside protection, but it aggravates the losses when the stock decreases in value.
Williamson is very interested in the total return swap. He asks Potter how much it would cost to enter into this transaction. Which of the following is the cost of the swap at inception?
Swaps are priced so that their value at inception is zero.
Williamson likes the characteristics of the swap arrangement in Table 1 but would like to consider the options in Table 3 before making an investment decision. Given Williamson's current situation which of the following option trades makes the most sense in the short-term (all options are on Reston stock)? A)
| Buy out of the money call options. |
|
B)
| Sell at the money call options. |
|
C)
| Buy at the money put options. |
|
Buying at the money put options greatly reduces Williamson's downside risk. Selling call options yields an option premium to the seller but does not deliver any downside protection and limits the upside potential of the portfolio.
作者: HuskyGrad2010 时间: 2012-4-3 14:40
Joel Franklin, CFA, has recently been promoted to junior portfolio manager for a large equity portfolio at Davidson Sherman (DS), a large multinational investment banking firm. The portfolio is subdivided into several smaller portfolios. In general, the portfolios are composed of U.S. based equities, ranging from medium to large-cap stocks. Currently, DS is not involved in any foreign markets. In his new position, he will now be responsible for the development of a new investment strategy that DS wants all of its equity portfolios to implement. The strategy involves overlaying option strategies on its equity portfolios. Recent performance of many of their equity portfolios has been poor relative to their peer group. The upper management at DS views the new option strategies as an opportunity to either add value or reduce risk.
Franklin recognizes that the behavior of an option’s value is dependent upon many variables and decides to spend some time closely analyzing this behavior. He took an options strategies class in graduate school a few years ago, and feels that he is fairly knowledgeable about the valuation of options using the Black-Scholes model. Franklin understands that the volatility of the underlying asset returns is one of the most important contributors to option value. Therefore, he would like to know when the volatility has the largest effect on option value. Upper management at DS has also requested that he further explore the concept of a delta neutral portfolio. He must determine how to create a delta neutral portfolio, and how it would be expected to perform under a variety of scenarios. Franklin is also examining the change in the call option's delta as the underlying equity value changes. He also wants to determine the minimum and maximum bounds on the call option delta. Franklin has been authorized to purchase calls or puts on the equities in the portfolio. He may not, however, establish any uncovered or “naked” option positions. His analysis has resulted in the information shown in Exhibits 1 and 2 for European style options.| Exhibit 1 |
| Input for European Options |
| Stock Price (S) | 100 |
| Strike Price (X) | 100 |
| Interest Rate (r) | 0.07 |
| Dividend Yield (q) | 0 |
| Time to Maturity (years) (t) | 1 |
| Volatility (Std. Dev.) (sigma) | 0.2 |
| Black-Scholes Put Option Value | $4.7809 |
| Exhibit 2 |
| European Option Sensitivities |
| Sensitivity | Call | Put |
| Delta | 0.6736 | −0.3264 |
| Gamma | 0.0180 | 0.0180 |
| Theta | −3.9797 | 2.5470 |
| Vega | 36.0527 | 36.0527 |
| Rho | 55.8230 | −37.4164 |
What does it mean to make an options portfolio delta neutral? The option portfolio: A)
| moves exactly in line with the stock price. |
|
B)
| moves exactly in the opposite direction with the stock price. |
|
C)
| is insensitive to price changes in the underlying security. |
|
The delta of the option portfolio is the change in value of the portfolio if the underlying stock price changes. A delta neutral option portfolio has a delta of zero. (Study Session 17, LOS 56.e)
Which of the following most accurately describes the sensitivity of the call option's delta to changes in the underlying asset’s price? The sensitivity to changes in the price of the underlying is the greatest when the call option is: |
|
C)
| it depends on the other inputs. |
|
When the option is at the money, delta is most sensitive to changes in the underlying asset’s price. (Study Session 17, LOS 56.f)
Which of the following most accurately describes when the call option delta reaches its minimum bound? The call option reaches its minimum bound when call option is: |
|
C)
| the option's delta has no minimum bound. |
|
When a call option is far out of the money its value is insensitive to changes in value of the underlying. This is because the chances that it is going to end up in the money at expiration are very small. (Study Session 17, LOS 56.e)
If the portfolio has 10,000 shares of the underlying stock and he wants to completely hedge the price risk using options, what kind of options should Franklin buy?
Buying put options will allow Franklin to completely hedge the stock price risk. (Study Session 17, LOS 56.e)
Compute the number of shares of stock necessary to create a delta neutral portfolio consisting of 100 long put options in Exhibit 2 and the stock.
This is simply −100 times the put option delta. Since each share has a delta of 1, we only need 32.64 shares (long) to create a delta neutral portfolio. (Study Session 17, LOS 56.e)
Compute the number of shares of stock necessary to create a delta neutral portfolio consisting of 100 long call options in Exhibit 2 and the stock.
This is simply −100 times the call option delta. Since each share has a delta of 1, we only need −67.36 (short) shares to create a delta neutral portfolio. (Study Session 17, LOS 56.e)
作者: HuskyGrad2010 时间: 2012-4-3 14:40
Which of the following is the best approximation of the gamma of an option if its delta is equal to 0.6 when the price of the underlying security is 100 and 0.7 when the price of the underlying security is 110?
The gamma of an option is computed as follows:
Gamma = change in delta/change in the price of the underlying = (0.7 – 0.6)/(110 – 100) = 0.01
作者: HuskyGrad2010 时间: 2012-4-3 14:41
Which of the following is the best approximation of the gamma of an option if its delta is equal to 0.6 when the price of the underlying security is 100 and 0.7 when the price of the underlying security is 110?
The gamma of an option is computed as follows:
Gamma = change in delta/change in the price of the underlying = (0.7 – 0.6)/(110 – 100) = 0.01
作者: HuskyGrad2010 时间: 2012-4-3 14:41
When an option’s gamma is higher: |
B)
| a delta hedge will perform more poorly over time. |
|
C)
| a delta hedge will be more effective. |
|
Gamma measures the rate of change of delta (a high gamma could mean that delta will be higher or lower) as the asset price changes and, graphically, is the curvature of the option price as a function of the stock price. Delta measures the slope of the function at a point. The greater gamma is (the more delta changes as the asset price changes), the worse a delta hedge will perform over time.
作者: HuskyGrad2010 时间: 2012-4-3 14:42
Gamma is the greatest when an option: |
B)
| is deep out of the money. |
|
|
Gamma, the curvature of the option-price/asset-price function, is greatest when the asset is at the money.
作者: HuskyGrad2010 时间: 2012-4-3 14:42
Two call options have the same delta but option A has a higher gamma than option B. When the price of the underlying asset increases, the number of option A calls necessary to hedge the price risk in 100 shares of stock, compared to the number of option B calls, is a: A)
| larger positive number. |
|
B)
| smaller (negative) number. |
|
C)
| larger (negative) number. |
|
For call options larger gamma means that as the asset price increases, the delta of option A increases more than the delta of option B. Since the hedge ratio for calls is – 1/delta, the number of calls necessary for the hedge is a smaller (negative) number for option A than for option B.
作者: HuskyGrad2010 时间: 2012-4-3 14:43
The value of a put option will be higher if, all else equal, the: A)
| exercise price is lower. |
|
B)
| underlying asset has positive cash flows. |
|
C)
| underlying asset has less volatility. |
|
Positive cash flows in the form of dividends will lower the price of the stock making it closer to being "in the money" which increases the value of the option as the stock price gets closer to the strike price.
作者: HuskyGrad2010 时间: 2012-4-3 14:43
Compared to the value of a call option on a stock with no dividends, a call option on an identical stock expected to pay a dividend during the term of the option will have a: A)
| lower value in all cases. |
|
B)
| lower value only if it is an American style option. |
|
C)
| higher value only if it is an American style option. |
|
An expected dividend during the term of an option will decrease the value of a call option.
作者: HuskyGrad2010 时间: 2012-4-3 14:44
Dividends on a stock can be incorporated into the valuation model of an option on the stock by: A)
| subtracting the present value of the dividend from the current stock price. |
|
B)
| adding the present value of the dividend to the current stock price. |
|
C)
| subtracting the future value of the dividend from the current stock price. |
|
The option pricing formulas can be adjusted for dividends by subtracting the present value of the expected dividend(s) from the current asset price.
作者: HuskyGrad2010 时间: 2012-4-3 14:44
In order to compute the implied asset price volatility for a particular option, an investor: A)
| must have a series of asset prices. |
|
B)
| must have the market price of the option. |
|
C)
| does not need to know the risk-free rate. |
|
In order to compute the implied volatility we need the risk-free rate, the current asset price, the time to expiration, the exercise price, and the market price of the option.
作者: HuskyGrad2010 时间: 2012-4-3 14:44
Which of the following methods is NOT used for estimating volatility inputs for the Black-Scholes model? A)
| Using long term historical data. |
|
B)
| Using exponentially weighted historical data. |
|
C)
| Models of changing volatility. |
|
The volatility is constant in the Black-Scholes model
作者: HuskyGrad2010 时间: 2012-4-3 14:44
Which of the following best describes the implied volatility method for estimated volatility inputs for the Black-Scholes model? Implied volatility is found: A)
| using historical stock price data. |
|
B)
| using the most current stock price data. |
|
C)
| by solving the Black-Scholes model for the volatility using market values for the stock price, exercise price, interest rate, time until expiration, and option price. |
|
Implied volatility is found by “backing out” the volatility estimate using the current option price and all other values in the Black-Scholes model.
作者: HuskyGrad2010 时间: 2012-4-3 14:45
Which of the following best explains the sensitivity of a call option's value to volatility? Call option values: A)
| increase as the volatility of the underlying asset increases because investors are risk seekers. |
|
B)
| increase as the volatility of the underlying asset increases because call options have limited risk but unlimited upside potential. |
|
C)
| are not affected by changes in the volatility of the underlying asset. |
|
A higher volatility makes it more likely that options end up in the money and can be exercised profitably, while the down side risk is strictly limited to the option premium.
作者: HuskyGrad2010 时间: 2012-4-3 14:46
Which of the following statements concerning vega is most accurate? Vega is greatest when an option is:
When the option is at the money, changes in volatility will have the greatest affect on the option value.
作者: HuskyGrad2010 时间: 2012-4-3 14:46
If we use four of the inputs into the Black-Scholes-Merton option-pricing model and solve for the asset price volatility that will make the model price equal to the market price of the option, we have found the: |
|
C)
| historical volatility. |
|
The question describes the process for finding the expected volatility implied by the market price of the option.
作者: HuskyGrad2010 时间: 2012-4-3 14:47
At time = 0, for a put option at exercise price (X) on a newly issued forward contact at FT (the forward price at time = 0), a portfolio with equal value could be constructed from being long in: A)
| the underlying asset, long a put at X, and short in a pure-discount risk-free bond that pays X – FT at option expiration. |
|
B)
| a risk-free pure-discount bond that pays FT – X at option expiration and long in a put at X. |
|
C)
| a call at X and long in a pure-discount risk-free bond that pays X – FT at option expiration. |
|
Utilizing the basic put/call parity equation, we're looking for a portfolio that is equal to the portfolio mentioned in the stem (a put option). The put-call parity equation is c0 + (X – FT) / (1+R)T = p0. Since (X – FT) / (1+R) is actually just the present value of the bond at expiration, the relationship can be simplified to long call + long bond = put.
作者: HuskyGrad2010 时间: 2012-4-3 14:47
Put-call parity for options on forward contracts at the initiation of the option where the forward price at that time (time=0) is FT, can best be expressed as: A)
| c0 + X / (1 + R)T − FT = p0. |
|
B)
| c0 + (X − FT) / (1 + R)T = p0. |
|
C)
| c0 − (X − FT) / (1 + R)T = p0. |
|
Put call parity for stocks (with discrete time discounting) is c0 + X / (1 + R)T − S0 = p0. Noting that for the forward contract on an asset with no underlying cash flows, S0 = FT / (1 + R)T, and substituting, we get c0 + (X − FT) / (1 + R)T = p0.
作者: HuskyGrad2010 时间: 2012-4-3 14:48
Which of the following would have the same value at t = 0 as an at-the-money call option on a forward contract priced at FT (the forward price at time = 0)? A)
| A put option, long the underlying asset, and short a risk-free bond that pays X-FT at option expiration. |
|
B)
| A put option, long the underlying asset, and short a risk-free bond that matures at X at option expiration. |
|
C)
| A put option on the forward at exercise price (X). |
|
Put-call parity for options on forward contracts is c0 + (X – FT) / (1+R)T = p0. Since X = FT for an at-the-money option, the put and the call have the same value for an at-the-money option.
作者: HuskyGrad2010 时间: 2012-4-3 14:48
Which of the following is a correct specification of put-call parity for options on futures?
Begin with put-call parity for a stock, 
and substitute 
作者: HuskyGrad2010 时间: 2012-4-3 14:49
Early exercise of in-the-money American options on: A)
| both futures and forwards is sometimes worthwhile. |
|
B)
| futures is sometimes worthwhile but never is for options on forwards. |
|
C)
| forwards is sometimes worthwhile but never is for options on futures. |
|
Early exercise of in-the-money American options on futures is sometimes worthwhile because the immediate mark to market upon exercise will generate funds that can earn interest. It is never worthwhile for options on forwards because no funds are generated until the settlement date of the forward contract.
作者: HuskyGrad2010 时间: 2012-4-3 14:49
Regarding deep in-the-money options on futures, it is: A)
| sometimes worthwhile to exercise both calls and puts early. |
|
B)
| sometimes worthwhile to exercise calls early but not puts. |
|
C)
| never worthwhile to exercise puts or calls early. |
|
If puts or calls on futures are significantly in-the-money it may be worthwhile to exercise them early to generate the cash from the immediate mark to market of the futures contract when the option is exercised.
作者: HuskyGrad2010 时间: 2012-4-3 14:49
Regarding deep in-the-money options on forwards, it is: A)
| never worthwhile to exercise puts or calls early. |
|
B)
| sometimes worthwhile to exercise calls early but not puts. |
|
C)
| sometimes worthwhile to exercise both calls and puts early. |
|
Unlike futures, forwards do not generate any cash at exercise even when they are deep in-the-money so there is no advantage to early exercise.
作者: HuskyGrad2010 时间: 2012-4-3 14:50
Which of the following statements is most accurate? A)
| American options on forwards are more valuable than comparable European options on forwards. |
|
B)
| European options on futures are more valuable than comparable American options on futures. |
|
C)
| American options on futures are more valuable than comparable European options on futures. |
|
Because of the mark-to-market feature of futures contracts, American options on futures are more valuable than comparable European options. The value of American and European options on forwards are the same.
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