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标题: [2008]Topic 20: Greek Letters相关习题 [打印本页]

作者: Baran    时间: 2009-6-26 10:28     标题: [2008]Topic 20: Greek Letters相关习题

AIM 1: Discuss and assess the risks associated with naked and covered option positions.

 

1、As an option approaches expiration, the value of rho for a put option:

A) decreases and tends toward zero.
 
B) decreases and tends toward negative infinity.
 
C) increases and tends toward infinity.
 
D) increases and tends toward zero.


作者: Baran    时间: 2009-6-26 10:29

The correct answer is D


Rho values for put options are always negative and approach zero as the option nears maturity.


作者: Baran    时间: 2009-6-26 10:38

 

AIM 3: Define delta hedging for an option, forward, and futures contracts.

 

1、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

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?

A) -1 and 1.


B) -1 and 0.


C) -1 and no maximum bound.


D) There are no minimum or maximum bounds.


作者: Baran    时间: 2009-6-26 10:39

 

The correct answer is B

 

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.


作者: Baran    时间: 2009-6-26 11:05

 

2、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

Which of the following is the best estimate of the change in the put option when the underlying equity increases by $1?

A) -$3.61.


B) -$0.33.


C) -$0.37.


D) $0.67.


作者: Baran    时间: 2009-6-26 11:05

 

The correct answer is B

 

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.
The incorrect value $0.67 represents the change in the call option.


作者: Baran    时间: 2009-6-26 11:06

 

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?

A) -0.3264.


B) 0.6736.


C) 0.0180.


D) 36.0527.


作者: Baran    时间: 2009-6-26 11:06

 

The correct answer is C


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.

The incorrect value 36.0527 is the put option's Vega.


作者: Baran    时间: 2009-6-26 11:07

 

3、Which of the following is the best interpretation of delta for an option? Delta is the change in the option price for:


A) an instantaneous change in interest rates.


B) an instantaneous change in price of the underlying stock.


C) a change in the time until expiration of the option.


D) an instantaneous change in the volatility of the underlying stock.


作者: Baran    时间: 2009-6-26 11:07

 

The correct answer is B

 

Delta is the slope of the price function of the call option payoff diagram.


作者: Baran    时间: 2009-6-26 11:07

 

4、An option dealer is delta hedging a short call position on a stock. As the stock price increases, in order to maintain the hedge, the dealer would most likely have to:


A) buy more shares of the stock. 


B) sell some the shares of the stock.


C) buy T-bills.


D) short T-bills.


作者: Baran    时间: 2009-6-26 11:08

 

The correct answer is A

 

As the value of the underlying increases, the delta of a call option increases. This means more of the underlying asset is needed to hedge the position.


作者: Baran    时间: 2009-6-26 11:08

 

5、To create a delta-neutral portfolio, an investor who has written 5,000 call options that have deltas equal to 0.5 will be:


A) short 2,500 shares in the underlying.


B) long 2,500 shares in the underlying and short 2,500 more options.


C) long 2,500 shares in the underlying.


D) short 2,500 shares in the underlying and be short 2,500 more options.


作者: Baran    时间: 2009-6-26 11:08

 

The correct answer is C

 

If the investor has written 5,000 call options, he then must go long 0.5 × 5,000 = 2,500 shares to create a delta neutral position since the delta of a share is 1.


作者: Baran    时间: 2009-6-26 11:09

 

AIM 4: Compute delta for an option.

 

1、The deltas of puts and calls are most sensitive to changes in the underlying when:


A) both calls and puts are deep in-the-money.


B) both puts and calls are deep out-of-the-money.


C) calls are deep out-of-the-money, but puts are deep in-the-money.


D) both calls and puts are at-the-money.


作者: Baran    时间: 2009-6-26 11:09

 

 The correct answer is D

 

Call and put deltas are the most sensitive to changes in the underlying security (i.e., gammas are largest) when the option is at-the-money.


作者: Baran    时间: 2009-6-26 11:09

 

2、Which of the following is FALSE?


The delta of forwards and futures is 1.

Gamma is largest when options are at-the-money.

Two problems using stop-loss trading on naked options are transaction costs and stock price uncertainty.

For a delta-neutral portfolio, although opposite in sign, theta can serve as a proxy for gamma.

A) II only.


B) I and III only.


C) II and IV only.


D) I only.


作者: Baran    时间: 2009-6-26 11:09

 

 The correct answer is D

 

The delta of forwards is one. The delta of futures is not usually one. Two problems using stop-loss trading on naked options are transaction costs and stock-price certainty. Gamma is largest when options are at-the-money. For a delta-neutral portfolio, although opposite in sign, theta can serve as a proxy for gamma.


作者: Baran    时间: 2009-6-26 11:12

 

The correct answer is A

 

The above diagram is for a long stock, long put strategy (portfolio insurance). The loss is limited to the cost of the option while the potential upside profit is unlimited. Note that the portfolio insurance payoff diagram is identical to the profit/loss diagram for a long call option, however a long call is not one of the answer choices.


作者: Baran    时间: 2009-6-26 11:12

 

AIM 8: Define, compute and discuss theta, gamma, vega, and rho for option positions.


1、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?


A) 0.00.


B) 0.01.


C) 0.10.


D) 1.00.


作者: Baran    时间: 2009-6-26 11:12

 

The correct answer is B

 

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: Baran    时间: 2009-6-26 11:12

 

2、How is the gamma of an option defined? Gamma is the change in the:


A) vega as the option price changes.


B) theta as the option price changes.


C) delta as the price of the underlying security changes.


D) option price as the underlying security changes.


作者: Baran    时间: 2009-6-26 11:13

 

The correct answer is C

 

Gamma is the rate of change in delta. It measures how fast the price sensitivity changes as the underlying asset price changes.


作者: Baran    时间: 2009-6-26 11:13

 

3、When an option’s gamma is higher:


A) delta will be lower.


B) delta will be higher. 


C) a delta hedge will perform more poorly over time. 


D) a delta hedge will be more effective. 


作者: Baran    时间: 2009-6-26 11:13

 

The correct answer is C

 

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: Baran    时间: 2009-6-26 11:13

 

4、Gamma is the greatest when an option:


A) is deep in the money. 


B) is deep out of the money. 


C) is at the money. 


D) has a shorter maturity.


作者: Baran    时间: 2009-6-26 11:14

 

The correct answer is C

 

Gamma, the curvature of the option-price/asset-price function, is greatest when the asset is at the money.


作者: Baran    时间: 2009-6-26 11:14

 

5、Call and put option values are most sensitive to changes in the volatility of the underlying when:


A) both calls and puts are deep in-the-money.


B) both puts and calls are deep out-of-the-money.


C) calls are deep out-of-the-money and puts are deep in-the-money.


D) both calls and puts are at-the-money.


作者: Baran    时间: 2009-6-26 11:15

 

The correct answer is D

 

Vega measures the sensitivity of the option value to changes in volatility. Vega is at a maximum when calls and put options are at-the-money.



作者: Baran    时间: 2009-6-26 11:16

 

AIM 9: Explain how to implement and maintain a gamma-neutral position.


1、Gamma-neutral hedging:


A) increases sensitivity to small changes in asset prices.


B) decreases sensitivity to small changes in asset prices. 


C) increases sensitivity to large changes in asset prices.


D) decreases sensitivity to large changes in asset prices. 


作者: Baran    时间: 2009-6-26 11:16

 

The correct answer is D

 

Gamma-neutral hedging is designed to mitigate the effect of large changes in asset prices on delta-neutral positions that are designed to protect against small changes in asset prices.


作者: Baran    时间: 2009-6-26 11:17

 

2、Which of the following is least accurate regarding a gamma hedge?


A) More frequent rebalancing of a gamma hedge should result in higher returns.


B) Gamma hedges require less frequent rebalancing than delta hedges.


C) The gamma increases with larger changes in the stock price.


D) Gamma measures the change in delta.



作者: Baran    时间: 2009-6-26 11:17

 

 The correct answer is A

 

Gamma measures the change in delta. Gamma becomes larger as the changes in stock price increase in absolute value. Gamma hedging requires less frequent rebalancing than delta hedging. Less frequent rebalancing in a gamma hedge can result in higher returns but also increases the position’s volatility.




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蔷薇老妈整形前后对比图

蔷薇老妈的成名、励志之路
微博上,“蔷薇老妈”不仅在字里行间炫耀着她的身材,还贴上风情艳照大秀自己的S身材,参加“快乐女声”、远嫁美国……精彩之至堪比熟女版的凤姐、芙蓉姐姐,在网上引起一阵又一阵热议。据悉,这是07年“蔷薇老妈”整形蜕变重拾已逝芳华后,励志之作。
下面我们来看看蔷薇老妈是如何一跃成名的。
“蔷薇老妈”原名周蔷玲,是武汉某毛纺厂一名退休女工,含辛茹苦单身抚养儿子20余年。自06年,儿子考上了华中科大同济医学院研究生后,为了排遣孤单,在儿子的帮助下学会了上网。谈吐不俗的周蔷玲很快结交网友多达1000余人。后来,在网友的建议下,她开始以“蔷薇老妈”的网名开起了博客。
“我的眼睛不大,曾经有同事说我眼睛是‘9点一刻’,我有点气,不恼。也有的同事说我眼睛无神,看人迷迷的,有点媚劲。”“这辈子,年轻时臀部和胸部让我自卑,现在却让我自豪,我现在知道了,这就是性感…… ”“蔷薇老妈”博文里的这些言辞一面世,就引起了不小的轰动。让人更为之震惊的是她娇艳的摸样,绝对联想不到她已56岁“高”龄。
据透露,她的“美魔女”样子可是经过现今风靡全球的抗衰老除皱术得来的。果真,天下没有丑女人,看你是否懂得改变。

许燕玲整形前后对比图

不远千里挑战南国“美魔女”
当“蔷薇老妈”得知,远在南国的“美魔女”——东方整形医院的童颜明星许艳玲名声在外,势头即将赶上她时,“蔷薇老妈”坐不住了,声称在26号会一会“素颜辣妈”许艳玲。
“蔷薇老妈”告诉笔者,挑战“素颜辣妈”是假,不过是网友们的夸大之词,但是前来谈心、交流是真。因为同样的经历、同样的不幸婚姻,同样的年龄让她觉得看到了自己的影子,希望用她们的励志经历告诉给更多人:中老年人也有追求美的权利。向世人再次证明,年龄不是问题,关键在你是否想改变,是否有勇气改变你的人生。
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更多精彩请关注,11月26日东方整形美容医院名媛盛典“大妈变大姐”网络红人蔷薇老妈见证会。“蔷薇老妈”及“素颜辣妈”现场为您现身说法如何逆转时空,重拾青春艳美;尊享马来西亚开运整形设计大师为您美来运转。
诚邀爱美的您光临盛会!





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