1.金融市场与产品练习题第54题这类delta hedge问题解题思路是什么?

 

Initially,the call option on Big Kahuna Inc.with 90 days to maturity trades at USD 1.40.The option has a delta of 0.5739.A dealer sells 200 call option contracts,and to delta-hedge the position,the dealer purchases 11,478 shares of the stock at the current market price of USD 100 per share.The following day,the prices of both the stock and the call option increase.Consequently,delta increases to 0.7040.To maintain the delta hedge,the dealer should

 

A.sell 2,602 shares

 

B.sell 1,493 shares

 

C.purchase 1,493 shares

 

D.purchase 2,602 shares

 

Answer:D

 

答疑；其实这种题目的本质是使原有头寸的线性风险能够被对冲头寸的收益所覆盖，在这道题中就相当于由股价变动引发的期权的价格变化（由delta反映）能够由购买或者卖出股票获得的收益所覆盖。可以参考handbook P322的公式进行理解，原理是一样的。这题的关键是200 call option contracts，而每个合约里面有100个期权，所以总共有200*100=20000个期权。

 

2.EVT中那个字母说明尾部形状，大于1肥尾，小于0细尾，而数量分析习题课第52题第二个说法，说这个字母为负的时候说明尾部比正态分布尾部消失的快，尾部细不就是消失得快吗？

 

Which of the following statements about Extreme Value Theory(EVT)and its application to value at risk are true?

 

I.EVT extends the Central Limit Theorem to the distribution of the tails of independent,identically distributed random variables drawn from an unknown distribution.

 

II.For empirical stock market data,the shape parameter in EVT is negative implying tails that disappear more rapidly than a normal distribution.

 

III.EVT can help avoid a shortcoming of the historical simulation method which may have difficulty calculating VaR reliably due to a lack of data in the tails.

 

IV.For empirical stock market data,standard value at risk estimates at the 95%confidence level are exceeded more often than 5%of the time and would therefore benefit from the use of extreme value theory.

 

A.I and III

 

B.II and IV

 

C.I,III and IV

 

D.III and IV

 

Answer:A

 

I.correct.Whereas the Central Limit Theorem concerns the distribution of the average of independent,identically distributed variables drawn from an unknown distribution,EVT deals with the distribution of the tails.

 

II.incorrect.The shape parameter in EVT for empirical stock market data is typically between 0.2 and 0.4,implying that the tails disappear more slowly than a normal distribution.

 

III.correct.Due to its reliance on historical data which may lack sufficient tail data(i.e.,extreme events),reliably calculating VaR with the historical simulation method can be difficult;EVT can help avoid this shortcoming.

 

IV．incorrect.For empirical stock market data,standard value at risk estimates at the 95%confidence level tend to be fairly accurate,and generally only becomes inaccurate at the 99.5%confidence level and beyond.

 

答疑：Shape parameter determines the speed at which the tail disappears.The normal distribution corresponds to shape parameter equals zero while typical financial data have a positive shape parameter which implies fat tails.(Handbook P364)。这道题的问题出在for empirical stock market data,应该是肥尾的现象，对应正的shape parameter。而后半句没有问题。

 

3.下题不理解，请教老师！

 

You are using key rate shifts to analyze the effect of yield changes on bond prices.Suppose that the 10-year yield has increased by 10 basis points and that this shock decreases linearly to zero for the 20-year yield.What is the effect of this shock on the 14-year yield?

 

A.increase of 0 basis points

 

B.increase of 4 basis points

 

C.increase of 6 basis points

 

D.increase of 10 basis points

 

Answer:C

 

Explanation:The 10 basis point shock to the 10-year yield is supposed to decline linearly to zero for the 20 year yield.Thus,the shock decreases by 1 basis point per year and will result in an increase of 6 basis points for the 14 year yield.

 

答疑：就是说10年期的利率增量为10bps，然后直线下降到20年0bps，相当于每年增量减少1bps，那14年的增量就是(10–4)bps。这题考察的是关键利率的移动技术，在强化班“估值与风险模型”PPT里面有讲到。

 

4.2010 FRM EXAMINATION PRACTICE EXAM第5题怎么做？

 

Sarah is a risk manager responsible for the fixed income portfolio of a large insurance company.The portfolio contains a 30-year zero coupon bond issued by the US Treasury(STRIPS)with a 5%yield.What is the bond';s DV01?

 

A.0.0161

 

B.0.0665

 

C.0.0692

 

D.0.0694

 

Answer:B

 

Explanation:The DV0I of a zero-coupon is

 

DVO1=30/100(1+y/2)2T+1100(1+5%/2)61=0.0665

 

答疑：此题答案有误，从题目中给出的信息应该计算的是零息债券的DV01，而解释中采用的是付息债券的计算方法。按题目考查的应该是零息债券麦考林久期的特性，麦考林久期、修正久期以及DV01之间的转换，建议以把握这些内容为主。

 

5.对于delta，put在in the money时应该趋近于-1？

 

答疑：对。The delta of an at-the-money put option is close to-0.5.Delta moves to-1 as the put goes deep in-the-money.It moves to 0 as the put goes deep out-of-the-money.

 

6.EXAMPLE 2.1:FRM EXAM 2009-QUESTION 2-3

 

An analyst gathered the following information about the return distributions for two portfolios during the same time period:

 

Portfolio

 

Skewness

 

Kurtosis

 

A

 

-1.6

 

1.9

 

B

 

0.8

 

3.2

 

The analyst states that the distribution for Portfolio A is more peaked than a normal distribution and that the distribution for Portfolio B has a long tail on the left side of the distribution.Which of the following is correct?

 

A.The analyst';s assessment is correct.

 

B.The analyst';s assessment is correct for Portfolio A and incorrect for Portfolio B.

 

C.The analyst';s assessment is not correct for Portfolio A but is correct for Portfolio B.

 

D.The analyst';s assessment is incorrect for both portfolios.

 

答案是否有问题？

 

答疑：A负偏（左偏）B正偏（右偏）；A低峰瘦尾B高峰肥尾；应该选择D

 

7.Nick Leeson’s now infamous trading strategies in 1994 and 1995 at Barings Bank focused on calculated bets on the Nikkei 225.Which of the following trading strategies did not contribute to the staggering losses that ultimately forced Barings into bankruptcy?

 

I.Long-long futures arbitrage.

 

II.Long straddle.

 

A.I only

 

B.II only

 

C.Both I and II

 

D.Neither I nor II

 

这个里面的long-long futures arbitrage是什么意思？

 

答疑：这个案例中尼克里森本来被允许进行的交易是在不同市场的关于期货的套利交易，也就是低买高卖，在一个市场上买入日经225的期货，在另外一个市场上卖出日经225的期货。因此他的交易策略应该是一个long-short futures arbitrage strategy，而最终为了弥补他前期的损失他采用了投机的策略，在不同的交易市场同时买入日经225的期货，从而变成了一个long-long futures的投机策略。此外他采用的是一个short straddle的策略，所以这道题应该选择B。

 

8.An investor enters a short position in a gold futures contract at$318.60.Each futures contract controls 100 troy ounces.The initial margin is$5,000 and the maintenance margin is$4,000.At the end of the first day the futures price rises to$329.22.Which of the following is the amount of the variation margin at the end of the first day?

 

A.$0

 

B.$62

 

C.$1,000

 

D.$1,062

 

与Handbook7.7作比较

 

答疑：卖出黄金期货合约，所以价格上升的时候亏钱。亏：（329.22-318.60）*100=1062，初始保证金为5000，亏损1062，此时保证金跌至4000以下，所以需要追加保证金。这道题问的是变动保证金也就是说需要补充回初始保证金的金额；所以追加的变动保证金就是亏的金额，即1062。而Handbook中7.7，题目中价格下跌的时候卖出期货合约是赚钱的。不用追加保证金。

 

9.A bank had entered into a 3-year interest rate swap for a notional amount of USD 300 million,paying a fixed rate of 7.5%per year and receiving LIBOR annually.Just after the payment was made at the end of the first year,the continuously compounded 1-year and 2-year annualized LIBOR rates were 7%per year and 8%per year,respectively.The value of the swap at that time was closest to which of the following choices?

 

A.USD-14 million

 

B.USD-4 million

 

C.USD 4 million

 

D.USD 14 million

 

这道题用折现的思想怎么做？

 

答案：C

 

答疑：这道题由于采用的是连续复利，所以在FRA定价中将7%和8%作为利率水平会产生误差，建议还是采用债券法的方法来进行计算，如果是一般复利，两种方法算出来应该是一样的。具体做法如下：

 

所以答案选择：C

 

10.（1）计算时间怎么使用计算器，详细些，比如2005-1-21~2008-8-8之间的时间间隔，详细说说输入的按钮；（2）先付年金怎么调的；（3）数据回归的方法。

 

答疑：1.[2nd][DATE]→输入时间，以小数点形式，如：12.10.2008输入12.1008→[ENTER]→[↓]→同样输入第二个日期→[↓]→[CPT]

 

2.[2nd][BGN]→[2nd][SET]，屏幕中会出现BGN字样。还原只要再做一次[2nd][SET]即可。FRM考试大多数情况下不用到期初情况。

 

3.[2nd][DATA]进入数据表输入，可输入两组数据。[2nd][STAT]可以查看输入的数据的均值、方差，线性回归参数等。

 

11.Roy Thomson,a global investment risk manager of FBN Bank,is assessing markets A and B using a two-factor model.In order to determine the covariance between markets A and B,Thomson developed the following factor covariance matrix for global assets:

 

Factor Covariance Matrix for Global Assets

 

Global Equity Factor

 

Global Bond Factor

 

Global Equity Factor

 

0.3543

 

-0.0132

 

Global Bond Factor

 

-0.0132

 

0.0089

 

Suppose the factor sensitivities to the global equity factor are 0.75 for market A and 0.45 for market B,and the factor sensitivities to the global bond factors are 0.2 for market A and 0.65 for market B.The covariance between market A and Market B is closest to:

 

A.-0.215

 

B.-0.113

 

C.0.113

 

D.0.215

 

答案：C

 

答疑：题目求的是：市场A与市场B的协方差，*9步我们先来寻找一下市场A与市场B的资产构成情况。

 

在本题中，如果记：Global Equity Factor为X,Global Bond Factor为Y，那么从“Factor Covariance Matrix for Global Assets”中可以看出：=0.0089,Cov(X,Y)=-0.0132。

 

由题目中的：“Suppose the factor sensitivities to the global equity factor are 0.75 for market A and 0.45 for market B,and the factor sensitivities to the global bond factors are 0.2 for market A and 0.65 for market B.”可以得出：

 

市场A=0.75X+0.2Y，市场B=0.45X+0.65Y，

 

从而Cov(A,B)=Cov(0.75X+0.2Y,0.45X+0.65Y)=0.75×0.45×Cov(X,Y)+(0.2×0.45+0.75×0.65)×Cov(X,Y)+0.2×0.65×Cov(Y,Y)=0.113

 

12.Assume the six-month LIBOR is 6%and its annualized volatility is 20%.Based on this information,the six-month LIBOR should not exceed which of the following with 95%confidence level within a year?

 

A.7.97%

 

B.7.60%

 

C.7.40%

 

D.7.02%

 

答疑：There is a 95%probability that LIBOR will be less than 6%+(1.65)(0.20)(6%)=7.98%.Thus,with a 95%confidence level,LIBOR should not exceed 7.90%.

 

注意：划线部分，一般的题目都是直接让关键值(1.65)乘以波动率(0.20)，而此处却是0.20乘以6%，主要是因为LIBOR的报价是依照差价报价；而这里的波动率20%也是在差价基础上求得的波动率，所以需要乘以20%。

 

13.Which of the following statements are true with respect to basis risk?

 

I.Basis risk arises in cross-hedging strategies but there is no basis risk when the underlying asset and hedge asset are identical.

 

II.Short hedge position benefits from unexpected strengthening of basis.

 

III.Long hedge position benefits from unexpected strengthening of basis.

 

A.I and II

 

B.I and III

 

C.II only

 

D.III only

 

答疑：主要的疑点在于II与III，首先需要知道short hedge position是指卖出期货合约来对冲，当期货价格相对下降时获利；long hedge position是指买入期货合约来对冲，当期货价格相对上升时获利。basis是指：现货-期货。

 

接下来是对“strengthening of basis”的理解：有几个理解角度，相对现货来说，期货价格下降的程度更大；或者说，相对于现货来说，期货价格涨幅低于现货价格。不论从哪一个角度出发，都是short hedge(卖空期货)时，从中受益。

 

14.Which of the following statements are TRUE?

 

I The convexity of a 10-year zero coupon bond is higher than the convexity of a 10-year,6%bond

 

II The convexity of a 10-year zero coupon bond is higher than the convexity of a 6%bond with a duration of 10 years.

 

III Convexity grows proportionately with the maturity of the bond

 

IV Convexity is always positive for all types of bonds.

 

V Convexity is always positive for“straight”bonds.

 

A.I only

 

B.I and II only

 

C.I and V only

 

D.II,III,and V only

 

答案：C

 

答疑：这道题目考察的是convexity的性质，All else equal,convexity increase for longer maturities,lower coupons,and lower yields。需注意的是，convexity是与到期日的平方成正比的。

 

重点关注II，10-year zero coupon bond久期也为10，a 6%bond with a duration of 10 years的到期日肯定大于久期10，所以II错误。

 

15.A dollar-based investor has portfolio consisting of$1 million in cash plus a position in 1,000 million Japanese yen.The distribution of dollar/yen exchange rate X has mean of

 

E(x)=0.01 and volatility of SD(x)=0.001.Which of the following is the expected value and standard deviation of portfolio?

 

A.$11 million and$1 million

 

B.$1 million and$11 million

 

C.$11 million and$2 million

 

D.Can not be calculated from above infromation.

 

答案：A

 

答疑：E(P)=E(1+1000*X)=1+1000*0.01=$11(million)

 

Var(P)=Var(1+1000*X)=10002×Var(X)=10002×（0.001）2=$1(million)

 

16.Assume that the 3-month and 6-month LIBOR spot rates are 4%and 5%respectively(continuously compounded).An investor enters into an FRA in which she will receive 8%(quarterly compounding)on a principal of$5,000,000 between months 3 and calculate the current value of an FRA.

 

A.$23,773

 

B.$24,773

 

C.$25,773

 

D.$26,773

 

答案：A

 

答疑：这道题目需严格注意，在计算过程中采取的利率形式。

 

*9步：求解远期利率。

 

第二步：利率形式转化。

 

第三步：计算FRA价值。

 

17.The market portfolio(M)contains the optimal allocation of only risky asseta and no risky assets.Let the S1 be the Sharpe ratio of this market portflio.There exists a risk-free asset.Initially,an investor is fully(100%)invested in M with a portfolio Sharpe ratio of S1.Subsequently,the investor borrows 30%at the risk-free rate,such that she is 130%invested in the market portfolio(M)where this leverage portfolio has a Sharpe ratio of S2。After the leverage(i.e.,borrowing at the riskfree rate to invest+30%in M,is the investor still on the efficient frontier and how do the Sharpe ratios?

 

A.No(no longer efficient),and S2<S1

 

B.No,but S2=S1

 

C.Yes(still efficient),but S2<S1

 

D.Yes and S2=S1

 

答案：D

 

答疑：

 

All portfolios on the CML have the same Sharpe ratio(市场组合的Sharpe ratio):the slope of the CML.

 

18.Gamma Industries,Inc.issues an inverse floater with a face value of USD 50,000,000 that pays a semiannual coupon of 11.50%minus LIBOR.Gamma Industries intends to execute an arbitrage strategy and earn a profit by selling the notes,using the proceeds to purchase a bond with a fixed semiannual coupon rate of 6.75%a year,and hedging the risk by entering into an appropriate swap.Gamma Industries receives a quote from a swap dealer with a fixed rate of 5.75%and a floating rate of LIBOR.What would be the most appropriate type of swap Gamma Industries,Inc.should enter into to hedge their risk?

 

A.Pay-fixed,receive-fixed

 

B.Pay-floating,receive-fixed swap

 

C.Pay-fix,receive-floating

 

D.The risk cannot be hedged with a swap

 

答案：B

 

The company has a floating outflow of and a fixed inflow of 6.75%.On the outflow,is the same as an inflow Pay-floating,Receive-fixed.Gamma Industries is exposed to interest rate fluctuations of LIBOR.Therefore,the appropriate swap would be a pay-floating,receive-fixed swap.

 

答疑：‘The company has a floating outflow of and a fixed inflow of 6.75%’，本质上：浮动利率方面，公司支付一个负的LIBOR，相当于收取一个正的LIBOR；固定利率方面：公司支付11.50%，收取6.75%，相当于支付4.75%。那么公司需要支付浮动利率，收取固定利率加以对冲，所以选择B。