FRM二級市場風(fēng)險易錯題
精選問答1
題干
In early 2000, a risk manager calculates the VAR for a technology stock fund based on the past three years of data. The strategy of the fund is to buy stocks and write out-of-the-money puts. The manager needs to compute VAR. Which of the following methods would yield results that are least representative of the risks inherent in the portfolio?
選項(xiàng)A
Historical simulation with full repricing
選項(xiàng)B
Delta-normal VAR assuming zero drift
選項(xiàng)C
Monte Carlo style VAR assuming zero drift with full repricing
選項(xiàng)D
Historical simulation using delta equivalents for all positions
答案解析
D is correct.
Because the portfolio has options, methods A or C based on full repricing would be appropriate. Next, recall that technology stocks had a big increase in price until March 2000.
From 1996 to 1999, the NASDAQ index went from 1,300 to 4,000. This creates a positive drift in the series of returns. So, historical simulation without an adjustment for this drift would bias the simulated returns upward, thereby underestimating VAR.
FRM考點(diǎn)解析：
這題首先是歷史背景情況，當(dāng)時科技股全在大漲，相對于大盤有更高的收益，即return drift upward，因此要對漂移進(jìn)行處理，否則這時候會計量的var會小。assuming zero drift就是不對漂移做處理。
解題思路
看提問情況，對B和D的疑問多一些，D選項(xiàng)用historical simulation在互聯(lián)網(wǎng)泡沫的背景下肯定是高估的，而delta normal的VAR至少還是假設(shè)了基礎(chǔ)資產(chǎn)收益呈正態(tài)分布的delta*(u-z*sigama)，相對而言，D比B高估的更直接。
精選問答2
題干
A trader buys an at-the-money call option with the intention of delta-hedging it to maturity. Which one of the following is likely to be the most profitable over the life of the option?
選項(xiàng)A
An increase in implied volatility
選項(xiàng)B
The underlying price steadily rising over the life of the option
選項(xiàng)C
The underlying price steadily decreasing over the life of the option
選項(xiàng)D
The underlying price drifting back and forth around the strike over the life of the option
答案解析
D is correct. An important aspect of the question is the fact that the option is held to maturity. Answer A is incorrect because changes in the implied volatility would change the value of the option, but this has no effect when holding to maturity.
The profit from the dynamic portfolio will depend on whether the actual volatility differs from the initial implied volatility. It does not depend on whether the option ends up in-the-money, so answers B and B are incorrect. The portfolio will be profitable if the actual volatility is small, which implies small moves around the strike price (answer D).
解題思路
首先要明白這題考的不是期權(quán)利潤而是動態(tài)對沖的理念。
delta對沖是瞬時對沖，要想一直實(shí)現(xiàn)對沖目的，就必須要動態(tài)對沖，即不斷根據(jù)標(biāo)的資產(chǎn)的價格變動，來調(diào)整對沖比例。如果標(biāo)的資產(chǎn)的真實(shí)波動率過大，即每次調(diào)整投資組合的幅度過大，就會帶來過大的成本。因此，真實(shí)波動率越小，動態(tài)投資組合的調(diào)整成本也越小，即越profitable。
ATM的option的delta大致是在0.5的，這時候gamma最大，一旦underlying上升或者下降，ITM或者OTM的option的delta都會有顯著變化，導(dǎo)致對沖的頭寸需要調(diào)整，增加成本，而一直圍繞行權(quán)價波動的話，delta一直是0.5左右，對沖的頭寸比較穩(wěn)定，成本就比較低。
精選問答3
題干
Trader A purchases a down-and-out call with a strike price of USD 100 and a barrier at USD 96 from Trader B. Both traders need to unwind their delta hedge at the barrier. Which trader is more at risk if there is a price gap (discontinuity) that prevents them from exiting the trade at the barrier?
選項(xiàng)A
Trader A has the bigger risk.
選項(xiàng)B
Trader B has the bigger risk.
選項(xiàng)C
They both have the same risk.
選項(xiàng)D
Neither trader has any risk because both are hedged.
答案解析
Each trader replicates dynamically the down-and-out call as a hedge. Trader B sold the option, so needs to replicate a long position in this call.
The hedge ratio for a down-and-out call resembles the usual one except that it has an abrupt discontinuity, dropping to zero below the barrier.
Just above the barrier, Trader B is long the asset in the amount of the hedge ratio (e.g., 0.4). When the price jumps down below the barrier, Trader B will be stuck with a large loss. Intuitively, this loss is the gain to Trader A, who has the opposite position.
解題思路
首先這個知識點(diǎn)是奇異期權(quán)的知識點(diǎn)，這里是barrier相當(dāng)于一個觸發(fā)價格，down-and-out call是一個看漲期權(quán)但是在基礎(chǔ)資產(chǎn)下降到一個特定價格(即barrier)時，這個期權(quán)就自動終止了，對于本題而言，barrier就是96。
然后考到的知識點(diǎn)是動態(tài)對沖。首先，雙方進(jìn)行該筆期權(quán)的交易，目的都是為了進(jìn)行套期保值對沖。所以，可以推出，A的投資組合是標(biāo)的資產(chǎn)的空頭+call的多頭。B的投資組合是標(biāo)的資產(chǎn)的多頭+call的空頭。
對于本題的down-and-out障礙期權(quán)，如果價格下跌觸碰到障礙水平，那么該障礙期權(quán)就將out失效。所以，此時，A就只剩下標(biāo)的資產(chǎn)的空頭，獲得資產(chǎn)下跌的收益，B將只剩下標(biāo)的資產(chǎn)的多頭，遭受損失。
精選問答4
題干
In May of 2005, several large hedge funds had speculative positions in the collateralized debt obligations (CDOs) tranches. These hedge funds were forced into bankruptcy due to the lack of understanding of correlations across tranches. Which of the following statements best describes the positions held by hedge funds at this time and the role of changing correlations? Hedge funds held a:
選項(xiàng)A
long equity tranche and short mezzanine tranche when the correlations in both tranches decreased.
選項(xiàng)B
short equity tranche and long mezzanine tranche when the correlations in both tranches increased.
選項(xiàng)C
short senior tranche and long mezzanine tranche when the correlation in the mezzanine tranche increased.
選項(xiàng)D
long mezzanine tranche and short equity tranche when the correlation in the mezzanine tranche decreased.
答案解析
D is correct. A number of large hedge funds were short on the CDO equity tranche and long on the CDO mezzanine tranche. Following the change in bond ratings for Ford and General Motors, the equity tranche spread increased dramatically.
This caused losses on the short equity tranche position. At the same time, the correlation decreased for CDOs in the mezzanine tranche, Which led to losses in the mezzanine tranche.
解題思路
這題要經(jīng)過兩次判斷，首先是違約率環(huán)境的高低，因?yàn)镸ezzanine層次的表現(xiàn)主要取決于違約率，違約率低的時候更像Senior，因此不太會遭受損失，相反則像Equity，因?yàn)楹芸赡茉馐軗p失。
之后就把mezzanine和判斷好的那邊打包一起看待。然后通過correlation判斷相對價值，相關(guān)性高的話senior相對也不怎么值錢了，相關(guān)性低的話senior會被保護(hù)的比較好。
精選問答5
題干
Which of the following statements about VAR estimation methods is wrong?
選項(xiàng)A
The delta-normal VAR method is more reliable for portfolios that implement portfolio insurance through dynamic hedging than for portfolios that implement portfolio insurance through the purchase of put options.
選項(xiàng)B
The full-valuation VAR method based on historical data is more reliable for large portfolios that contain significant option-like investments than the delta-normal VAR method.
選項(xiàng)C
The delta-normal VAR method can understate the true VAR for stock portfolios when the distribution of the return of the stocks has high kurtosis.
選項(xiàng)D
Full-valuation VAR methods based on historical data take into account nonlinear relationships between risk factors and security prices.
答案解析
Full-valuation methods are more precise for portfolios with options, so answers B and D are correct. The delta-normal VAR understates the risk when distributions have fat tails, so answer C is correct.
Answer A is indeed wrong. The delta-normal method will be poor for outright positions in options, or their dynamic replication.
解題思路：
delta-normal的方式是一階導(dǎo)數(shù)，如果按照泰勒公式展開還有二階導(dǎo)、三階導(dǎo)等等，二階導(dǎo)即Gamma，一般認(rèn)為到了二階導(dǎo)，大部分的價值變動的因素已經(jīng)被涵蓋，根據(jù)成本效益原則就不再去求三階甚至以上的了，但是full valuation是包含了全部的影響因素，因而在計量風(fēng)險方面比delta-normal要準(zhǔn)確。
很多同學(xué)對A為什么錯有疑問，A選項(xiàng)之所以不對，是因?yàn)閯討B(tài)對沖需要不斷的調(diào)整資產(chǎn)組合，會產(chǎn)生更多的成本。不符合資產(chǎn)收益率服從正態(tài)分布的前提條件
精選問答6
題干
Assume the profit/loss distribution for XYZ is normally distributed with an annual mean of $20 million and a standard deviation of $10 million. The 5% VaR is calculated and interpreted as which of the following statements?
選項(xiàng)A
5% probability of losses of at least $3.50 million.
選項(xiàng)B
5% probability of earnings of at least $3.50 million.
選項(xiàng)C
95% probability of losses of at least $3.50 million.
選項(xiàng)D
95%probability of earnings of at least $3.50 million.
答案解析
D is correct. The value at risk calculation at 95% confidence is: -20 million + 1.65 x 10 million = -$3.50 million. Since the expected loss is negative and VaR is an implied negative amount, the interpretation is that XYZ will earn less than +$3.50 million with 5% probability, which is equivalent to XYZ earning at least $3.50 million with 95% probability.
解題思路：
這題中是VaR的一種特殊情況。一般而言，我們提到VaR首先想到的肯定是最大損失，這是因?yàn)槲覀兤綍r的期望收益(u)跟)0考得很近，比如下圖紅色線條為0，那么減去1.96倍標(biāo)準(zhǔn)差的值必然小于0(在紅線左側(cè))。
但是極端情況，比如期望收益特別大，但是標(biāo)準(zhǔn)差特別小，比如黃色線條是0，你會發(fā)現(xiàn)即便減去1.96倍標(biāo)準(zhǔn)差，結(jié)果還是正數(shù)，此時就由我們平時的最大損失變成了最小收益的概念。
一個比較繞口但是容易理解的思路是，正常計算VaR，通常我們得到一個小于0的數(shù)字，取絕對值，我們把它叫做最大損失，但是現(xiàn)在計算完了是大于0的，我們可以說最大損失是-3.5million，損失是負(fù)數(shù)便是收益，所以換言之，最小收益是3.5million.
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精選問答7
題干
What is the effect on the value of a callable convertible bond of a decrease in interest rate volatility and stock price volatility?
選項(xiàng)A
An increase in value due to both interest rate volatility and stock price volatility
選項(xiàng)B
An increase and decrease in value, respectively
選項(xiàng)C
A decrease and increase in value, respectively
選項(xiàng)D
A decrease in value due to both
答案解析
A decrease in stock price volatility decreases the value of the equity conversion option and thus the convertible bond price.
A decrease in interest rate volatility decreases the value of the interest rate call option. Because the bond investor is short the interest rate option, this increases the value of the convertible.
解題思路：
callable convertible bond=bond - call option on bond + call option on stock
可贖回可轉(zhuǎn)債的一般性態(tài)仍是債券，發(fā)行人有權(quán)利將其贖回，這一般是當(dāng)債券的利率變低(也就是債券價格變高)時贖回，所以，這個贖回權(quán)，既可以看成是債券價格的call option，也可以看成是利率的put option。
當(dāng)利率的波動率下跌時，無論是債券的call option，還是利率的put option，它們的價格都會下降，而又因?yàn)槟闶莝hort，所以對于callable convertible bond，價格是上升的。
或者換個角度來理解，callable convertible bond可以拆分為callable bond再加一個convertible的權(quán)利(也就是call option on stock)，這樣要理解利率波動性對整個callable convertible bond的影響，只要看利率對callable bond的影響就可以了。
站在發(fā)行人的角度來理解：利率的波動性下降，那利率下降的可能性、幅度就不會那么大，那我就需要一個“弱一點(diǎn)的”保護(hù)，所以callable bond的價格就可以更高一些了，因?yàn)楸Ｗo(hù)越強(qiáng)，對發(fā)行人而言，價格就得越便宜。
精選問答8
題干
Which of the following is most accurate with respect to delta-normal VAR?
選項(xiàng)A
The delta-normal method provides accurate estimates of VAR for assets that can be expressed as a linear or nonlinear combination of normally distributed risk factors.
選項(xiàng)B
The delta-normal method provides accurate estimates of VAR for options that are near or at-the-money and close to expiration.
選項(xiàng)C
The delta-normal method provides estimates of VAR by generating a covariance matrix and measuring VAR using relatively simple matrix multiplication.
選項(xiàng)D
The delta-normal method provides accurate estimates of VAR for options and other derivatives over ranges even if deltas are unstable.
答案解析
The delta-normal approach will perform poorly with nonlinear payoffs, so answer A is false. Similarly, the approach will fail to measure risk properly for options if the delta changes, which is the case for at-the-money options, so answers B and D are false.
解題思路：
這題主要是問C選項(xiàng)的，協(xié)方差矩陣可以衡量多個資產(chǎn)的方差，收益率也可以用矩陣(向量)來表示。只要他們的聯(lián)合分布是服從正態(tài)分布，即使是多元正態(tài)分布，那也可以像計算單個資產(chǎn)的VaR一樣來計量多個資產(chǎn)的VaR
精選問答9
題干
Which of the following statements is not an advantage of spectral risk measures over expected shortfall? Spectral risk measures:
選項(xiàng)A
consider a manager’s aversion to risk.
選項(xiàng)B
are a special case of expected shortfall measures.
選項(xiàng)C
have the ability to modify the risk measure to reflect an investor's specific risk aversion.
選項(xiàng)D
have better smoothness properties when weighting observations.
答案解析
Spectral risk measures consider aversion to risk and offer better smoothness properties.
Expected shortfall is a special case of spectral risk measures.
解題思路：
譜風(fēng)險度量的方法如下圖，confidence level越大，即損失越大，它分配的權(quán)重也就越大。一直到40%，它都有分配權(quán)重。而ES法中，它只計算了尾部個別幾個數(shù)據(jù)的平均值。所以譜分析法的權(quán)重分配更平滑一些。
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