国际视野：修正波动率指数:降低波动率指数受操纵的影响（2）

Fix the vix:Reducing Manipulation in the Volatility Index（2）
修正波动率指数:降低波动率指数受操纵的影响（2）
Numerical Results
数值结果
Totest and demonstrate the results of applying our proposed modifications of theVIX calculation, we refer to the CBOE white paper. As a method of clearlyarticulating this calculation mechanism, the white paper gives details of thenearly 150 option quotes and strikes for a specific example.
为了测试和演示我们提出的VIX计算修改的结果，我们参考CBOE白皮书。作为明确阐述此计算机制的一种方法，白皮书详细介绍了近150个期权报价和执行价的具体示例。
Thewhite paper, moreover, produces numerical values at each step of thecalculation to enable readers to reproduce the methods. For this study, weplaced all the relevant information for the “Near-Term Strike” in a spreadsheetand confirmed the calculations. This data then forms the basis for our remarksand observations in this section.
此外，白皮书在计算的每一步都会产生数值，以便读者重现这些方法。在这项研究中，我们将“近期执行价”的所有相关信息放在电子表格中，并确认了计算结果。这些数据构成了本节我们的评论和观察的基础。
Letus note that, on almost all days, the VIX calculation incorporates options thatare both somewhat shorter in maturity than 30 days (“near-term”) and longerthan 30 days (“next-term”). For our purpose, we use only the “near-term.” Wematch precisely the white paper’s value for near-term square of the volatility(0.01846292).
让我们注意到，在几乎所有的日子里，VIX计算都包含了期限比30天（“近期”）短和30天（“下一期”）更长的期权。就我们的目的而言，我们只使用“近期期权”。我们恰恰匹配了白皮书波动率近期平方的价值（0.01846292）。
Thecorresponding near-term volatility, which CBOE calls “VIN,” is 13.59 (to twodecimal places after multiplication by 100). The VIX result of the white papercombines the “next-term” series of option values with the “near-term” values togive an overall VIX of 13.69. But we do not use this combined VIX in ourdiscussion. We focus instead on the VIN.
CBOE称之为“VIN”的相应近期波动率为13.59（乘以100后小数点后两位）。白皮书的VIX结果将“下一期”系列期权价值与“近期”价格结合起来，给出13.69的总体VIX。但我们在讨论中不使用这个组合的VIX。我们专注于VIN。
Figure1 below shows the prices of all options of the CBOE example as a function ofoption strike price. The price graph peaks at the strike of 1960 because thisis where the transition from put options (at lower strikes) and call options(at higher strikes) occurs. (Referring to equation (2), this transition strikelevel of 1960 is the K0 of the CBOE VIX calculation.) The forward value of theSPX equity index is 1962.9 – deliberately close to the transition strike valueof 1960.
下图1显示了作为期权执行价格函数的CBOE例子的所有期权价格。价格图在1960年的执行价达到峰值，因为这是从看跌期权（在较低的执行价）和看涨期权（在较高的执行价）过度的地方。（参照公式（2），这个1960年的过渡执行价水平是CBOEVIX计算的K0。）SPX股票指数的远期价值是1962.9--故意接近1960年的过渡期执行价。

For clarity, we emphasize that the quotes tothe left of the peak in Figure 1 pertain to out-of-the-money put options, whilethe quotes to the right of the peak are for out-of-the-money call options. Thisfigure also defines the range of strike prices (from 1370 to 2125) that theCBOE employed in its VIX calculation.
为了能够更清楚，我们强调图1中峰值左边的报价属于虚值看跌期权，而峰值右边的报价则为虚值看涨期权。该图还定义了CBOE在VIX计算中采用的执行价格范围（从1370到2125）。
Observation:VIX is sensitive to the range of strike prices
观察：VIX对执行价格的范围敏感
As we have noted, an approximation of theVIX calculation of equation (2) omits the term

for all options with insufficient price quotes. Ideally, all options withstrike prices Ki, from zero to “infinity,” should contribute to (2).
正如我们已经注意到的，方程（2）的VIX计算的近似值省略了报价不足的所有期权

的部分。理想情况下，从零到“无穷大”的所有期权行权价格Ki都应该对（2）有所贡献。
In our CBOE white paper example of thissection, the calculation includes no quotes with strike prices less than 1370.Figure 2 (below) shows a portion of available option quotes of the white paperexample. (Of the five columns, the strike price is at the far left while theput option bid and offer sit in the fourth and fifth columns, respectively. Thesecond and third columns hold quotes for in-the-money call options, but theseare not relevant for our purposes.)
在本节的CBOE白皮书示例中，计算中不包含执行价格低于1370的报价。图2（以下）显示了白皮书示例的可用选项报价的一部分。（在五栏中，执行价位于最左边，而看跌期权竞价及报价分别位于第四栏及第五栏;第二栏及第三栏则保留实值认购期权的报价，但这些栏目与我们的目的不相关。）

By the rules of the calculation, the loweststrike price of included options is 1370. Yet on a different day or at adifferent time on the same day, this lowest eligible strike price will change.
根据计算规则，包含期权的最低执行价格为1370.然而，在另一天或同一天的不同时间，此最低合格执行价格将发生变化。

As one plausible example, imagine that thelowest included option is at the strike value of 1530. We choose this value dueto the listed quotes of the CBOE white paper near this value (see Figure 3).The data description of the columns of Figure 3 matches that of the earlierFigure 2.
作为一个合理的例子，想象一下最低的包含期权的执行价格为1530.由于CBOE白皮书的上的报价接近此值，我们选择此值（见图3）。图3中列的数据描述与前面的图2中的数据描述相匹配。

The bid-side strength for put options isweak in this range. We consider it entirely possible that two consecutivestrikes would show no bids at sporadic moments or intervals. Furthermore, aparticipant with manipulative intent could offer aggressively at these weakpoints to eliminate bids and thereby shorten the range of options that equation(2) employs in the VIX calculation.
在这个范围内，看跌期权的竞价方力量很弱。我们认为完全可能的是，连续两次执行价在零星的时刻或间隔内不会显示竞价。此外，具有操纵意图的参与者可以积极地利用这些弱点以消除竞价，从而缩短方程式（2）在VIX计算中使用的选项范围。
By eliminating the range of options from the1370 strike to the 1530 strike, the calculated volatility would fall from13.59% to 13.39%. This is a meaningful deviation for holders of VIX futures andoptions contracts. We consider this deviation of 20 bps to be plausible, andpossibly typical, rather than an extreme case. By a different measure ofdeviation, for example, Griffin and Shams find a mean disparity of 31 bps.
通过消除从1370点到1530点的多个期权，计算波动率将从13.59％下降到13.39％。这对于VIX期货和期权合约持有人来说是一个有意义的偏差。我们认为20bps的偏差是合理的，可能是典型的，而不是极端的情况。例如，通过不同的偏差度量，格里芬和沙姆斯发现平均差距为31个基点。
Observation:Simple calculation of the Put Tail is not meaningful
观察：简单计算看跌尾部没有意义
The weakness of the equation (2) VIXcalculation is that it fails to include all possible strike prices from zero to“infinity.” This failing is understandable. When there are no market bids fordeep out-of-the-money options, the most evident approximation is to treat theomitted options as having zero or near-zero value.
等式（2）VIX计算的弱点在于它没有包括从零到“无穷大”的所有可能的行使价格。这种失败是可以理解的。当没有深度虚值期权的市场竞价时，最明显的近似是将省略的期权视为具有零或接近零值。
However, as with any measurement that isboth consequential to market players and for which there are clear rules, theapproximation invites manipulation. It is conceivable that participants willsend bids or offers to attempt to lengthen or shorten the range of eligibleoption strike prices.
然而，如同任何与市场参与者相关并且有明确规则的测量一样，近似值会引起操纵。可以想象的是，参与者将发送买价或卖价来尝试延长或缩短符合条件的期权执行价格的范围。
An apparent solution, or at least amitigant, to this possible manipulation is to add to equation (2) calculationsfor the Put and Call tails. Thus, if one VIN calculation has a range beginningat the 1370 strike, for example, while a later calculation begins at 1530strike, then separate calculations for the tails (“zero to 1370” and “zero to1530,” respectively) could compensate for the 20 bps deviation.
对于这种可能的操纵，一个明显的解决方案或者至少是一个缓解措施是将公式（2）加入看跌和看涨尾部的计算。因此，例如，如果一个VIN计算的范围从1370开始，那么后面的计算在1530开始时开始，然后单独计算尾部（分别为“零至1370”和“零至1530”）可以补偿20bps的偏差。
But a first attempt to create thismitigation fails. In applying equation (3a) for the Put Tail for the 1370strike with the 13.59% volatility, we find a mitigating “correction term” thatis exceedingly small (less than one part in 1027). Even for the correction ofthe tail range to 1530 strike, the calculated correction is just one part in1015.
但是，首次尝试创建这种缓解措施却失败了。在对1370执行价的看跌尾部的13.59％波动率应用方程（3a）时，我们发现一个改善的“修正项”非常小（小于1027中的一部分）。即使将尾部距离修正为1530，计算出的修正值也只是1015的一部分。
A fair conclusion might be that theapproximations of omitting these strike ranges (from zero to 1370 and 1370 to1530) is entirely reasonable. There’s a flaw in this logic, however. The VINresult of 13.59% (of this numerical example) is misleading in the sense thatthe value 13.59% is a blended outcome. The proper volatility of tail ranges –such as zero to 1370 and zero to 1530 – is higher than the “blended” VINresult. Hence, we defer the tail calculation momentarily.
一个公平的结论可能是，省略这些执行价范围（从零到1370和1370到1530）的近似值是完全合理的。然而，这个逻辑有一个缺陷。13.59％（这个数字例子）的VIN结果是误导性的，13.59％的值是混合结果。尾部范围的正确挥发性- 例如从零到1370和零到1530- 高于“混合”VIN结果。因此，我们暂时推迟尾部计算。
Observation:Volatility skew is critically important to VIX calculation
观察：波动偏移对VIX计算至关重要
Just as Figure 1 shows the SPX option quotesas a function of strike price, Figure 4 shows the implied volatility wecalculate from the quotes.
正如图1所示，SPX期权报价是执行价格的函数，图4显示了我们通过报价计算的隐含波动率。

Our equations (1a-b), (3a-b), and (4a-b) –as well as the framework of DDKZ and the CBOE VIX calculation of equation (2) –all treat volatility σ as if it is independent of strike price. (DDKZ didinclude analyses of volatility skew. As they assumed small deviations, theirresults are not applicable to this case in which the linear increase in putoption volatility is large relative to the VIX value.)
我们的方程（1a-b），（3a-b）和（4a-b）以及DDKZ的框架和方程（2）的CBOEVIX计算- 都处理波动σ，就好像它独立于执行价。（DDKZ确实包括了波动率偏度分析，因为他们假设了小的偏差，所以他们的结果不适用于这种情况，即看跌期权波动率的线性增长相对于VIX值较大）。
Rather than being a mistake in light of theclear strike-dependence in Figure 4, this single-volatility jargon is theconvention of the VIX. It’s easier to have a single, weighted-averagevolatility as an index. Yet in any deeper analysis, such as a tailextrapolation, incorporating the strike-dependence of implied volatility isnecessary.
鉴于图4中清晰的走势依赖性，这种单波动性术语不是VIX的惯例。以单一的加权平均波动率作为指标更容易。然而，在任何更深入的分析中，如尾部外推，都需要引入隐含波动率对执行价的依赖。
For the put options (at strike values lessthan 1960), the implied volatility is visually linear. The put volatility is~12% at the K0 of 1960, and rises to ~50% at the end of the range at 1370.
对于看跌期权（在执行价低于1960），隐含波动率在视觉上呈线性。K0在1960时，看跌波动率约为12％，并在1370范围内上涨至约50％。
Call options behave differently. Though thecall option implied volatility is not constant, neither is it linear orstriking in its deviation from its value at K0. It also bears noting that therange of call options is restricted. The range extends only 10% or so above K0,whereas the put option range goes 30% below K0.
看涨期权的表现有所不同。虽然看涨期权的隐含波动率并不是恒定的，但它既不是线性的，在K0上的偏差也不显著。同时值得注意的是，看涨期权的范围受到限制。范围仅延伸K0以上10％左右，而看跌期权的范围比K0低30％。
Observation:It is necessary to incorporate skew into the Put Tail calculation
观察：有必要将偏差纳入看跌尾部计算
Given the clear linear skew of impliedvolatility for put options, it is not appropriate to apply equation (3a) tocalculate the Put Tail. Even if we choose a constant volatility equal to thatof the lowest strike option (i.e., ~50% at the 1370 strike), the calculationwould arguably underestimate the tail.
鉴于看跌期权的隐含波动率存在明显的线性偏差，将方程（3a）用于计算看跌期权是不合适的。即使我们选择与最低行权价期权相等的不变波动率（即在1370的时候约为50％），该计算也可以低估尾部。
This type of procedure – setting a uniformtail volatility based on an illiquid, far out-of-the-money option – might alsomake the VIX more susceptible to manipulation. Instead, we create a Put Tailcalculation that assumes a linear volatility skew. We write again the equations(5a-c) we showed earlier, with a specification now in (5c) for the volatilityas a function of x (equivalent to K/F):

这种类型的程序- 基于流动差的深度虚值期权设定均匀的尾部波动率- 也可能使VIX更易于操纵。相反，我们创建了一个假设线性波动率偏离的看跌尾部计算。我们再次写出我们前面显示的方程（5a-c），现在（5c）中的波动率是x的函数（相当于K/ F）：

We observe the volatility parameters σ0 andα (11.8% and -1.16, respectively) in the data of Figure 4. We then performnumerical integration of (5a-c), with L = 1370, F = 1962.9 and T = 0.0683486,and find that the Put Tail calculation increases the VIN to 13.68% from 13.59%(roughly a 0.7% adjustment).
我们观察图4数据中的波动率参数σ0和α（分别为11.8％和-1.16）。然后我们进行（5a-c）的数值积分，其中L= 1370，F= 1962.9和T= 0.0683486，发现看跌尾部计算将VIN从13.59％（约0.7％调整）提高到13.68％。
Separate calculation of the Call Tail doesnot require this numerical integration. Rather, given the absence of a markedlinear skew for call options, we apply equations (4a-b) and find the VIN risesan additional two basis points to 13.70%.
看涨尾部的单独计算不需要这种数值积分。相反，考虑到对于看涨期权没有显著的线性偏差，我们应用方程（4a-b）并且发现VIN增加了两个基点至13.70％。
Observation:Inclusion of the Put Tail mitigates VIX sensitivity
观察：包含看跌尾巴可以改善VIX灵敏度
Returning to our earlier observation thatthe VIX calculation is sensitive to the range of strike prices of equation (2),we noted that modifying the lowest strike from 1370 to 1530 decreased the VINfrom 13.59% to 13.39%. The procedure of adding the Put Tail to the calculationlargely eliminates this sensitivity.
回到我们先前观察到的VIX计算对方程（2）的执行价格范围敏感的情况，我们注意到将最低从1370改为1530使得VIN从13.59％降至13.39％。在计算中添加看跌的过程很大程度上消除了这种敏感性。
As we noted just above, adding both the Putand Call Tails to the CBOE white paper example gives a VIN of 13.70% (ratherthan 13.59%). Modifying this example to make 1530, as opposed to 1370, thelowest strike price, our calculated VIN is 13.65% (rather than 13.39%).
正如我们前面提到的，将CBOE白皮书示例中的看跌和看涨尾部添加给出了13.70％的VIN（而不是13.59％）。修改这个例子使得1530，而不是1370，最低的行使价，我们计算的VIN是13.65％（而不是13.39％）。
Consequently, the tendency of incidental ordeliberate changes to the range of eligible options to affect VIX calculationsfalls greatly when we include the tail contributions. This smaller 5 bpsdifference (13.65% versus 13.70%) stems from the difference between actualoption quotes and the linear skew approximation in the range of 1370 to 1530.The traditional VIX calculation of equation (2) assumes a zero value, ratherthan the linear skew, for this 1370-1530 range.
因此，当我们包括尾部贡献时，偶然或故意改变影响VIX计算的合格选项的范围会大大降低。这个较小的5bps差异（13.65％对13.70％）源于实际期权报价与1370至1530范围内的线性偏差近似值之间的差异。方程（2）的传统VIX计算假定为零值，而不是线性偏斜，在1370-1530范围内。
Relevancefor the VIX of Individual Equities
个股VIX的相关性
As the CBOE white paper notes, the CBOEoffers VIX indices on individual equities (IBM, Alphabet, Amazon, Apple andGoldman Sachs), as well as on a variety of other equity indices andcommodity-based ETFs. Similar to the “conventional” VIX referencing the SPX,the VIX calculation for individual equities employs equation (2). Yet it’slikely that option quotes in single stocks cover a smaller range of strikeprices. With a more restricted range, the VIX approximations produce greatererrors.
正如CBOE白皮书所指出的那样，CBOE提供个人股票（IBM，Alphabet，Amazon，Apple和GoldmanSachs）以及各种其他股票指数和基于商品的ETF的VIX指数。与引用SPX的“常规”VIX类似，单个股票的VIX计算采用公式（2）。然而，单个股票的期权报价很可能会覆盖较小范围的执行价格。由于范围更加有限，VIX近似值会产生更大的误差。
Consider the VIX for the equity of Apple(“VXAPL”). Review of quotes for Apple call and put options at a CBOE websiteshows that the put range for purposes of VIX calculation is less than 25% ofthe equity’s forward value. (We reviewed option bids for contracts expiring onthe third Friday of the next full month. We searched for consecutive zero bidsto determine the minimum put option strike.)
考虑苹果的VIX（“VXAPL”）。在芝加哥商业交易所网站上查看苹果看涨和看跌期权的报价表明，用于VIX计算的看跌期权价格范围低于远期价值的25％。（我们审查了下个月第三个周五到期的合约期权投标，我们搜索了连续的零投标以确定最低看跌期权的执行价。）
The data for IBM options is comparable. Weexpect, therefore, that fixing the VIX methodology approximations provides asimilar or greater benefit to these single stock VIX variants.
IBM期权的数据是可比的。因此，我们期望修正VIX方法近似值可为这些单股VIX变体提供类似或更大的益处。
PartingThoughts
最后的想法
VIX is a popular, successful and importantelement of U.S. financial markets. In its disclosure to the public, the CBOEwhite paper is open and clear in its calculation methodology.
VIX是美国金融市场中流行，成功和重要的元素。在向公众公开的信息中，CBOE白皮书的计算方法是公开和明确的。
This article highlights the theory behindthe VIX and finds three approximations that may leave the VIX prone to error orvulnerable to manipulation: (I) the second term of equation (2) expressing theCBOE method; (II) the implied integral approximation of the first term ofequation (2); and (III) the absence of Put Tail and Call Tail contributions inthis same first term of equation (2).
本文重点介绍了VIX背后的理论，并发现了三种可能使VIX容易出错或易于操纵的近似值：（I）表达CBOE方法的第二项式（2）;（II）等式（2）的第一项的隐含积分近似;和（III）在方程（2）的相同第一项中没有看跌尾部和看涨尾部贡献。
We propose that CBOE remove or improve theapproximations, and we demonstrate how to do so. The inaccuracy of theseapproximations is not large, in an absolute sense, for typical scenarios. Yetthe small differences we find – almost exclusively in the third approximationpertaining to tail contributions – may be meaningful to the VIX futures andfutures options participants.
我们建议CBOE删除或改进近似值，我们将演示如何这样做。在绝对意义上，这些近似值的不准确性对于典型场景并不大。然而，我们发现的小差异- 几乎完全属于与尾部贡献相关的第三近似- 对VIX期货和期货期权参与者可能有意义。
Beyond the possibility of inadvertentdeviations of the VIX, our strong interest also lies with potentialmanipulation. Fixing the VIX calculation, again especially with regard to thetail contributions, will make the VIX less susceptible to manipulation.
除了无意中偏离VIX的可能性之外，我们强烈的兴趣还在于潜在的操纵。固定VIX计算，特别是关于尾部贡献，将使VIX不易被操纵。
As we describe for the tail calculations,our improved approximation does add new elements to the calculation. We need,for example, a numerical integration for the Put Tail, and suggest a FixedPoint iteration technique to incorporate the Call Tail.
正如我们描述尾部计算一样，我们的改进近似确实为计算添加了新元素。例如，我们需要对看跌尾部进行数值积分，并建议使用固定点迭代技术来计算看涨尾部。
What’s more, we implicitly require anumerical estimate of the linear skew parameters for out-of-the-money putoptions. The improved calculation must have code to implement all three elements.But these additions are not onerous and do not prohibit continuous updates ofthe VIX.
更重要的是，我们隐含地要求对虚值看跌期权的线性偏差参数进行数值估计。改进的计算必须具有实现所有三个元素的代码。但是这些增加并不繁琐，并且不禁止VIX的不断更新。