Background

Risk Navigator (SM) has two Adjusted Vega columns that you can add to your report pages via menu Metrics → Position Risk...: "Adjusted Vega" and "Vega x T^-1/2". A common question is what is our in-house time function that is used in the Adjusted Vega column and what is the aim of these columns. VR(T) is also generally used in our Stress Test or in the Risk Navigator custom scenario calculation of volatility index options (i.e VIX).

Abstract

Implied volatilities of two different options on the same underlying can change independently of each other. Most of the time the changes will have the same sign but not necessarily the same magnitude. In order to realistically aggregate volatility risk across multiple options into a single number, we need an assumption about relationship between implied volatility changes. In Risk Navigator, we always assume that within a single maturity, all implied volatility changes have the same sign and magnitude (i.e. a parallel shift of volatility curve). Across expiration dates, however, it is empirically known that short term volatility exhibits a higher variability than long term volatility, so the parallel shift is a poor assumption. This document outlines our approach based on volatility returns function (VR(T)). We also describe an alternative method developed to accommodate different requests.

VR(T) time decay

We applied the principal component analysis to study daily percentage changes of volatility as a function of time to maturity. In that study we found that the primary eigen-mode explains approximately 90% of the variance of the system (with second and third components explaining most of the remaining variance being the slope change and twist). The largest amplitude of change for the primary eigenvector occurs at very short maturities, and the amplitude monotonically decreases as time to expiration increase. The following graph shows the main eigenvector as a function of time (measured in calendar days). To smooth the numerically obtained curve, we parameterize it as a piecewise exponential function.

Functional Form: Amplitude vs. Calendar Days

To prevent the parametric function from becoming vanishingly small at long maturities, we apply a floor to the longer term exponential so the final implementation of this function is:

where bS=0.0180611, a=0.365678, bL=0.00482976, and ^T*=55.7 are obtained by fitting the main eigenvector to the parametric formula.

Inverse square root time decay

Another common approach to standardize volatility moves across maturities uses the factor 1/√T. As shown in the graph below, our house VR(T) function has a bigger volatility changes than this simplified model.

Time function comparison: Amplitude vs. Calendar Days

Adjusted Vega columns

Risk Navigator (SM) reports a computed Vega for each position; by convention, this is the p/l change per 1% increase in the volatility used for pricing.  Aggregating these Vega values thus provides the portfolio p/l change for a 1% across-the-board increase in all volatilities – a parallel shift of volatility.

However, as described above a change in market volatilities might not take the form of a parallel shift.  Empirically, we observe that the implied volatility of short-dated options tends to fluctuate more than that of longer-dated options.  This differing sensitivity is similar to the "beta" parameter of the Capital Asset Pricing Model.  We refer to this effect as term structure of volatility response.

By multiplying the Vega of an option position with an expiry-dependent quantity, we can compute a term-adjusted Vega intended to allow more accurate comparison of volatility exposures across expiries. Naturally the hoped-for increase in accuracy can only come about if the adjustment we choose turns out to accurately model the change in market implied volatility.

We offer two parametrized functions of expiry which can be used to compute this Vega adjustment to better represent the volatility sensitivity characteristics of the options as a function of time to maturity. Note that these are also referred as 'time weighted' or 'normalized' Vega.

Adjusted Vega

A column titled "Vega Adjusted" multiplies the Vega by our in-house VR(T) term structure function. This is available any option that is not a derivative of a Volatility Product ETP. Examples are SPX, IBM, VIX but not VXX.

Vega x T^-1/2

A column for the same set of products as above titled "Vega x T^-1/2" multiplies the Vega by the inverse square root of T (i.e. 1/√T) where T is the number of calendar days to expiry.

Aggregations

Cross over underlying aggregations are calculated in the usual fashion given the new values. Based on the selected Vega aggregation method we support None, Straight Add (SA) and Same Percentage Move (SPM). In SPM mode we summarize individual Vega values multiplied by implied volatility. All aggregation methods convert the values into the base currency of the portfolio.

Custom scenario calculation of volatility index options

Implied Volatility Indices are indexes that are computed real-time basis throughout each trading day just as a regular equity index, but they are measuring volatility and not price. Among the most important ones is CBOE's Marker Volatility Index (VIX). It measures the market's expectation of 30-day volatility implied by S&P 500 Index (SPX) option prices. The calculation estimates expected volatility by averaging the weighted prices of SPX puts and calls over a wide range of strike prices.

The pricing for volatility index options have some differences from the pricing for equity and stock index options. The underlying for such options is the expected, or forward, value of the index at expiration, rather than the current, or "spot" index value. Volatility index option prices should reflect the forward value of the volatility index (which is typically not as volatile as the spot index). Forward prices of option volatility exhibit a "term structure", meaning that the prices of options expiring on different dates may imply different, albeit related, volatility estimates.

For volatility index options like VIX the custom scenario editor of Risk Navigator offers custom adjustment of the VIX spot price and it estimates the scenario forward prices based on the current forward and VR(T) adjusted shock of the scenario adjusted index on the following way.

• Let S0 be the current spot index price, and
• S1 be the adjusted scenario index price.
• If F0 is the current real time forward price for the given option expiry, then
• F1 scenario forward price is F1 = F0 + (S1 - S0) x VR(T), where T is the number of calendar days to expiry.

For complex, multi-leg options positions comprising two or more legs, TWS might not track all changes to this position, e.g. a vertical spread where the short leg is assigned and the user re-writes the same leg the next day, or if the user creates a the position over multiple trades, or if the order is not filled as a native combination at the exchange.

简介

到期前行使股票看涨期权通常不会带来收益，因为：

• 这会导致剩余期权时间价值的丢失；
• 需要更大的资金投入以支付股票交割；并且
• 会给期权持有人带来更大的损失风险。

尽管如此，对于有能力满足更大资金或借款要求并能承受更大下行市场风险的账户持有人来说，提前行权行使美式看涨期权可获取即将分配的股息。

背景

看涨期权持有者无权获取底层股票的股息，因为该股息属于股息登记日前的股票持有人所有。 其他条件相同，股价应该下降，降幅与除息日的股息保持一致。期权定价理论提出看涨期权价格將反映预期股息的折扣价格，看涨期权价格也可能在除息日下跌。最可能促成该情境与提前行权决定的条件如下：

1. 期权为深度价内期权且delta值为100；

2. 期权几乎没有时间价值；

3. 股息相对较高且除息日在期权到期日之前。

举例

为阐述这些条件对提前行权决定的影响，假设账户的多头现金余额为$9,000美元，且持有行使价为$90.00美元的ABC多头看涨头寸，10天后到期。 ABC当前的成交价为$100.00美元，每股股息为$2.00美元，明天是除息日。再假设期权价格与股票价格动向相同，且在除息日下跌的幅度均为股息金额。

这里，我们将检查行权决定，目的是维持100股delta头寸并使用两种期权价格假设（假设一个为平价，一个高于平价）最大化总资产。

情境1：期权价格为平价 - $10.00美元
如果期权以平价交易，提前行权可维持delta头寸并可避免股票除息交易时多头期权价值遭受损失，从而保护资产。在这里现金收入被全数用于以行使价购买股票，期权权利金就此丧失并且股票（扣除股息）与应收股息会记入账户。如果您想通过在除息日前卖出期权并买入股票来达到同样的效果，请记得考虑佣金/价差：

情境2：期权价格高于平价 - $11.00美元
如果期权以高于平价的价格交易，提前行权获取股息则可能并不会带来收益。在此情景中，提前行权可能会导致期权时间价值损失$100美元，而卖出期权买入股票在扣除佣金之后收益情况也可能不如不采取行动。在这里，可取的行动为无行动。

请注意：考虑到空头期权边被行权的可能性，持有作为价差组成部分之多头看涨头寸的账户持有人应格外注意不行使多头期权边的风险。请注意，空头看涨期权的被行权会导致空头股票头寸，且在股息登记日前持有空头股票头寸的持有人有义务向股票的借出者支付股息。此外，清算所行权通知处理周期不支持提交响应被行权的行权通知。

例如，假设SPDR S&P 500 ETF Trust (SPY)的信用看涨（熊市）价差包括100张13年3月到期行使价为$146美元的空头合约，以及100张13年3月到期行使价为$147美元的多头合约。在13年3月14日，SPY Trust宣布每股股息为$0.69372美元，并且会在13年4月30日向13年3月19日前登记的股东支付。因为美国股票的结算周期为3个工作日，想要获取股息，交易者需要在13年1月14日之前买入股票或行使看涨期权，因为该日期一过，股票便开始除息交易。

13年3月14日，距离期权到期只剩一个交易日，平价成交的两张期权合约每张合约的最大风险为$100美元，100张合约则为$10,000美元。但是，未能行使多头合约以获取股息以及未能避免空头合约被其他想要获取股息的交易者行权会使每张合约产生额外$67.372美元的风险，如果所有空头看涨合约都被行权，则所有头寸总风险为$6,737.20美元。如下表所示，如果空头期权边没有被行权，则13年3月15日确定最终的合约结算价格时，最大风险仍为每张合约$100美元。

请注意，如果您的账户符合美国871(m)预扣税要求，则除息日前平仓头多期权头寸并在除息日后重新建仓可能会带来收益。

有关如何提交提前行权通知的信息，请查看网站。

上述内容仅作信息参考，不构成任何推荐、交易建议，也不代表提前行权会成功或适合所有客户或交易。账户持有人应咨询税务专家以确定提前行权可能带来的税务影响，并应格外注意以多头股票头寸替换多头期权头寸的潜在风险。

General overview of the Option Strategy Lab