## 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

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.

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.

## 在哪里可以了解更多有关期权的信息？

- 期权行业协会有关研讨会、视频和教学材料的信息；

- 基本期权问题，如期限定义和产品信息；

- 策略和操作性问题（包括特定交易头寸和策略）解答。

## 到期前行使看涨期权的注意事项

• 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。

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。

 情境1 账户组成部分 起始余额 提前行权 无行动 卖期权& 买股票 现金 \$9,000 \$0 \$9,000 \$0 期权 \$1,000 \$0 \$800 \$0 股票 \$0 \$9,800 \$0 \$9,800 应收股息 \$0 \$200 \$0 \$200 总资产 \$10,000 \$10,000 \$9,800 \$10,000减去佣金/价差

 情境2 账户组成部分 起始余额 提前行权 无行动 卖期权& 买股票 现金 \$9,000 \$0 \$9,000 \$100 期权 \$1,100 \$0 \$1,100 \$0 股票 \$0 \$9,800 \$0 \$9,800 应收股息 \$0 \$200 \$0 \$200 总资产 \$10,100 \$10,000 \$10,100 \$10,100减去佣金/价差

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。

 日期 SPY收盘价 13年3月行使价为\$146的看涨期权 13年3月行使价为\$147的看涨期权 2013年3月14日 \$156.73 \$10.73 \$9.83 2013年3月15日 \$155.83 \$9.73 \$8.83

## Considerations for Exercising Call Options Prior to Expiration

INTRODUCTION

Exercising an equity call option prior to expiration ordinarily provides no economic benefit as:

• It results in a forfeiture of any remaining option time value;
• Requires a greater commitment of capital for the payment or financing of the stock delivery; and
• May expose the option holder to greater risk of loss on the stock relative to the option premium.

Nonetheless, for account holders who have the capacity to meet an increased capital or borrowing requirement and potentially greater downside market risk, it can be economically beneficial to request early exercise of an American Style call option in order to capture an upcoming dividend.

BACKGROUND

As background, the owner of a call option is not entitled to receive a dividend on the underlying stock as this dividend only accrues to the holders of stock as of its dividend Record Date. All other things being equal, the price of the stock should decline by an amount equal to the dividend on the Ex-Dividend date. While option pricing theory suggests that the call price will reflect the discounted value of expected dividends paid throughout its duration, it may decline as well on the Ex-Dividend date.  The conditions which make this scenario most likely and the early exercise decision favorable are as follows:

1. The option is deep-in-the-money and has a delta of 100;

2. The option has little or no time value;

3. The dividend is relatively high and its Ex-Date precedes the option expiration date.

EXAMPLES

To illustrate the impact of these conditions upon the early exercise decision, consider an account maintaining a long cash balance of \$9,000 and a long call position in hypothetical stock “ABC” having a strike price of \$90.00 and time to expiration of 10 days. ABC, currently trading at \$100.00, has declared a dividend of \$2.00 per share with tomorrow being the Ex-Dividend date. Also assume that the option price and stock price behave similarly and decline by the dividend amount on the Ex-Date.

Here, we will review the exercise decision with the intent of maintaining the 100 share delta position and maximizing total equity using two option price assumptions, one in which the option is selling at parity and another above parity.

SCENARIO 1: Option Price At Parity - \$10.00
In the case of an option trading at parity, early exercise will serve to maintain the position delta and avoid the loss of value in long option when the stock trades ex-dividend, to preserve equity. Here the cash proceeds are applied in their entirety to buy the stock at the strike, the option premium is forfeited and the stock (net of dividend) and dividend receivable are credited to the account.  If you aim for the same end result by selling the option prior to the Ex-Dividend date and purchasing the stock, remember to factor in commissions/spreads:

 SCENARIO 1 Account Components Beginning Balance Early Exercise No Action Sell Option & Buy Stock Cash \$9,000 \$0 \$9,000 \$0 Option \$1,000 \$0 \$800 \$0 Stock \$0 \$9,800 \$0 \$9,800 Dividend Receivable \$0 \$200 \$0 \$200 Total Equity \$10,000 \$10,000 \$9,800 \$10,000 less commissions/spreads

SCENARIO 2: Option Price Above Parity - \$11.00
In the case of an option trading above parity, early exercise to capture the dividend may not be economically beneficial. In this scenario, early exercise would result in a loss of \$100 in option time value, while selling the option and buying the stock, after commissions, may be less beneficial than taking no action. In this scenario, the preferable action would be No Action.

 SCENARIO 2 Account Components Beginning Balance Early Exercise No Action Sell Option & Buy Stock Cash \$9,000 \$0 \$9,000 \$100 Option \$1,100 \$0 \$1,100 \$0 Stock \$0 \$9,800 \$0 \$9,800 Dividend Receivable \$0 \$200 \$0 \$200 Total Equity \$10,100 \$10,000 \$10,100 \$10,100 less commissions/spreads

NOTE: Account holders holding a long call position as part of a spread should pay particular attention to the risks of not exercising the long leg given the likelihood of being assigned on the short leg.  Note that the assignment of a short call results in a short stock position and holders of short stock positions as of a dividend Record Date are obligated to pay the dividend to the lender of the shares. In addition, the clearinghouse processing cycle for exercise notices does not accommodate submission of exercise notices in response to assignment.

As example, consider a credit call (bear) spread on the SPDR S&P 500 ETF Trust (SPY) consisting of 100 short contracts in the March '13 \$146 strike and 100 long contracts in the March '13 \$147 strike.  On 3/14/13, with the SPY Trust declared a dividend of \$0.69372 per share, payable 4/30/13 to shareholders of record as of 3/19/13. Given the 3 business day settlement time frame for U.S. stocks, one would have had to buy the stock or exercise the call no later than 3/14/13 in order receive the dividend, as the next day the stock began trading Ex-Dividend.

On 3/14/13, with one trading day left prior to expiration, the two option contracts traded at parity, suggesting maximum risk of \$100 per contract or \$10,000 on the 100 contract position. However, the failure to exercise the long contract in order to capture the dividend and protect against the likely assignment on the short contracts by others seeking the dividend created an additional risk of \$67.372 per contract or \$6,737.20 on the position representing the dividend obligation were all short calls assigned.  As reflected on the table below, had the short option leg not been assigned, the maximum risk when the final contract settlement prices were determined on 3/15/13 would have remained at \$100 per contract.

 Date SPY Close March '13 \$146 Call March '13 \$147 Call March 14, 2013 \$156.73 \$10.73 \$9.83 March 15, 2013 \$155.83 \$9.73 \$8.83

Please note that if your account is subject to tax withholding requirements of the US Treasure rule 871(m), it may be beneficial to close a long option position before the ex-dividend date and re-open the position after ex-dividend.

The above article is provided for information purposes only as is not intended as a recommendation, trading advice nor does it constitute a conclusion that early exercise will be successful or appropriate for all customers or trades. Account holders should consult with a tax specialist to determine what, if any, tax consequences may result from early exercise and should pay particular attention to the potential risks of substituting a long option position with a long stock position.