VR(T) time decay and term adjusted Vega columns in Risk Navigator (SM)
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.
オプションに関する追加情報の入手元
米国の取引所で取引されるすべての証券オプションの中央決済機関であるオプション清算会社（OCC）は、個人投資家とリテール証券ブローカー用の学習ニーズの答えるコールセンターを設置しています。コールセンターではOCCで決済されるオプション商品に関する下記の様な質問や問題に対応しています:
セミナーやビデオ、またその他の学習機能に関するOptions Industry Councilによる情報、
言葉の定義や商品情報など、オプションに関する基本的な質問、
特定の取引ポジションやストラテジーなどを含める運用関連の質問。
コールセンターへは1-800-OPTIONSの番号でお問い合わせください。対応時間は月曜日から木曜日の午前8時から午後5時（中部標準時）、および金曜日の午前8時から午後4時（中部標準時）までとなっております。また毎月、満期の金曜日は午後5時（中部標準時）までとなっております。
満期前のコールオプション権利行使について
概要
満期前の株式コールオプションの権利行使は、以下の点により通常は経済的利点がありません:
- オプション・タイムバリューの喪失;
- 株式受け渡しへの支払い、借入にかかる費用の増加;
- オプションプレミアムに関連する株式損失リスクの増加
しかしながら、支払や借入、ダウンサイドのマーケットリスク増加に対応できる口座保有者にとって、アメリカンスタイルのコールオプションの早期権利行使は支払予定の配当金の受け取りが可能なため、経済的利点が発生する可能性があります。
背景
配当金は割り当て日の株式保有者に対して支払われるため、コールオプションの保有者への支払はありません。その他の条件が同じ場合、株価は配当落ち日に配当金分、下落します。オプション価格理論上、コールオプションの価格は限月中、配当金支払い見込みが反映されたものとなりますが、配当落ち日に下落することがあります。この様なシナリオで早期権利行使が好まれる条件は以下のような場合です：
1. オプションがディープ・イン・ザ・マネーでデルタが100
2. オプションのタイムバリューが少額、もしくはゼロ
3. 配当金が比較的高く、配当落ち日がオプション満期日の前にある
例
早期権利行使の影響を例にして見てみます。口座の現金残高が$9,000であり、架空の株式ABCのロング・コール（権利行使価格$90.00）が10日後に満期を迎えると仮定します。ABCの株価は$100.00、配当金額は1株あたり$2.00、権利落ち日は明日であるとします。また、オプション価格と株価の動向は類似しており、権利落ち後に配当金額分下落すると仮定します。
100株のデルタポジションを、パリティ価格でのオプションの売却とパリティ価格以上でのオプションの売却の二つのオプション価格をを想定し、資産の最大化を査定します。
シナリオ1：オプション価格がパリティの時 - $10.00
パリティ価格でオプションが取引されている場合、株式が配当落ち後に資金に変わる際、早期権利行使でポジションデルタを維持し、ロングオプションの損失を避けます。現金は権利行使価格で株式を購入するために使用され、オプションプレミアムは喪失し、株式および配当金受取は口座に反映されます。これは配当落ち日の前にオプションを売却し、株式を購入することで達成できますが、手数用およびスプレッドも考慮してください:
| シナリオ1 | ||||
|
口座 バランス |
当初 バランス |
早期 権利行使 |
アクション 無し |
オプション売 &株式買 |
| 現金 | $9,000 | $0 | $9,000 | $0 |
| オプション | $1,000 | $0 | $800 | $0 |
| 株式 | $0 | $9,800 | $0 | $9,800 |
| 配当金受取 | $0 | $200 | $0 | $200 |
| 資産合計 | $10,000 | $10,000 | $9,800 | $10,000 |
シナリオ2：オプション価格がパリティ価格より上の時 - $11.00
パリティ価格以上でオプションが取り引きされる場合、配当金確保のための早期権利行使には経済的利点がない場合があります。このシナリオでは早期権利行使によりオプション・タイムバリューに$100の損失が発生しますが、オプションを売却して株式を購入した場合、手数料を差し引いた結果、アクションを取らない方がかえって良い可能性もあります。この場合、最も好まれるアクションはオプションを取らないことです。
| シナリオ2 | ||||
|
口座 バランス |
当初 バランス |
早期 権利行使 |
アクション 無し |
オプション売 |
| 現金 | $9,000 | $0 | $9,000 | $100 |
| オプション | $1,100 | $0 | $900 | $0 |
| 株式 | $0 | $9,800 | $0 | $9,800 |
| 配当金受取 | $0 | $200 | $0 | $200 |
| 資産合計 | $10,100 | $10,000 | $9,900 | $10,100 |
.jpg){kind=small}
注意: ロングコール・ポジションをスプレッド取引の一部として保有している場合には、ショートポジションの割り当てが発生した際に、ロングポジションの権利行使を行わないリスクに特に注意を払う必要があります。割り当て日に株式ショートポジションを保有していて、ショートコールの割り当てが発生した場合、株式の貸し手に配当金を支払う義務が発生しますのでご注意ください。また、権利行使処理を行うクリアリング機関は、割り当ての対応として権利行使処理を行いません。
例：SPDR S&P 500 ETF Trust（SPY）のショート100枚（March '13 権利行使$146）とロング100枚（March '13 権利行使$147）を含むクレジットコール（ベア）スプレッドを例にとって考えて見ます。2013年3月14日に1株あたり$0.69372の配当金が発表されたとします。2013年3月19日付で、20134月30日支払の配当金支払見込みが株主の口座に反映されました。米国株式の3営業日決済により、配当金を受け取るには2013年3月14日までに株式の購入、もしくはコールオプションの権利行使を行わなければなりません。それ以降は配当落ち株式となります。
2013年3月14日には満期まで取引日が1日しか残されていません。二つのオプションはパリティ価格で取引され、1枚当たり$100、または100枚のポジションに$10,00の最大リスクが伴います。しかし、配当金の受け取りと、配当金受取を見込んだ、相手方の割り当てからの保護としてのロングコールの権利行使に失敗するとショートコール割り当てに対する配当金支払い義務が発生し、1枚当たり$67.372、または100枚のポジションに$6,737.20の追加のリスクが発生します。以下の表の通り、ショートコール・オプションの割り当てが発生しなかった場合、ポジションの最終決済価格が2013年3月15日に決定し、最大損失リスクは1枚あたり$100に留まります。
| 期日 | SPY 終値 | March '13 $146 コール | March '13 $147 コール |
| March 14, 2013 | $156.73 | $10.73 | $9.83 |
| March 15, 2013 | $155.83 | $9.73 | $8.83 |
お持ちのお口座が米国債ルール871（ｍ）により米国源泉徴収税の対象となっている場合、権利落ち日前に一度ロングポジションをクローズし、権利落ち日後に再度オープンされることにより、効果的に作用する可能性があります。
早期権利行使通知の提出方法については弊社ウェブサイトをご確認ください。
このページは情報提供のみを目的とするものであり、推奨や取引アドバイスではなく、早期権利行使がすべてのお客様やトレーダーに適していることを結論付けるものではありません。口座を保有されるお客様は専門の税理士にご相談の上、早期権利行使による課税への影響をご確認ください。また、ロングポジションの変わりに株式ポジションを保有するにあたって発生する可能性のあるリスクにご注意ください。
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.
For information regarding how to submit an early exercise notice please click here.
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.
Where can I receive additional information on options?
The Options Clearing Corporation (OCC), the central clearinghouse for all US exchange traded securities option, operates a call center to serve the educational needs of individual investors and retail securities brokers. The resource will address the following questions and issues related to OCC cleared options products:
- Options Industry Council information regarding seminars, video and educational materials;
- Basic options-related questions such as definition of terms and product information;
- Responses to strategic and operational questions including specific trade positions and strategies.
The call center can be reached by dialing 1-800-OPTIONS. The hours of operation are Monday through Thursday from 8 a.m. to 5 p.m. (CST) and Friday from 8 a.m. to 4 p.m. (CST). Hours for the monthly expiration Friday will be extended to 5 p.m. (CST).
