as in my own trad-ing approach.
• Delta. Change in the price (premium) of an option relative to the price
change of the underlying security.
• Gamma. Change in the delta of an option with respect to the change
in price of its underlying security.
• Theta. Change in the price of an option with respect to a change in its
time to expiration.
• Vega. Change in the price of an option with respect to its change in
volatility.
Each of these risk measurements contains speciﬁc important tradinginformation. As you become more acquainted with the various aspects ofoptions trading, you will ﬁnd more and more uses for each of them. Forexample, they each make a unique contribution to an option’s premium.The two most important components of an option’s premium are intrinsicvalue and time value (extrinsic value).
In an effort to understand the elements that inﬂuence the value of anoption,  various  option  pricing  models  were  created,  including  Black-Scholes and Cox-Rubinstein. To comprehend the Greeks, we must under-stand that they are derived from these types of theoretical pricing models.The values that are needed as inputs into the option pricing models are re-lated to the Greeks. However, the inputs for the models are not the Greeksthemselves. A common mistake among options traders is to refer to vegaas  implied  volatility.  When  we  refer  to  the  Greeks,  we  are  talking  aboutrisk that will ultimately affect the option’s price. Therefore, a more accu-rate description of vega would be the option’s price sensitivity to impliedvolatility changes.
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The concept of the option’s delta seems to be the ﬁrst Greek that everyonelearns.  It’s  basically  a  measure  of  how  much  the  value  of  an  option  willchange given a change in the underlying stock. When the strike price ofthe option is close or at-the-money, the delta of the option will be around50 for long call options and –50 for long put options. In the case of a calloption, the option’s delta could be higher if the value of the stock has ex-ceeded  the  option’s  strike  price  signiﬁcantly.  If  our  call  option’s  strikeprice were much higher than the price of the shares, the value of the deltawould be smaller. For example, if XYZ stock is trading for $50 per shareand I own the $60 strike price call, my delta may be around 30. Recall thatdelta is computed using an option pricing model. It will vary based on thedifference  between  the  stock  price  and  the  strike  price  of  the  option  aswell as the time left until expiration. In this case, let’s assume the delta ofthis option is 30, or .30. Therefore, my position will theoretically make $30for every $1 increase in XYZ stock based on the option’s delta.
There are many ways that traders can use the delta, or hedge ratio, intheir options analysis. A very basic way to use delta is in hedging a sharesposition. Let’s suppose that I have 500 shares of XYZ and that I want topurchase some puts to protect my position. Most traders would purchaseﬁve at-the-money puts. This creates a synthetic call position. The idea isthat the trader can exercise the puts if the market moves against him. Inthis  respect  the  purchased  options  become  like  insurance  for  the  stocktrader. However, there is another way to look at this scenario. If I have 500shares of XYZ stock, I can hedge the delta of the stock by purchasing 10 ofthe XYZ at-the-money puts. Since the delta of each share of stock is 1 andthe delta of each at-the-money put is –50, I would need 10 puts to hedgethe deltas of the long stock position. The results are similar to a straddle.
Example
Long 500 shares of XYZ Delta = +500Delta = –500Long 10 ATM puts
Net delta
0
Gamma
Gamma tells us how fast the delta of the option changes for every 1 pointmove  in  the  underlying  stock.  For  this  reason,  some  traders  refer  togamma as the delta of the delta. However, gamma is different from delta inthat it is always expressed as a positive number regardless of whether itrelates to a put or a call. If the price of the stock increases $1 and the deltaincreases or decreases by a value of 15 then the gamma is 15. Remember,
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we  are  using  our  option  pricing  model  to  make  this  determination.  An-other interesting characteristic of gamma is that it is largest for the at-the-money  options.  This  means  that  the  deltas  for  the  at-the-money  optionsare more sensitive to a change in the price of the underlying stock.
While I have been talking about delta and gamma in relation to the un-derlying stock price, it is important to note that they are also inﬂuencedby time and volatility. Statistical (or historical) volatility is a measure ofthe ﬂuctuation of the underlying stock. As I have already noted, delta is ameasure of how the options price will change when the underlying stockchanges. Therefore, the delta of the options will be generally higher for ahigher-volatility  stock  versus  a  lower-volatility  stock.  This  is  due  to  thefact  that  the  stock’s  volatility  and  the  option’s  delta  are  related  to  themovement of the stock. Also, ITM and ATM option deltas fall faster thanOTM options as they approach expiration.
Theta
The theta of an option is a measure of the time decay of an option. Thetacan also be deﬁned as the amount by which the price of an option exceedsits  intrinsic  value.  Generally  speaking,  theta  decreases  as  an  option  ap-proaches expiration. Theta is one of the most important concepts for a be-ginning option trader to understand for it basically explains the effect oftime on the premium of the options that have been purchased or sold. Theless time that an option has until expiration, the faster that option is goingto lose its value. Theta is a way of measuring the rate at which this value islost. The further out in time you go, the smaller the time decay will be foran option. Therefore, if you want to buy an option, it is advantageous topurchase  longer-term  contracts.  If  you  are  using  a  strategy  that  proﬁtsfrom time decay, then you will want to be short the shorter-term optionsso that the loss in value due to time decay happens quickly.
Since  an  option  loses  value  as  time  passes,  theta  is  expressed  as  anegative number. For example, an option (put or call) with a theta of –.15will lose 15 cents per day. As noted earlier, time decay is not linear. Forthat  reason,  options  with  less  time  until  expiration  will  have  a  higher(negative) theta than those with only a few days of life remaining.
Vega
Vega tells  us  how  much  the  price  of  the  options  will  change  for  every  1percent change in implied volatility. So, if we purchased the XYZ optionfor $100 and its vega is 20, we can expect the cost of the option to increaseby  $20  when  implied  volatility  moves  up  by  1  percent.  Vega  tends  to  behighest  for  options  that  are  at-the-money  and  decreases  as  the  option
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reaches  its  expiration  date.  It  is  interesting  to  note  that  vega  does  notshare the correlation to the stock’s ﬂuctuation that delta and gamma do.This  is  because  vega  is  dependent  on  the  measure  of  implied  volatilityrather than statistical volatility. This is an important distinction for traderswho like to trade options straddles.
We all know that there is time value associated with the value of anoption. The rate at which the option’s time premium depreciates on a dailybasis is called theta. It is typically highest for at-the-money options and isexpressed as a negative value. So, if I have an option that has a theta of–.50, I can expect the value of my option to decrease 50 cents per day untilthe option’s expiration. This characteristic of the option’s time premiumhas particular interest to the trader of credit spreads.
ASSESSING THE RISKS
As options traders become more experienced with creating spreads, theyshould  become  more  aware  of  the  types  of  risks  involved  with  eachspread. To reach this level of trading competence, options traders shouldcombine  the  values  of  the  Greeks  used  to  create  the  optimal  optionsspread. The result will allow the trader to more accurately assess the risksof any given options spread.
Understanding  the  relative  impact  of  the  Greeks  on  positions  youhold is indispensable. Here are six of the more salient mathematical rela-tionships of these Greek variables:
1. The delta of an at-the-money option is about 50. Out-of-the-money op-tions  have  smaller  deltas  and  they  decrease  the  farther  out-of-the-money  you  go.  In-the-money  options  have  greater  deltas  and  theyincrease the farther in-the-money you go. Call deltas are positive andput deltas are negative.
2. When you sell options, theta is positive and gamma is negative. Thismeans you make money through time decay, but price movement isundesirable. So proﬁts you’re trying to earn through option time de-cay when you sell puts and calls may never be realized if the stockmoves quickly in price. Also, rallies in price of the underlying assetwill  cause  your  overall  position  to  become  increasingly  delta-shortand to lose money. Conversely, declines in the underlying asset pricewill  cause  your  position  to  become  increasingly  delta-long  and  tolose money.
3. When you buy options, theta is negative and gamma is positive. Thismeans you lose money through time decay but price movement is de-sirable. So proﬁts you’re attempting to earn through volatile moves of
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the  underlying  stock  may  never  be  realized  if  time  decay  causeslosses. Also, rallies in price result in your position becoming increas-ingly delta-long and declines result in your position becoming increas-ingly delta-short. 
4. Theta and gamma increase as you get close to expiration, and they’regreatest  for  at-the-money  options.  This  means  the  stakes  grow  ifyou’re short at-the-money because either the put or the call can easilybecome  in-the-money  and  move  point-for-point  with  the  equity.  Youcan’t adjust quickly enough to accommodate such a situation.
5. When  you  sell  options,  vega  is  negative.  This  means  if  impliedvolatility  increases,  your  position  will  lose  money,  and  if  it  de-creases,  your  position  will  make  money.  When  you  buy  options,vega is positive, so increases in implied volatility are proﬁtable anddecreases are unproﬁtable.
6. Vega is greatest for options far from expiration. Vega becomes less ofa factor while theta and gamma become more signiﬁcant as optionsapproach expiration.
TIME DECAY’S EFFECT ON STRATEGY SELECTION
Since the rate of time decay varies from one options contract to the next,the strategist generally wants to know how time is impacting the overall po-sition. For instance, a straddle, which involves the purchase of both a putand a call, can lose signiﬁcant value due to time decay. In addition, giventhat the rate of time decay is not linear, straddles using short-term optionsare generally not advised. Instead, longer-term options are more suitable forstraddles and the trade should be closed well before (30 days or more be-fore) expiration. (The straddle strategy is explored in Chapter 8.)
Some strategies, on the other hand, can use time decay to the optiontrader’s advantage. Have you ever heard that 85 percent of options expireworthless? While that is probably not entirely true, a large number of op-tions do expire worthless each month. As a result, some traders prefer tosell, rather than buy, options because, unlike the option buyers, the optionseller beneﬁts from forces of time decay.
The simplest strategy that attempts to proﬁt from time decay is thecovered call—or buying shares of XYZ Corp. and selling XYZ calls. Per-haps  a  better  alternative,  however,  is  the  calendar  spread  (see  Chapter10). This type of spread involves purchasing a longer-term call option ona stock and selling a shorter-term call on the same stock. The goal is tohold the long-term option while the short-term contract loses value at afaster rate due to the nonlinear nature of time decay. There are a numberof different ways to construct calendar spreads. Some diagonal spreads,
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butterﬂies, and condors are examples of other strategies that can beneﬁtfrom  the  loss  of  time  value.  In  each  instance,  the  strategist  is  generallynot  interested  in  seeing  the  underlying  asset  make  a  dramatic  movehigher  or  lower,  but  rather  seeing  the  underlying  stock  trade  within  arange while time decay eats away at the value of its options.
VOLATILITY REVISITED
Volatility can be deﬁned as a measurement of the amount by which an un-derlying asset is expected to ﬂuctuate in a given period of time. It is one ofthe  most  important  variables  in  options  trading,  signiﬁcantly  impactingthe  price  of  an  option’s  premium  as  well  as  contributing  heavily  to  anoption’s time value.
As previously mentioned, there are two basic kinds of volatility: im-plied and historical (statistical). Implied volatility is computed using theactual market prices of an option and one of a number of pricing models(Black-Scholes for shares and indexes; Black for futures). For example, ifthe market price of an option increases without a change in the price ofthe underlying instrument, the option’s implied volatility will have risen.Historical volatility is calculated by using the standard deviation of under-lying asset price changes from close-to-close of trading going back 21 to23 days or some other predetermined period. In more basic terms, histori-cal  volatility  gauges  price  movement  in  terms  of  past  performance.  Im-plied volatility approximates how much the marketplace thinks prices willmove. Understanding volatility can help you to choose and implement theappropriate  option  strategy.  It  holds  the  key  to  improving  your  markettiming as well as helping you to avoid the purchase of overpriced optionsor the sale of underpriced options.
In basic terms, volatility is the speed of change in the market. Somepeople refer to it as confusion in the market. I prefer to think of it as in-surance. If you were to sell an insurance policy to a 35-year-old who drivesa basic Honda, the stable driver and stable car would equal a low insur-ance premium. Now, let’s sell an insurance policy to an 18-year-old, freshout of high school with no driving record. Furthermore, let’s say he’s dri-ving a brand-new red Corvette. His policy will cost more than the policyfor the Honda. The 18-year-old lives in a state of high volatility!
The term vega represents the measurement of the change in the priceof the option in relation to the change in the volatility of the underlying as-set. As the option moves quicker within time, we have a change in volatil-ity:  Volatility  moves  up.  If  the  S&P’s  volatility  was  sitting  just  below  17,perhaps now it’s at 17.50. You can equate that .50 rise to an approximate 3percent increase in options. Can options increase even if the price of the
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underlying  asset  moves  nowhere?  Yes.  This  frequently  happens  in  thebonds market just before the government issues the employment reporton the ﬁrst Friday of the month. Before the Friday report is released, de-mand  causes  option  volatility  to  increase.  After  the  report  is  issued,volatility usually reverts to its normal levels. In general, it is proﬁtable tobuy options in low volatility and sell them during periods of high volatility.When trading options, you can use a computer to look at various indi-cators to assess whether an option’s price is abnormal when compared tothe movement of the underlying asset. This abnormality in price is causedmostly by an option’s implied volatility, or perception of the future move-ment of the asset. Implied volatility is a computed value calculated by us-ing an option pricing model for volume, as well as strike price, expirationdate, and the price of the underlying asset. It matches the theoretical op-tion price with the current market price of the option. Many times, optionprices reﬂect higher or lower option volatility than the asset itself.
The best thing about implied volatility is that it is very cyclical; that is,it tends to move back and forth within a given range. Sometimes it may re-main high or low for a while, and at other times it might reach a new highor  low.  The  key  to  utilizing  implied  volatility  is  in  knowing  that  when  itchanges direction, it often moves quickly in the new direction. Buying op-tions when the implied volatility is high causes some trades to end up los-ing even when the price of the underlying asset moves in your direction.You can take advantage of this situation by selling options and receivingtheir premium as a credit to your account instead of buying options. Forexample, if you buy an option on IBM when the implied volatility is at ahigh you may pay $6.50 for the option. If the market stays where it is, theimplied volatility will drop and the option may then be priced at only $4.75with this drop in volatility.
I generally search the computer for price discrepancies that indicatethat an option is very cheap or expensive compared to its underlying as-set. When an option’s actual price differs from the theoretical price by anysigniﬁcant  amount,  I  take  advantage  of  the  situation  by  purchasing  op-tions with low volatility and selling options with high volatility, expectingthe prices to get back in line as the expiration date approaches.
To place a long volatility trade, I want the volatility to increase. I lookfor a market where the implied volatility for the ATM options has droppeddown  toward  its  historic  lows.  Next,  I  wait  for  the  implied  volatility  toturn around and start going back up.
In its most basic form, volatility means change. It can be summed upjust like that. Markets that move erratically—such as the energy markets intimes of crisis, or grains in short supply—command higher option premi-ums than markets that lag. I look at volatilities on a daily basis and manytimes ﬁnd options to be priced higher than they should be. This is known
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as volatility skew. Most option pricing models give the trader an edge inestimating an option’s worth and thereby identifying skew. Computers arean invaluable resource in searching for these kinds of opportunities.
For  example,  deeply  OTM  options  tend  to  have  higher  impliedvolatility levels than ATM options. This leads to the overpricing of OTMoptions based on a volatility scale. Increased volatility of OTM optionsoccurs for a variety of reasons. Many traders prefer to buy two $5 op-tions than one $10 option because they feel they are getting more bangfor their buck. What does this do to the demand for OTM options? It in-creases that demand, which increases the price, which creates volatilityskew. These skews are another key to ﬁnding proﬁtable option strategyopportunities.
Although I strongly recommend using a computer to accurately de-termine volatility prices, there are a couple of techniques available forpeople  who  do  not  have  a  computer.  One  way  is  to  compare  the  S&Pagainst the Dow Jones Industrial Average. You can analyze this relation-ship simply by watching CNBC, looking at Investor’s Business Daily orThe  Wall  Street  Journal,  or  going  online  to  consult  our  web  site(www.optionetics.com). A 1-point movement in the S&P generally cor-responds to 8 to 10 points of movement in the Dow. For example, if theDow drops 16 to 20 points, but the S&P is still moving up a point, thenS&P  volatility  is  increasing.  On  the  other  side,  if  you  are  consistentlygetting 1-point movement in the S&P for more than 10 points movementin the Dow, then volatility on the S&P is decreasing. This is one way todetermine volatility.
Another way to determine volatility is to check out the range of the mar-kets you wish to trade. The range is the difference between the high for thecycle and the low value for whatever cycle you wish to study (daily, weekly,etc.). For example, try charting the daily range of a stock and then keep arunning average of this range. Then, compare this range to the range of theDow or the S&P. If the range of your stock is greater than the Dow/S&P aver-age, then volatility is increasing; if the range is less than the average, thenvolatility is decreasing. Determining the range or checking out the Dow/S&Prelationship are two ways of gauging volatility with or without a computer.You can use these techniques to your advantage to determine whether youshould be buying, initiating a trade, or just waiting. Remember, option pricescan change quite dramatically between high and low volatility.
TYPES OF VOLATILITY RE-EXAMINED
In order to stack the odds in your favor when developing options strate-gies, it is important to clearly distinguish between two types of volatility:
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implied and historical. Implied volatility (IV) as we have already noted, isthe measure of volatility that is embedded in an option’s price. In addition,each options contract will have a unique level of implied volatility that canbe  computed  using  an  option  pricing  model.  All  else  being  equal,  thegreater an underlying asset’s volatility, the higher the level of IV. That is,an underlying asset that exhibits a great deal of volatility will command ahigher option premium than an underlying asset with low volatility.
To understand why a volatile stock will command a higher option pre-mium, consider buying a call option on XYZ with a strike price of 50 andexpiration in January (the XYZ January 50 call) during the month of De-cember. If the stock has been trading between $40 and $45 for the past sixweeks, the odds of the option rising above $50 by January are relativelyslim. As a result, the XYZ January 50 call option will not carry much value.But say the stock has been trading between $40 and $80 during the pastsix weeks and sometimes jumps $15 in a single day. In that case, XYZ hasexhibited  relatively  high  volatility,  and  therefore  the  stock  has  a  betterchance  of  rising  above  $50  by  January.  A  call  option,  which  gives  thebuyer the right to purchase the stock at $50 a share, will have better oddsof being in-the-money and as a result will command a higher price if thestock has been exhibiting higher levels of volatility.
Options  traders  understand  that  stocks  with  higher  volatility  have  agreater  chance  of  being  in-the-money  at  expiration  than  low-volatilitystocks. Consequently, all else being equal, a stock with higher volatility willhave more expensive option premiums than a low-volatility stock. Mathe-matically,  the  difference  in  premiums  between  the  two  stocks  owes  to  adifference in implied volatility—which is computed using an option pricingmodel  like  the  one  developed  by  Fischer  Black  and  Myron  Scholes,  theBlack-Scholes model. Furthermore, IV is generally discussed as a percent-age. For example, the IV of the XYZ January 50 call is 25 percent. Impliedvolatility of 20 percent or less is considered low. Extremely volatile stockscan have IV in the triple digits.
Sometimes traders and analysts attempt to gauge whether the impliedvolatility  of  an  options  contract  is  appropriate.  For  example,  if  the  IV  istoo high given the underlying asset’s future volatility, the options may beoverpriced and worth selling. On the other hand, if IV is too low given theoutlook for the underlying asset, the option premiums may be too low, orcheap, and worth buying. One way to determine whether implied volatilityis high or low at any given point in time is to compare it to its past levels.For example, if the options of an underlying asset have IV in the 20 to 25percent range during the past six months and then suddenly spike up to 50percent, the option premiums have become expensive.
Statistical  volatility  (SV)  can  also  offer  a  barometer  to  determinewhether  an  options  contract  is  cheap  (IV  too  low)  or  expensive  (IV  too
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high). Since SV is computed as the annualized standard deviation of pastprices over a period of time (10, 30, 90 days), it is considered a measure ofhistorical volatility because it looks at past prices. If you don’t like math,statistical  volatility  on  stocks  and  indexes  can  be  found  on  various  websites like the Optionetics.com Platinum site. SV is a tool for reviewing thepast volatility of a stock or index. Like implied volatility, it is discussed interms of percentages.
Comparing the SV to IV can offer indications regarding the appropri-ateness of the current option premiums. If the implied volatility is signiﬁ-cantly  higher  than  the  statistical  volatility,  chances  are  the  options  areexpensive. That is, the option premiums are pricing in the expectations ofmuch higher volatility going forward when compared to the underlying as-set’s actual volatility in the past. When implied volatility is low relative tostatistical volatility, the options might be cheap. That is, relative to the as-set’s  historical  volatility,  the  IV  and  option  premiums  are  high.  Savvytraders attempt to take advantage of large differences between historicaland  implied  volatility.  In  later  chapters,  we  will  review  some  strategiesthat show how.
APPLYING THE GREEKS
Now the challenge for the options strategist and particularly the delta neu-tral trader is to effectively interpret and manage these Greeks not only attrade initiation but also throughout the life of the position. The ﬁrst thingto  realize  is  that  changes  in  the  underlying  instrument  cause  changes  inthe delta, which then impact all the other Greeks.
Keep  these  three  rules  in  mind  when  evaluating  the  Greeks  of  your
position.
1. The deltas of out-of-the-money options are smaller and they continueto  decrease  as  you  go  further  out-of-the-money.  When  the  optionsstrategist purchases options, theta is negative and gamma is positive.In  this  situation  the  position  would  lose  money  through  time  decaybut price movement has a positive impact.
2. When  the  trader  sells  option  premium  then  theta  is  positive  andgamma is negative. This position would make money through time de-cay but price movement would have a negative effect. Also, theta andgamma both increase, the closer the position gets to expiration. In ad-dition, theta and gamma are larger for at-the-money options.
3. When  selling  options,  vega  is  negative;  if  implied  volatility  rises  theposition  loses  money  and  if  it  declines  the  position  makes  money.
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Conversely,  when  a  trader  purchases  options,  vega  is  positive,  so  arise  in  implied  volatility  is  proﬁtable  and  decreases  have  a  negativeimpact on the position’s proﬁtability.
To be an effective options strategist, particularly if you want to makedelta neutral type adjustments, understanding these Greek basics and beingable to apply them to your options strategies is paramount to your success.These  very  important  measurement  tools  coupled  with  the  position’s  riskgraph can provide the necessary information that will allow you to consis-tently execute proﬁtable trades and send your equity curve sharply upward.
CONCLUSION
Many times when you have conversations with options traders, you willnotice  that  they  refer  to  the  delta,  gamma,  vega,  or  theta  of  their  posi-tions. This terminology can be confusing and sometimes intimidating tothose who have