Getting To Know The Greeks

Getting To Know The "Greeks" by John Summa,CTA, PhD, Founder of OptionsNerd.com andTradingAgainstTheCrowd.com (Contact Author | Biography) Email Article Print Comments Reprints

An option's price can be influenced by a number of factors. These factors can either help or hurt traders, depending on the type of options positions they have established. To become a successful option trader, it is essential to understand what factors influence the price of an option, which requires learning about the so-called "Greeks" - a set of risk measures that indicate how exposed an option is to time-value decay, implied volatility and changes in the underlying price of the commodity. In this article, I we'll look at four "Greek" risk measures - delta, theta, vega and gamma - and explain their importance. But first, let's review some related option characteristics that will help you better understand the Greeks. (For background reading, see the Options Basics Tutorial.) Influences on an Option's Price Figure 1 lists the major influences on both a call and put option's price. The plus or minus sign indicates an option's price direction resulting from a change in one of the price variables in Figure 1. For example, taking call options and looking at the impact of a change in implied volatility shows that when there is a rise in implied volatility, there is an increase in the price of an option, all other things remaining the same.

Options

Increase in Volatility

Decrease in Volatility

Increase in Time to Expiration

Decrease in Time to Expiration

Increase in the Underlying

Decrease in the Underlying

Calls

+

-

+

-

+

-

Puts

+

-

+

-

-

+

Figure 1: Major influences on an option's price Bear in mind that results will differ depending on whether you long or short an option. Naturally, if you long a call option, a rise in implied volatility will be favorable because rising implied volatility typically gets priced into the option premium. If you establish a short call option position, on the other hand, a rise in implied volatility will have an inverse (or negative sign) effect. The writer of a naked option, be it a put or a call, would therefore not benefit from a rise in volatility because writers want the price of the option to decline. (To learn more, see Implied Volatility: Buy Low And Sell High and The Price-Volatility Relationship: Avoiding Negative Surprises.) Figures 2 and 3 below present the same variables, but in terms of long and short call options (Figure 2) and long and short put options (Figure 3). Note that a decrease in implied volatility, a reduced time to expiration and a fall in the price of the underlying will benefit the short call holder. At the same time, an increase in volatility, a greater time remaining on the option and a rise in the underlying will benefit the long call holder. Figure 3 shows that a short put holder benefits from a decrease in implied volatility, a reduced time remaining until expiration and a rise in the price of the underlying. Meanwhile, an increase in implied volatility, a greater time remaining until expiration and a decrease in the price of the underlying will benefit the long put holder.

Call Increase in Options Volatility

Decrease in Volatility

Increase in Time to Expiration

Decrease in Time to Expiration

Increase in the Underlying

Decrease in the Underlying

Long

+

-

+

-

+

-

Short

-

+

-

+

-

+

Figure 2: Major influences on a short and long call option's price Increase Decrease Increase Decrease Increase in Decrease Put in in in Time to in Time to the in the Options Volatility Volatility Expiration Expiration Underlying Underlying Long

+

-

+

-

-

+

Short

-

+

-

+

+

-

Figure 3: Major influences on a short and long put option's price Because interest rates play a negligible role in a position during the life of most option trades, we will be excluding this price variable from the discussion. It is worth noting, however, that higher interest rates make call options more expensive and put options less expensive, all other things remaining the same. This summary of the influences on option price provides a nice backdrop for an examination of the risk measures used to gauge the degree to which an option's price is influenced by these price variables. Let's now take a look at how the Greeks permit us to project changes in an option's price. The Greeks Figure 4, below, contains four major risk measures - our so-called Greeks - all of which a trader should take into account before taking any option position. Because the Greeks are actually represented by letters of the Greek language alphabet, let's take them in alphabetical order. Delta Delta is a measure of the change in an option's price (premium of an option) resulting from a change in the underlying security (i.e. stock) or commodity (i.e. futures contract). The value of delta ranges from -100 to 0 for puts and 0 to 100 for calls (here delta has been multiplied by 100 to shift the decimal). Puts have a negative delta because they have what is called a "negative relationship" to the underlying: put premiums fall when the underlying rises, and vice versa. Call options, on the other hand, have a positive relationship to the price of the underlying: if the underlying rises, so does the premium on the call, provided there are no changes

in other variables like implied volatility and time remaining until expiration. And if the price of the underlying falls, the premium on a call option, provided all other things remain constant, will decline. An at-the-money option has a delta value of approximately 50 (0.5 without the decimal shift), which means the premium will rise or fall by half a point with a one-point move up or down in the underlying. For example, if an at-the-money wheat call option has a delta of 0.5, and if wheat makes a 10-cent move higher (which is a large move), the premium on the option will increase by approximately 5 cents (0.5 x 10 = 5), or $250 (each cent in premium is worth $50). Vega

Theta

Delta

Gamma

Measures Impact of a Change in Volatility

Measures Impact of a Change in Time Remaining

Measures Impact of a Change in the Price of Underlying

Measures the Rate of Change of Delta

Figure 4: The major "Greeks". As the option gets further in the money, delta approaches 100 on a call and -100 on a put, which means that at these extremes there is a one-for-one relationship between changes in the option price and changes in the price of the underlying. In effect, at delta values of -100 and 100, the option behaves like the underlying in terms of price changes. This occurs with little or no time value, as most of the value of the option is intrinsic. We'll come back to the concept of time value below when we discuss theta. (To take a more advanced look at delta, see Going Beyond Simple Delta: Understanding Position Delta.) Three things to keep in mind with delta: 1.

Delta tends to increase as you get closer to expiration for near or at-the-money options.

2.

Delta is not a constant, a fact related to gamma, our next risk measurement, which is a measure of the rate of change of delta given a move by the underlying.

3.

Delta is subject to change given changes in implied volatility.

Gamma Gamma, also known as the "first derivative of delta", measures delta's rate of change. Figure 5 shows how much delta changes following a one-point change in the price of the underlying. This is a simple concept to grasp. When call options are deep out of the money, they generally have a small delta. This is because changes in the underlying bring about only tiny changes in the price of the option. As the call option gets closer to the money, resulting from a continued rise in the price of the underlying, the delta gets larger.

Actual Risk Measures for Short December S&P 500 930 Call Option

Figure 5: Created using OptionVue 5 Options Analysis Software

In Figure 5, delta is rising as we read the figures from left to right, and it is shown with values for gamma at different levels of the underlying. The column showing profit/loss (P/L) of -200 represents the at-the-money strike of 930, and each column represents a one-point change in the underlying. As you can see, at-the-money gamma is -0.79, which means that for every one-point move of the underlying, delta will increase by exactly 0.79 (for both delta and gamma the decimal has been shifted two digits by multiplying by 100). If you move right to the next column (which represents a one-point move higher to 931 from 930), you can see that delta is -53.13 (an increase of . 79 from -51.34). As you can see, delta rises as this short call option gets into the money, and the negative sign means that the position is losing because it is a short position (in other words, the position delta is negative). Therefore, with a negative delta of -51.34, the position will lose 0.51 (rounded) points in premium with the next one-point rise in the underlying. (To read more about a gamma-related strategy, see Gamma-Delta Neutral Option Spreads.) There are some additional points to keep in mind about gamma:

1.

Gamma is smallest for deep out-of-the-money and deep in-the money options.

2.

Gamma is highest when the option gets near the money

3.

Gamma is positive for long options and negative for short options (as seen above in Figure 5 with our example of a short call).

Theta Theta is not used much by traders, but it is an important conceptual dimension. Theta measures the rate of decline of time-premium resulting from the passage of time. In other words, an option premium that is not intrinsic value will decline at an increasing rate as expiration nears. Figure 6 shows theta values at different time intervals for an S&P 500 Dec at-the-money call option. The strike price is 930. As you can see, theta increases as expiration gets closer (T+25 is expiration). At T+19, which is six days before expiration, theta has reached 93.3, which in this case tells us that the option is now losing $93.30 per day, up from $45.40 per day at T+0, when we hypothetically opened the position.

-

T+0

T+6

T+13

T+19

Theta

45.4

51.85

65.2

93.3

Figure 6 : Theta values for short S&P Dec 930 call option Some additional points about theta to consider when trading: 1.

Theta can be very high for out-of-the-money options if they contain a lot of implied volatility.

2.

Theta is typically highest for at-the-money options

3.

Theta will increase sharply in the last few weeks of trading and can severely undermine a long option holder's position, especially if implied volatility is on the decline at the same time.

Vega Vega, our fourth and final risk measure, quantifies risk exposure to implied volatility changes. Vega tells us approximately how much an option price will increase or decrease given an increase or decrease in the level of implied volatility. Option sellers benefit from a fall in implied volatility, and it's just the reverse for option buyers. Referring to Figure 5 above, you can see that the short call has a negative vega, which tells us that the position will gain if implied volatility falls (hence the inverse relationship indicated by the negative sign). The value of vega itself indicates by how much the position will gain in this case. Using at-the-money vega, which is -96.94, we know that for each percentagepoint drop in implied volatility, a short call position will gain by $96.94. (For a trading strategy that relies on vega, see An Option Strategy For Trading Market Bottoms.) Conversely, if there should be a 1% rise in implied volatility, the position would lose $96.94. Additional points to keep in mind regarding vega include the following: 1.

Vega can increase or decrease even without price changes of the underlying because implied volatility is the level of expected volatility.

2.

Vega can increase from quick moves of the underlying, especially if there is a big drop in the stock market, or if there is a sudden upward burst in a commodity like coffee after a reported frost in Brazil.

3.

Vega falls as the option gets closer to expiration.

Conclusion While this overview is only an intermediate-level discussion of delta, gamma, theta and vega (an advanced level analysis would involve mathematical nuances which are not practical in terms of trading), it should nonetheless help clarify not only how the price of an option is influenced by changes in the underlying, the time to expiration and the implied volatility, but also how we measure the impact of these variables on an option's price. Finally, always remember that there is a risk of loss when trading futures and options. Trade only with risk capital.

Going Beyond Simple Delta: Understanding Position Delta by John Summa,CTA, PhD, Founder of OptionsNerd.com andTradingAgainstTheCrowd.com (Contact Author | Biography) Email Article Print Comments Reprints

The article Getting to Know the Greeksdiscusses risk measures such as delta, gamma, theta and vega, which are summarized in figure 1 below. This article takes a closer look at delta as it relates to actual and combined positions - known as position delta - a very important concept for option sellers. Below I begin with a quick review of the risk measure delta, and then proceed to explaining position delta, including an example of what it means to be position-delta neutral. Simple Delta Let's review some basic concepts before jumping right into position delta. Delta is one of four major risk measures used by option traders, all of which are outlined in figure 1 below. Delta measures the degree to which an option is exposed to shifts in the price of the underlying asset (i.e. stock) or commodity (i.e. futures contract). Values range from +1.0 to –1.0 (or +100 to –100, depending on the convention employed). For example, if you buy a call or a put option that is just out of the money (i.e. the strike price of the option is above the price of the underlying if the option is a call and below the price of the underlying if the option is a put), then the option will always have a delta value that is somewhere between 1.0 and –1.0. Generally speaking an at-the-moneyoption usually has a delta at approximately 0.5 or -0.5. Vega Measures the impact of a change in volatility.

Theta Measures the impact of a change in time remaining.

Delta Measures impact of a change in the price of underlying.

Gamma Measures the rate of change of delta.

Figure 1 - Delta and the other "Greeks". Figure 2 contains some hypothetical values for S&P 500 call options that are at, out and in the money (in all these cases I am using long options). Call delta values range from 0 to 1.0, while put delta values range from 0 to –1.0. As you can see, the at-the-money call option (strike price at 900) in figure 2 has a 0.5 delta, while the out-of-the-money (strike price at 950) call option has a 0.25 delta and the in-the-money (strike at 850) has a delta value of 0.75. Keep in mind that these call delta values are all positive because we are dealing with long call options, a point to which we will return later. If these were puts, the same values would have a negative sign attached to them. This reflects the fact that put options increase in value when the underlying asset price falls. (An inverse relationship is indicated by the negative delta sign.) You will see below, when we look at short option positions and the concept of position delta, that the story gets a bit more complicated. Strikes 950 900 850 Note: We are assuming that the underlying S&P 500 is trading at 900

Delta 0.25 0.5 0.75

Figure 2 - Hypothetical S&P 500 long call options. At this point you might be wondering what these delta values are telling you. Let me offer an example to help illustrate the concept of simple delta and the meaning of these values. If an S&P 500 call option has a delta of 0.5 (for a near or at-the-money option), a one-point move (which is worth $250) of the underlying futures contract would produce a 0.5 (or 50%) change (worth $125) in the price of the call option. A delta value of 0.5, therefore, tells you that for every $250 change in value of the underlying futures, the option changes in value by about $125. If you were long this call option and the S&P 500 futures move up by one point, your call option would gain approximately $125 in value, assuming no other variables change in the short run. We say "approximately" because as the underlying moves, delta will change as well. (To understand this relationship, Getting to know the Greeks.) Be aware that as the option gets further in the money, delta approaches 1.00 on a call and –1.00 on a put. At these extremes there is a near or actual one-for-one relationship

between changes in the price of the underlying and subsequent changes in the option price. In effect, at delta values of –1.00 and 1.00, the option mirrors the underlying in terms of price changes. Also bear in mind that this simple example assumes no change in other variables like the following: (1) delta tends to increase as you get closer to expiration for near or at-themoney options; (2) delta is not a constant, a concept related to gamma, another risk measurement, which is a measure of the rate of change of delta given a move by the underlying; (3) delta is subject to change given changes in implied volatility. Long vs. Short Options and Delta As a segue into looking at position delta, let me say a few words about how short and long positions change the picture somewhat. First, the negative and positive signs for values of delta mentioned above do not tell the full story. As indicated in figure 3 below, if you are long a call or a put (that is, you purchased them to open these positions), then the put will be delta negative and the call delta positive; however, our actual position will determine the delta of the option as it appears in our portfolio. Note how the signs are reversed for short put and short call. Long Call Delta Positive

Short Call Delta Negative

Long Put Delta Negative

Short Put Delta Positive

Figure 3 - Delta signs for long and short options. The delta sign in your portfolio for this position will be positive, not negative. This is because the value of the position will increase if the underlying increases. Likewise, if you are short a call position, you will see that the sign is reversed. The short call now acquires a negative delta, which means that if the underlying rises, the short call position will lose value. This is getting us closer to an actual discussion of position delta. Position Delta Position delta can be understood by reference to the idea of a hedge ratio. Delta is in effect a hedge ratio because it tells us how many options contracts are needed to hedge a long or short position in the underlying. It is a very easy concept to grasp. For example, if an at-the-money call option has a delta value of approximately 0.5 - which means that there is a 50% chance the option will end in the money and a 50% chance it will end out of the money - then this delta tells us that it would take two at-the-money call options to hedge one short contract of the underlying. In other words, you need two long call options to hedge one short futures contract. (Two long call options x delta of 0.5 = position delta of 1.0, which equals one short futures position). This means that a onepoint rise in the S&P 500 futures (a loss of $250), which you are short, will be offset by a one-point (2 x $125 = +$250) gain in the value of the two long call options. In this example we would say that we are position-delta neutral. By changing the ratio of calls to number of positions in the underlying, we can turn this position delta either positive or negative. For example, if are bullish we might add another long call, so we are now delta positive because our overall strategy is set to gain if the futures rise. We would have three long calls with delta of 0.5 each, which means we have a net long position delta by +0.5. On the other hand, if we are bearish, we could reduce our long calls to just one, which we would now make us net short position delta. This means that we are net short the futures by -0.5. Conclusion This article explains the concept of simple delta and then proceeds to explain how position delta is a measure of how net long or net short the underlying you are when taking into account your entire portfolio of options (and futures).

Gamma-Delta Neutral Option Spreads by Daniel McNulty (Contact Author | Biography) Email Article Print Comments Reprints

Have you ever found that strategies that make full use the decay of an option's theta are attractive but you can't stand the risk associated with them? At the same time, conservative strategies like covered-call writing or synthetic covered-call writing can be too restrictive. Thegamma-delta neutral spread may just be the best middle ground to these concerns when searching for a way to exploit time decay while neutralizing the effect of price actions on your position's value. In this article, we'll introduce you to this strategy. Learning Greek In order to understand the application of this strategy as explained here, knowledge of the basic Greek measures associated with options is essential. This inherently means that the reader must also be familiar with options and their characteristics. (To learn more, readGetting To Know The "Greeks" and Using The Greeks To Understand Options.) Theta Theta is the rate of decay in an option's value that can be attributed to the passage of one day's time. With this spread, we will be exploiting the decay of theta to our advantage to extract a profit from the position. Of course, many other spreads do this, but as you'll discover, by hedging the net gamma and the net delta of our position, we can safely keep our position direction neutral in terms of price. (For further reading, check out Option Spread Strategies.) The Strategy For our purposes, we will be using a ratio write call strategy as our core position. In these examples, we will be buying options at a lower strike price than that at which they are sold. For example, this means if we are buying the calls with a $30 strike price, we will be selling the calls at a strike price of $35. Of course, we will not be just performing a regular ratio write call strategy - we will be adjusting the ratio at which we buy and sell options to materially eliminate the net gamma of our position. We know that in a ratio write options strategy, more options are written than are purchased. This means that some options are sold "naked". This is inherently risky. The risk here is, that if the stock rallies enough, the position will lose money as a result of the unlimited exposure to the upside with the naked options. By reducing the net gamma to a value close to zero, we eliminate the risk that the delta will shift significantly (assuming only a very short time frame). (For related reading, see Naked Call Writing and To Limit Or Go Naked, That Is The Question.) Neutralizing the Gamma To effectively neutralize the gamma, we first need to find the ratio at which we are going to buy and write. Instead of going through a system of equations models to find the ratio, you can quickly figure out the gamma neutral ratio by doing the following. 1.

Find the gamma of each option.

2.

To find the number you will buy, take the gamma of the option you are selling and round it to three decimal places and multiply it by 100.

3.

To find the number you will sell take the gamma of the option you are buying, round it to three decimal places and multiply it by 100.

For example, if we have our $30 call with a gamma of 0.126 and our $35 call with a gamma of 0.095, we would buy 95 $30 calls and sell 126 $35 calls. Remember this is per share, and each option represents 100 shares. •

Buying 95 calls with a gamma of 0.126 is a gamma of 1,197 (9,500*0.126)

•

Selling 126 calls with a gamma of -0.095 (negative because we're selling them) is a gamma of -1,197 (12,600*-0.095).

This adds up to a net gamma of 0. Because the gamma is usually not a nicely rounded to three decimal places, your actual net gamma might vary by about 10 points around 0, but because we are dealing with such large numbers, these variations of actual net gamma are not material and will not affect a good spread. Neutralizing the Delta Now that we have the gamma neutralized we will need to make the net delta zero. If our $30 calls have a delta of 0.709 and our $35 calls have a delta of 0.418, we can calculate the following. •

95 calls bought with a delta of 0.709 is 6,735.5

•

126 calls sold with a delta of -0.418 (negative because we're selling them) is -5,266.8

This results in a net delta of positive 1,468.7. To make this net delta very close to zero, we can short 1,469 shares of the underlying stock. This is because each share of stock has a delta of 1. This adds -1,469 to the delta making it -0.3, very close to zero. Because you can not short parts of a share, -0.3 is as close as we can get the net delta to zero. Again, like we stated in the gamma, because we are dealing with large numbers, this will not be materially large enough to affect the outcome of a good spread. Examining the Theta Now that we have our position effectively price neutral, let's examine its profitability. The $30 calls have a theta of -0.018 and the $35 calls have a theta of -0.027. This means: •

95 calls bought with a theta of -0.018 is -171

•

126 calls sold with a theta of 0.027 (positive because we're selling them) is 340.2

This results in a net theta of 169.2. This can be interpreted as your position making $169.2 per day. Because option behavior isn't adjusted on a daily basis, you'll have to hold your position roughly a week before you'll be able to notice these changes and profit from them. Profitability Without going through all the margin requirements and net debits and credits, the strategy we've detailed would require about $32,000 in capital to set up. If you held this position for five days, you could expect to make $846. This is 2.64% on the capital needed to set this up - a pretty good return for five days. In most real life examples, you'll find a position held for five days would yield about 0.5-0.7%. This may not seem like a lot until you annualize 0.5% in five days - this represents a 36.5% return per year. Possible Drawbacks There are a few risks associated with this strategy. First of all, you'll need low commissionsto be able to make a profit. This is why it is very important to have a very low commission broker. Very large price moves can also throw this out of whack. If held for a week, a required adjustment to the ratio and the delta hedge is not probable; if held for a longer time, the price of the stock will have more time to move in one direction. Changes in implied volatility, which are not hedged here, can result in dramatic changes in the position's value. Although we have eliminated the relative a day-to-day price movements, we are faced with another risk: an increased exposure to changes in implied volatility. Over the short time horizon of a week, changes in volatility should play a small role in your overall position. This doesn't mean you shouldn't keep your eyes on it though! (To learn more, check out The ABCs Of Option Volatility.) Conclusion We can see that the risk of ratio writes can be brought down by mathematically hedging certain characteristics of the options we are dealing with along with adjusting our position in the underlying common stock. By doing this, we can profit from the theta decay in the written options. Although this strategy is attractive to most investors, it can only be functionally executed by market professionals due to the high commission costs associated with it. by Daniel McNulty, (Contact Author | Biography)

An Option Strategy for Trading Market Bottoms by John Summa,CTA, PhD, Founder of OptionsNerd.com andTradingAgainstTheCrowd.com (Contact Author | Biography) Email Article Print Comments Reprints

High volatility associated with stock-market bottoms offers options traders tremendous profit potential if the correct trading setups are deployed; however, many traders are familiar with only option buying strategies, which unfortunately do not work very well in an environment of high volatility. Buying strategies - even those using bull and bear debit spreads - are generally poorly priced when there is high implied volatility. When abottom is finally achieved, the collapse in high-priced options following a sharp drop in implied volatility strips away much of the profit potential. So even if you are correct in timing a market bottom, there may be little to no gain from a big reversal move following a capitulation sell-off. Through a net options selling approach, there is a way around this problem. Here we'll look at a simple strategy that profits from falling volatility, offers a potential for profit regardless of market direction and requires little up-front capital if used with options on futures. Finding the Bottom Trying to pick a bottom is hard enough, even for savvy market technicians. Oversoldindicators can remain oversold for a long time, and the market can continue to trade lower than expected. The decline in the broad equity market measures in 2009 offers a case in point. The correct option selling strategy, however, can make trading a market bottom considerably easier. The strategy we'll examine here has little or no downside risk, thus eliminating the bottom-picking dilemma. This strategy also offers plenty of upside profit potential if the market experiences a solid rally once you are in your trade. More important, though, is the added benefit that comes with a sharp drop in implied volatility, which typically accompanies a capitulation reversal day and a follow-through multiweek rally. By getting short volatility, or short vega, the strategy offers an additional dimension for profit. (For background

reading, see Implied Volatility: Buy Low And Sell High.) Shorting Vega The CBOE Volatility Index (VIX) uses the implied volatilities of a wide range of S&P 500 Index options to show the market's expectation of 30-day volatility. A high VIX means that options have become extremely expensive because of increased expected volatility, which gets priced into options. This presents a dilemma for buyers of options - whether of puts or calls - because the price of an option is so affected by implied volatility that it leaves traders long vega just when they should be short vega. (For related reading, see Volatility's Impact On Market Returns.) Vega is a measure of how much an option price changes with a change in implied volatility. If, for instance, implied volatility drops to normal levels from extremes and the trader is long options (hence long vega), an option's price can decline even if the underlying moves in the intended direction. When there are high levels of implied volatility, selling options is, therefore, the preferred strategy, particularly because it can leave you short vega and thus able to profit from an imminent drop in implied volatility; however, it is possible for implied volatility to go higher (especially if the market goes lower), which leads to potential losses from still higher volatility. By deploying a selling strategy when implied volatility is at extremes compared to past levels, we can at least attempt to minimize this risk. Reverse Calendar Spreads To capture the profit potential created by wild market reversals to the upside and the accompanying collapse in implied volatility from extreme highs, the one strategy that works the best is called a reverse call calendar spread. (To read about spreads in more depth, seeOption Spread Strategies.) Normal calendar spreads are neutral strategies, involving selling a near-term option and buying a longer-term option, usually at the same strike price. The idea here is to have the market stay confined to a range so that the near-term option, which has a higher theta (the rate of time-value decay), will lose value more quickly than the long-term option. Typically, the spread is written for a debit (maximum risk). But another way to use calendar spreads is to reverse them - buying the near-term and selling the long-term, which works best when volatility is very high. The reverse calendar spread is not neutral and can generate a profit if the underlying makes a huge move in either direction. The risk lies in the possibility of the underlying going nowhere, whereby the short-term option loses time value more quickly than the long-term option, which leads to a widening of the spread - exactly what is desired by the neutral calendar spreader. Having covered the concept of a normal and reverse calendar spread, let's apply the latter to S&P call options. Reverse Calendar Spreads in Action At volatile market bottoms, the underlying is least likely to remain stationary over the near term, which is an environment in which reverse calendar spreads work well; furthermore, there is a lot of implied volatility to sell, which, as mentioned above, adds profit potential. The details of our hypothetical trade are presented in Figure 1 below. S&P 500 Reverse Call Calendar Spread Long

Short

1 1

Strike/Option

Month/Yr

Price

Value

850 Call

Oct’02

56

-14,000

850 Call

Dec’02

79.20

+19,800

Net Credit

+5,800

Figure 1: Theoretical prices with the S&P 500 trading at 850 with 61 days to near-term expiration of the October 850 call. The trade is constructed using S&P 500 call options on futures. Initial SPAN margin requirement is $935. Assuming Dec S&P 500 futures are trading at 850 after what we determine is a capitulation day sell-off, we would buy one 850 Oct call for 56 points in premium (-$14,000) and simultaneously sell one 850 Dec call for 79.20 points in premium ($19,800), which leaves a net credit of $5,800 before any commission or fees. A reliable broker who can place a limit order using a limit price on the spread should enter this order. The plan of a reverse calendar call spread is to close the position well ahead of expiration of the near-term option (Oct expiry). For this example, we will look at profit/loss while assuming that we hold the position 31 days after entering it, exactly 30 days before expiration of the Oct 850 call option in our spread. Should the position be held open until the expiration of the shorter-term option, the maximum loss for this trade would be slightly more than $7,500. To keep potential losses limited, however, the trader should close out this trade no less than a month before expiration of the near-term option. If, for instance, this position is held no more than 31 days, maximum losses would be limited to $1,524, provided there is no change in implied volatility levels and Dec S&P futures don't trade lower than 550. Maximum profit is meanwhile limited to $5,286 if the underlying S&P futures rise substantially to 1050 or above. To get a better idea of the potential of our reverse call spread, see Figure 2, below, which contains the profit and loss levels within a range of prices from 550 to 1150 of the underlying Dec S&P futures. (Again we are assuming that we are 31 days into the trade.) In column 1, the losses rise to $974 if the S&P is at 550, so downside risk is limited should the market bottom turn out to be false. Note that there is a small profit potential on the downside at near-term expiry if the underlying futures drop far enough. Upside potential, meanwhile, is significant, especially given the potential for a drop in volatility, which we show in columns 2 (5% drop) and 3 (10% drop). S&P 500 Reverse Call Calendar Spread – Profit/Loss Decemb (1) Profit/Loss No Change (2) Profit/Loss -5% Change (3) Profit/Loss -10% er S&P in IV in IV Change in IV Futures 550 -971 -149 +651 650 -649 +301 +1251 750 -1399 -399 +601 850 -1524 +624 +301 950 +1701 +2851 +4001 1050 +5251 +5776 +5826 1150 +5826 +5826 +5286

Figure 2: Profit levels given different levels of implied volatility and levels of December S&P futures 31 days into our trade. IV represents implied volatility of the options trading on the December S&P 500 futures contract. If, for example, the Dec S&P futures run up to 950 with no change in volatility, the position would show a profit of $1,701. If, however, there is an associated 5% drop in implied volatility with this rally, the profit would increase to $2,851. Finally, if we factor a 10% drop in volatility into the same 100-point rally in the December futures, profit would increase to $4,001. Given that the trade requires just $935 in initial margin, the percentage return on capital is quite large: 182%, 305% and 428% respectively. Should, on the other hand, volatility increase, which might happen from continued decline of the underlying futures, the losses of different time intervals outlined above could be significantly higher. While the reverse calendar spread may or may not be profitable, it may not be suitable to all investors. Conclusion A reverse calendar spreads offers an excellent low-risk (provided you close the position before expiration of the shorter-term option) trading setup that has profit potential in both directions. This strategy, however, profits most from a market that is moving fast to the upside associated with collapsing implied volatility. The ideal time for deploying reverse call calendar spreads is, therefore, at or just following stock market capitulation, when huge moves of the underlying often occur rather quickly. Finally, the strategy requires very little upfront capital, which makes it attractive to traders with smaller accounts.

The Greeks: What They Are and How to Use Them all lessons

The Greeks have given us feta cheese, philosophy, mathematics, and the Oedipal complex. They also tell us how much risk our option positions have. There are ways of estimating the risks associated with options, such as the risk of the stock price moving up or down, implied volatility moving up or down, or how much money is made or lost as time passes. They are numbers generated by mathematical formulas. Collectively, they are known as the "greeks", because most use Greek letters as names. Each greek estimates the risk for one variable: delta measures the change in the option price due to a change in the stock price, gamma measures the change in the option delta due to a change in the stock price, theta measures the change in the option price due to time passing, vega measures the change in the option price due to volatility changing, and rho measures the change in the option price due to a change in interest rates.

Delta The first and most commonly used greek is "delta". For the record, and contrary to what is frequently written and said about it, delta is NOT the probability that the option will expire ITM. Simply, delta is a number that measures how much the theoretical value of an option will change if the underlying stock moves up or down $1.00. Positive delta means that the option position will rise in value if the stock price rises, and drop in value if the stock price falls. Negative delta means that the option position will theoretically rise in value if the stock price falls, and theoretically drop in value if the stock price rises. The delta of a call can range from 0.00 to 1.00; the delta of a put can range from 0.00 to –1.00. Long calls have positive delta; short calls have negative delta. Long puts have negative delta; short puts have positive delta. Long stock has positive delta; short stock has negative delta. The closer an option's delta is to 1.00 or –1.00, the more the price of the option responds like actual long or short stock when the stock price moves. So, if the XYZ Aug 50 call has a value of $2.00 and a delta of +.45 with the price of XYZ at $48, if XYZ rises to $49, the value of the XYZ Aug 50 call will theoretically rise to $2.45. If XYZ falls to $47, the value of the XYZ Aug 50 call will theoretically drop to $1.55. If the XYZ Aug 50 put has a value of $3.75 and a delta of -.55 with the price of XYZ at $48, if XYZ rises to $49, the value of the XYZ Aug 50 put will drop to $3.20. If XYZ falls to $47, the value of the XYZ Aug 50 put will rise to $4.30. Now, these numbers assume that nothing else changes, such as a rise or fall in volatility or interest rates, or time passing. Changes in any one of these can change delta, even if the price of the stock doesn't change. Note that the delta of the XYZ Aug 50 call is .45 and the delta of the Aug 50 put is -.55. The sum of their absolute values is 1.00 (|.45| + |-.55| = 1.00). This is true for every call and put at every strike. The intuition behind this is that long stock has a delta of +1.00. Synthetic long stock is long a call and short a put at the same strike in the same month. Therefore, the delta of a long call plus the delta of a short put must equal the delta of long stock. In the case of the XYZ Aug call and put, .45 + .55 = 1.00. Remember, a short put has a positive delta. (Note: delta can be calculated with different formulas, which won't be discussed here. Using the Black-Scholes model for European-style options, the sum of the absolute values of the call and put is 1.00. But using other models for American-style options and under certain circumstances, the sum of the absolute values of the call and put can be slightly less or slightly more than 1.00.) You can add, subtract, and multiply deltas to calculate the delta of a position of options and stock. The position delta is a way to see the risk/reward characteristics of your position in terms of shares of stock, and it's how thinkorswim presents it to you on the Position Statement on the Monitor page. The calculation is very straightforward. Position delta = option theoretical delta * quantity of option contracts * number of shares of stock per option contract. (The number of shares of stock per option contract in the U.S. is usually 100 shares. But it can be more or less, due to stock splits or mergers.) thinkorswim performs this calculation for each option in your position, then adds them together for each stock. So, if you are long 5 of the XYZ Aug 50 calls, each with a delta of +.45, and short 100 shares of XYZ stock, you will have a position delta of +125. (Short 100 shares of stock = -100 deltas, long 5 calls with delta +.45, with 100 shares of stock per contract = +225. –100 + 225 = +125) A way to interpret this delta is that if the price of XYZ rises $1, you will theoretically make $125. If XYZ falls $1, you will theoretically lose $125. IMPORTANT: These numbers are theoretical. In reality, delta is accurate for only very small changes in the stock price. Nevertheless, it is still a very useful tool for a $1.00 change, and is a good way to evaluate your risk. An ATM option has a delta close to .50. The more ITM an option is, the closer its delta is to 1.00 (for calls) or –1.00 (for puts). The more OTM and option is, the closer its delta is to 0.00.

Delta is sensitive to changes in volatility and time to expiration. The delta of ATM options is relatively immune to changes in time and volatility. This means an option with 120 days to expiration and an option with 20 days to expiration both have deltas close to .50. But the more ITM or OTM an option is, the more sensitive its delta is to changes in volatility or time to expiration. Fewer days to expiration or a decrease in volatility push the deltas of ITM calls closer to 1.00 (-1.00 for puts) and the deltas of OTM options closer to 0.00. So an ITM option with 120 days to expiration and a delta of .80 could see its delta grow to .99 with only a couple days to expiration without the stock moving at all. The delta of an option depends largely on the price of the stock relative to the strike price. Therefore, when the stock price changes, the delta of the option changes. That's why gamma is important. Top

Gamma Gamma is an estimate of how much the delta of an option changes when the price of the stock moves $1.00. As a tool, gamma can tell you how "stable" your delta is. A big gamma means that your delta can start changing dramatically for even a small move in the stock price. Long calls and long puts both always have positive gamma. Short calls and short puts both always have negative gamma. Stock has zero gamma because its delta is always 1.00 – it never changes. Positive gamma means that the delta of long calls will become more positive and move toward +1.00 when the stock prices rises, and less positive and move toward 0.00 when the stock price falls. It means that the delta of long puts will become more negative and move toward –1.00 when the stock price falls, and less negative and move toward 0.00 when the stock price rises. The reverse is true for short gamma. For example, the XYZ Aug 50 call has a delta of +.45, and the XYZ Aug 50 put has a delta of -.55, with the price of XYZ at $48.00. The gamma for both the XYZ Aug 50 call and put is .07. If XYZ moves up $1.00 to $49.00, the delta of the XYZ Aug 50 call becomes +.52 (+.45 + ($1 * .07), and the delta of the XYZ Aug 50 put becomes -.48 (-.55 + ($1 * .07). If XYZ drops $1.00 to $47.00, the delta of the XYZ Aug 50 call becomes +.38 (+.45 + (-$1 * . 07), and the delta of the XYZ Aug 50 put becomes -.62 (-.55 + (-$1 * .07). Position gamma measures how much the delta of a position changes when the stock price moves $1.00. Position gamma is calculated much in the same way as position delta. In the Position Statement on the Monitor page, thinkorswim takes the gamma of each option in your position, multiplies it by the number of contracts and the number of shares of stock per option contract, then adds them together. Just as delta changes, so does gamma. If you were to look at a graph of gamma versus the strike prices of the options, it would look like a hill, the top of which is very near the ATM strike. Gamma is highest for ATM options, and is progressively lower as options are ITM and OTM. This means that the delta of ATM options changes the most when the stock price moves up or down. Let's look at a deep ITM call option (delta near 1.00), an ATM call option (delta near .50), and an OTM call option (delta near .10). If the stock rises, the value of the ITM call will increase the most because it acts most like stock. Even though the ITM call has positive gamma, its delta really doesn't get much closer to 1.00 than before the stock rose. The value of the OTM call will also increase, and its delta will probably increase as well, but it will still be a long way from 1.00. The value of the ATM option increases, and its delta changes the most. That is, its delta is moving closer to 1.00 much quicker than the delta of the OTM call. Practically speaking, the ATM call can provide a good balance of potential profit if the stock rises versus loss if the stock falls. The OTM call will not make as much money if the stock rises, and the ITM will lose more money if the stock falls. Judging how gamma changes as time passes and volatility changes depends on whether the option is ITM, ATM or OTM. Time passing or a decrease in volatility acts as if it's "pulling up" the top of the hill on the graph of gamma,

and making the slope away from the top steeper. What happens is that the ATM gamma increases, but the ITM and OTM gamma decreases. The gamma of ATM options is higher when either volatility is lower or there are fewer days to expiration. But if an option is sufficiently OTM or ITM, the gamma is also lower when volatility is lower or there are fewer days to expiration. What this all means to the option trader is that a position with positive gamma is relatively safe, that is, it will generate the deltas that benefit from an up or down move in the stock. But a position with negative gamma can be dangerous. It will generate deltas that will hurt you in an up or down move in the stock. But all positions that have negative gamma are not all dangerous. For example, a short straddle and a long ATM butterfly both have negative gamma. But the short straddle presents unlimited risk if the stock price moves up or down. The long ATM butterfly will lose money if the stock price moves up or down, but the losses are limited to the total cost of the butterfly. Gamma is a good reason to look at a profit/loss graph of your position over a wide range of possible stock prices. The thinkorswim Analysis page will help you see how risky a negative gamma position might be. Top

Theta Theta, a.k.a. time decay, is an estimate of how much the theoretical value of an option decreases when 1 day passes and there is no move in either the stock price or volatility. Theta is used to estimate how much an option's extrinsic value is whittled away by the always-constant passage of time. The theta for a call and put at the same strike price and the same expiration month are not equal. Without going into detail, the difference in theta between calls and puts depends on the cost of carry for the underlying stock. When the cost of carry for the stock is positive (i.e. dividend yield is less than the interest rate) theta for the call is higher than the put. When the cost of carry for the stock is negative (i.e. dividend yield is greater than the interest rate) theta for the call is lower than the put. Long calls and long puts always have negative theta. Short calls and short puts always have positive theta. Stock has zero theta – its value is not eroded by time. All other things being equal, an option with more days to expiration will have more extrinsic value than an option with fewer days to expiration. The difference between the extrinsic value of the option with more days to expiration and the option with fewer days to expiration is due to theta. Therefore, it makes sense that long options have negative theta and short options have positive theta. If options are continuously losing their extrinsic value, a long option position will lose money because of theta, while a short option position will make money because of theta. But theta doesn't reduce an option's value in an even rate. Theta has much more impact on an option with fewer days to expiration than an option with more days to expiration. For example, the XYZ Oct 75 put is worth $3.00, has 20 days until expiration and has a theta of -.15. The XYZ Dec 75 put is worth $4.75, has 80 days until expiration and has a theta of -.03. If one day passes, and the price of XYZ stock doesn't change, and there is no change in the implied volatility of either option, the value of the XYZ Oct 75 put will drop by $0.15 to $2.85, and the value of the XYZ Dec 75 put will drop by $0.03 to $4.72. Theta is highest for ATM options, and is progressively lower as options are ITM and OTM. This makes sense because ATM options have the highest extrinsic value, so they have more extrinsic value to lose over time than an ITM or OTM option. The theta of options is higher when either volatility is lower or there are fewer days to expiration. If you think about gamma in relation to theta, a position of long options that has the highest positive gamma also has the highest negative theta. There is a trade-off between gamma and theta. Think of long gamma as the stuff that provides the power to a position to make money if the stock price starts to move big (think of a long straddle). But theta is the price you pay for all that power. The longer the stock price does not move big, the more theta will hurt your position.

Position theta measures how much the value of a position changes when one day passes. Position theta is calculated much in the same way as position delta, but instead of using the number of shares of stock per option contract, theta uses the dollar value of 1 point for the option contract. (The dollar value of 1 point in an option contract for U.S. equities is usually $100, but can be different due to stock splits.) thinkorswim takes the theta of each option in your position, multiplies it by the number of contracts and the value of 1 point for the option contract, then adds them together. Top

Vega Vega (the only greek that isn't represented by a real Greek letter) is an estimate of how much the theoretical value of an option changes when volatility changes 1.00%. Higher volatility means higher option prices. The reason for this is that higher volatility means a greater price swings in the stock price, which translates into a greater likelihood for an option to make money by expiration. Long calls and long puts both always have positive vega. Short calls and short puts both always have negative vega. Stock has zero vega – it's value is not affected by volatility. Positive vega means that the value of an option position increases when volatility increases, and decreases when volatility decreases. Negative vega means that the value of an option position decreases when volatility increases, and increases when volatility decreases. Let's look at the XYZ Aug 50 call again. It has a value of $2.00 and a vega of +.20 with the volatility of XYZ stock at 30.00%. If the volatility of XYZ rises to 31.00%, the value of the XYZ Aug 50 call will rise to $2.20. If the volatility of XYZ falls to 29.00%, the value of the XYZ Aug 50 call will drop to $1.80. Vega is highest for ATM options, and is progressively lower as options are ITM and OTM. This means that the value of ATM options changes the most when the volatility changes. The vega of ATM options is higher when either volatility is higher or there are more days to expiration. Position vega measures how much the value of a position changes when volatility changes 1.00%. Position vega is calculated much in the same way as position theta. thinkorswim takes the vega of each option in your position, multiplies it by the number of contracts and the dollar value of 1 point for the option contract, then adds them together. Top

Rho Rho is an estimate of how much the theoretical value of an option changes when interest rates move 1.00%. The rho for a call and put at the same strike price and the same expiration month are not equal. Rho is one of the least used greeks. When interest rates in an economy are relatively stable, the chance that the value of an option position will change dramatically because of a drop or rise in interest rates is pretty low. Nevertheless, we'll describe it here for your edification. Long calls and short puts have positive rho. Short calls and long puts have negative rho. How does this happen? The cost to hold a stock position is built into the value of an option. It all has to do with the idea of an option being a substitute of sorts for a stock position. For example, if you think the stock of XYZ is going to rise, you could buy 100 shares of XYZ for $4800, or you could buy 2 of the XYZ Aug 50 calls for $400. (2 XYZ Aug 50 calls would give me a position delta of +90 — pretty close to the XYZ stock position delta of +100.) As you can see, you would have to spend about 12X the amount spent on the options that you would spend on the stock. That means that you

would have to borrow money or take cash out of an interest-bearing account to buy the stock. That interest cost is built into the call option's value. The more expensive it is to hold a stock position, the more expensive the call option. An increase in interest rates increases the value of calls and decreases the value of puts. A decrease in interest rates decreases the value of calls and increases the value of puts. Back to the XYZ Aug 50 calls. They have a value of $2.00 and a rho of +.02 with XYZ at $48.00 and interest rates at 5.00%. If interest rates increase to 6.00%, the value of the XYZ Aug 50 calls would increase to $2.02. If interest rates decrease to 4.00%, the value of the XYZ Aug 50 calls would decrease to $1.98. Top