Chapter 9
Binary Events
To this point, this book has highlighted unpredictable implied volatility (IV) expansions and their impact on short premium portfolios. However, traders can expect acertain class of IV expansions and contractions with near certainty. These expected volatility dynamics are the result of
binary events
. Abinary event is aknown
upcoming event affecting aspecific asset (or group of assets) that is
anticipated
to create alarge price move. Though price variance is
expected
to increase, it may or may not actually do so depending on the outcome of the binary event.
1
Some examples of binary events include company earnings reports (motivating earnings trades), new product announcements, oil market reports, elections, and Federal Reserve announcements pertaining to the broader market.
Because the date of the anticipated price swing is known, there is typically significant demand for contracts expiring on or after the binary event for that underlying asset. This increased demand results in an increase in the asset's IV, which usually contracts back to nonevent levels immediately after the outcome is known. This trend is shown in
Figure 9.1
.
The impact from abinary event volatility expansion differs from that of unexpected periods of market volatility because the options approaching binary events are priced to reflect the expectation of large moves in the underlying. However, the high credits and immediate volatility contractions that often result from binary events do not necessarily translate into higher (or even likely) profits for short premium positions. This is because the
magnitude
of the price move following the outcome of the binary event is unpredictable, and it may meet or diverge from expectations. On average, the market response to abinary event tends to be quite large, causing the short options strategies that capitalize on these conditions to be
highly
volatile and not necessarily profitable in the long run. This phenomenon also follows from the efficient market hypothesis (EMH), as the well‐understood nature of binary events challenges any consistent edge for these types of strategies.
There is no strong evidence that buying or selling premium around binary events provides aconsistent edge with respect to probability of profit (POP) or average profit and loss (P/L) because alot of the IV overstatement edge is lost in the large moves following abinary event. However, binary event trades are avery time‐efficient use of capital because volatility contractions happen more rapidly and predictably than in more regular market conditions. Binary event trades may also be attractive to risk‐tolerant traders as asource of market engagement. During earnings season, asingle week may present up to 20 high‐risk/high‐reward opportunities for earnings trades. Binary event trades can also be educational for new traders wanting to learn how to adjust positions in rapidly changing, high volatility conditions outside of sell‐offs. These types of trades, as they take place under unique circumstances, are structured and managed differently than typical core or supplemental positions.
Figure 9.1
IV indexes for different stocks from 2017–2020. Assets include (a) AMZN (Amazon stock) and (b) AAPL (Apple stock).
Option Strategies for Binary Events
Because binary event trades are highly volatile and have no strong evidence of along‐term statistical edge, they should only occupy spare portfolio capital and their position size should be kept
exceptionally
small. For example, if atrader'susual position size for an AAPL strangle is afive‐lot (five calls and five puts, each written for 100 shares of stock), then an AAPL earnings strangle may comprise aone‐ or two‐lot. Additionally, underlyings for binary event trades are typically stocks, with quarterly earnings reports being the most common type of binary event. Binary event trades take place over much shorter timescales than more typical trades and must be carefully monitored. Earnings trades, for example, are typically opened the day before earnings and closed the day following earnings. This strategy limits downside risk and capitalizes on the majority of the volatility contraction, which tends to occur immediately after the binary event.
The long‐term success of binary event trades is difficult to verify because there are relatively few occurrences, resulting in high statistical uncertainty. AAPL, for example, has only reported earnings roughly 100 times since the mid 1990s. The Federal Reserve holds press conferences just eight times per year, and large‐scale elections take place once every two or four years. For trading strategies not built around earnings, there are thousands of data points and the statistics are more representative of long‐term expectations (the central limit theorem at work). Therefore, working with this small number of data points can yield an
idea
of how binary events trades have performed in the past, but they should be taken with a

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re are thousands of data points and the statistics are more representative of long‐term expectations (the central limit theorem at work). Therefore, working with this small number of data points can yield an
idea
of how binary events trades have performed in the past, but they should be taken with alarge grain of salt.
Tables 9.1
–
9.3
demonstrate how earnings trades for three different tech companies have performed over 15 years.
There is clearly significant variability in strategy performance for these three different underlyings. To reiterate, high statistical uncertainty makes it difficult to make definitive conclusions about the success of earnings trades, but some consistent trends are observable. Earnings trades are highly sensitive to changes in time. This is evidenced by the significant differences in the per‐trade statistics further from the earnings announcement and demonstrates why binary event trades must be closely monitored. The
majority
of earnings trades are usually profitable, but do not necessarily average aprofit in the long term because of the high per‐trade standard deviation. Per‐trade variance and tail exposure also tend to increase the longer the trade is held, indicating why these types of trades should be relatively short term. This is why generally, binary event trades, such as earning trades, are closed the day following the binary event.
Table 9.1
Statistics for 45 days to expiration (DTE) 16
AAPL strangles from 2005–2020. Trades are opened the day before an earnings report and closed either one, five, 10, or 20 days after earnings.
AAPL Strangle Statistics (2005–2020)
Day Position Is Closed Relative to Earnings
POP
Average P/L
Standard Deviation of P/L
Conditional Value at Risk (CVaR) (5%)
Day After
72%
$85
$203
–$405
5 Days After
70%
$43
$400
–$1,027
10 Days After
61%
$60
$408
–$1,025
20 Days After
56%
−$34
$660
–$1,976
Table 9.2
Statistics for 45 DTE 16
AMZN strangles from 2005–2020. Trades are opened the day before an earnings report and closed either one, five, 10, or 20 days after earnings.
AMZN Strangle Statistics (2005–2020)
Day Position Is Closed Relative to Earnings
POP
Average P/L
Standard Deviation of P/L
CVaR (5%)
Day After
65%
$99
$803
–$1,927
5 Days After
65%
$85
$842
–$2,154
10 Days After
72%
$1
$1,446
–$4,416
20 Days After
76%
$78
$1,540
–$4,477
Table 9.3
Statistics for 45 DTE 16
GOOGL strangles from 2005–2020. Trades are opened the day before an earnings report and closed either one, five, 10, or 20 days after earnings.
GOOGL Strangle Statistics (2005–2020)
Day Position Is Closed Relative to Earnings
POP
Average P/L
Standard Deviation of P/L
CVaR (5%)
Day After
75%
–$60
$1,320
–$4,639
5 Days After
67%
–$113
$1,358
–$4,724
10 Days After
65%
–$122
$1,275
–$3,675
20 Days After
71%
–$2
$1,584
–$4,909
Takeaways
Abinary event is aknown upcoming event affecting aspecific asset (or group of assets) that is anticipated to create alarge price move. This anticipation creates demand for options contracts expiring on or after the binary event and an increase in the IV of the asset. IV typically contracts back to nonevent levels immediately after the outcome is known.
The high credits and immediate volatility contractions resulting from binary events do not necessarily translate to large or consistent short premium profits because the magnitude of the market response is unpredictable. Binary events trades are generally highly volatile and undependable sources of profit but can be used for market engagement or an educational experience for new traders.
Binary event trades should only occupy spare portfolio capital and their position size should be kept
exceptionally
small. Binary event trades should also take place over much shorter timescales than more typical trades, and they must be carefully monitored.
Note
1
The term binary is used to describe systems that can exist in one of two possible states (on/off, yes/no). In this context, abinary event is atype of event where price changes either remain within expectations or exceed expectations.


Chapter 10
Conclusion and Key Takeaways
Successful traders do not rely on luck. Rather, the long‐term success of traders depends on their ability to obtain aconsistent, statistical edge from the tools, strategies, and information available. This book introduces the core concepts of options trading and teaches new traders how to capitalize on the versatility and capital efficiency of options in apersonalized and responsible way. Options are fairly complicated instruments, but this book aims to lessen the learning curve by focusing on the most essential aspects of applied options trading. The detailed framework laid out in this book can be summarized succinctly in the following key takeaways:
Implied volatility (IV) is aproxy for the sentiment of market risk derived from supply and demand for financial insurance. When options prices increase, IV increases; when options prices decrease, IV decreases. IV gives the perceived magnitude of future movements and is not directional. It can also be used to approximate the one
standard deviation expected price range of an asset (although this does not take strike skew into account). The CBOE Volatility Index (VIX) is meant to track the IV for the S&P 500 and is used as aproxy for the perceived risk of the broader market. The VIX, like all volatility signals, is assumed to revert down following significant expansions, which indicates some statistical validity in making downward directional assumptions about volatility once it is inflated.
Compared to long premium strategies, short premium strategies yield more consistent profits and have the long‐term statistical advantage. The trade‐off for receiving consistent profits is exposure to large and sometimes undefined losses, which is why the two most important goals of ashort premium trader are to profit consistently enough to cover moderate, more likely losses and to construct aportfolio that can survive unlikely extreme losses.
The profitability of short options strategies depends on having alarge number of occurrences to reach positive statistical averages, aconsequence of the law of large numbers and the central limit theorem. At minimum, approximately 200 occurrences are needed for the average profit and loss (P/L) of astrategy to converge to long‐term profit targets and more is better.
Extreme losses for short premium positions are highly unlikely but typically happen when price swings in the underlying are large while the expected move range is tight (low IV). Because large price movements in low IV are rare and difficult to reliably model, the most effective way to reduce this exposure is to trade short premium once IV is elevated.
Although high volatility environments are ideal for short premium positions, short premium positions have high probability of profits (POPs) and some statistical edge in all volatility environments. Additionally, because volatility is low the majority of the time, trading short options strategies in
all
IV environments allows traders to profit more consistently and increases the number of occurrences. To manage exposure to outlier risk when adopting this strategy, it is essential to maintain small position sizes and limit the amount of capital allocated to short premium positions, especially when IV is
low. This strategy can be further improved by scaling the amount of capital allocated to short premium according to the current market conditions.
VIX Range
Maximum Portfolio Allocation
0–15
25%
15–20
30%
20–30
35%
30–40
40%
40+
50%
Buying power reduction (BPR) is the amount of portfolio capital required to place and maintain an option trade. The BPR for long options is merely the cost of the contract, and the BPR for short options is meant to encompass at least 95% of the potential losses for exchange‐traded fund (ETF) underlyings and 90% of the potential losses for stock underlyings. BPR is used to evaluate short premium risk on atrade‐by‐trade basis in two ways: BPR is afairly reliable metric for the worst‐case loss of an undefined risk position, and BPR is used to determine if aposition is appropriate for aportfolio based on its buying power. Adefined risk trade should not occupy more than 5% of portfolio buying power and an undefined risk trade should not occupy more than 7%, with exceptions allowed for small accounts. The formulae for BPR are complicated and specific to the type of strategy, but the BPR for short strangles is approximately 20% of the price of the underlying. BPR can be used to compare the risk for variations of the same strategy (e.g., strangle on underlying Aversus strangle on underlying B), but it cannot be used to compare risk for strategies with different risk profiles (e.g., strangle on underlying Aversus iron condor on underlying A).
Traders trade according to their personal profit goals, risk tolerances, and market beliefs, but it is generally good practice to be aware of the following:
Only trade underlyings with liquid options markets to minimize illiquidity

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(e.g., strangle on underlying Aversus iron condor on underlying A).
Traders trade according to their personal profit goals, risk tolerances, and market beliefs, but it is generally good practice to be aware of the following:
Only trade underlyings with liquid options markets to minimize illiquidity risk.
The choice of underlying is somewhat arbitrary, but it'simportant to select an underlying with an appropriate level of risk. Stock underlyings tend to be higher‐risk, higher‐reward than ETF underlyings. This means stock underlyings present high IV opportunities more frequently, but they have more tail loss exposure and are more expensive to trade.
Choose acontract duration that is an efficient use of buying power, allows for consistency, offers areasonable number of occurrences, has manageable P/Lswings throughout the duration, and has moderate ending P/Lvariability. Durations between 30 and 60 days are suitable for most traders.
Compared to defined risk trades, undefined risk trades have higher POPs, higher profit potentials, unlimited downside risk, and higher BPRs. High‐POP defined risk trades, such as wide iron condors, have comparable risk profiles to undefined risk trades while also offering protection from extreme losses. Such trades can be better suited for low IV conditions compared to undefined risk trades and are allowed to occupy undefined risk portfolio capital.
Contracts with higher deltas are higher-risk, higher-reward than contracts with lower deltas. When trading premium, consider contracts between 10Δ and 40Δ, which is large enough to achieve areasonable amount of growth but small enough to have manageable P/Lswings and ending P/Lvariability.
When choosing amanagement strategy, the primary factors to consider are convenience and consistency, capital allocation preferences, desired number of occurrences, per‐trade average P/L, and per‐trade exposure. Early‐managed positions have lower per‐trade P/Ls but less tail risk than positions held to expiration. Because managing early also accommodates more occurrences and averages more P/Lper day, closing positions prior to expiration and redeploying capital to new positions is generally amore efficient use of capital compared to extracting more value from an existing position.
If managing according to days to expiration (DTE), consider closing trades around the contract duration midpoint to achieve adecent amount of long‐term profit and justify the tail loss exposure.
If managing an undefined position according to aprofit target, choosing atarget between 50% and 75% of the initial credit allows for reasonable profits while also reducing the potential magnitude of outlier losses. Choosing aprofit target that is too low reduces average P/L, and choosing aprofit target that is too high does little to mitigate outlier risk. Profit targets for defined risk positions can be lower because they are generally less volatile.
If combining strategies, managing undefined risk contracts at either 50% of the initial credit or halfway through the contract duration generally achieves reasonable, consistent profits and moderates outlier risk.
If implementing astop loss, amid‐range stop loss threshold of at least −200% of the initial credit is practical. If the stop loss is too small (−50% for example), losses are more likely since options have significant P/Lvariance, although they often recover. It'salso important to note that stop losses do not guarantee amaximum loss in cases of rapid price movements, so stop losses are typically paired with another management strategy unless trading passively. Stop losses are generally not suitable for defined risk strategies.
Maintaining the capital allocation guidelines is crucial for limiting tail exposure and achieving areasonable amount of long‐term growth:
The amount of portfolio buying power allotted to short premium positions, such as short strangles and short iron condors, should range from 25% to 50% depending on the current market volatility, with the remaining capital either kept in cash or alow‐risk passive investment. [refer to Takeaway 5].
Of the amount allocated to short premium, at least 75% should be reserved for undefined risk trades (with less than 7% of portfolio buying power allocated to asingle position) and at most 25% reserved for defined risk strategies (with less than 5% of portfolio buying power allocated to asingle position) [refer to Takeaway 6].
Generally speaking, at most 25% of the capital allocated to short premium should go toward supplemental positions, or higher-risk, higher-reward trades that are tools for market engagement. The remainder should go toward core positions or trades with high POPs and moderate P/Lvariation that offer consistent profits and reasonable outlier exposure.
Diversifying the underlyings of an options portfolio (i.e., trading acollection of assets with low correlations) is one of the most essential diversification tools for portfolio risk management, particularly outl


nd moderate P/Lvariation that offer consistent profits and reasonable outlier exposure.
Diversifying the underlyings of an options portfolio (i.e., trading acollection of assets with low correlations) is one of the most essential diversification tools for portfolio risk management, particularly outlier risk management. Strategy diversification and duration diversification, though not as essential as underlying diversification, are other straightforward risk management techniques.
The Greeks form aset of risk measures that quantify different dimensions of exposure for options. Each contract has its own specific set of Greeks, but some Greeks are additive across positions with different underlyings. Consequently, these metrics can be used to summarize the various sources of risk for aportfolio and guide adjustments. Two key Greeks are beta‐weighted delta (
) and theta (
). Beta‐weighted delta represents the amount of directional exposure aposition has relative to some index rather than the underlying itself. Theta represents the expected decrease in an option'svalue per day.
neutral portfolios are attractive to investors because they are relatively insensitive to directional moves in the market and profit from changes in IV and time.
Because short‐premium traders consistently profit from time decay, the total theta across positions gives areliable estimate for the expected daily growth of ashort options portfolio. The minimum theta ratio (
) for an options portfolio should range from 0.05% to 0.1% and should not exceed 0.2% because this indicates excessive risk. If aportfolio is not meeting these theta ratio guidelines, then the positions should be adjusted as follows:
If aproperly allocated, well‐diversified portfolio is
neutral but does not provide asufficient amount of theta, then the positions in the portfolio should be reevaluated. In this case, perhaps some defined risk trades should be replaced with undefined risk trades or undefined risk positions be rolled to higher deltas. New delta neutral positions can also be added, such as strangles and iron condors, for example. IV and theta are also highly correlated, meaning that higher IV underlyings could also be considered if theta is too low. These measures can be reversed if the portfolio has too much theta exposure while being
neutral.
If the theta ratio is too low (<0.1%) and the portfolio is not
neutral, then either existing positions should be re‐centered or tightened or new short premium positions should be added.
If the
is too large and positive (bullish), then add new negative
positions (e.g., add short calls on positive beta underlyings or add short puts on negative beta underlyings).
If the
is too large and negative (bearish), then add new positive
positions (e.g., add short puts on positive beta underlyings).
If the theta ratio is too large (>0.2%) and the portfolio is not
neutral, then either existing positions should be re‐centered or widened or short premium positions should be removed.
If the
is too large and positive (bullish), then remove positive
positions (e.g., remove short puts on positive beta underlyings).
If the
is too large and negative (bearish), then remove negative
positions (e.g., remove short calls on positive beta underlyings).
If aproperly allocated, well‐diversified portfolio provides asufficient amount of theta but is not βΔ neutral, then existing positions should be reevaluated. For example, skewed positions could be closed and re‐centered or replaced with new delta-neutral positions that offer comparable amounts of theta.
Binary event trades, such as trades around quarterly earnings reports, should be traded cautiously, only occupy spare portfolio capital, and their position size should be kept exceptionally small. Binary event trades must be carefully monitored and typically take place over much shorter timescales than more typical trades. They are often opened the day before abinary event and closed the day after.
Options trading is not for everyone. However, for traders who are prepared to understand the complex risk profiles of options, comfortable accepting acertain level of exposure, and willing to commit the time to active trading, short premium strategies can offer aprobabilistic edge and the potential to profit in any type of market. There is no “right” way to trade these instruments; all traders have unique profit goals and risk tolerances. It is our hope that this book will guide traders to make informed decisions that best align with their personal objectives.


Appendix
I. The Logarithm, Log‐Normal Distribution, and Geometric Brownian Motion,
with contributions from Jacob Perlman
For the following section, let
be the initial value of some asset or collection of assets and
the value at time
. Given the goals of investing, the most obvious statistic to evaluate an investment or portfolio is the profit or loss:
. However, according to the efficient market hypothesis (EMH), assets should be judged relative to their initial size, represented using returns,
.
The returns of the asset from time 0 to time
can also be written in terms of each individual return over that time frame. More specifically, for an integer
, if
then the returns,
, can be split into atelescoping
1
product.
(A.1)
The EMH states that each term in this product should be independent and similarly distributed. The central limit theorem, and many other powerful tools in probability theory, concern long
sums
of independent random variables. To apply these tools to this telescoping product of random variables, it first must be converted into asum of random variables. Logarithms offer aconvenient way to accomplish this.
Logarithmic functions are aclass of functions with wide applications in science and mathematics. Though there are several equivalent definitions, the simplest is as the inverse of exponentiation. If
and
are positive numbers, and
, then
(read as “the log base
of
”) is the number such that
. For example,
can be equivalently written as
.
The choice of base is largely arbitrary, only affecting the logarithm by aconstant multiple. If
is another possible base, then
. In mathematics, the most common choice is Euler'sconstant, aspecial number:
. Using this constant as abase results in the
natural logarithm
, denoted
. The justification for this choice largely comes down to notational convenience, such as when taking derivatives:
. In this example, as
, using
avoids the accumulation of cumbersome and not particularly meaningful constant factors.
As
, logarithms have the useful property
2
given by:
(A.2)
This property transforms the telescoping product given above into asum of small independent pieces, given by the following equation:
(A.3)
The central limit theorem states that if arandom variable is made by adding together many independently random pieces, then the result will be normally distributed. One can, therefore, conclude that log returns are normally distributed. Observe the following:
(A.4)
This suggests that stock prices follow alog‐normal distribution or adistribution where the logarithm of arandom variable is normally distributed. Within the context of Black‐Scholes, this implies that stock log‐returns evolve as Brownian motion (normally distributed), and stock prices evolve as geometric Brownian motion (log‐normally distributed). The log‐normal distribution is more appropriate to describe stock prices because the log‐normal distribution cannot have negative values and is skewed according to the volatility of price, as shown in the comparisons in
Figure A.1
.
II. Expected Range, Strike Skew, and the Volatility Smile
The majority of this book refers to expected range approximated with the following equation:
(A.5)
For astock trading at current price
with volatility
and risk‐free rate
, the Black‐Scholes theoretical
price range at afuture time
for this asset is given by the following equation:
(A.6)
The equation in (
A.5
) is avalid approximation of this formula when
is small, which follows from the mathematical relation
. Generally speaking, (
A.5
) is avery rough approximation for expected range, and it becomes less accurate in high volatility conditions, when
is larger.
Though (
A.5
) still yields areasonable, back‐of‐the‐envelope estimate for expected range, the one standard deviation expected move range is calculated on most trading platforms according to the following:
(A.7)
Figure A.1
Comparison of the log‐normal distribution (a) and the normal distribution (b). The mean and standard deviation of the normal distribution are the exponentiated parameters of the log‐normal distribution.
According to the EMH, this is simply the expected future price displacement, i.e., price of at‐the‐money (ATM) straddle, with additional terms (prices of near ATM strangles) to counterbalance the heavy tails pulling the expected value beyond the central 68%. To see how this formula compares with the (
A.5
) approximation, consider the statistics in
Table A.1
.
Table A.1
Expected 30‐day price range approximations for an underlying with aprice of $100 and implied volatility (IV) of 20%. According to the Black‐Scholes model, the per‐share prices for the 30‐day options are $4.58 for the straddle, $3.64 for the strangle one strike from ATM, and $2.85 for the strangle two strikes from ATM.
30‐Day Expected Price Range Comparison
Equation (A.5)
Equation (A.7)
$5.73
$4.13
Compared to
Equation (A.5)
,
Equation (A.7)
is amore attractive way to calculate expected range on trading platforms because it is computation

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rangle one strike from ATM, and $2.85 for the strangle two strikes from ATM.
30‐Day Expected Price Range Comparison
Equation (A.5)
Equation (A.7)
$5.73
$4.13
Compared to
Equation (A.5)
,
Equation (A.7)
is amore attractive way to calculate expected range on trading platforms because it is computationally simpler and independent of arigid mathematical model. However, neither of these expected range calculations take
skew
into account.
When comparing contracts across the options chain, an interesting phenomenon commonly observed is the
volatility smile
. According to the Black‐Scholes model, options with the same underlying and duration should have the same implied volatility, regardless of strike price (as volatility is aproperty of the underlying). However, because the market values each contract differently and implied volatility is derived from from options prices, the implied volatilities across strikes often vary. Avolatility smile appears when the implied volatility is lowest for contracts near ATM and increases as the strikes move further out‐of‐the‐money (OTM). Similarly, avolatility smirk (also known as volatility skew) is aweighted volatility smile, where the options with lower strikes tend to have higher IV than options with higher strikes. The opposite of the volatility smirk is described as forward skew, which is relatively rare, having occurred, for example, with GME in early 2021. For an example of volatility skew, consider the SPY 30 days to expiration (DTE) OTM option data shown in
Figure A.2
.
Figure A.2
Volatility curve for OTM 30 DTE SPY calls and puts, collected on November 15, 2021, after the close.
The volatility curve in
Figure A.2
is clearly asymmetric around the ATM strike, with the options with lower strikes (OTM puts) having higher IVs than options with higher strikes (OTM calls). This type of curve is useful for analyzing the perceived value of OTM contracts. Compared to ATM volatility, OTM puts are generally overvalued while OTM calls are generally undervalued until very far OTM (near $510). This suggests that traders are willing to pay ahigher premium to protect against downside risk compared to upside risk.
This is an example of put skew, aconsequence of put contracts further from ATM being perceived as equivalently risky as call contracts closer to ATM.
Table A.2
reproduces data from
Chapter 5
.
Table A.2
Data for 16
SPY strangles with different durations from April 20, 2021. The first row is the distance between the strike for a 16
put and the price of the underlying for different DTEs (i.e., if the price of the underlying is $100 and the strike for a 16
put is $95, then the put distance is [$100 – $95]/$100 = 5%). The second row is the distance between the strike for a 16
call and the price of the underlying for different contract durations.
16
SPY Option Distance from ATM
Option Type
15 DTE
30 DTE
45 DTE
Put Distance
3.9%
6.5%
8.0%
Call Distance
2.4%
3.9%
4.9%
This skew results from market fear to the
downside
, meaning the market fears larger extreme moves to the downside more than extreme moves to the upside. According to the EMH, the skew has already been priced into the current value of the underlying. Hence, the put skew implies that the market views large moves to the downside as more likely than large moves to the upside but small moves to the upside as being the most likely outcome overall. For agiven duration, the strikes for the 16
puts and calls approximately correspond to the one standard deviation expected range of that asset over that time frame. For example, since SPY was trading at approximately $413 on April 20, 2021, the 30‐day expected price move to the upside was $16 and the expected price move to the downside was $27 according to the 16
options.
III. Conditional Probability
Conditional probability is mentioned briefly in this book, but it is an interesting concept in probability theory worthy of ashort discussion. Conditional probability is the probability that an event will occur, given that another event occurred. Consider the following examples:
Given that the ground is wet, what is the probability that it rained?
Given that the last roll of afair die was six, what is the probability that the next roll will also be asix?
Given that SPY had an up day yesterday, what is the probability it will have an up day tomorrow?
Analyzing probabilities conditionally looks at the likelihood of agiven outcome within the context of known information. For events
and
the conditional probability
(read as the probability of
, given
) is calculated as follows:
(A.8)
where
is the probability that event
occurs and
is the probability that
and
occur. For example, suppose
is the event that it rains on any given day and
(20% chance of rain). Suppose
is the event that there is atornado on any given day, there is a 1% chance of atornado occurring on any given day, and tornados never happen without rain, meaning that
. Therefore, given that it is arainy day, we have the following


ent that it rains on any given day and
(20% chance of rain). Suppose
is the event that there is atornado on any given day, there is a 1% chance of atornado occurring on any given day, and tornados never happen without rain, meaning that
. Therefore, given that it is arainy day, we have the following probability that atornado will appear:
In other words, atornado is five times more likely to appear if it is raining than under regular circumstances.
IV. The Kelly Criterion,
derivation courtesy of Jacob Perlman
The Kelly Criterion is aconcept from information theory and was originally created to analyze signal transmission through noisy communication channels. It can be used to determine the optimal theoretical bet size for arepeated game, presuming the odds and payouts are known. The Kelly bet size is the fraction of the bankroll that maximizes the expected long‐term growth rate of the game, more specifically the logarithm of wealth. For agame with probability
of winning
and aprobability
of losing 1 (the full wager), the Kelly bet size is given as follows:
(A.9)
This is the theoretically optimal fraction of the bankroll to maximize the expected growth rate of the game. Abrief justification for this formula follows from the paper listed in Reference 4.
Consider agame with probability
of winning
and aprobability
of losing the full wager. If aplayer has
in starting wealth and bets afraction of that wealth,
, on this game, the player'sgoal is to choose avalue of
that maximizes their wealth growth after
bets.
If the player has
wins and
losses in the
plays of this game, then:
Over many bets of this game, the log‐growth rate is then given by the following:
following from the law of large numbers
The bet size that maximizes the long‐term growth rate corresponds to
.
The Kelly Criterion can also be applied to asset management to determine the theoretically optimal allocation percentage for atrade with known (or approximated) probability of profit (POP) and edge. More specifically, for an option with agiven duration and POP, the optimal fraction of the bankroll to allocate to this trade is approximately:
(A.10)
where
is the risk‐free rate and
is the duration of the trade in years. The derivation for this equation is outlined as follows:
For agame with probability
of winning
and aprobability
of losing 1 unit, the expected change in bankroll after one play is given by
.
For an investment of time
with the risk‐free rate given by
, the expected change in value is estimated by
, derived from the future value of the game with continuous compounding. Assuming that
is small, then
.
For the bet to be fairly priced, the change in the bankroll should also equal
. Therefore, if
, the odds for this trade can be estimated as
.
Using this value for
in the Kelly Criterion formula, one arrives at the following:
This then yields the approximate optimal proportion of bankroll to allocate to agiven trade, substituting
for
and POP for
.
Notes
1
So called because adjacent numerators and denominators cancel, allowing the long product to be collapsed like atelescope.
2
Stated abstractly, logarithms are the group homomorphisms between
and
.


Glossary of Common Tickers, Acronyms, Variables, and Math Equations
Ticker
Full Name
SPY
SPDR S&P 500
XLE
Energy Select Sector SPDR Fund
GLD
SPDR Gold Trust
QQQ
Invesco QQQ ETF (NASDAQ‐100)
TLT
iShares 20+ Year Treasury Bond ETF
SLV
iShares Silver Trust
FXE
Euro Currency ETF
XLU
Utilities ETF
AAPL
Apple Stock
GOOGL
Google Stock
IBM
IBM Stock
AMZN
Amazon Stock
TSLA
Tesla Stock
VIX
CBOE Volatility Index (implied volatility for the S&P 500)
GVZ
CBOE Gold Volatility Index
VXAPL
CBOE Equity VIX On Apple
VXAZN
CBOE Equity VIX On Amazon
VXN
CBOE NASDAQ‐100 Volatility Index
Acronym
Full Name
NYSE
New York Stock Exchange
ETF
Exchange‐Traded Fund
DTE
Days to Expiration
EMH
Efficient Market Hypothesis
ITM
In‐the‐Money
OTM
Out‐of‐the‐Money
ATM
At‐the‐Money
P/L
Profit and Loss
IV
Implied Volatility
VaR
Value at Risk
CVaR
Conditional Value at Risk
POP
Probability of Profit
BPR
Buying Power Reduction
IVP
IV Percentile
IVR
IV Rank
NFT
Non‐Fungible Tokens
Variable Symbol
Variable Name/Definition
Spot/stock price: the price of the underlying
Contract price: the price of the option, noting that
Cis used if the contract is acall and
Pis used in the case of puts
Strike price: the price at which the holder of an option can buy or sell an asset on or before afuture date
Risk‐free rate of return: the theoretical rate of return of ariskless asset
Mean: the central tendency of adistribution
Standard deviation: the spread of adistribution; also used as ameasure of uncertainty or risk
Volatility: the standard deviation of log‐returns for an asset; akey input in options pricing
Delta: the expected change in an option'sprice given a $1 increase in the price of the underlying
Gamma: the expected change in an option'sdelta given a $1 change in the price of the underlying
Theta: the expected time decay of an option'sextrinsic value in dollars per day
Beta: the volatility of the stock relative to that of the overall market
Beta‐weighted delta: the expected change in an option'sprice given a $1 change in some reference index
Equation Number
Equation
1.1
Simple Returns
1.2
Log Returns
1.3
Long Call P/L
1.4
Long Put P/L
1.5
Population Mean
1.6
Expected Value
1.7
Population Variance
1.8
Variance
1.9
Skew
1.15
Delta
1.16
Gamma
1.17
Theta
1.18
Population Covariance
1.19
Covariance
1.20
Correlation Coefficient
1.21
Additive Property of Variance
1.22
Beta
2.1
±1σ Expected Range Approximation (%)
2.2
±1σ Expected Range Approximation ($)
3.1
IV Percentile (IVP)
3.2
IV Rank (IVR)
4.1
Short Put BPR
4.2
Short Call BPR
4.3
Short Strangle BPR
5.1
Short Iron Condor BPR
8.1
Approximate Kelly Allocation Percentage

eBooks\The Unlucky Investor_s Guide to Options Trading\The Unlucky Investor_s Guide to Options Trading.epub#section:b02.xhtml


s,
37
negative covariance,
36
positive covariance,
35
Cumulative horizontal displacement,
25f
Cumulative horizontal displacements,
24
Daily P/Ls, standard deviation,
150f
Daily returns, distribution,
39f
,
40f
Days to expiration (DTE), management usage,
118
–120
Defined risk, selection,
102
–104
Defined risk strategies,
152
maximum loss,
102
BPR, usage,
84
limitation,
59
P/Ltargets, attainment,
153
POP, usage,
110
portfolio allocation,
103t
risk, comparison,
89
selection,
94
short premium allocation,
173
stop losses, unsuitability,
173
undefined risk strategies, comparison,
102t
,
109
avoidance,
110
Delta (Δ),
31
–32,
106t
basis,
112
beta‐weighted delta,
38
,
144
–145,
148
,
174
contract delta,
33
–35,
79
,
111
,
114
implied volatility (IV), equivalence,
87
contract risk,
172
contract usage,
32
directional exposure measurement,
111
drift,
145
level,
114
magnitude,
32
,
114
negative delta/positive delta,
33
neutral position,
33
,
61
,
146
,
156
neutral positions
profit,
111
normalization,
145
perceived risk measure,
112
–113
raw delta, comparisons (impossibility),
146
–147
re‐centering,
114
scaling up,
114
selection,
94
,
111
–115
optimum, factors,
114
sensitivity,
146
sign,
32
value, range,
32
Derivatives, gamma comparison (impossibility),
146
–147
Deterministic price trends,
28
–29
Dice rolls, histogram,
13
,
14f
,
17f
,
19f
,
21f
Directional assumption, selection,
94
,
104
–110
Directional exposure (measurement), delta (usage),
111
Directional risk, degree (measure),
32
Distributions
asymmetry (measure), skew (usage),
65
mean (histogram),
17e
normal distribution,
22f
,
44
–45,
180f
skew,
16
–18,
20
,
39
statistics, understanding,
21
–22
Diversification,
136
–144,
158
effectiveness, understanding,
137
–138
time diversification,
151
tools,
173
Dividends payment, avoidance,
23
Downside risk
amount, preference,
104
limitation, absence,
102
Downside skew,
112
Early‐managed contracts,
126
,
129
occurrences allowance,
80
Early‐managed portfolio, losses,
129
Early‐management strategies,
80
Earnings dates, marking,
54f
Earnings report,
115
dates,
53
,
96
,
102
impact,
43
inclusion,
163
quarterly earnings report (single‐company factors),
52
,
166
,
175
Efficient market hypothesis (EMH),
11
–13,
177
–178,
183
binary events, relationship,
164
forms,
11
–12,
104
–105
interpretation,
104
,
116
Equities
implied volatility indexes,
54f
pricing/bid‐ask spread/volume data,
95t
trading,
137
,
140
European call options,
29
European options, expiration (payoff),
29
European‐style option (price evolution), Black‐Scholes equation (relationship),
23
Events
outcomes,
44
–45
sampling, probability distribution (usage),
72
Exchange‐traded funds (ETFs),
5
–7,
36
,
157
BPR, historical effectiveness,
84
correlations,
157t
diversification,
53
–54,
134
historical risk, approximation,
63
IV overstatement rates,
46
market ETFs,
139
–142,
145
,
157
volatility assets, correlation,
139
skewed returns distribution,
22
stability,
98
underlyings,
95
,
135
,
137
,
172
advantages/disadvantages,
96t
losses,
84
,
171
strangles, usage,
156
usage,
97
volatility profiles, differences,
96
Expected move cones,
44
,
45f
,
60f
Expected move range,
179
Expected price range,
45f
Expected range,
58
,
179
–183
adjustment,
68
–69
calculation,
43
–45,
47
,
179
,
181
estimates,
44
increase,
115
short strike prices, relationship,
111
tightness,
51
underlying price expected range,
60
–61
External events, outlier underlying moves/IV expansions (relationship),
58
Financial derivative, options (comparison),
7
Financial insurance, risk‐reward trade‐off,
47
Gamma (
Γ
),
31
,
33
comparison, impossibility,
146
–147
increase,
79
magnitude,
33
risk,
146
–147
Gaussian distribution (bell curve),
20
Geometric Brownian motion, Black‐Scholes model (relationship),
27
–28
GLD returns, SPY returns (contrast),
36f
Greeks,
31
–35.
See also
Delta
;
Gamma
;
Sigma
;
Theta
assumptions,
35
balance,
148
option Greeks,
38
portfolio Greeks,
160
maintenance,
133
,
144
–147
risk measures,
174
Heteroscedasticity,
26
High implied volatility (high IV)
short premium trading,
75
trading,
66
–72,
75
Histogram
daily returns/prices,
27f
dice rolls,
14f
,
17f
,
19f
,
21f
Historical distribution,
73f
Historical P/Ldistribution,
62f
,
64f
,
71f
Historical returns, standard deviation,
28
,
30
Historical tail risk, estimation,
65
Historical volatility,
21
–22,
38
increase,
42
market historical volatility,
69
representation,
43
stock historical volatility,
30
underlying historical volatility,
31
usage,
30
,
63
Historic risk, estimation,
21
–22
Horizontal displacements, distribution,
26f
Hurricane insurance
price, proportion,
47
sellers, strategic room,
47
–48
Idiosyncratic risk,
137
Illiquid asset, example,
94
Illiquidity risk, minimization,
171
Implied volatility (IV),
3
,
38
,
41
,
83
,
169
–170,
181
basis,
48
BPR
comparison,
66
inverse volatility,
87
contract delta, equivalence,
87
contraction,
50
conversion,
66
correlation,
42
–43
decrease/increase,
42
–43,
87
,
89
derivation,
44
–45
differences,
42t
en

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liquidity risk, minimization,
171
Implied volatility (IV),
3
,
38
,
41
,
83
,
169
–170,
181
basis,
48
BPR
comparison,
66
inverse volatility,
87
contract delta, equivalence,
87
contraction,
50
conversion,
66
correlation,
42
–43
decrease/increase,
42
–43,
87
,
89
derivation,
44
–45
differences,
42t
environments, short option strategies (trading),
76
expansion,
58
,
122
–123,
163
indexes,
53f
,
54f
,
165f
indication,
30
IV‐derived price range,
44
–45
long‐term baseline reversion,
52
metric, importance,
30
–31
overstatement,
46
,
46t
,
164
profits,
58
rates,
47
peak, increase,
49
price range forecast,
47
ranges,
76t
realized risk measurement,
46
reversion,
51
–54
signals, capacity,
51
source,
23
SPY annualized implied volatility (tracking),
48
SPY implied volatility,
69f
standard deviation range,
43
tracking,
48
trading (volatility trading concept),
58
underlying IV,
60
,
86
,
88f
usage,
41
,
134
Implied volatility percentile (IVP),
66
–68
Implied volatility rank (IVR),
68
Increments, distribution,
26f
Insider trading,
12
,
105
In‐the‐money (ITM),
99
contract description,
9
ITM put, price,
10
long calls ITM,
33
movement,
34
–35
options, directional risk,
114
positions,
34
relationship,
32
Iron condors,
105
,
110t
,
151
cap, long wings,
106
,
108
drawdowns, experience,
151
narrow wings, POP (presence),
109
neutral SPY strategies,
151
profit potential,
109
representation,
107f
risk,
110
short iron condor BPR,
108
short iron condors, range,
134
,
173
statistical comparison,
109t
underlying strangle, contrast,
171
wide iron condors,
116
,
172
wide wings, inclusion (trading),
109
wings, inclusion,
106
Kelly Criterion
application,
185
buying power percentage,
153
derivation,
184
–186
formula,
186
heuristic derivation,
160
uncorrelated bets,
154
Law of large numbers,
72
,
170
,
185
Liquidity
importance, understanding,
94
net liquidity,
89t
options liquidity,
94
–95
portfolio net liquidity,
104
,
145
theta ratio/net portfolio liquidity,
145
Log‐normal distribution,
177
comparison,
180f
skew,
179
stock prices, relationship,
179
Log returns
equation,
7
standard deviation,
23
Long call,
32
,
34
addition,
106
directional assumption,
8t
option, price,
32
,
111
P/L,
10
–11
position,
33
profit potential,
90
Long premium
contracts, impact,
84
positions, profit yield (comparison),
12
strategies,
58
,
170
trade,
8
Long put,
32
,
34
addition,
106
directional assumption,
8t
option, price,
111
P/L,
10
–11
position,
33
Long stock,
32
Long strikes,
207f
Loss
incurring, probability,
113t
targets,
122t
,
123
Low‐loss targets, attainment,
122
Management
P/Ltarget, usage,
120
–124
techniques,
123
–126,
152
timeline, usage,
118
–119
Management strategies,
121t
,
122t
,
158
average daily P/Land average duration,
121t
impact/comparison,
79
–80,
80t
,
102
long‐term risks,
126
,
129
performance, scenarios (impact),
126
qualitative comparison,
125t
selection,
172
usage,
117
–118
Management time, selection,
119
,
129
Margin, BPR (contrast),
84
Market
conditions, risk/return expectations,
35
exposure,
98
frictionlessness,
23
historical volatility,
69
implied volatility (IV),
76t
,
152
perceived uncertainty,
46
trader beliefs,
171
–172
uncertainty sentiment, IV tracking,
48
volatility amounts, differences,
89t
Market ETFs,
139
–142,
145
,
157
historic correlations,
141t
,
143t
percentage,
138t
volatility assets, correlation,
139
Market risk
sentiment, IV proxy,
42
sentiment, IV proxy (usage),
169
–170
Maximum per‐trade BPR, limitation,
134
Mean (moment),
14
–15
Middle ground contract duration,
101
Mid‐range stop loss,
123
,
130
,
173
Moments,
14
–22
Near‐the‐money options, gamma (increase),
79
Negative covariance,
36
Non‐dividend‐paying stock, trading,
30
–31
Non‐fungible tokens (NFTs),
5
Normal distribution
comparison,
180f
mean/standard deviation,
180f
plot,
22f
standard deviation range,
44
–45
Occurrences,
62f
,
71f
,
101
compound occurrences, loss potential,
142
concentration,
20
consistency,
117
density,
64
early‐managed contract allowance,
80
final P/L, correspondence,
61
goal,
125
–126,
151
increase,
117
P/Ldistribution,
39
reduction/increase,
72
,
89
,
124
standard deviation range,
64
VIX level, contrast,
67
Occurrences, number,
58
,
72
–76,
99
attainment,
99
compromise, absence,
151
increase,
81
,
101
–102,
118
presence,
15
,
126
,
129
trade‐off,
119
volatility trading concept,
58
Off‐diagonal entries,
141t
Options,
5
–6
buying, profit,
57
–58
capital efficiency, BPR (relationship),
90
demand,
42
–43
fair price (estimation), Black‐Scholes model (usage),
30
,
42
financial derivative, comparison,
7
Greeks,
38
illiquidity, risk,
94
leverage, effects (clarity),
90
liquidity,
94
–95
market, liquidity,
98
P/Lstandard deviation, usage,
73
P/Lstatistics,
97t
price, BPR (inverse correlation),
87
profitability,
10
risk, visualization,
59
–63
traders, assumptions,
11
types,
7
underlyings, sample,
98t
Options trading,
84
,
97
,
102
,
169
,
175
casinos, usage,
1
–2
diversification, importance,
136
ETF underlyings, usage,
97
gamma, awareness (importan


cs,
97t
price, BPR (inverse correlation),
87
profitability,
10
risk, visualization,
59
–63
traders, assumptions,
11
types,
7
underlyings, sample,
98t
Options trading,
84
,
97
,
102
,
169
,
175
casinos, usage,
1
–2
diversification, importance,
136
ETF underlyings, usage,
97
gamma, awareness (importance),
33
implied volatility
metric,
30
–31
reversion,
51
learning curve/math knowledge,
3
option theory, transition,
90
profitability, option pricing (impact),
11
quantitative options trading,
3
retail options trading, assets (suitability),
94
risk management, relationship,
2
usage, market performance,
12
Outlier losses,
142
capital exposure, limitation,
134
probability,
141t
Outlier risk
carrying, avoidance,
103
reduction,
76
Out‐of‐the‐money (OTM)
contract description,
9
positions,
34
volatility curve,
182f
Over‐the‐counter (OTC) options,
7
Passive investment, daily performance statistics,
146t
Passive traders,
125
Perceived risk (measurement), delta (usage),
112
–113
Personal profit goals,
171
–172
Per‐trade allocation percentage,
158t
Per‐trade standard deviation,
158
,
166
Per‐trade statistics, differences,
166
Per‐trade variance,
167
P/Ltargets, attainment,
153
Portfolio
averages, variance,
74f
backtest performance statistics,
159t
concentration excess, avoidance,
77
construction,
12
,
156
–160
cumulative P/L,
152f
delta skew,
145
expected loss, CVaR estimate,
40
Greeks,
149
,
160
maintenance,
133
,
144
–147
net liquidity,
89t
,
104
,
146
passive investment, daily performance statistics,
146t
performance,
159f
comparison,
139f
P/Laverages,
74f
POP‐weighted portfolio,
157
–158
risk management, diversification tools,
173
statistical analysis,
153t
Portfolio allocation,
109
defined/undefined risk strategies,
103t
guidelines, usage,
89
,
104
,
134
percentages,
137
–138,
138t
,
154
,
154t
position sizing, relationship,
75
–79
scaling,
77
,
156
strategies, comparison,
77
usage,
103
volatility trading concept,
58
Portfolio buying power,
83
,
89
,
134
allotment/allocation,
134
–135,
154
–155,
157
defined risk position occupation,
118
expected profit,
146
undefined risk strategy occupation,
110
usage,
99
,
109
,
117
Portfolio capital
allocation
control,
81
guidelines, market IV (impact),
76t
amount, BPR (relationship),
171
diversification,
152
investment,
127f
,
128f
Portfolio management,
3
,
93
,
149
back‐of‐the‐envelope tactics,
133
beta (
β
) metric, importance,
38
capital allocation,
134
–136
capital balancing, POP (usage),
153
–156
concepts,
133
construction,
156
–160
diversification, usage,
136
–144,
149
–153
portfolio Greeks, maintenance,
144
–147
position sizing,
134
–136
simplification,
101
Positional capital allocation, quantitative approach,
153
Positions
core position statistics,
158t
delta drift,
145
delta level,
114
expected loss, CVaR estimate,
40
intrinsic value,
9
ITM, relationship,
32
long side/short side, adoption,
8
management,
118
P/Lcorrelation, reduction,
150
POP‐weighting,
156
profiting, likelihood,
104
sizing
capital allocation, relationship,
134
–136
portfolio allocation, relationship,
75
–79
volatility trading concept,
58
Positive covariance,
35
Premium sellers, profit,
50
Premium, trading,
172
Price dynamics
Black‐Scholes model approximation,
24
Brownian motion, comparison,
25
–26
Price predictability (limitation), EMH implications,
105
Probabilistic system, probability distribution,
14
Probability distribution,
13
–22
asymmetry,
16
events sampling,
72
Gaussian distribution (bell curve),
20
mean (moment),
14
–15
normal distribution,
20
skew (moment),
16
–22
variance (moment),
15
–16
Probability of profit (POP),
89
,
164
,
185
asset weighting,
149
buying power, allocation percentages,
154t
capital, balancing,
153
–156
decrease,
114
dependence,
77
heuristic,
160
IV ranges,
76t
level, elevation,
61
,
90
,
105
,
108
–109,
120
,
123
–124,
151
percentage,
62
,
73
,
109
POP‐weighted allocation,
158
POP‐weighted portfolio,
157
–158,
159t
POP‐weight scaling method,
156
positions, POP‐weighting,
156
profit potential, differences,
103
selection,
2
statistics,
80t
,
97t
,
153t
trade‐off,
63
trades, level (elevation),
134
usage,
110
,
149
,
153
,
185
–186
weights, usage,
155
yield,
121
Product indifference,
97
–98
Profitability, considerations,
8t
Profit and loss (P/L)
average daily P/L,
121t
average P/L,
76t
,
164
averages,
74f
cumulative P/L,
152f
daily P/Ls, standard deviation,
150f
distribution skew,
62
–63
expectations,
135
frequency,
124
historical distribution,
73f
historical P/Ldistribution,
62f
,
64f
,
71f
IV ranges,
76t
per‐day standard deviation,
150
standard deviation,
134
,
153t
,
157
carrying,
120
–121
core position usage,
156
–157
reduction,
118
–119,
122
,
126
trade‐offs,
124
usage,
63
–65,
74
–75,
80t
,
99
,
100t
,
123
swings,
79
,
97
magnitude,
98
tolerance,
97
–98
Profit potential, POP
differences,
103
level, elevation,
151
Profit targets,
104
,
120t
,
123
Put options,
9
Put prices, differences,
98t
Put skew,
112
Puts (option type),
7
QQQ
returns, SPY returns (contrast),
36f
,
37
str


4
–75,
80t
,
99
,
100t
,
123
swings,
79
,
97
magnitude,
98
tolerance,
97
–98
Profit potential, POP
differences,
103
level, elevation,
151
Profit targets,
104
,
120t
,
123
Put options,
9
Put prices, differences,
98t
Put skew,
112
Puts (option type),
7
QQQ
returns, SPY returns (contrast),
36f
,
37
strangles, outlier losses,
142
Quantitative options trading,
3
Quarterly earnings report (single‐company factors),
52
,
166
,
175
Random variable, probability distribution,
13
Realized moves, IV overstatement,
46t
Realized risk (measurement), IV (usage),
46
Realized volatility, IV overstatement,
46
Reference index, usage,
144
Relative volatility, metrics,
66
–68
Retail options trading, assets (suitability),
94
Returns
distributions skews,
22
past volatility/future volatility,
43
standard deviation,
21
–22
usage,
63
Risk
approximation,
30
categories,
137
measures,
38
–40
minimization, liquidity (impact),
95
reduction, trade‐by‐trade basis,
117
sentiment, measure,
30
–31
tolerances,
171
–172
trade‐off,
12
Risk‐free rate,
29
approximation,
154
value, usage,
154
–155
Risk management,
2
–3,
37
,
140
,
156
importance,
51
strategy/technique,
136
,
151
,
158
,
174
Risk‐reward trade‐off,
59
Sector exposure,
98
Sector‐specific risk,
96
Sell‐offs
2020 sell‐off, performances (2017‐2021),
78f
volatility conditions,
164
Semi‐strong EMH,
12
,
104
–105
Short call,
32
,
34
addition,
147
,
175
BPR,
86
directional assumption,
8t
P/L,
10
–11
position,
33
removal,
175
short put, pairing,
33
strike,
60
undefined risk,
59
Short‐call/put BPR,
86
Short iron condors, range,
134
,
173
Short options
P/Ldistribution skew,
63
trading, capital requirements,
90
Short option strategies,
106t
profitability, factors,
170
trading,
76
,
170
Short premium
allocation,
173
capital allocation, scaling up,
76
positions, losses (unlikelihood),
170
risk (evaluation), BPR (usage),
83
strategies, POP trade‐off,
63
traders, profit,
51
Short premium trading,
48
,
114
benefits,
68
implied volatility
elevation, impact,
71
importance,
59
mechanics,
57
risk‐reward trade‐off,
59
Short put,
34
addition,
147
,
174
–175
BPR,
86
bullish strategy,
32
directional assumption,
8t
position,
33
removal,
175
strike,
60
Short strangles, POP level (elevation),
61
Short strike prices, expected range (relationship),
111
Short volatility trading,
83
Sigma (
σ
),
15
Single‐company factors,
52
–53
Single company risk factors, impact,
46
Skew,
68
amount, consideration,
71
contextualization,
65
distribution skew,
16
–18,
20
,
39
log‐normal distribution skew,
179
magnitude, decrease,
72
moment,
16
–22
P/Ldistribution skew,
62
–63
portfolio delta skew,
145
pure number,
17
reduction,
71
–72
returns distribution skews,
22
strike skew,
111
–112,
179
–183
tail skew, usage,
39
usage,
65
–66
volatility skew (volatility smirk),
181
SPDR S&P 500 (SPY)
annualized implied volatility, tracking,
48
daily returns distribution,
39f
,
40f
expected move cone,
45f
expected price ranges,
44
histogram, daily returns/prices,
27f
implied volatility (IV),
69f
,
70f
iron condors, wings (inclusion),
107t
–110t
neutral SPY strategies,
151
price,
112f
change,
60f
,
78f
trends,
24
returns, QQQ/TLT/GLD returns (contrast),
36f
,
37
trading level,
183
SPDR S&P 500 (SPY) strangles,
64f
,
73f
BPR loss,
85f
data (2005‐2021),
88f
,
89t
deltas (differences), statistical comparison,
113t
durations, differences,
183t
example,
107t
initial credits,
108t
management
statistics,
119t
,
120t
,
122t
,
124t
,
125t
strategies, comparison,
80t
outlier losses,
142
P/Lper‐day standard deviation,
150
stability,
63
VIX level labeling,
69f
Standard deviation,
20
–21
daily P/Ls, standard deviation,
150
estimates,
16
expected move range,
179
expected range,
60
strikes, correspondence,
183
histogram,
17f
historical returns, standard deviation,
28
,
30
indication,
16
interpretation,
18
–19,
64
–65
log returns, standard deviation,
23
normal distribution usage,
180f
per‐trade standard deviation,
158
,
166
P/Lper‐day standard deviation,
150
P/Lstandard deviation,
63
–65,
74
–75,
80t
,
99
,
100t
,
134
,
153t
,
157
carrying,
120
–121
reduction,
118
–119,
122
,
126
trade‐offs,
124
usage,
123
probabilities,
22f
range, sigma (
σ
),
37
,
43
–45,
64
–65
representation,
15
returns, standard deviation,
21
–22
sigma (
σ
),
15
usage,
63
–65
Steady‐state value,
48
Stocks,
5
–6
historical risk, approximation,
63
historical volatility,
30
IV overstatement rates,
46
liquidity,
94
–95
log returns,
23
options, trading,
96
–97
prices
differences,
98t
log‐normal distribution, relationship,
179
skewed returns distributions,
22
stock‐specific binary events,
156
trading,
90
,
179
margin, usage,
84
underlyings
advantages/disadvantages,
96t
trading,
135
volatility profiles, differences,
96
Stop loss,
122
application,
129
implementation,
122
,
130
,
173
mid‐range stop loss,
123
,
130
,
173
threshold, usage,
122
–123
usage,
123
–125
Straddles
ATM straddle, price,
181
trades, BPR result,
87t
Strangles,
105
buyer assumption,
61
drawdowns, experience,
151
durations, differen


ifferences,
96
Stop loss,
122
application,
129
implementation,
122
,
130
,
173
mid‐range stop loss,
123
,
130
,
173
threshold, usage,
122
–123
usage,
123
–125
Straddles
ATM straddle, price,
181
trades, BPR result,
87t
Strangles,
105
buyer assumption,
61
drawdowns, experience,
151
durations, differences,
101t
magnitude,
65
management strategies,
123
neutral SPY strategies,
151
P/Ldistributions, skew/tail losses,
71
–72
sale, BPR requirement,
86
seller, profit,
61
short strangle BPR,
86
statistics,
167t
management,
122t
,
136t
trades, examples,
87t
trading, effects,
142
usage,
156
Strategy‐specific factors,
152
–153
Strike skew,
111
–112,
179
–183
Strikes
long strikes,
107f
prices, comparison,
114t
range,
104
standard deviation, expected range (correspondence),
183
Strong EMH,
12
,
104
–105
Supplemental positions,
134
Swaptions,
5
Systemic risk,
137
Tail exposure
limitation, capital allocation guidelines (maintenance),
173
magnitude,
98
Tail losses
CVaR sensitivity,
40
reduction,
71
–72
Tail risk,
83
,
103
,
121
,
145
acceptance,
57
carrying,
58
,
62
,
120
elimination,
122
–123
exposure,
102
,
135
historical tail risk, estimation,
65
increase,
97
,
108
–109,
119
inherent tail risk, justification,
121
mitigation,
135
–136
negative tail risk,
65
,
72
,
80
Tail skew, usage,
39
Theta (
Θ
),
31
,
34
,
144
,
174
additivity,
145
–147
ratio, size (reaction),
147
theta ratio/net liquidity,
174
theta ratio/net portfolio liquidity,
145
Time diversification,
151
TLT returns, SPY returns (contrast),
36f
Trade‐by‐trade basis,
79
–80,
117
,
125
–126
Trade‐by‐trade performance, comparison,
118
Trade‐by‐trade risk tolerances,
119
–120
Trades
BPR,
98
bullish directional exposure,
90t
management,
3
,
80
–81,
117
–118
strategies, usage,
101
maximum loss, reduction,
108
Trades, construction,
93
asset universe, selection,
94
–95
contract duration, selection,
99
–102
defined risk, selection,
102
–104
delta, selection,
111
–115
directional assumption, selection,
104
–110
procedure,
94
undefined risk, selection,
102
–104
underlying, selection,
96
–98
Trading
engagement, preferences,
124
mechanics,
48
platforms, usage,
179
,
181
strategies,
129
,
166
Uncertainty sentiment, IV tracking,
48
Undefined risk
capital allocation, sharing,
110
selection,
102
–104
Undefined risk strategies,
59
,
152
BPR, relationship,
84
,
103
defined risk strategies, comparison,
102t
,
109
avoidance,
110
downside risk, limitation (absence),
102
gain, limitation,
84
loss, limitation (absence),
59
,
84
management, focus,
118
P/Ltargets, attainment,
153
portfolio allocation,
103t
risk, comparison,
89
selection,
94
short premium allocation,
173
trader compensation,
103
Underlying
historical volatility,
31
increase,
42
implied volatility (IV),
98
option underlyings, sample,
98t
selection,
94
,
96
–98
strangle, iron condor (contrast),
171
Underlying price
BPR function,
88f
expected range,
60
–61,
181t
Upside skew,
112
Value at risk (VaR).
See
Conditional value at risk
CVaR, contrast,
40
distribution statistic,
39
inclusion,
39f
,
40f
Variance
moment,
15
–16
per‐trade variance,
166
–167
Volatility
curve,
182
expansions,
50
–51
forecast,
43
realized volatility, IV overstatement,
46
reversion,
105
smile,
179
–183
smirk (volatility skew),
181
trading,
41
,
44
–48,
58
Volatility assets, market ETFs (correlation),
139
Volatility index (VIX) (CBOE volatility index),
51
,
60
,
78f
2008 sell‐off,
50
,
63
2020 sell‐off,
50
,
63
,
77
,
78f
comparison,
89
contraction,
50
contracts, acceleration,
49
correlations,
141t
expansion,
48
increase,
54
IVP labeling,
67f
levels,
127f
,
128f
differences,
103t
SPY strangles, labeling,
69f
long‐term average,
67
,
69
long‐term behavior,
66
lull/expansion/contraction,
49
occurrences, relationship,
71f
phases,
49f
range,
48
,
66
,
72
,
171
reduction/increase,
69
,
134
frequency,
75t
spikes, causes,
50
states,
48
–51
valuation,
135
VXAZN
IVP values labeling,
67f
level,
66
Weak EMH,
11
,
29
,
31
,
104
–105
Wide iron condors,
116
,
172
Wide wings, usage,
109
Wiener process,
29
Black‐Scholes model, relationship,
23
–26
increments, distribution,
26f
Wings,
105


Foreword
The past several years have brought about aresurgence in market volatility and options volume unlike anything that has been seen since the close of the twentieth century. As markets have become more interdependent, interrelated, and international, the U.S. listed option markets have solidified their place as the most liquid and transparent venue for risk management and hedging activities of the world’slargest economy. Technology, competition, innovation, and reliability have become the hallmarks of the industry, and our customer base has benefited tremendously from this ongoing evolution.
However, these advances can be properly tapped only when the users of the product continue to expand their knowledge of the options product and its unique features. Education has always been the driver of growth in our business, and it will be the steward of the next generation of options traders. Dan Passarelli’snew and updated book
Trading Option Greeks
is anecessity for customers and traders alike to ensure that they possess the knowledge to succeed and attain their objectives in the high-speed, highly technical arena that the options market has become.
The retail trader of the past has given way to anew retail trader of the present—one with an increased level of technology, support, capital treatment, and product selection. The impact of the staggering growth in such products as the CBOE Holdings’ VIX options and futures, and the literally dozens of other products tied to it, have made the volatility asset class anew, unique, and permanent pillar of today’soption markets.
Dan’supdated book continues his mission of supporting, preparing, and reinforcing the trader’sunderstanding of pricing, volatility, market terminology, and strategy, in away that few other books have been able. Using aperspective forged from years as an options market maker, professional trader, and customer, Dan has once again provided aresource for those who wish to know best how the option markets behave today, and how they are likely to continue to behave in the future. It is important to understand not only what happens in the options space, but also
why
it happens. This book is intended to provide those answers. Iwish you all the best in your trading!
William J. Brodsky
Chairman and CEO Chicago Board Options Exchange

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Preface
I’ve always been fascinated by trading. When Iwas young, I’dsee traders on television, in their brightly colored jackets, shouting on the seemingly chaotic trading floor, and I’dmarvel at them. What awonderful job that must be! These traders seemed to me to be very different from the rest of us. It’sall so very esoteric.
It is easy to assume that professional traders have closely kept secrets to their ways of trading—something that secures success in trading for them, but is out of reach for everyone else. In fact, nothing could be further from the truth. If there are any “secrets” of professional traders, this book will expose them.
True enough, in years past there have been some barriers to entry to trading success that did indeed make it difficult for nonprofessionals to succeed. For example, commissions, bid-ask spreads, margin requirements, and information flow all favored the professional trader. Now, these barriers are gone. Competition among brokers and exchanges—as well as the ubiquity of information as propagated on the Internet—has torn down those walls. The only barrier left between the Average Joe and the options pro is that of knowledge. Those who have it will succeed; those who do not will fail.
To be sure, the knowledge held by successful traders is not that of what will happen in the future; it is the knowledge of how to manage the uncertainty. No matter what our instincts tell us, we do not know what will happen in the future with regard to the market. As Socrates put it, “The only true wisdom is in knowing you know nothing.” The masters of option trading are masters of managing the risk associated with what they don’tknow—the risk of uncertainty.
As an instructor, I’ve talked to many traders who were new to options who told me, “Imade atrade based on what Ithought was going to happen. Iwas right, but my position lost money!” Choosing the right strategy makes all the difference when it comes to mastery of risk management and ultimate trading success. Knowing which option strategy is the right strategy for agiven situation comes with knowledge and experience.
All option strategies are differentiated by their unique risk characteristics. Some are more sensitive to directional movement of the underlying asset than others; some are more affected by time passing than others. The exact exposure positions have to these market influences determines the success of individual trades and, indeed, the long-term success of the trader who knows how to exploit these risk characteristics. These option-value sensitivities can be controlled when atrader understands the option greeks.
Option greeks are metrics used to measure an option’ssensitivity to influences on its price. This book will provide the reader with an understanding of these metrics, to help the reader truly master the risk of uncertainty associated with option trading.
Successful traders strive to create option positions with risk-reward profiles that benefit them the most in agiven situation. Atrader’sobjectives will dictate the right strategy for the right situation. Traders can tailor aposition to fit aspecific forecast with respect to the time horizon; the degree of bullishness, bearishness, neutrality, or volatility in the underlying stock; and the desired amount of leverage. Furthermore, they can exploit opportunities unique to options. They can trade option greeks. This opens the door to many new opportunities.
A New Direction
Traders, both professional and retail, need ways to act on their forecasts without the constraints of convention. “Get long, or do nothing” is no longer aviable business model for people active in the market. “Up is good; down is bad” is burned into traders’ minds from the beginning of their market education. This concept has its place in the world of investing, but becoming an active trader in the option market requires thinking in anew direction.
Market makers and other expert option traders look at the market differently from other traders. One fundamental difference is that these traders trade all four directions: up, down, sideways, and volatile.
Trading Strategies
Buying stock is atrading strategy that most people understand. In practical terms, traders who buy stock are generally not concerned with the literal ownership stake in acorporation, just the opportunity to profit if the stock rises. Although it’simportant for traders to understand that the price of astock is largely tied to the success or failure of the corporation, it’sessential to keep in mind exactly what the objective tends to be for trading astock: to profit from changes in its price. Abullish position can also be taken in the options market. The most basic example is buying acall.
Abearish position can be taken by trading stock or options, as well. If traders expect the value of astock they own to fall, they will sell the stock. This eliminates the risk of losses from the stock’sfalling. If the traders do not own the stock that they

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et. The most basic example is buying acall.
Abearish position can be taken by trading stock or options, as well. If traders expect the value of astock they own to fall, they will sell the stock. This eliminates the risk of losses from the stock’sfalling. If the traders do not own the stock that they think will decline, they can take amore active stance and short it. The short-seller borrows the stock from aparty that owns it and then sells the borrowed shares to another party. The goal of selling stock short is to later repurchase the shares at alower price before returning the stock to its owner. It is simply reversing the order of “buy low/sell high.” The risk is that the stock rises and shares have to be bought at ahigher price than that at which they were sold. Although shorting stock can lead to profits when the market cooperates, in the options market, there are alternative ways to profit from falling prices. The most basic example is buying aput.
Atrader can use options to take abullish or bearish position, given adirectional forecast. Sideways, nontrending stocks and their antithesis, volatile stocks, can be traded as well. In the later market conditions, profit or loss can be independent of whether the stock rises or falls. Opportunity in option trading is not necessarily black and white—not necessarily up and down. Option trading is nonlinear. Consequently, more opportunities can be exploited by trading options than by trading stock.
Option traders must consider the time period in question, the volatility expected during this period, interest rates, and dividends. Along with the stock price, these factors make up the dynamic components of an option’svalue. These individual factors can be isolated, measured, and exploited. Incremental changes in any of these elements as measured by option greeks provide opportunity for option traders. Because of these other influences, direction is not the only tradable element of aforecast. Time, volatility, interest rates—these can all be traded using option greeks. These factors and more will all be discussed at great length throughout this book.
This Second Edition of
Trading Option Greeks
This book addresses the complex price behavior of options by discussing option greeks from both atheoretical and apractical standpoint. There is some tactical discussion throughout, although the objective of this book is to provide education to the reader. This book is meant to be less ahow-to manual than ahow-come tutorial.
This informative guide will give the retail trader alook inside the mind of aprofessional trader. It will help the professional trader better understand the essential concepts of his craft. Even the novice trader will be able to apply these concepts to basic options strategies. Comprehensive knowledge of the greeks can help traders to avoid common pitfalls and increase profit potential.
Much of this book is broken down into adiscussion of individual strategies. Although the nuances of each specific strategy are not relevant, presenting the material this way allows for adiscussion of very specific situations in which greeks come into play. Many of the concepts discussed in asection on one option strategy can be applied to other option strategies.
As in the first edition of
Trading Option Greeks
, Chapter 1 discusses basic option concepts and definitions. It was written to be areview of the basics for the intermediate to advanced trader. For newcomers, it’sessential to understand these concepts before moving forward.
Adetailed explanation of option greeks begins in Chapter 2. Be sure to leave abookmark in this chapter, as you will flip to it several times while reading the rest of the book and while studying the market thereafter. Chapter 3 introduces volatility. The same bookmark advice can be applied here, as well. Chapters 4 and 5 explore the minds of option traders. What are the risks they look out for? What are the opportunities they seek? These chapters also discuss direction-neutral and direction-indifferent trading. The remaining chapters take the reader from concept to application, discussing the strategies for nonlinear trading and the tactical considerations of asuccessful options trader.
New material in this edition includes updated examples, with more current price information throughout many of the chapters. More detailed discussions are also included to give the reader adeeper understanding of important topics. For example, Chapter 8 has amore elaborate explanation of the effect of dividends on option prices. Chapter 17 of this edition has new material on strategy selection, position management, and adjusting, not featured in the first edition of the book.


CHAPTER 1
The Basics
To understand how options work, one needs first to understand what an option is. An option is acontract that gives its owner the right to buy or the right to sell afixed quantity of an underlying security at aspecific price within acertain time constraint. There are two types of options: calls and puts. Acall gives the owner of the option the right to buy the underlying security. Aput gives the owner of the option the right to sell the underlying security. As in any transaction, there are two parties to an option contract—abuyer and aseller.
Contractual Rights and Obligations
The option buyer is the party who owns the right inherent in the contract. The buyer is referred to as having along position and may also be called the holder, or owner, of the option. The right doesn’tlast forever. At some point the option will expire. At expiration, the owner may exercise the right or, if the option has no value to the holder, let it expire without exercising it. But he need not hold the option until expiration. Options are transferable—they can be traded intraday in much the same way as stock is traded. Because it’suncertain what the underlying stock price of the option will be at expiration, much of the time this right has value before it expires. The uncertainty of stock prices, after all, is the raison d’être of the option market.
Along position in an option contract, however, is fundamentally different from along position in astock. Owning corporate stock affords the shareholder ownership rights, which may include the right to vote in corporate affairs and the right to receive dividends. Owning an option represents strictly the right either to buy the stock or to sell it, depending on whether it’sacall or aput. Option holders do not receive dividends that would be paid to the shareholders of the underlying stock, nor do they have voting rights. The corporation has no knowledge of the parties to the option contract. The contract is created by the buyer and seller of the option and made available by being listed on an exchange.
The party to the contract who is referred to as the option seller, also called the option writer, has ashort position in the option. Instead of having aright to take aposition in the underlying stock, as the buyer does, the seller incurs an obligation to potentially either buy or sell the stock. When atrader who is long an option exercises, atrader with ashort position gets
assigned
. Assignment means the trader with the short option position is called on to fulfill the obligation that was established when the contract was sold.
Shorting an option is fundamentally different from shorting astock. Corporations have aquantifiable number of outstanding shares available for trading, which must be borrowed to create ashort position, but establishing ashort position in an option does not require borrowing; the contract is simply created. The strategy of shorting stock is implemented statistically far less frequently than simply buying stock, but that is not at all the case with options. For every open long-option contract, there is an open short-option contract—they are equally common.
Opening and Closing
Traders’ option orders are either opening or closing transactions. When traders with no position in aparticular option buy the option, they buy to open. If, in the future, the traders wish to eliminate the position by selling the option they own, the traders enter asell to close order—they are closing the position. Likewise, if traders with no position in aparticular option want to sell an option, thereby creating ashort position, the traders execute asell-to-open transaction. When the traders cover the short position by buying back the option, the traders enter abuy-to-close order.
Open Interest and Volume
Traders use many types of market data to make trading decisions. Two items that are often studied but sometimes misunderstood are volume and open interest. Volume, as the name implies, is the total number of contracts traded during atime period. Often, volume is stated on aone-day basis, but could be stated per week, month, year, or otherwise. Once anew period (day) begins, volume begins again at zero. Open interest is the number of contracts that have been created and remain outstanding. Open interest is arunning total.
When an option is first listed, there are no open contracts. If Trader Aopens along position in anewly listed option by buying aone-lot, or one contract, from Trader B, who by selling is also opening aposition, acontract is created. One contract traded, so the volume is one. Since both parties opened aposition and one contract was created, the open interest in this particular option is one contract as well. If, later that day, Trader Bcloses his short position by buying one contract from Trader C, who had no position to start with, the volume is now two contracts for that day, but open interest is still one. Only one contract exists; it was traded twice. I

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his particular option is one contract as well. If, later that day, Trader Bcloses his short position by buying one contract from Trader C, who had no position to start with, the volume is now two contracts for that day, but open interest is still one. Only one contract exists; it was traded twice. If the next day, Trader Cbuys her contract back from Trader A, that day’svolume is one and the open interest is now zero.
The Options Clearing Corporation
Remember when Wimpy would tell Popeye, “I’ll gladly pay you Tuesday for ahamburger today.” Did Popeye ever get paid for those burgers? In acontract, it’svery important for each party to hold up his end of the bargain—especially when there is money at stake. How does atrader know the party on the other side of an option contract will in fact do that? That’swhere the Options Clearing Corporation (OCC) comes into play.
The OCC ultimately guarantees every options trade. In 2010, that was almost 3.9 billion listed-options contracts. The OCC accomplishes this through many clearing members. Here’show it works: When Trader Xbuys an option through abroker, the broker submits the trade information to its clearing firm. The trader on the other side of this transaction, Trader Y, who is probably amarket maker, submits the trade to his clearing firm. The two clearing firms (one representing Trader X’sbuy, the other representing Trader Y’ssell) each submit the trade information to the OCC, which “matches up” the trade.
If Trader Ybuys back the option to close the position, how does that affect Trader Xif he wants to exercise it? It doesn’t. The OCC, acting as an intermediary, assigns one of its clearing members with acustomer that is short the option in question to deliver the stock to Trader X’sclearing firm, which in turn delivers the stock to Trader X. The clearing member then assigns one of its customers who is short the option. The clearing member will assign the trader either randomly or first in, first out. Effectively, the OCC is the ultimate counterparty to both the exercise and the assignment.
Standardized Contracts
Exchange-listed options contracts are standardized, meaning the terms of the contract, or the contract specifications, conform to acustomary structure. Standardization makes the terms of the contracts intuitive to the experienced user.
To understand the contract specifications in atypical equity option, consider an example:
Buy 1 IBM December 170 call at 5.00
Quantity
In this example, one contract is being purchased. More could have been purchased, but not less—options cannot be traded in fractional units.
Option Series, Option Class, and Contract Size
All calls or puts of the same class, the same expiration month, and the same strike price are called an
option series
. For example, the IBM December 170 calls are aseries. Options series are displayed in an option chain on an online broker’suser interface. An option chain is afull or partial list of the options that are listed on an underlying.
Option class
means agroup of options that represent the same underlying. Here, the option class is denoted by the symbol IBM—the contract represents rights on International Business Machines Corp. (IBM) shares. Buying one contract usually gives the holder the right to buy or to sell 100 shares of the underlying stock. This number is referred to as
contract size
. Though this is usually the case, there are times when the contract size is something other than 100 shares of astock. This situation may occur after certain types of stock splits, spin-offs, or stock dividends, for example. In the minority of cases in which the one contract represents rights on something besides 100 shares, there may be more than one class of options listed on astock.
Afairly unusual example was presented by the Ford Motor Company options in the summer of 2000. In June 2000, Ford spun off Visteon Corporation. Then, in August 2000, Ford offered shareholders achoice of converting their shares into (a) new shares of Ford plus $20 cash per share, (b) new Ford stock plus fractional shares with an aggregate value of $20, or (c) new Ford stock plus acombination of more new Ford stock and cash. There were three classes of options listed on Ford after both of these changes: Frepresented 100 shares of the new Ford stock; XFO represented 100 shares of Ford plus $20 per share ($2,000) plus cash in lieu of $1.24; and FOD represented 100 shares of new Ford, 13 shares of Visteon, and $2,001.24.
Sometimes these changes can get complicated. If there is ever aquestion as to what the underlying is for an option class, the authority is the OCC. Alot of time, money, and stress can be saved by calling the OCC at 888-OPTIONS and clarifying the matter.
Expiration Month
Options expire on the Saturday following the third Friday of the stated month, which in this case is December. The final trading day for an option is commonly the day before expiration—here, the third Friday of December. There are usually at least


OPTIONS and clarifying the matter.
Expiration Month
Options expire on the Saturday following the third Friday of the stated month, which in this case is December. The final trading day for an option is commonly the day before expiration—here, the third Friday of December. There are usually at least four months listed for trading on an option class. There may be atotal of six months if Long-Term Equity AnticiPation Securities
®
or LEAPS
®
are listed on the class. LEAPS can have one year to about two-and-a-half years until expiration. Some underlyings have one-week options called Weeklys
SM
listed on them.
Strike Price
The price at which the option holder owns the right to buy or to sell the underlying is called the strike price, or exercise price. In this example, the holder owns the right to buy the stock at $170 per share. There is method to the madness regarding how strike prices are listed. Strike prices are generally listed in $1, $2.50, $5, or $10 increments, depending on the value of the strikes and the liquidity of the options.
The relationship of the strike price to the stock price is important in pricing options. For calls, if the stock price is above the strike price, the call is in-the-money (ITM). If the stock and the strike prices are close, the call is at-the-money (ATM). If the stock price is below the strike price the call is out-of-the-money (OTM). This relationship is just the opposite for puts. If the stock price is below the strike price, the put is in-the-money. If the stock price and the strike price are about the same, the put is at-the-money. And, if the stock price is above the put strike, it is out-of-the-money.
Option Type
There are two types of options: calls and puts. Calls give the holder the right to buy the underlying and the writer the obligation to sell the underlying. Puts give the holder the right to sell the underlying and the writer the obligation to buy the underlying.
Premium
The price of an option is called its premium. The premium of this option is $5. Like stock prices, option premiums are stated in dollars and cents per share. Since the option represents 100 shares of IBM, the buyer of this option will pay $500 when the transaction occurs. Certain types of spreads may be quoted in fractions of apenny.
An option’spremium is made up of two parts: intrinsic value and time value. Intrinsic value is the amount by which the option is in-the-money. For example, if IBM stock were trading at 171.30, this 170-strike call would be in-the-money by 1.30. It has 1.30 of intrinsic value. The remaining 3.70 of its $5 premium would be time value.
Options that are out-of-the-money have no intrinsic value. Their values consist only of time premium. Sometimes options have no time value left. Options that consist of only intrinsic value are trading at what traders call
parity
. Time value is sometimes called
premium over parity
.
Exercise Style
One contract specification that is not specifically shown here is the exercise style. There are two main exercise styles: American and European. American-exercise options can be exercised, and therefore assigned, anytime after the contract is entered into until either the trader closes the position or it expires. European-exercise options can be exercised and assigned only at expiration. Exchange-listed equity options are all American-exercise style. Other kinds of options are commonly European exercise. Whether an option is American or European has nothing to with the country in which it’slisted.
ETFs, Indexes, and HOLDRs
So far, we’ve focused on equity options—options on individual stocks. But investors have other choices for trading securities options. Options on baskets of stocks can be traded, too. This can be accomplished using options on exchange-traded funds (ETFs), index options, or options on holding company depositary receipts (HOLDRs).
ETF Options
Exchange-traded funds are vehicles that represent ownership in afund or investment trust. This fund is made up of abasket of an underlying index’ssecurities—usually equities. The contract specifications of ETF options are similar to those of equity options. Let’slook at an example.
One actively traded optionable ETF is the Standard & Poor’s Depositary Receipts (SPDRs or Spiders). Spider shares and options trade under the symbol SPY. Exercising one SPY call gives the exerciser along position of 100 shares of Spiders at the strike price of the option. Expiration for ETF options typically falls on the same day as for equity options—the Saturday following the third Friday of the month. The last trading day is the Friday before. ETF options are American exercise. Traders of ETFs should be aware of the relationship between the price of the ETF shares and the value of the underlying index. For example, the stated value of the Spiders is about one tenth the stated value of the S&P 500. The PowerShares QQQ ETF, representing the Nasdaq 100, is about one fortieth the stated value of the Nasdaq 100.
I


e relationship between the price of the ETF shares and the value of the underlying index. For example, the stated value of the Spiders is about one tenth the stated value of the S&P 500. The PowerShares QQQ ETF, representing the Nasdaq 100, is about one fortieth the stated value of the Nasdaq 100.
Index Options
Trading options on the Spiders ETF is aconvenient way to trade the Standard & Poor’s (S&P) 500. But it’snot the only way. There are other option contracts listed on the S&P 500. The SPX is one of the major ones. The SPX is an index option contract. There are some very important differences between ETF options like SPY and index options like SPX.
The first difference is the underlying. The underlying for ETF options is 100 shares of the ETF. The underlying for index options is the numerical value of the index. So if the S&P 500 is at 1303.50, the underlying for SPX options is 1303.50. When an SPX call option is exercised, instead of getting 100 shares of something, the exerciser gets the ITM cash value of the option times $100. Again, with SPX at 1303.50, if a 1300 call is exercised, the exerciser gets $350—that’s 1303.50 minus 1300, times $100. This is called
cash settlement
.
Many index options are European, which means no early exercise. At expiration, any long ITM options in atrader’sinventory result in an account credit; any short ITMs result in adebit of the ITM value times $100. The settlement process for determining whether a European-style index option is in-the-money at expiration is alittle different, too. Often, these indexes are a.m. settled. A.m.-settled index options will have actual expiration on the conventional Saturday following the third Friday of the month. But the final trading day is the Thursday before the expiration day. The final settlement value of the index is determined by the opening prices of the components of the index on Friday morning.
HOLDR Options
Like ETFs, holding company depositary receipts also represent ownership in abasket of stocks. The main difference is that investors owning HOLDRs retain the ownership rights of the individual stocks in the fund, such as the right to vote shares and the right to receive dividends. Options on HOLDRs, for all intents and purposes, function much like options on ETFs.
Strategies and At-Expiration Diagrams
One of the great strengths of options is that there are so many different ways to use them. There are simple, straightforward strategies like buying acall. And there are complex spreads with creative names like jelly roll, guts, and iron butterfly. Aspread is astrategy that involves combining an option with one or more other options or stock. Each component of the spread is referred to as aleg. Each spread has its own unique risk and reward characteristics that make it appropriate for certain market outlooks.
Throughout this book, many different spreads will be discussed in depth. For now, it’simportant to understand that all spreads are made up of acombination of four basic option positions: buy call, sell call, buy put, and sell put. Understanding complex option strategies requires understanding these basic positions and their common, practical uses. When learning options, it’shelpful to see what the option’spayout is if it is held until expiration.
Buy Call
Why buy the right to buy the stock when you can simply buy the stock? All option strategies have trade-offs, and the long call is no different. Whether the stock or the call is preferable depends greatly on the trader’sforecast and motivations.
Consider along call example:
Buy 1 INTC June 22.50 call at 0.85.
In this example, atrader is bullish on Intel (INTC). He believes Intel will rise at least 20 percent, from $22.25 per share to around $27 by June expiration, about two months from now. He is concerned, however, about downside risk and wants to limit his exposure. Instead of buying 100 shares of Intel at $22.25—atotal investment of $2,225—the trader buys 1 INTC June 22.50 call at 0.85, for atotal of $85.
The trader is paying 0.85 for the right to buy 100 shares of Intel at $22.50 per share. If Intel is trading below the strike price of $22.50 at expiration, the call will expire and the total premium of 0.85 will be lost. Why? The trader will not exercise the right to buy the stock at a $22.50 if he can buy it cheaper in the market. Therefore, if Intel is below $22.50 at expiration, this call will expire with no value.
However, if the stock is trading above the strike price at expiration, the call can be exercised, in which case the trader may purchase the stock below its trading price. Here, the call has value to the trader. The higher the stock, the more the call is worth. For the trade to be profitable, at expiration the stock must be trading above the trader’sbreak-even price. The break-even price for along call is the strike price plus the premium paid—in this example, $23.35 per share. The point here is that if the call is exercised, the effective purchase price