t risk profiles (e.g., strangle on underlying A versus 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's important 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 a contract duration that is an efficient use of buying power, allows for consistency, offers a reasonable number of occurrences, has manageable P/L swings throughout the duration, and has moderate ending P/L variability. 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 a reasonable amount of growth but small enough to have manageable P/L swings and ending P/L variability.
When choosing a management 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/L per day, closing positions prior to expiration and redeploying capital to new positions is generally a more 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 a decent amount of long‐term profit and justify the tail loss exposure.
If managing an undefined position according to a profit target, choosing a target between 50% and 75% of the initial credit allows for reasonable profits while also reducing the potential magnitude of outlier losses. Choosing a profit target that is too low reduces average P/L, and choosing a profit 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 a stop loss, a mid‐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/L variance, although they often recover. It's also important to note that stop losses do not guarantee a maximum 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 a reasonable 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 a low‐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 a single position) and at most 25% reserved for defined risk strategies (with less than 5% of portfolio buying power allocated to a single 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/L variation that offer consistent profits and reasonable outlier exposure.
Diversifying the underlyings of an options portfolio (i.e., trading a collection of assets with low correlations) is one of the most essential diversification tools for p


core positions or trades with high POPs and moderate P/L variation that offer consistent profits and reasonable outlier exposure.
Diversifying the underlyings of an options portfolio (i.e., trading a collection 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 a set 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 a portfolio and guide adjustments. Two key Greeks are beta‐weighted delta (
) and theta (
). Beta‐weighted delta represents the amount of directional exposure a position has relative to some index rather than the underlying itself. Theta represents the expected decrease in an option's value 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 a reliable estimate for the expected daily growth of a short 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 a portfolio is not meeting these theta ratio guidelines, then the positions should be adjusted as follows:
If a properly allocated, well‐diversified portfolio is
neutral but does not provide a sufficient 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 a properly allocated, well‐diversified portfolio provides a sufficient 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 a binary 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 a certain level of exposure, and willing to commit the time to active trading, short premium strategies can offer a probabilistic 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.

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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 a telescoping
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 a sum of random variables. Logarithms offer a convenient way to accomplish this.
Logarithmic functions are a class 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 a constant multiple. If
is another possible base, then
. In mathematics, the most common choice is Euler's constant, a special number:
. Using this constant as a base 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 a sum of small independent pieces, given by the following equation:
(A.3)
The central limit theorem states that if a random 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 a log‐normal distribution or a distribution where the logarithm of a random 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 a stock trading at current price
with volatility
and risk‐free rate
, the Black‐Scholes theoretical
price range at a future time
for this asset is given by the following equation:
(A.6)
The equation in (
A.5
) is a valid approximation of this formula when
is small, which follows from the mathematical relation
. Generally speaking, (
A.5
) is a very rough approximation for expected range, and it becomes less accurate in high volatility conditions, when
is larger.
Though (
A.5
) still yields a reasonable, 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 a price 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 a more attractive way to calculate expected range on trading platforms becau

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e, $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 a more attractive way to calculate expected range on trading platforms because it is computationally simpler and independent of a rigid 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 a property 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. A volatility 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, a volatility smirk (also known as volatility skew) is a weighted 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 a higher premium to protect against downside risk compared to upside risk.
This is an example of put skew, a consequence 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 a given 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 a short 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 a fair die was six, what is the probability that the next roll will also be a six?
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 a given 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 a tornado on any given day, there is a 1% chance of a tornado occurring on any given day, and tornados never happen without rain, meaning that
. Therefore, given that it is


. 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 a tornado on any given day, there is a 1% chance of a tornado occurring on any given day, and tornados never happen without rain, meaning that
. Therefore, given that it is a rainy day, we have the following probability that a tornado will appear:
In other words, a tornado 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 a concept 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 a repeated 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 a game with probability
of winning
and a probability
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. A brief justification for this formula follows from the paper listed in Reference 4.
Consider a game with probability
of winning
and a probability
of losing the full wager. If a player has
in starting wealth and bets a fraction of that wealth,
, on this game, the player's goal is to choose a value 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 a trade with known (or approximated) probability of profit (POP) and edge. More specifically, for an option with a given 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 a game with probability
of winning
and a probability
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 a given trade, substituting
for
and POP for
.
Notes
1
So called because adjacent numerators and denominators cancel, allowing the long product to be collapsed like a telescope.
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
C
is used if the contract is a call and
P
is 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 a future date
Risk‐free rate of return: the theoretical rate of return of a riskless asset
Mean: the central tendency of a distribution
Standard deviation: the spread of a distribution; also used as a measure of uncertainty or risk
Volatility: the standard deviation of log‐returns for an asset; a key input in options pricing
Delta: the expected change in an option's price given a $1 increase in the price of the underlying
Gamma: the expected change in an option's delta given a $1 change in the price of the underlying
Theta: the expected time decay of an option's extrinsic 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's price 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

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Index
Active management,
117
,
125
–126,
150
capital allocation, relationship,
79
–81
volatility trading concept,
58
Active traders,
125
Active trading,
2
–3,
123
,
149
Advanced portfolio management,
3
,
133
,
149
advanced diversification,
149
–153
capital (balancing), POP (usage),
153
–156
market/underlying IV, consideration (absence),
156
portfolio construction,
156
–160
positions, POP‐weighting,
156
straddle, price,
181
strategy diversification,
151
American options, exercise,
7
Assets
combination,
38
correlation history,
138t
illiquid asset,
94
management, Kelly Criterion (application),
185
portfolio, trading,
137
rates, experience,
52t
realized moves, IV overstatement,
46t
trading,
94
,
173
–174
universe, selection,
94
–95
volatility,
31
weighting, POP (usage),
149
At‐the‐money (ATM)
contract description,
9
positions,
33
straddle,
181
strikes,
32
closeness,
99
–100,
146
Autocorrelation,
26
Backtest period, usage,
158
Beta (
β
)
beta‐weighted delta,
38
,
144
–145,
148
,
174
index, assumption,
145
metric,
38
Bid‐ask spread,
94
,
95t
Binary events,
3
EMH, relationship,
164
examples,
163
option strategies,
166
–167
outcomes, predictability (absence),
164
premium, buying/selling result (evidence),
164
stock‐specific binary events,
156
trades,
166
–167,
175
volatility expansion, impact,
164
Black‐Scholes assumptions, usage,
44
–45
Black‐Scholes equation,
29
European‐style option price evolution, relationship,
23
Black‐Scholes model,
22
–31,
35
,
42
,
179
,
181t
assumptions,
23
,
27
,
35
,
44
,
45
Brownian motion (Wiener process), relationship,
23
–26
geometric Brownian motion, relationship,
27
–28
mathematical definition, derivation,
111
mechanics,
30
options fair price, estimate,
30
,
42
price dynamics approximation,
24
Black‐Scholes option pricing formalism, impact,
22
–23
Black‐Scholes options pricing model,
44
–45
Black‐Scholes theoretical price range,
179
Brownian motion
Black‐Scholes model, relationship,
23
–26
cumulative horizontal displacements,
24
–25,
25f
particle, 2D position,
25f
price dynamics, comparison,
25
–26
stock log‐returns, evolution,
179
Bullish directional exposure,
90t
Buying power, allocation percentages,
154t
Buying Power Reduction (BPR),
3
,
66
,
83
,
153
average BPR comparison,
110t
,
113t
calculation,
84
capital
coverage,
89
requirement, correspondence,
85
–86
definition, variation,
84
historical effectiveness,
84
IV
comparison,
66
inverse relationship,
87
loss percentage,
85f
margin, contrast,
84
maximum per‐trade BPR, limitation,
134
option price, inverse correlation,
87
options, capital efficiency (relationship),
90
portfolio capital amount,
171
reduction, impact,
89
result,
87t
short call/put BPR,
86
short strangle BPR,
86
stock margin, option counterpart,
84
undefined risk strategies, relationship,
84
underlying IV function/underlying price function,
88f
understanding, importance,
90
usage,
83
,
89
variables, dependence,
85
–86
Call option,
7
,
9
strike prices, comparison,
114t
trading level (determination), Black‐Scholes model (usage),
31
Call skew,
112
Call strikes, comparison,
111
–112
Capital
balancing, POP (usage),
153
–156
requirement, BPR (correspondence),
85
–86
Capital allocation
amounts, differences,
78f
control,
81
efficiency, active management (relationship),
79
–81
estimate, risk‐free rate value (usage),
154
–155
guidelines,
133
,
152
,
155
,
156
maintenance,
149
,
173
market IV, impact,
76t
violation, avoidance,
155
position sizing, relationship,
134
–136
positional capital allocation, quantitative approach,
153
preferences,
129
,
131
,
172
proportions, estimation,
155
risk management techniques, incorporation (impact),
158
scaling,
79
,
134
short premium capital allocation, scaling up,
76
undefined risk capital allocation, sharing,
110
undefined risk strategy, usage,
110
volatility trading concept,
58
Capital at risk (comparison), BPR (usage),
89
Casinos
long‐run statistical advantage,
58
,
72
–73
options trading,
1
–2
CBOE volatility index.
See
Volatility index
Central limit theorem,
20
,
72
,
166
,
170
,
178
Company exposure,
98
Company‐specific risk,
96
Company‐specific uncertainty,
43
Conditional probability,
183
–184
calculation,
184
usage,
142
Conditional value at risk (CVaR),
63
inclusion,
40f
metric,
38
usage,
39
–40,
65
–66,
123
VaR, contrast,
40
Contract delta,
33
–35,
79
,
111
,
114
implied volatility (IV), equivalence,
87
risk,
172
Contract duration
middle ground contract duration,
101
selection,
94
,
99
–102
trading, importance,
151
Contracts
average daily P/L and average duration,
121t
extrinsic value, decrease,
34
prices, differences,
42t
Core positions,
134
P/L standard deviations, presence,
156
–157
Covariance
correlation, relationship,
35
–38
measures,
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
D

eBooks\The Unlucky Investor_s Guide to Options Trading\The Unlucky Investor_s Guide to Options Trading.epub#section:b04.xhtml


res,
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/L targets, 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/L distribution,
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,
42


4
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
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/L target, usage,
120
–124
techniques,
123
–126,
152
timeline, usage,
118
–119
Management strategies,
121t
,
122t
,
158
average daily P/L and 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/L distribution,
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/L standard deviation, usage,
73
P/L statistics,
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


statistics,
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/L targets, 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/L averages,
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/L correlation, 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/L distribution,
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),


ge,
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
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/L distribution 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/L distribution 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/L per‐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/L per‐day standard deviation,
150
P/L standard 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
d


tility 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, differences,
101t
magnitude,
65
management strategies,
123
neutral SPY strategies,
151
P/L distributions, 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/L targets, 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


CHAPTER 1
The Basics
To understand how options work, one needs first to understand what an option is. An option is a contract that gives its owner the right to buy or the right to sell a fixed quantity of an underlying security at a specific price within a certain time constraint. There are two types of options: calls and puts. A call gives the owner of the option the right to buy the underlying security. A put 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—a buyer and a seller.
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 a long position and may also be called the holder, or owner, of the option. The right doesn’t last 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’s uncertain 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.
A long position in an option contract, however, is fundamentally different from a long position in a stock. 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’s a call or a put. 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 a short position in the option. Instead of having a right to take a position in the underlying stock, as the buyer does, the seller incurs an obligation to potentially either buy or sell the stock. When a trader who is long an option exercises, a trader with a short 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 a stock. Corporations have a quantifiable number of outstanding shares available for trading, which must be borrowed to create a short position, but establishing a short 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 a particular 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 a sell to close order—they are closing the position. Likewise, if traders with no position in a particular option want to sell an option, thereby creating a short position, the traders execute a sell-to-open transaction. When the traders cover the short position by buying back the option, the traders enter a buy-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 a time period. Often, volume is stated on a one-day basis, but could be stated per week, month, year, or otherwise. Once a new 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 a running total.
When an option is first listed, there are no open contracts. If Trader A opens a long position in a newly listed option by buying a one-lot, or one contract, from Trader B, who by selling is also opening a position, a contract is created. One contract traded, so the volume is one. Since both parties opened a position and one contract was created, the open interest in this particular option is one contract as well. If, later that day, Trader B closes 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. Onl

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contract was created, the open interest in this particular option is one contract as well. If, later that day, Trader B closes 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 C buys her contract back from Trader A, that day’s volume 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 a hamburger today.” Did Popeye ever get paid for those burgers? In a contract, it’s very important for each party to hold up his end of the bargain—especially when there is money at stake. How does a trader know the party on the other side of an option contract will in fact do that? That’s where 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’s how it works: When Trader X buys an option through a broker, the broker submits the trade information to its clearing firm. The trader on the other side of this transaction, Trader Y, who is probably a market maker, submits the trade to his clearing firm. The two clearing firms (one representing Trader X’s buy, the other representing Trader Y’s sell) each submit the trade information to the OCC, which “matches up” the trade.
If Trader Y buys back the option to close the position, how does that affect Trader X if he wants to exercise it? It doesn’t. The OCC, acting as an intermediary, assigns one of its clearing members with a customer that is short the option in question to deliver the stock to Trader X’s clearing 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 a customary structure. Standardization makes the terms of the contracts intuitive to the experienced user.
To understand the contract specifications in a typical 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 a series. Options series are displayed in an option chain on an online broker’s user interface. An option chain is a full or partial list of the options that are listed on an underlying.
Option class
means a group 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 a stock. 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 a stock.
A fairly 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 a choice 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 a combination of more new Ford stock and cash. There were three classes of options listed on Ford after both of these changes: F represented 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 a question as to what the underlying is for an option class, the authority is the OCC. A lot 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 befor


CC. A lot 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 four months listed for trading on an option class. There may be a total 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 a penny.
An option’s premium 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’s listed.
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 a fund or investment trust. This fund is made up of a basket of an underlying index’s securities—usually equities. The contract specifications of ETF options are similar to those of equity options. Let’s look 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 a long 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, rep


iday 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.
Index Options
Trading options on the Spiders ETF is a convenient way to trade the Standard & Poor’s (S&P) 500. But it’s not 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 a trader’s inventory result in an account credit; any short ITMs result in a debit of the ITM value times $100. The settlement process for determining whether a European-style index option is in-the-money at expiration is a little 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 a basket 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 a call. And there are complex spreads with creative names like jelly roll, guts, and iron butterfly. A spread is a strategy that involves combining an option with one or more other options or stock. Each component of the spread is referred to as a leg. 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’s important to understand that all spreads are made up of a combination 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’s helpful to see what the option’s payout 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’s forecast and motivations.
Consider a long call example:
Buy 1 INTC June 22.50 call at 0.85.
In this example, a trader 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—a total investment of $2,225—the trader buys 1 INTC June 22.50 call at 0.85, for a total 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’s break-even price. The break-even price for a long call is the strike price plus the premium paid—in thi


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’s break-even price. The break-even price for a long 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 of the stock upon exercise is $23.35. The stock is literally bought at the strike price, which is $22.50, but the premium of 0.85 that the trader has paid must be taken into account.
Exhibit 1.1
illustrates this example.
EXHIBIT 1.1
Long Intel call.
Exhibit 1.1
is an at-expiration diagram for the Intel 22.50 call. It shows the profit and loss, or P&(L), of the option if it is held until expiration. The X-axis represents the prices at which INTC could be trading at expiration. The Y-axis represents the associated profit or loss on the position. The at-expiration diagram of any long call position will always have this same hockey-stick shape, regardless of the stock or strike. There is always a limit of loss, represented by the horizontal line, which in this case is drawn at −0.85. And there is always a line extending upward and to the right, which represents effectively a long stock position stemming from the strike.
The trade-offs between a long stock position and a long call position are shown in
Exhibit 1.2
.
EXHIBIT 1.2
Long Intel call vs. long Intel stock.
The thin dotted line represents owning 100 shares of Intel at $22.25. Profits are unlimited, but the risk is substantial—the stock
can
go to zero. Herein lies the trade-off. The long call has unlimited profit potential with limited risk. Whenever an option is purchased, the most that can be lost is the premium paid for the option. But the benefit of reduced risk comes at a cost. If the stock is above the strike at expiration, the call will always underperform the stock by the amount of the premium.
Because of this trade-off, conservative traders will sometimes buy a call rather than the associated stock and sometimes buy the stock rather than the call. Buying a call can be considered more conservative when the volatility of the stock is expected to rise. Traders are willing to risk a comparatively small premium when a large price decline is feared possible. Instead, in an interest-bearing vehicle, they harbor the capital that would otherwise have been used to purchase the stock. The cost of this protection is acceptable to the trader if high-enough price advances are anticipated. In terms of percentage, much higher returns
and losses
are possible with the long call. If the stock is trading at $27 at expiration, as the trader in this example expected, the trader reaps a 429 percent profit on the $0.85 investment ([$27 − 23.35] / $0.85). If Intel is below the strike price at expiration, the trader loses 100 percent.
This makes call buying an excellent speculative alternative. Those willing to accept bigger risk can further increase returns by purchasing more calls. In this example, around 26 Intel calls—representing the rights on 2,600 shares—can be purchased at 85 cents for the cost of 100 shares at $22.25. This is the kind of leverage that allows for either a lower cash outlay than buying the stock—reducing risk—or the same cash outlay as buying the stock but with much greater exposure—creating risk in pursuit of higher returns.
Sell Call
Selling a call creates the obligation to sell the stock at the strike price. Why is a trader willing to accept this obligation? The answer is option premium. If the position is held until expiration without getting assigned, the entire premium represents a profit for the trader. If assignment occurs, the trader will be obliged to sell stock at the strike price. If the trader does not have a long position in the underlying stock (a naked call), a short stock position will be created. Otherwise, if stock is owned (a covered call), that stock is sold. Whether the trader has a profit or a loss depends on the movement of the stock price and how the short call position was constructed.
Consider a naked call example:
Sell 1 TGT October 50 call at 1.45
In this example, Target Corporation (TGT) is trading at $49.42. A trader, Sam, believes Target will continue to be trading below $50 by October expiration, about two months from now. Sam sells 1 Target two-month 50 call at 1.45, opening a short position in that series.
Exhibit 1.3
will help explain the expected payout of this naked call position if it is held until expiration.
EXHIBIT 1.3
Naked Target call.
If TGT is trading below the exercise price of 50, the call will expire worthless. Sam keeps the 1.45 premium, and the obligation to sell the stock ceases to exist. If Target is trading above the strike price, the call will be in-the-money. The higher the stock is above the strike price, the more intrinsic value the call will have. As a seller, Sam wants the call to have litt


ess. Sam keeps the 1.45 premium, and the obligation to sell the stock ceases to exist. If Target is trading above the strike price, the call will be in-the-money. The higher the stock is above the strike price, the more intrinsic value the call will have. As a seller, Sam wants the call to have little or no intrinsic value at expiration. If the stock is below the break-even price at expiration, Sam will still have a profit. Here, the break-even price is $51.45—the strike price plus the call premium. Above the break-even, Sam has a loss. Since stock prices can rise to infinity (although, for the record, I have never seen this happen), the naked call position has unlimited risk of loss.
Because a short stock position may be created, a naked call position must be done in a margin account. For retail traders, many brokerage firms require different levels of approval for different types of option strategies. Because the naked call position has unlimited risk, establishing it will generally require the highest level of approval—and a high margin requirement.
Another tactical consideration is what Sam’s objective was when he entered the trade. His goal was to profit from the stock’s being below $50 during this two-month period—not to short the stock. Because equity options are American exercise and can be exercised/assigned any time from the moment the call is sold until expiration, a short stock position cannot always be avoided. If assigned, the short stock position will extend Sam’s period of risk—because stock doesn’t expire. Here, he will pay one commission shorting the stock when assignment occurs and one more when he
buys back
the unwanted position. Many traders choose to close the naked call position before expiration rather than risk assignment.
It is important to understand the fundamental difference between buying calls and selling calls. Buying a call option offers limited risk and unlimited reward. Selling a naked call option, however, has limited reward—the call premium—and unlimited risk. This naked call position is not so much bearish as
not bullish
. If Sam thought the stock was going to zero, he would have chosen a different strategy.
Now consider a covered call example:
Buy 100 shares TGT at $49.42
Sell 1 TGT October 50 call at 1.45
Unlimited
and
risk
are two words that don’t sit well together with many traders. For that reason, traders often prefer to sell calls as part of a spread. But since spreads are strategies that involve multiple components, they have different risk characteristics from an outright option. Perhaps the most commonly used call-selling spread strategy is the covered call (sometimes called a
covered write
or a
buy-write
). While selling a call naked is a way to take advantage of a “not bullish” forecast, the covered call achieves a different set of objectives.
After studying Target Corporation, another trader, Isabel, has a neutral to slightly bullish forecast. With Target at $49.42, she believes the stock will be range-bound between $47 and $51.50 over the next two months, ending with October expiration. Isabel buys 100 shares of Target at $49.42 and sells 1 TGT October 50 call at 1.45. The implications for the covered-call strategy are twofold: Isabel must be content to own the stock at current levels, and—since she sold the right to buy the stock at $50, that is, a 50 call, to another party—she must be willing to sell the stock if the price rises to or through $50 per share.
Exhibit 1.4
shows how this covered call performs if it is held until the call expires.
EXHIBIT 1.4
Target covered call.
The solid kinked line represents the covered call position, and the thin, straight dotted line represents owning the stock outright. At the expiration of the call option, if Target is trading below $50 per share—the strike price—the call expires and Isabel is left with a long position of 100 shares
plus
$1.45 per share of expired-option premium. Below the strike, the buy-write always outperforms simply owning the stock by the amount of the premium. The call premium provides limited downside protection; the stock Isabel owns can decline $1.45 in value to $47.97 before the trade is a loser. In the unlikely event the stock collapses and becomes worthless, this limited downside protection is not so comforting. Ultimately, Isabel has $47.97 per share at risk.
The trade-off comes if Target is above $50 at expiration. Here, assignment will likely occur, in which case the stock will be sold. The call can be assigned before expiration, too, causing the stock to be
called away
early. Because the covered call involves this obligation to sell the sock at the strike price, upside potential is limited. In this case, Isabel’s profit potential is $2.03. The stock can rise from $49.42 to $50—a $0.58 profit—plus $1.45 of option premium.
Isabel does not want the stock to decline too much. Below $47.97, the trade is a loser. If the stock rises too much, the stock is sold prematurely and upside opportuni


In this case, Isabel’s profit potential is $2.03. The stock can rise from $49.42 to $50—a $0.58 profit—plus $1.45 of option premium.
Isabel does not want the stock to decline too much. Below $47.97, the trade is a loser. If the stock rises too much, the stock is sold prematurely and upside opportunity is lost. Limited reward and unlimited risk. (Technically, the risk is not unlimited—the stock can only go to zero. But if the stock drops from $49.42 to zero in a short time, the risk will certainly feel unlimited.) The covered call strategy is for a neutral to moderately bullish outlook.
Sell Put
Selling a put has many similarities to the covered call strategy. We’ll discuss the two positions and highlight the likenesses. Chapter 6 will detail the nuts and bolts of why these similarities exist.
Consider an example of selling a put:
Sell 1 BA January 65 put at 1.20
In this example, trader Sam is neutral to moderately bullish on Boeing (BA) between now and January expiration. He is not bullish enough to buy BA at the current market price of $69.77 per share. But if the shares dropped below $65, he’d gladly scoop some up. Sam sells 1 BA January 65 put at 1.20. The at-expiration diagram in
Exhibit 1.5
shows the P&(L) of this trade if it is held until expiration.
EXHIBIT 1.5
Boeing short put.
At the expiration of this option, if Boeing is above $65, the put expires and Sam retains the premium of $1.20. The obligation to buy stock expires with the option. Below the strike, put owners will be inclined to exercise their option to sell the stock at $65. Therefore, those short the put, as Sam is in this example, can expect assignment. The break-even price for the position is $63.80. That is the strike price minus the option premium. If assigned, this is the effective purchase price of the stock. The obligation to buy at $65 is fulfilled, but the $1.20 premium collected makes the purchase effectively $63.80. Here, again, there is limited profit opportunity ($1.20 if the stock is above the strike price) and seemingly unlimited risk (the risk of potential stock ownership at $63.80) if Boeing is below the strike price.
Why would a trader short a put and willingly assume this substantial risk with comparatively limited reward? There are a number of motivations that may warrant the short put strategy. In this example, Sam had the twin goals of profiting from a neutral to moderately bullish outlook on Boeing and buying it if it traded below $65. The short put helps him achieve both objectives.
Much like the covered call, if the stock is above the strike at expiration, this trader reaches his maximum profit potential—in this case 1.20. And if the price of Boeing is below the strike at expiration, Sam has ownership of the stock from assignment. Here, a strike price that is lower than the current stock level is used. The stock needs to decline in order for Sam to get assigned and become long the stock. With this strategy, he was able to establish a target price at which he would buy the stock. Why not use a limit order? If the put is assigned, the effective purchase price is $63.80 even if the stock price is above this price. If the put is not assigned, the premium is kept.
A consideration every trader must make before entering the short put position is how the purchase of the stock will be financed in the event the put is assigned. Traders hoping to acquire the stock will often hold enough cash in their trading account to secure the purchase of the stock. This is called a
cash-secured put
. In this example, Sam would hold $6,380 in his account in addition to the $120 of option premium received. This affords him enough free capital to fund the $6,500 purchase of stock the short put dictates. More speculative traders may be willing to buy the stock on margin, in which case the trader will likely need around 50 percent of the stock’s value.
Some traders sell puts without the intent of ever owning the stock. They hope to profit from a low-volatility environment. Just as the short call is a not-bullish stance on the underlying, the short put is a not-bearish play. As long as the underlying is above the strike price at expiration, the option premium is all profit. The trader must actively manage the position for fear of being assigned. Buying the put back to close the position eliminates the risk of assignment.
Buy Put
Buying a put gives the holder the right to sell stock at the strike price. Of course, puts can be a part of a host of different spreads, but this chapter discusses the two most basic and common put-buying strategies: the long put and the protective put. The long put is a way to speculate on a bearish move in the underlying security, and the protective put is a way to protect a long position in the underlying security.
Consider a long put example:
Buy 1 SPY May 139 put at 2.30
In this example, the Spiders have had a good run up to $140.35. Trader Isabel is looking for a 10 percent correction in SPY between now and the e


ecurity, and the protective put is a way to protect a long position in the underlying security.
Consider a long put example:
Buy 1 SPY May 139 put at 2.30
In this example, the Spiders have had a good run up to $140.35. Trader Isabel is looking for a 10 percent correction in SPY between now and the end of May, about three months away. She buys 1 SPY May 139 put at 2.30. This put gives her the right to sell 100 shares of SPY at $139 per share.
Exhibit 1.6
shows Isabel’s P&(L) if the put is held until expiration.
EXHIBIT 1.6
SPY long put.
If SPY is above the strike price of 139 at expiration, the put will expire and the entire premium of 2.30 will be lost. If SPY is below the strike price at expiration, the put will have value. It can be exercised, creating a short position in the Spiders at an effective price of $136.70 per share. This price is found by subtracting the premium paid, 2.30, from the strike price, 139. This is the point at which the position breaks even. If SPY is below $136.70 at expiration, Isabel has a profit. Profits will increase on a tick-for-tick basis, with downward movements in SPY down to zero. The long put has limited risk and substantial reward potential.
An alternative for Isabel is to short the ETF at the current price of $140.35. But a short position in the underlying may not be as attractive to her as a long put. The margin requirements for short stock are significantly higher than for a long put. Put buyers must post only the premium of the put—that is the most that can be lost, after all.
The margin requirement for short stock reflects unlimited loss potential. Margin requirements aside, risk is a very real consideration for a trader deciding between shorting stock and buying a put. If the trader expects high volatility, he or she may be more inclined to limit upside risk while leveraging downside profit potential by buying a put. In general, traders buy options when they expect volatility to increase and sell them when they expect volatility to decrease. This will be a common theme throughout this book.
Consider a protective put example:
This is an example of a situation in which volatility is expected to increase.
Own 100 shares SPY at 140.35
Buy 1 SPY May139 put at 2.30
Although Isabel bought a put because she was bearish on the Spiders, a different trader, Kathleen, may buy a put for a different reason—she’s bullish but concerned about increasing volatility. In this example, Kathleen has owned 100 shares of Spiders for some time. SPY is currently at $140.35. She is bullish on the market but has concerns about volatility over the next two or three months. She wants to protect her investment. Kathleen buys 1 SPY May 139 put at 2.30. (If Kathleen bought the shares of SPY and the put at the same time, as a spread, the position would be called a married put.)
Kathleen is buying the right to sell the shares she owns at $139. Effectively, it is an insurance policy on this asset.
Exhibit 1.7
shows the risk profile of this new position.
EXHIBIT 1.7
SPY protective put.
The solid kinked line is the protective put (put and stock), and the thin dotted line is the outright position in SPY alone, without the put. The most Kathleen stands to lose with the protective put is $3.65 per share. SPY can decline from $140.35 to $139, creating a loss of $1.35, plus the $2.30 premium spent on the put. If the stock does not fall and the insuring put hence does not come into play, the cost of the put must be recouped to justify its expense. The break-even point is $142.65.
This position implies that Kathleen is still bullish on the Spiders. When traders believe a stock or ETF is going to decline, they sell the shares. Instead, Kathleen sacrifices 1.6 percent of her investment up front by purchasing the put for $2.30. She defers the sale of SPY until the period of perceived risk ends. Her motivation is not to sell the ETF; it is to hedge volatility.
Once the anticipated volatility is no longer a concern, Kathleen has a choice to make. She can let the option run its course, holding it to expiration, at which point it will either expire or be exercised; or she can sell the option before expiration. If the option is out-of-the-money, it may have residual time value prior to expiration that can be recouped. If it is in-the-money, it will have intrinsic value and maybe time value as well. In this situation, Kathleen can look at this spread as two trades—one that has declined in price, the SPY shares, and one that has risen in price, the put. Losses on the ETF shares are to some degree offset by gains on the put.
Measuring Incremental Changes in Factors Affecting Option Prices
At-expiration diagrams are very helpful in learning how a particular option strategy works. They show what the option’s price will ultimately be at various prices of the underlying. There is, however, a caveat when using at-expiration diagrams. According to the Options Industry Council, most options are closed before they reach expira