American and European options. American options can be exercised at any point prior to expiration, and European options can only be exercised on the expiration date.
1
Because American options are generally more popular than European options and offer more flexibility, this book focuses on American options.
The most basic types of options are calls and puts. American
calls
give the holder the right to
buy
the underlying asset at a certain price within a given time frame, and American
puts
give the holder the right to
sell
the underlying asset. The contract parameters must be specified prior to opening the trade and are listed below:
The underlying asset trading at the spot price, or the current per share price
.
The number of underlying shares. One option usually covers 100 shares of the underlying, known as a one lot.
The price at which the underlying shares can be bought or sold prior to expiration. This price is called the strike price
.
The expiration date, after which the contract is worthless. The time between the present day and the expiration date is the contract's duration or days to expiration (DTE).
Note that the price of the option is commonly denoted as
C
for calls,
P
for puts, and
V
if the type of contract is not specified. Options traders may buy or sell these contracts, and the conditions for profitability differ depending on the choice of position. The purchaser of the contract pays the option premium (current market price of the option) to adopt the
long
side of the position. This is also known as a long premium trade. The seller of the contract receives the option premium to adopt the
short
side of the position, thus placing a short premium trade. The choice of strategy corresponds to the directional assumption of the trader. For calls and puts, the directional assumption is either bullish, assuming the underlying price will increase, or bearish, assuming the underlying price will decrease. The directional assumptions and scenarios for profitability for these contracts are summarized in the following table.
Table 1.1
The definitions, conditions for profitability, and directional assumptions for long/short calls/puts.
Call
Put
Long
Purchase the right to buy an underlying asset
at the strike price
prior to the expiration date.
Profits increase as the price of the underlying increases above the strike price
.
Directional assumption: Bullish
Purchase the right to sell an underlying asset
at the strike price
prior to the expiration date.
Profits increase as the price of the underlying decreases below the strike price
.
Directional assumption: Bearish
Short
Sell the right to buy an underlying asset
at the strike price
prior to the expiration date.
Profits increase as the price of the underlying decreases below the strike price
.
Directional assumption: Bearish
Sell the right to sell an underlying asset
at the strike price
prior to the expiration date.
Profits increase as the price of the underlying increases above the strike price
.
Directional assumption: Bullish
The relationship between the strike price and the current price of the underlying determines the
moneyness
of the position. This is equivalently the
intrinsic value
of a position, or the value of the contract if it were exercised immediately. Contracts can be described as one of the following, noting that options cannot have negative intrinsic value:
In‐the‐money (ITM): The contract would be profitable if it was exercised immediately and thus has intrinsic value.
Out‐of‐the‐money (OTM): The contract would result in a loss if it was exercised immediately and thus has no intrinsic value.
At‐the‐money (ATM): The contract has a strike price equal to the price of the underlying and thus has no intrinsic value.
The intrinsic value of a position is based entirely on the type of position and the choice of strike price relative to the price of the underlying:
Call options
Intrinsic Value = Either
(stock price – strike price) or 0
ITM:
OTM:
ATM:
Put options
Intrinsic Value = Either
or 0
ITM:
OTM:
ATM:
For example, consider a 45 DTE put contract with a strike price of $100:
Scenario 1 (ITM): The underlying price is $95. In this case, the intrinsic value of the put contract is $5 per share.
Scenario 2 (OTM): The underlying price is $105. In this case, the put contract has no intrinsic value.
Scenario 3 (ATM): The underlying price is $100. In this case, the put is also considered to have no intrinsic value.
The value of an option also depends on speculative factors, driven by supply and demand. The
extrinsic
value of the option is the difference
between the current market price for the option and the intrinsic value of the option. Again, consider a 45 DTE put contract with a strike price of $100 on an underlying with a current price per share of $105. Suppose that, due to a period of recent market turbulence, investors are fearful the underlying price will crash within the next 45 days and create a demand for these OTM put contracts. The surge in


DTE put contract with a strike price of $100 on an underlying with a current price per share of $105. Suppose that, due to a period of recent market turbulence, investors are fearful the underlying price will crash within the next 45 days and create a demand for these OTM put contracts. The surge in demand inflates the price of the put contract to $10 per share. Therefore, because the put contract has no intrinsic value but has a market price of $10, the extrinsic value of the contract is $10 per share. If, instead, the price of the underlying is $95 and the price of the ITM put is still $10 per share, then the contract will have $5 in intrinsic value and $5 in extrinsic value.
The profitability of an option ultimately depends on both intrinsic and extrinsic factors, and it is calculated as the difference between the intrinsic value of an option and the cost of the contract. Mathematically, profit and loss (P/L) approximations for long calls and puts at exercise are given by the following equations:
2
(1.3)
(1.4)
where the max function simply outputs the larger of the two values. For instance,
equals 1 while
equals 0. The P/Ls for the corresponding short sides are merely
Equations (1.3)
and
(1.4)
multiplied by –1. Following is a sample trade that applies the long call profit formula.
Example trade: A call with 45 DTE duration is traded on an underlying that is currently priced at $100
. The strike price is $105
and the long call is currently valued at $100 per one lot ($1 per share).
Scenario 1: The underlying increases to $105 by the expiration date.
Long call P/L:
Short call P/L: +$100.
Scenario 2: The underlying increases to $110 by the expiration date.
Long call P/L:
Short call P/L: –$400.
Scenario 3: The underlying decreases to $95 by the expiration date.
Long call P/L:
Short call P/L: +$100.
The trader adopting the long position pays the seller the option premium upfront and profits when the intrinsic value exceeds the price of the contract. The short trader profits when the intrinsic value remains below the price of the contract, especially when the position expires worthless (no intrinsic value). The extrinsic value of an option generally decreases over the duration of the contract, as uncertainty around the underlying price and uncertainty around the profit potential of the option decrease. As a position nears expiration, the price of an option converges toward its intrinsic value.
Options pricing clearly plays a large role in options trading. To develop an intuitive understanding around how options are priced, understanding the mathematical assumptions around market efficiency and price dynamics is critical.
The Efficient Market Hypothesis
Traders must make a number of assumptions prior to placing a trade. Options traders must make directional assumptions about the price of the underlying over a given time frame: bearish (expecting price to decrease), bullish (expecting price to increase), or neutral (expecting price to remain relatively unchanged). Options traders also must make assumptions about the current value of an option. If options contracts are perceived as overvalued, long positions are less likely to profit. If options contracts are perceived as undervalued, short positions are less likely to profit. These assumptions about underlying and option price dynamics are a personal choice, but traders can formulate consistent assumptions by referring to the efficient market hypothesis (EMH). The EMH states that instruments are traded at a fair price, and the current price of an asset reflects some amount of available information. The hypothesis comes in three forms:
Weak EMH: Current prices reflect all past price information.
Semi‐strong EMH: Current prices reflect all publicly available information.
Strong EMH: Current prices reflect all possible information.
No variant of the EMH is universally accepted or rejected. The form that a trader assumes is subjective, and methods of market analysis available are limited depending on that choice. Proponents of the strong EMH posit that investors benefit from investing in low‐cost passive index funds because the market is unbeatable. Opponents believe the market is beatable by exploiting inefficiencies in the market. Traders who accept the weak EMH believe technical analysis (using past price trends to predict future price trends) is invalidated, but fundamental analysis (using related economic data to predict future price trends) is still viable. Traders who accept the semi‐strong EMH assume fundamental analysis would not yield systematic success but trading according to private information would. Traders who accept the strong EMH maintain that even insider trading will not result in consistent success and no exploitable market inefficiencies are available to anyone.
This book focuses on highly liquid markets, and inefficiencies are assumed to be minimal. More specifically, this book assumes a semi‐strong form of the EMH. Rather than constructing po

eBooks\The Unlucky Investor_s Guide to Options Trading\The Unlucky Investor_s Guide to Options Trading.epub#section:c01.xhtml


er trading will not result in consistent success and no exploitable market inefficiencies are available to anyone.
This book focuses on highly liquid markets, and inefficiencies are assumed to be minimal. More specifically, this book assumes a semi‐strong form of the EMH. Rather than constructing portfolios according to forecasts of future price trends, the purpose of this text is to demonstrate how trading options according to current market conditions and directional
volatility
assumptions (rather than price assumptions) has allowed options sellers to consistently outperform the market.
This “edge” is not the result of some inherent market inefficiency but rather a trade‐off of risk. Recall the example long call trade from the previous section. Notice that there are more scenarios in which the short trader profits compared to the long trader. Generally, short premium positions are more likely to yield a profit compared to long premium positions. This is because options are assumed to be priced efficiently and scaled according to the perceived risk in the market, meaning that long positions only profit when the underlying has large directional moves outside of expectations. As these types of events are uncommon, options contracts go unused the majority of the time and short premium positions profit more often than long positions. However, when those large,
unexpected moves
do
occur, the short premium positions are subject to potentially massive losses. The risk profiles for options are complex, but they can be intuitively represented with probability distributions.
Probability Distributions
To better understand the risk profiles of short options, this book utilizes basic concepts from probability theory, specifically random variables and probability distributions. Random variables are formal stand‐ins for uncertain quantities. The probability distribution of a random variable describes possible values of that quantity and the likelihood of each value occurring. Generally, probability distributions are represented by the symbol
, which can be read as “the probability that.” For example,
. Random variables and probability distributions are tools for working with probabilistic systems (i.e., systems with many unpredictable outcomes), such as stock prices. Although future outcomes cannot be precisely predicted, understanding the distribution of a probabilistic system makes it possible to form expectations about the future, including the uncertainty associated with those expectations.
Let's begin with an example of a simple probabilistic system: rolling a pair of fair, six‐sided dice. In this case, if
represents the sum of the dice, then
is a random variable with 11 possible values ranging from 2 to 12. Some of these outcomes are more likely than others. Since, for instance, there are more ways to roll a sum of 7 ([1,6], [2,5], [3,4], [4,3], [5,2], [6,1]) than a sum of 10 ([4,6], [5,5], [6,4]), there is a higher probability of rolling a 7 than a 10. Observing that there are 36 possible rolls ([1,1], [1,2], [2,1], etc.) and that each is equally likely, one can use symbols to be more precise about this:
The distribution of
can be represented elegantly using a histogram. These types of graphs display the frequency of different outcomes, grouped according to defined ranges. When working with measured data, histograms are used to estimate the true underlying
probability distribution of a probabilistic system. For this fair dice example, there will be 11 bins, corresponding to the 11 possible outcomes. This histogram is shown below in
Figure 1.1
, populated with data from 100,000 simulated dice rolls.
Figure 1.1
A histogram for 100,000 simulated rolls for a pair of fair dice. This diagram shows the likelihood of each outcome occurring according to this simulation (e.g., the height of the bin ranging from 6.5 to 7.5 is near 17%, indicating that 7 occurred nearly 17% of the time in the 100,000 trials).
Distributions like the ones shown here can be summarized using quantitative measures called
moments
.
3
The first two moments are mean and variance.
Mean
(first moment): Also known as the average and represented by the Greek letter
(mu), this value describes the central tendency of
a distribution. This is calculated by summing all the observed outcomes
together and dividing by the number of observations
:
(1.5)
For distributions based on statistical observations with
a sufficiently large number of occurrences
, the mean corresponds to the expected value of that distribution. The expected value of a random variable is the weighted average of outcomes and the anticipated average outcome over future trials. The expected value of a random variable
, denoted
, can be estimated using statistical data and
Equation (1.5)
,
or
if the unique outcomes (
) and their respective probabilities
are known, then the expected value can also be calculated using the following formula:
(1.6)
In the dice sum example, represented with random va


a random variable
, denoted
, can be estimated using statistical data and
Equation (1.5)
,
or
if the unique outcomes (
) and their respective probabilities
are known, then the expected value can also be calculated using the following formula:
(1.6)
In the dice sum example, represented with random variable
, the possible outcomes (2, 3, 4, …, 12) and the probability of each occurring (2.78%, 5.56%, 8.33%, …, 2.78%) are known, so the expected value can be determined as follows:
The theoretical long‐term average sum is seven. Therefore, if this experiment is repeated many times, the mean of the observations calculated using
Equation (1.5)
should yield an output close to seven.
Variance
(second moment): This is the measure of the spread, or variation, of the data points from the mean of the distribution. Standard deviation, represented with by the Greek letter
(sigma), is the square root of variance and is commonly used as a measure of uncertainty (equivalently, risk or volatility). Distributions with more variance are wider and have more uncertainty around future outcomes. Variance is calculated according to the following:
4
(1.7)
When a large portion of data points are dispersed far from the mean, the variance of the entire set is large, and uncertainty on measurements from that system is significant. The variance of a random variable
X
, denoted
(
X
), can also be calculated in terms of the expected value,
[
X
]:
(1.8)
For the dice sum random variable,
D
, the possible outcomes (2, 3, 4, …, 12) and the probability of each occurring (2.78%, 5.56%, 8.33%, …, 2.78%) are known, so the variance of this experiment is as follows:
This equation indicates that the spread of the distribution for this random variable is around 5.84 and the uncertainty (standard deviation) is approximately 2.4 (shown in
Figure 1.2
).
One can compare these theoretical estimates for the mean and standard deviation of the dice sum experiment to the values measured from statistical data. The calculated first and second moments from the simulated dice roll experiment are plotted in
Figure 1.2
for comparison.
Obtaining a distribution average near 7.0 makes intuitive sense because 7 is the most likely sum to roll out of the possible outcomes. The standard deviation indicates that the uncertainty associated with that expected value is near 2.4. Inferring from the shape of the distribution, which has most of the probability mass concentrated near the center, one can conclude that on any given roll the outcome will most likely fall between five and nine.
The distribution just shown is symmetric about the mean, but probability distributions are often asymmetric. To quantify the degree of asymmetry for a distribution, the third moment is used.
Skew
(third moment): This is a measure of the asymmetry of a distribution. A distribution's skew can be positive, negative, or zero and depends on whether the tail to the right of the mean is larger (positive skew), to the left is larger (negative skew), or equal on both sides (zero skew). Unlike mean and standard deviation, which have units defined by the random variable, skew is a pure number that quantifies the degree of asymmetry according to the following formula:
(1.9)
Figure 1.2
A histogram for 100,000 simulated dice rolls with fair dice. Included is the mean of the distribution (solid line) and the standard deviation of the distribution on either side of the mean (dotted line), both calculated using the observations from the simulated experiment. The average of this distribution was 7.0 and the standard deviation was 2.4, consistent with the theoretical estimates.
The concept of skew and its applications can be best understood with a modification to the dice rolling example. Suppose that the dice are biased rather than fair. Let's consider two scenarios: a pair of unfair dice with a small number bias (two and three more likely) and a pair of unfair dice with a large number bias (four and five more likely).
The probabilities of each number appearing on each die for the different cases are shown in
Table 1.2
.
Table 1.2
The probability of each number appearing on each die in the three different scenarios, one fair and two unfair.
Probability of Number Appearing on Each Die
Die Number
Fair
Unfair (Small Number Bias)
Unfair (Large Number Bias)
1
16.67%
10%
10%
2
16.67%
30%
10%
3
16.67%
30%
10%
4
16.67%
10%
30%
5
16.67%
10%
30%
6
16.67%
10%
10%
When rolling the
fair
pair and plotting the histogram of the possible sums, the distribution is symmetric about the mean and has a skew of zero. However, the distributions when rolling the unfair dice are skewed, as shown in
Figures 1.3
(a) and (b).
The skew of a distribution is classified according to where the majority of the distribution mass is concentrated. Remember that the positive side is to the right of the mean and the negative side is to the left. The histogram in
Figure 1.3
(a) has a longer tail on the positive side and has the most mass concentrated on


tion is classified according to where the majority of the distribution mass is concentrated. Remember that the positive side is to the right of the mean and the negative side is to the left. The histogram in
Figure 1.3
(a) has a longer tail on the positive side and has the most mass concentrated on the negative side of the mean: This is an example of
positive
skew (skew = 0.45). The histogram in
Figure 1.3
(b) has a longer tail on the negative side and has the majority of the mass concentrated on the positive side of the mean: This is an example of
negative
skew (skew = –0.45).
When a distribution has skew, the interpretation of standard deviation changes. In the example with fair dice, the expected value of the experiment is
2.4, suggesting that any given trial will most likely have an outcome between five and nine. This is a valid interpretation because the distribution is symmetric about the mean and most of the distribution mass is concentrated around it. However, consider the distribution in the unfair example with the large number bias. This distribution has a mean of 7.8 and a standard deviation of 2.0, naively suggesting that the outcome will most likely be between six and nine with the outcomes on either side being equally probable. However, because the majority of the occurrences are concentrated on the positive side of the mean (roughly 60% of occurrences), the uncertainty is not symmetric. This concept will be discussed in more detail in a later chapter, as distributions of financial instruments are commonly skewed, and there is ambiguity in defining risk under those circumstances.
Figure 1.3
(a) A histogram for 100,000 simulated dice rolls with unfair dice, biased such that smaller numbers (2 and 3) are more likely to appear on each die. (b) A histogram for 100,000 simulated dice rolls with unfair dice, biased such that larger numbers (4 and 5) are more likely to appear on each die.
Mathematicians and scientists have encountered some probability distributions repeatedly in theory and applications. These distributions have, in turn, received a great deal of study. Assuming the underlying distribution of an experiment resembles a well known form can often greatly simplify statistical analysis. The normal distribution (also known as the Gaussian distribution or the bell curve) is arguably one of the most well‐known probability distributions and foundational in quantitative finance. It describes countless different real‐world systems because of a result known as the central limit theorem. This theorem says, roughly, that if a random variable is made by adding together many independently random pieces, then, regardless of what those pieces are, the result will be normally distributed. For example, the distribution in the two‐dice example is fairly non‐normal, being relatively triangular and lacking tails. If one considered the sum of more and more dice, each of which is an independent random variable, the distribution would gradually take on a bell shape. This is shown in
Figure 1.4
.
The normal distribution is a symmetric, bell‐shaped distribution, meaning that equidistant events on either side of the center are equally likely and the skew is zero. The distribution is centered around the mean, and outcomes further away from the mean are less likely. The normal distribution has the intriguing property that 68% of occurrences fall within
of the mean, 95% of occurrences are within
of the mean, and 99.7% of occurrences are within
of the mean.
Figure 1.5
plots a normal distribution.
These probabilities can be used to roughly contextualize distributions with similar geometry. For example, in the fair dice pair model, the expected value of the fair dice experiment was 7.0, and the standard deviation was 2.4. With the assumption of normality, one would infer there is roughly a 68% chance that future outcomes will fall between five and nine. The true probability is 66.67% for this random variable, indicating that the normality assumption is not exactly correct but can be used for the purposes of approximation. As more dice are added to the example, this approximation becomes increasingly accurate.
Figure 1.4
A histogram for 100,000 simulated rolls with a group of fair, six‐sided dice numbering (a) 2, (b) 4, or (c) 6.
Understanding distribution statistics and the properties of the normal distribution is incredibly useful in quantitative finance. The expected return of a stock is usually estimated by the mean return, and the historic risk is estimated with the standard deviation of returns (historical volatility). Stock log returns are also widely assumed to be normally distributed. Although, this is only approximately true because the overwhelming majority of stocks and ETFs have skewed returns distributions.
5
Regardless, this normality estimation provides a quantitative framework for expectations around future price moments. This approximation also simplifies mathematical models of price dynamics and optio


tely true because the overwhelming majority of stocks and ETFs have skewed returns distributions.
5
Regardless, this normality estimation provides a quantitative framework for expectations around future price moments. This approximation also simplifies mathematical models of price dynamics and options pricing, the most notable of which is the Black‐Scholes model.
Figure 1.5
A detailed plot of the normal distribution and the corresponding probabilities at each standard deviation mark.
The Black‐Scholes Model
The Black‐Scholes options pricing formalism revolutionized options markets when it was published in 1973. It provided the first popular quantitative framework for estimating the fair price of an option according to the contract parameters and the characteristics of the underlying. The Black‐Scholes equation models the price evolution of a European‐style option
(an option that can only be exercised at expiration) within the context of the broader financial market. The corresponding Black‐Scholes formula uses this equation to estimate the theoretical price of that option according to its parameters.
It's important to note that the purpose of this Black‐Scholes section is
not
to elucidate the underlying mathematics of the model, which can be quite complicated. The output of the model is merely a theoretical value for the fair price of an option. In practice, an option's price typically deviates from this value because of market speculation and supply and demand, which this model does not take into account. Rather, it is essential to have at least a superficial grasp of the Black‐Scholes model to understand (1) the foundational assumptions of financial markets and (2) where implied volatility (a gauge for the market's
perception
of risk) comes from.
The Black‐Scholes model is based on a set of assumptions related to the dynamics of financial assets and the market as a whole. The assumptions are as follows:
The market is frictionless (i.e., there are no transaction fees).
Cash can be borrowed and lent in any amount, even fractional, at the risk‐free rate (the theoretical rate of return of an investment with no risk, a macroeconomic variable assumed to be constant).
There is no arbitrage opportunity (i.e., profits in excess of the risk‐free rate cannot be made without risk).
Stocks can be bought and sold in any amount, even fractional amounts.
Stocks do not pay dividends.
6
Stock log returns follow Brownian motion with constant drift and volatility (the theoretical mean and standard deviation of annual log returns).
A Brownian motion, or a Wiener process, is a type of stochastic process or a system that experiences random fluctuations as it evolves with time. Traditionally used to describe the positional fluctuations of a
particle suspended in fluid at thermal equilibrium,
7
a standard Wiener process (denoted
W
(
t
)) is mathematically defined by the conditions in the grey box. The mathematical definition can be overlooked if preferred, as the intuition behind the mathematics is more crucial for understanding the theoretical foundation of options pricing and follows after.
(i.e., the process initially begins at location 0).
is almost surely continuous.
The increments of
, defined as
where
, are normally distributed with mean 0 and variance
(i.e., the steps of the Wiener process are normally distributed with constant mean of 0 and variance of
).
Disjoint increments of
are independent of one another (i.e., the current step of the process is not influenced by the previous steps, nor does it influence the subsequent steps).
Simplified, a Wiener process is a process that follows a random path. Each step in this path is probabilistic and independent of one another. When disjoint steps of equal duration are plotted in a histogram, that distribution is normal with a constant mean and variance. Brownian motion dynamics are driven by this underlying process. These conditions can be best understood visually, which will also demonstrate why this assumption appears in the development of the Black‐Scholes model as an approximation for price dynamics.
Figures 1.6
and
1.7
illustrate the characteristics of Brownian motion, and
Figure 1.8
illustrates the dynamics of SPY from 2010–2015
8
for the purposes of comparison.
The price trends of SPY in
Figure 1.8
(b) appear fairly similar to the Brownian motion cumulative horizontal displacements shown in
Figure 1.6
(c). The daily returns for SPY are more prone to outlier moves compared to the horizontal displacements of Brownian motion but share some characteristics. The symmetric geometry of the SPY returns histogram bears resemblance to the fairly normal distribution of horizontal displacements, with the tails of the distribution being more prominent as a result of the history of large price moves.
Figure 1.6
(a) The 2D position of a particle in a fluid, moving with Brownian motion. The particle begins at a coordinate of
and drifts to a new location over 1,000 steps. (b) The horizon


ements, with the tails of the distribution being more prominent as a result of the history of large price moves.
Figure 1.6
(a) The 2D position of a particle in a fluid, moving with Brownian motion. The particle begins at a coordinate of
and drifts to a new location over 1,000 steps. (b) The horizontal displacements
9
of the particle (i.e., the movements of the particle along the X‐axis over 1,000 steps). (c) The cumulative horizontal displacement of the particle over 1,000 steps.
Similarities are clear between price dynamics and Brownian motion, but this remains a highly simplified model of price dynamics. In reality, stock log returns are not normal and are typically skewed to the upside or downside, depending on the specific underlying. Additionally, the drift and volatility of a stock are not directly observable, and it cannot be experimentally confirmed whether or not these variables are constant. Stock volatility approximated with historical return data is rarely constant with time (a phenomenon known as heteroscedasticity). Stock returns are also not typically independent of one another across time (a phenomenon known as autocorrelation), which is a requirement for this model.
Figure 1.7
The distribution of the horizontal displacements of the particle over 1,000 steps. As characteristic of a Wiener process, the increments are normally distributed, have a mean of zero and variance
(which equals 1 in this case). This figure indicates that horizontal step sizes between –1 and 1 are most common, and step sizes with a larger magnitude than 1 are less common.
Figure 1.8
The (a) daily returns, (b) price, and (c) daily returns histogram for SPY from 2010–2015.
Although the normality assumption is not entirely accurate, making this simplification allows the development of the rest of this theoretical framework shown in the gray box. The formalism in the gray box is supplemental material for the mathematically inclined. The interpretation of the math, which is more significant, follows after. It should be noted that the Black‐Scholes model technically assumes that stock prices follow
geometric Brownian motion
, which is more accurate because price movements cannot be negative. Geometric Brownian motion is a slight modification of Brownian motion and requires that the logarithm of the signal follow Brownian motion rather than
the signal itself. As it relates to price dynamics, this suggests that the log returns are normally distributed with constant drift (return rate) and volatility.
10
For the price of a stock that follows a geometric Brownian motion, the dynamics of the asset price can be represented with the following stochastic differential equation:
11
(1.10)
where
is the price of the stock at time
t
,
is the Wiener process at time
,
is a drift rate, and
is the volatility of the stock. The drift rate and volatility of the stock are assumed to be constant, and it's important to reiterate that neither of these variables are directly observable. These constants can be approximated using the average return of a stock and the standard deviation of historical returns, but they can never be precisely known.
The equation states that each stock price increment
is driven by a predictable amount of drift (with expected return
) and some amount of random noise
. In other words, this equation has two components: one that models
deterministic
price trends
and one that models probabilistic price fluctuations
. The important takeaway from this observation is that inherent uncertainty is in the price of stock, represented with the contributions from the
Wiener process. Because the increments of a Wiener process are independent of one another, it also is common to assume that the weak EMH holds at minimum, in addition to the normality of log returns.
Using this equation as a basis for the derivation, assuming a riskless options portfolio must earn the risk‐free rate, and rearranging terms, the Black‐Scholes equation follows:
(1.11)
where
is the price of a European call (with a dependence on
and
),
is the price of the stock (with a dependence on
),
is the risk‐free rate, and
is the volatility of the stock. The Black‐Scholes formula can be calculated by solving the Black‐Scholes equation according to boundary conditions given by the payoff at expiration of European options. The formula, which provides the value of a European call option for a non‐dividend‐paying stock, is given by the following equation:
(1.12)
where
is the value of the standard normal cumulative distribution function at
and similarly for
,
T
is the time that the option will expire (
is the duration of the contract),
is the price of the stock at time
t
,
K
is the strike price of the option, and
and
are given by the following:
(1.13)
(1.14)
where
is the volatility of the stock. If the equations seem gross, it's because they are.
Again, the purpose of this section is not to describe the underlying mechanics of the Black‐Scholes model in detail. Rather,
Equatio


strike price of the option, and
and
are given by the following:
(1.13)
(1.14)
where
is the volatility of the stock. If the equations seem gross, it's because they are.
Again, the purpose of this section is not to describe the underlying mechanics of the Black‐Scholes model in detail. Rather,
Equations (1.10)
through
(1.14)
are included to emphasize three important points.
There is inherent uncertainty in the price of stock. Stock price movements are also assumed to be independent of one another and log‐normally distributed.
12
An estimate for the fair price of an option can be calculated according to the price of the stock, the volatility of the stock, the risk‐free rate, the duration of the contract, and the strike price.
The volatility of a stock, which plays an important role in estimating the risk of an asset and the valuation of an option, cannot be directly observed. This suggests that the “true risk” of an instrument can never be exactly known. Risk can only be approximated using a metric, such as historical volatility or the standard deviation of the historical returns over some timescale, typically matching the duration of the contract. Other than using a past‐looking metric, such as historical volatility to estimate the risk of an asset, one can also infer the risk of an asset from the price of its options.
As stated previously, the Black‐Scholes model only gives a
theoretical
estimate for the fair price of an option. Once the contract is traded on the options market, the price of the contract is often driven up or down depending on speculation and perceived risk. The deviation of an option's price from its theoretical value as a result of these external factors is indicative of
implied volatility
. When initially valuing an option, the historical volatility of the stock has been priced into the model. However, when the price of the option trades higher or lower than its theoretical value, this indicates that the
perceived
volatility of the underlying deviates from what is estimated by historical returns.
Implied volatility may be the most important metric in options trading. It is effectively a measure of the
sentiment
of risk for a given underlying according to the supply and demand for options contracts. For an example, suppose a non‐dividend‐paying stock currently trading at $100 per share has a historical 45‐day returns volatility of 20%. Suppose its call option with a 45‐day duration and a strike price of $105 is trading at
$2 per share. Plugging these parameters into the Black‐Scholes model, this call option should theoretically be trading at $1 per share. However, demand for this position has increased the contract price significantly. For the model to return a call price of $2 per share, the volatility of this underlying would have to be 28% (assuming all else is constant). Therefore, although the historical volatility of the underlying is only 20%, the perceived risk of that underlying (i.e., the implied volatility) is actually 28%.
To conclude, the primary purpose of this section was not to dive into the math of the Black‐Scholes. These concepts were, instead, introduced to justify the following axioms that are foundational to this book:
Profits cannot be made without risk.
Stock log returns have inherent uncertainty and are assumed to follow a normal distribution.
Stock price movements are independent across time (i.e., future price changes are independent of past price changes, requiring a minimum of the weak EMH).
Options can theoretically be priced fairly based on the price of the stock, the volatility of the stock, the risk‐free rate, the duration of the contract, and the strike price.
The volatility of an asset cannot be directly observed, only estimated using metrics like historical volatility or implied volatility.
The Greeks
Other than implied volatility, the Greeks are the most relevant metrics derived from the Black‐Scholes model. The Greeks are a set of risk measures, and each describes the sensitivity of an option's price with respect to changes in some variable. The most essential Greeks for options traders are delta
, gamma
, and theta
.
Delta
is one of the most important and widely used Greeks. It is a first‐order
13
Greek that measures the expected change in the option
price given a $1 increase in the price of the underlying (assuming all other variables stay constant). The equation is as follows:
(1.15)
where
V
is the price of the option (a call or a put) and
S
is the price of the underlying stock, noting that ∂ is the partial derivative. The value of delta ranges from –1 to 1, and the sign of delta depends on the type of position:
Long stock:
is 1.
Long call and short put:
is between 0 and 1.
Long put and short call:
is between –1 and 0.
For example, the price of a long call option with a delta of 0.50 (denoted 50
because that is the total
for a one lot, or 100 shares of underlying) will increase by approximately $0.50 per share when the price of the underlying increa


is between 0 and 1.
Long put and short call:
is between –1 and 0.
For example, the price of a long call option with a delta of 0.50 (denoted 50
because that is the total
for a one lot, or 100 shares of underlying) will increase by approximately $0.50 per share when the price of the underlying increases by $1. This makes intuitive sense because a long stock, a long call, and a short put are all bullish strategies, meaning they will profit when the underlying price increases. Similarly, because long puts and short calls are bearish, they will take a loss when the underlying price increases.
Delta has a sign and magnitude, so it is a measure of the
degree
of
directional risk
of a position. The sign of delta indicates the direction of the risk, and the magnitude of delta indicates the severity of exposure. The larger the magnitude of delta, the larger the profit and loss potential of the contract. This is because positions with larger deltas are closer to/deeper ITM and more sensitive to changes in the underlying price. A contract with a delta of 1.0 (100
) has maximal directional exposure and is maximally ITM. 100
options behave like the stock price, as a $1 increase in the underlying creates a $1 increase in the option's price per share. A contract with a delta of 0.0 has no directional exposure and is maximally OTM. A 50
contract is defined as having the ATM strike.
14
Because delta is a measure of directional exposure, it plays a large role when hedging directional risks. For instance, if a trader currently has a 50
position on and wants the position to be relatively insensitive to
directional moves in the underlying, the trader could offset that exposure with the addition of 50 negative deltas (e.g., two 25
long puts). The composite position is called delta neutral.
Gamma
is a second‐order Greek and a measure of the expected change in the option
delta
given a $1 change in the underlying price. Gamma is mathematically represented as follows:
(1.16)
As with delta, the sign of gamma depends on the type of position:
Long call and long put:
.
Short call and short put:
.
In other words, if there is a $1 increase in the underlying price, then the delta for all long positions will become more positive, and the delta for all short positions will become more negative. This makes intuitive sense because a $1 increase in the underlying pushes long calls further ITM, increasing the directional exposure of the contract, and it pushes long puts further OTM, decreasing the inverse directional exposure of the contract and bringing the negative delta closer to zero. The magnitude of gamma is highest for ATM positions and lower for ITM and OTM positions, meaning that delta is most sensitive to underlying price movements at –50
and 50
.
Awareness of gamma is critical when trading options, particularly when aiming for specific directional exposure. The delta of a contract is typically transient, so the gamma of a position gives a better indication of the long‐term directional exposure. Suppose traders wanted to construct a delta neutral position by pairing a short call (negative delta) with a short put (positive delta), and they are considering using 20
or 40
contracts (all other parameters identical). The 40
contracts are much closer to ATM (50
) and have more profit potential than the 20
positions, but they also have significantly more gamma risk and are less likely to remain delta neutral in the long term. The optimal choice would then depend on how much risk traders are willing to accept and their profit goals. For traders with high profit goals and a large enough account to handle the large P/L swings and loss potential of the trade, the 40
contracts are more suitable.
Theta
is a first‐order Greek that measures the expected P/L changes resulting from the decay of the option's extrinsic value (the difference between the current market price for the option and the intrinsic value of the option) per day. It is also commonly referred to as the time decay of the option. Theta is mathematically represented as follows:
(1.17)
where
V
is the price of the option (a call or a put) and
t
is time. The sign of theta depends on the type of position and is opposite gamma:
Long call and long put:
.
Short call and short put:
.
In other words, the time decay of the extrinsic value decreases the value of the long position and increases the value of the short position. For instance, a long call with a theta of –5 per one lot is expected to decline in value by $5 per day. This makes intuitive sense because the holders of the contract take gradual losses as their asset depreciates with time, a result of the value of the option converging to its intrinsic value as uncertainty dissipates. Because the extrinsic value of a contract decreases with time, the short side of the position profits with time and experiences positive time decay. The magnitude of theta is highest for ATM options and lower for ITM and OTM positions, all else constant.
There


insic value as uncertainty dissipates. Because the extrinsic value of a contract decreases with time, the short side of the position profits with time and experiences positive time decay. The magnitude of theta is highest for ATM options and lower for ITM and OTM positions, all else constant.
There is a trade‐off between the gamma and theta of a position. For instance, a long call with the benefit of a large, positive gamma will also be subjected to a large amount of negative time decay. Consider these examples:
Position 1:
A 45 DTE, 16
call with a strike price of $50 is trading on a $45 underlying. The long position has a gamma of 5.4 and a theta of –1.3.
Position 2:
A 45 DTE, 44
call with a strike price of $50 is trading on a $49 underlying. The long position has a gamma of 7.9 and a theta of –2.2.
Compared to the first position, the second position has more gamma exposure, meaning that the contract delta (and the contract price) is more sensitive to changes in the underlying price and is more likely to
move ITM. However, this position also comes with more theta decay, meaning that the extrinsic value also decreases more rapidly with time.
To conclude this discussion of the Black‐Scholes model and its risk measures, note that the outputs of all options pricing models should be taken with a grain of salt. Pricing models are founded on simplified assumptions of real financial markets. Those assumptions tend to become less representative in highly volatile market conditions when potential profits and losses become much larger. The assumptions and Greeks of the Black‐Scholes model can be used to form reasonable expectations around risk and return
in most market conditions
, but it's also important to supplement that framework with model‐free statistics.
Covariance and Correlation
Up until now we have discussed trading with respect to a single position, but quantifying the relationships between multiple positions is equally important. Covariance quantifies how two signals move relative to their means with respect to one another. It is an effective way to measure the variability between two variables. For one signal,
X
, with observations
and mean
, and another,
Y
, with observations
and mean
the covariance between the two signals is given by the following:
(1.18)
Represented in terms of random variables
X
and
Y,
this is equivalent to the following:
15
(1.19)
Simplified, covariance quantifies the tendency of the linear relationship between two variables:
A
positive
covariance indicates that the high values of one signal coincide with the high values of the other and likewise for the low values of each signal.
A
negative
covariance indicates that the high values of one signal coincide with the low values of the other and vice versa.
A covariance of zero indicates that no linear trend was observed between the two variables.
Covariance can be best understood with a graphical example. Consider the following ETFs with daily returns shown in the following figures: SPY (S&P 500), QQQ (Nasdaq 100), and GLD (Gold), TLT (20+ Year Treasury Bonds).
Figure 1.9
(a) QQQ returns versus SPY returns. The covariance between these assets is 1.25, indicating that these instruments tend to move similarly. (b) TLT returns versus SPY returns. The covariance between these assets is –0.48, indicating that they tend to move inversely of one another. (c) GLD returns versus SPY returns. The covariance between these assets is 0.02, indicating that there is not a strong linear relationship between these variables.
Covariance measures the direction of the linear relationship between two variables, but it does not give a clear notion of the
strength
of that relationship. Because the covariance between two variables is specific to the scale of those variables, the covariances between two sets of pairs are not comparable. Correlation, however, is a normalized covariance that indicates the direction
and
strength of the linear relationship, and it is also invariant to scale. For signals
with standard deviations
and covariance
, the correlation coefficient
(rho) is given by the following:
(1.20)
The correlation coefficient ranges from –1 to 1, with 1 corresponding to a perfect positive linear relationship, –1 corresponding to a perfect negative linear relationship, and 0 corresponding to no measured linear relationship. Revisiting the example pairs shown in
Figure 1.9
, the strength of the linear relationship in each case can now be evaluated and compared.
For
Figure 1.9
(a), QQQ returns versus SPY returns, the correlation between these assets is 0.88, indicating a strong, positive linear relationship. For
Figure 1.9
(b), TLT returns versus SPY returns, the correlation between these assets is –0.43, indicating a moderate, negative linear relationship. And for
Figure 1.9
(c), GLD returns versus SPY returns, the correlation between these assets is 0.00, indicating no measurable linear relationship between these variables. According to the correlation


the correlation between these assets is –0.43, indicating a moderate, negative linear relationship. And for
Figure 1.9
(c), GLD returns versus SPY returns, the correlation between these assets is 0.00, indicating no measurable linear relationship between these variables. According to the correlation values for the pairs shown, the strongest linear relationship is between SPY and QQQ because the magnitude of the correlation coefficient is largest.
The correlation coefficient plays a huge role in portfolio construction, particularly from a risk management perspective. Correlation quantifies the relationship between the directional tendencies of two assets. If portfolio assets have highly correlated returns (either positively or negatively), the portfolio is highly exposed to directional risk. To understand how correlation impacts risk, consider the additive property of variance. For two random variables
with individual variances
and covariance
, the
combined
variance is given by the following:
(1.21)
When combining two assets, the overall impact on the uncertainty of the portfolio depends on the uncertainties of the individual assets as well as the covariance between them. Therefore, for every new position that occupies additional portfolio capital, the covariance will increase portfolio uncertainty (high correlation), have little effect on portfolio uncertainty (correlation near zero), or reduce portfolio uncertainty (negative correlation).
Additional Measures of Risk
This chapter has introduced several measures for risk including historical volatility, implied volatility, and the option Greeks. Two additional metrics are worth noting and will appear throughout this text: beta
and conditional
value at risk (CVaR). Beta is a measure of systematic risk and specifically quantifies the volatility of the stock relative to that of the overall market, which is typically estimated with a reference asset, such as SPY. Given the market's returns,
, a stock with returns
has the following beta:
(1.22)
The volatility of a stock relative to the market can then be evaluated according to the following:
: The asset tends to move more than the market. (For example, if the beta of a stock is 1.5, then the asset will tend to move $1.50 for every $1 the market moves.)
: The asset movements tend to match those of the market.
: The asset is less volatile than the market. (For example, if the beta of a stock is 0.5, then the asset will be 50% less volatile than the market.)
: The asset has no systematic risk (market risk).
: The asset tends to move inversely to the market as a whole.
This metric is essential for portfolio management, where it is used in the formulation of beta‐weighted delta. This will be covered in more detail in
Chapter 7
.
Value at risk (VaR) is another distribution statistic that is especially useful when dealing with heavily skewed distributions. VaR is an estimate of the potential losses for a portfolio or position over a given time frame at a specific likelihood level based on historical behavior. For example, a position with a daily VaR of –$100 at the 5% likelihood level can expect to lose $100 (or more) in a single day at most 5% of the time. This means that the bottom 5% of occurrences on the historical daily P/L distribution are –$100 or worse. For a visualization, see the historical daily returns distribution for SPY in
Figure 1.10
.
Figure 1.10
SPY daily returns distribution from 2010–2021. Included is the VaR at the 5% likelihood level, indicating that SPY lost at most 1.65% of its value on 95% of all days.
For strategies with significant negative tail skew, VaR gives a numerical estimate for the extreme loss potential according to past tendencies. To place more emphasis on the negative tail of a distribution and determine a more extreme loss estimate, traders may use CVaR, otherwise
known as expected shortfall. CVaR is an estimate for the expected loss of portfolio or position if the extreme loss threshold (VaR) is crossed. This is calculated by taking the average of the distribution losses past the VaR benchmark. To see how VaR and CVaR compare for SPY returns, refer to
Figure 1.11
.
Figure 1.11
SPY daily returns distribution from 2010–2021. Included are VaR and CVaR at the 5% likelihood level. A CVaR of 2.7% indicates that SPY can expect an average daily loss of roughly 2.7% on the worst 5% of days.
The choice between using VaR and CVaR depends on the risk profile of the portfolio or position considered. CVaR is more sensitive to tail losses and provides a metric that is more conservative from the perspective of risk, which is more suitable for the kind of instruments focused on in this book.
Notes
1
In liquid markets, which will be discussed in
Chapter 5
, American and European options are mathematically very similar.
2
The future value of the option should be used, but for simplicity, this approximates the future value as the current price of the option. The future value of the option premium is


liquid markets, which will be discussed in
Chapter 5
, American and European options are mathematically very similar.
2
The future value of the option should be used, but for simplicity, this approximates the future value as the current price of the option. The future value of the option premium is the current value of the option multiplied by the time‐adjusted interest rate factor.
3
Population calculations are used for all the moments introduced throughout this chapter.
4
This is the sum of the squared differences between each data point and the distribution mean, normalized by the number of data points in the set.
5
The skew of the returns distribution is also used to estimate the directional risk of an asset. The fourth moment (kurtosis) quantifies how heavy the tails of a returns distribution are and is commonly used to estimate the outlier risk of an asset.
6
Dividends can be accounted for in variants of the original model.
7
This application of Wiener processes as well as their use in financial mathematics are due to them arising as the scaling limit of simple random walk. A simple random walk is a discrete process that takes independent
steps with probability
. The scaling limit is reached by shrinking the size of the steps while speeding up their rate in such a way that the process neither sits at its initial location nor runs off to infinity immediately.
8
Note that, unless stated or shown otherwise, the date ranges throughout this book generally end on the first of the final year. For the range shown here, the data begins on January 1, 2010 and ends on January 1, 2015.
9
Displacement along the X‐axis is the difference between the current horizontal location of the particle and the previous horizontal location of the particle for each step.
10
Simple returns will also be approximated as normally distributed throughout this book. Although this is not explicitly implied by the Black‐Scholes model, it is a fair and intuitive approximation in most cases because the difference between log returns and simple returns is typically negligible on daily timescales.
11
d
is a symbol used in calculus to represent a mathematical derivative. It equivalently represents an infinitesimal change in the variable it's applied to.
dS
(
t
) is merely a very small, incremental movement of the stock price at time
t
. ∂ is the partial derivative, which also represents a very small change in one variable with respect to variations in another.
12
The log function and log‐normal distribution are both covered in the appendix.
13
Order refers to the number of mathematical derivatives taken on the price of the option. Delta has a single derivative of
V
and is first‐order. Greeks of second‐order are reached by taking a derivative of first‐order Greeks.
14
In practice, the strike and underlying prices for 50Δ contracts tend to differ
slightly
due to strike skew.
15
The covariance of a variable with itself (e.g., Cov(X, X)) is merely the variance of the signal itself.


Chapter 2
The Nature of Volatility Trading and Implied Volatility
Traders often hedge against periods of extreme market volatility (either to the upside or the downside) using options. Options are effectively financial insurance, and they are priced according to similar principles as other forms of insurance. Premiums increase or decrease according to the
perceived
risk of a given underlying (a result of supply and demand for those contracts), just as the cost of hurricane insurance increases or decreases depending on the perceived risk of hurricanes in a given area. To quantify the perceived risk in the market, traders use implied volatility (IV).
Implied volatility is the value of volatility that would make the current market price for an option be the fair price for that option in a
given model, such as Black‐Scholes.
1
When options prices
increase
(i.e., there is more demand for insurance), IV increases accordingly, and when options prices decrease, IV decreases. IV is, thus, a proxy for the
sentiment
of market risk as it relates to supply and demand for financial insurance. IV gives the perceived
magnitude
of expected price movements; it is not directional.
2
Table 2.1
gives a numerical example.
Table 2.1
Two underlyings with the same price and put contracts on each underlying with identical parameters (number of shares, put strike, contract duration). The contract prices differ, indicating that these two instruments have different implied volatilities.
45‐day Put Contract
Underlying A
Underlying B
Underlying Price
$101
$101
Strike Price
$100
$100
Contract Price
$10
$5
The price of the put is around 10% of the stock price for underlying A and 5% of the stock price for underlying B. This suggests that there is more perceived uncertainty associated with the price of underlying A compared to underlying B. Equivalently, this indicates that the anticipated magnitude of future moves in the underlying price is larger for underlying A compared to underlying B.
Demand for options tends to increase when the historical volatility of an underlying increases unexpectedly, particularly with large moves to the downside. This means that IV tends to be positively correlated with historical volatility and negatively correlated with price. However, there are exceptions to this rule, as IV is based on the perceived risk and not on historical risk directly. IV may increase due to factors that are not directly
related to price movements, such as company‐specific uncertainty (earnings reports, silly tweets from the CEO) or larger‐scale macroeconomic uncertainty (political conflict, proposed legislative measures). This also means that volatility profiles vary significantly from instrument to instrument, which will be discussed more later in the chapter.
Similar to historical volatility, IV gives a one standard deviation range of annual returns for an instrument. Though historical volatility represents the realized
past volatility of returns
, IV is the approximation for
future volatility of returns
because it is based on how the market is using options to hedge against future price changes. While each option for an underlying has its own implied volatility, the “overall” IV of an asset is normally calculated from 30‐day options and is a rough annualized volatility forecast.
3
Example: An asset has a price of $100 and an IV of 0.10 (10%). Therefore, the asset is expected to move about 10% to the upside or the downside by the end of the following year. This means the ending price will most likely be between $90 and $110.
The volatility forecast can also be scaled to approximate the expected price across days, weeks, months, or longer. The equations used to calculate the expected price ranges of an asset over some forecasting period are given below.
4
(2.1)
(2.2)
These estimates of expected range will be used to formulate options strategies in future chapters. The time frame for the expected range is often scaled to match the contract duration. Most examples in this book will have a duration of 45 days to expiration (DTE) (or 33 trading days), so implied volatilities are typically multiplied by 0.35 to ensure forecasts match the duration of the contract.
The expected move cone is helpful to visualize this likely price range for an instrument according to market speculation. The width of the cone is calculated using
Equation (2.2)
and scales with the IV of the underlying. More specifically, the cones are wider in higher volatility environments and narrower when volatility is low and the expected range is tighter. Consider the expected move cones shown in
Figure 2.1
, corresponding to the expected price ranges for SPY.
Figure 2.1
(c) shows the realized price trajectory for SPY in December 2019, which stayed within its expected price range for the majority of the 45‐day duration. Prices tend to stay within their expected range more often than not, and the assumptions of the Black‐Scholes model can be used to develop a theoretical e

eBooks\The Unlucky Investor_s Guide to Options Trading\The Unlucky Investor_s Guide to Options Trading.epub#section:c02.xhtml


ows the realized price trajectory for SPY in December 2019, which stayed within its expected price range for the majority of the 45‐day duration. Prices tend to stay within their expected range more often than not, and the assumptions of the Black‐Scholes model can be used to develop a theoretical estimate for how often that should be.
Trading Volatility
An inconceivable number of factors affect prices in financial markets, which makes precisely forecasting price movements extremely difficult. Arguably, the most reliable way to form expectations around future price trends is using statistics from past price data and financial models. IV is derived from current options prices and the Black‐Scholes options pricing model, meaning that the Black‐Scholes assumptions can be used to add statistical context to the expected price range. More specifically, one can infer the likelihood of a stock price remaining within its IV‐derived price range because stock returns are assumed to be normally distributed. The one standard deviation range of the normal distribution encompasses 68.2% of event outcomes, so there is theoretically a 68.2% chance the price of an equity lands within its expected range. This probability can also be generalized over any timescale using
Equation (2.1)
.
Figure 2.1
(a) The 45‐day expected move cone for SPY in early 2019. The price of SPY was roughly $275, and the IV was around 19%, corresponding to a 45‐day expected price range of ±6.7% (
Equation (2.1)
) or ±$18 (
Equation (2.2)
). (b) The 45‐day expected move cone for SPY when IV was 12%. (c) The same expected move cone as (b) with the realized price over 45 days.
Example: An asset has a price of $100 and an IV of 0.10 (10%). The asset price is expected to remain between $90 and $110 by the end of the following year with 68% certainty. Equivalently, the asset price is expected to remain between $96 and $104 58 days from today with 68% certainty (calculated using
Equation (2.2)
).
However, historical data show that perceived uncertainty in the market (IV) tends to overstate the realized underlying price move more often than theory suggests. Though theory predicts that IV should overstate the realized move roughly only 68% of the time, market IV (estimated using the IV for SPY) overstated the realized move 87% of the time between 2016 and 2021. This means the price for SPY stayed within its expected price range more often than estimated. Realized moves were larger just 13% of the time, indicating that IV rarely understates the realized risk in the market. The
exact
degree to which IV tends to overstate realized volatility depends on the instrument. For example, consider the IV overstatement rates of the stocks and exchange‐traded funds (ETFs) in
Table 2.2
.
Table 2.2
IV overstatement of realized moves for six assets from 2016–2021. Assets include SPY (S&P 500 ETF), GLD (gold commodity ETF), SLV (silver commodity ETF), AAPL (Apple stock), GOOGL (Google stock), AMZN (Amazon stock).
Volatility Data (2016–2021)
Asset
IV Overstatement Rate
SPY
87%
GLD
79%
SLV
89%
AAPL
70%
GOOGL
79%
AMZN
77%
Different assets are more or less prone to stay within their expected move range depending on their unique risk profile. Stocks are subject to single‐company risk factors and tend to be more volatile. ETFs, which contain a variety of assets, are inherently diversified and tend to be less prone to dramatic price swings. For example, the S&P 500 includes Apple, but it also includes around 499 other companies. This means that a tech‐sector specific event will have a bigger impact on APPL compared to SPY. Commodities like gold and silver also tend to be less volatile than individual stocks, meaning they are less prone to spikes in IV and
have more predictable returns. Although the IV overstatement rates differ between instruments, one can conclude that
fear
of large price moves is usually greater than realized price moves in the market. So, how exactly can options traders capitalize on this knowledge of IV and IV overstatement?
Let's revisit the example of hurricane insurance. The price for hurricane insurance is proportional to the expected cost of potential hurricane damage in the area. These prices are based on historical hurricane activity and forecasts of future events, which may underestimate, overestimate, or match the realized outcomes. People who
sell
hurricane insurance initially collect premiums, with the value depending on the perceived risk of home damage. During uneventful hurricane seasons, most policies go unused, and insurers keep the majority of premiums initially collected. In the unlikely event that hurricane damage is
significantly
worse than expected in an area dense with policyholders, insurers take very large losses. Insurance companies essentially make small, consistent profits the majority of the time while being exposed to large, infrequent losses.
Financial insurance carries a similar risk‐reward trade‐off as sellers make small, consistent


in an area dense with policyholders, insurers take very large losses. Insurance companies essentially make small, consistent profits the majority of the time while being exposed to large, infrequent losses.
Financial insurance carries a similar risk‐reward trade‐off as sellers make small, consistent profits most of the time but run the risk of large losses in extreme circumstances. IV yields an approximate price range forecast for a given underlying with 68% certainty. This means there is a 68% chance that the calls with strikes at the upper end of the expected range and puts with strikes at the lower end will both expire with no intrinsic value. For example, if traders sold one call and one put with strikes along the expected move cone, they would theoretically profit with 68% certainty. If the underlying price were to move unexpectedly to the upside or the downside, however, the traders may take substantial losses.
Unlike sellers of hurricane insurance, options sellers have more room to strategize and more control over their risk‐reward profile. Premium sellers can choose when to sell insurance and how to construct contracts most likely to be profitable. Because IV is a proxy for the demand for options and the inflation of premium, it can be used to identify opportune times to sell insurance. Additionally, because IV can be used to estimate the most likely price range for a specific asset, premium sellers can use IV to structure those positions so they likely expire worthless,
like in the previous example. Options sellers (or short premium traders) have the long‐term statistical advantage over options buyers, with the trade‐off of exposure to unlikely, potentially significant losses. Because of that long‐term statistical advantage, short premium trading is the focus of this book, with the next chapter detailing the mechanics of trading based on implied volatility.
The States of VIX
SPY is frequently used as a proxy for the broader market. It is also a baseline underlying for the short options strategies in this book because it is highly diversified across market sectors and has minimal idiosyncratic risk factors. The CBOE Volatility Index (VIX) is meant to track the annualized IV for SPY and is derived from 30‐day index options. As SPY is a proxy for the broader market, the VIX, therefore, gauges the perceived risk of the broader market. For context, from 1990 to 2021, the VIX ranged from roughly 10 to a peak of just over 80 in March 2020 during the COVID‐19 pandemic.
5
Unlike equities, whose prices typically drift from their starting values over time, IV tends to revert back to a long‐term value following a cyclic trend. This is because equities are used to estimate the perceived value of a company, sector, or commodity, but IV tracks the uncertainty sentiment of the market, which can only stay elevated for so long. During typical bull market conditions, the VIX hovers at a relatively low value at or below its average of 18.5. This is known as a lull state. When market uncertainty rapidly increases for whatever reason, often in response to large sudden price changes, the VIX expands and spikes far above its steady‐state value. Once the market adjusts to the new volatility conditions or the volatile conditions dissipate, the VIX gradually contracts back to a lull state. To see an example of this cycle, refer to
Figure 2.2
.
Figure 2.2
The three phases of the VIX, using data from early 2017 to late 2018.
When comparing how often the VIX is in each state, one finds the following approximate rates:
Lull (70%): IV consistently remains below or near its long‐term average. This state occurs when market prices trend upward gradually and market uncertainty is consistently low.
Expansion (10%): IV expansion usually follows a prolonged lull period and is marked by expanding market uncertainty and typically large price moves in the underlying equity.
Contraction (20%): IV contraction follows an expansion and is marked by a continued decline in IV. A contraction turns into a lull when IV reverts back to its long‐term average.
Lull periods are most common and tend to be much longer than the average expansion or contraction period. Since 2000, the average lull period was more than three times the length of the average expansion or contraction. When expansions do happen, the higher the IV peak, the faster the VIX contracts. For example, according to data from 2005 to 2020, when the VIX contracted from 20 to 16 points (20% decrease), it
took an average of 75.3 trading days to do so. However, when the VIX contracted from 70 to 56 points (also a 20% decrease), it only took an average of four trading days.
Spikes in the VIX are generally caused by unprecedented market or worldwide events. For example, the VIX reached over 80 in November 2008 during the peak of the worldwide financial crisis and hit its all‐time high of 82.69 in March 2020 during the COVID‐19 pandemic. The VIX peak of 2020 was especially unprecedented as the


enerally caused by unprecedented market or worldwide events. For example, the VIX reached over 80 in November 2008 during the peak of the worldwide financial crisis and hit its all‐time high of 82.69 in March 2020 during the COVID‐19 pandemic. The VIX peak of 2020 was especially unprecedented as the first major spike due to COVID‐19 happened on February 28, 2020 when the VIX hit 40.11. This VIX high in 2020 had not been reached since February 2018, and it followed a 96‐day lull. On March 16, 2020, the VIX hit 82.69, making the 2020 VIX expansion one of the most rapid ever recorded.
Though contraction periods tend to be longer than expansions but much shorter than lulls, fairly long contractions tend to follow major sell‐offs or corrections. For example, the VIX contraction following the 2008 sell‐off lasted well over a year, and the contraction following the 2020 sell‐off lasted more than 10 months. This is normally because it takes time for the market (and specific subsectors) to revert to regular conditions following such broad macroeconomic shocks.
Premium sellers can potentially profit in any type of market, whether it be during volatility expansions (bearish), contractions (bullish/neutral), or lulls (neutral) if adopting an appropriate strategy for the volatility conditions. Generally, the most favorable trading state for selling premium is when IV contracts. This is because IV contracts when premium prices deflate, meaning that options traders who sold positions in high IV
6
are able to buy identical positions back in low IV at a lower price, thus profiting from the difference. Volatility expansions, on the other hand, have the potential to generate significant losses for short premium traders.
Volatility expansions tend to occur when there are large movements in the underlying price and uncertainty increases, causing options on that underlying to become more expensive. If traders sell premium during
an expansion period once IV is
already elevated
, then the traders can capitalize on higher premium prices and the increased likelihood of a volatility contraction. However, if traders sell premium during a lull period, when the expected range is tight, and volatility
transitions
into an expansion period, then those traders will likely take large losses from the underlying price moving far outside the expected range. Additionally, to close their positions early, traders must buy back their options for more than they received in initial credit and incur a loss from the difference.
Short premium traders can profit in any type of market, but the risk of significant losses for short premium traders is highest when volatility is
low
. Unexpected transitions from a volatility lull to an expansion do not happen often, but when they do happen, they can be detrimental to an account. It is still necessary to trade during these low‐IV periods because IV spends the majority of the time in this state, but risk management during this period is crucial. These risk management techniques will be outlined in the upcoming chapters.
This cyclic trend (lull, expansion, contraction, lull) is easily observable when looking at a relatively stable volatility index, such as the VIX. However, this trend, which we will describe as IV reversion, is present in some capacity for
all
IV signals.
IV Reversion
Certain types of signals tend to revert back to a long‐term value following a significant divergence. Although this concept cannot be empirically proven or disproven, the reversion of IV is a core assumption in options trading.
7
The reversion dynamics and the minimum IV level vary across instruments, but reversion is assumed to be present in
all
IV signals to some extent. To understand this, first consider the probability of large magnitude returns for four assets with different risk profiles: SPY, GLD, AAPL, and AMZN. A comparison of these probabilities is shown in
Table 2.3
.
Table 2.3
Rates thats different assets experienced daily returns larger than 1%, 3%, and 5% in magnitude. For example, there is a 22% chance that SPY returns more than 1% or less than –1% in a single day (according to past data).
Probability of Surpassing Daily Returns Magnitude (2015–2021)
Asset
> 1% Magnitude
> 3% Magnitude
> 5% Magnitude
SPY
22%
3%
0.8%
GLD
19%
1%
0.1%
AAPL
43%
9%
2%
AMZN
45%
10%
3%
Compared to assets like SPY and GLD, AMZN and AAPL are more volatile. These tech stocks experience large daily returns roughly three times as often as SPY and roughly 10 times as often as GLD. Each of these assets is subject to unique risk factors, but all are expected to have reverting IV signals nonetheless.
Figure 2.3
shows these volatility profiles graphically.
Figure 2.3
demonstrates how IV has tended to revert back to a long‐term baseline for each of the different assets, and it also demonstrates that elevated uncertainty is
unsustainable
in financial markets. Events may occur that spark fear in the market and drive up the demand for insurance, but as f


Figure 2.3
demonstrates how IV has tended to revert back to a long‐term baseline for each of the different assets, and it also demonstrates that elevated uncertainty is
unsustainable
in financial markets. Events may occur that spark fear in the market and drive up the demand for insurance, but as fear inevitably dissipates and the market adapts to the new conditions, IV deflates back down. This phenomenon has significant implications for short options traders. As stated in
Chapter 1
, it is controversial whether directional price assumptions are statistically valid or not as trading according to pricing forecasts has never been proven to consistently outperform the market. IV is assumed to eventually revert down following inflations from its stable volatility state unlike asset prices, which drift from their initial value with time. The timescale for these contractions is unpredictable, but this nonetheless indicates some statistical validity to make downward directional assumptions about volatility once it is elevated.
Figure 2.3
also shows how volatility profiles vary greatly across instruments. More volatile assets like Apple and Amazon stocks have higher IV averages, twice that of SPY and gold in this case, and experience expansion events more often. Single‐company factors, such as quarterly earnings reports, pending mergers, acquisitions, and executive changes can all cause volatility spikes not seen in diversified assets and portfolios. However, this increased volatility also comes with higher credits and more volatility contraction opportunities for premium sellers.
8
For an example of how the propensity for expansions and contractions differs between stocks with earnings and a diversified ETF, refer to
Figures 2.4
(a)–(c). Marked are the earnings report dates for each stock or the date when the company reported its quarterly profits (after‐tax net income).
Figure 2.3
IV indexes for different assets with their respective averages (dashed) from 2015–2021. Assets include (a) SPY (S&P 500 ETF), (b) GLD (gold commodity ETF), (c) AAPL (Apple stock), and (d) AMZN (Amazon stock).
Figure 2.4
Implied volatility indexes for different equities from 2017–2020 with earnings dates marked (if applicable). Assets include (a) AMZN (Amazon stock), (b) AAPL (Apple stock), and (c) SPY (S&P 500 ETF).
With tech stocks like AMZN and AAPL, it's common for IV to increase sharply prior to earnings and contract almost immediately afterward. The previous graphs show that sharp IV expansions happen less frequently with a more diversified market ETF, such as SPY. These figures indicate that when SPY does experience a volatility expansion, it generally takes much longer to contract. From 2017 to 2020, the VIX only rose above 35 two times and, in both situations, took roughly half a month to contract down to its original level. Meanwhile, volatility levels of AMZN and AAPL rose above 40 many times and even had a few spikes above 50, or in the case of AMZN, almost 60.
Takeaways
IV is a proxy for the sentiment of market risk derived from supply and demand. When options prices increase, IV increases; when options prices decrease, IV decreases. IV also gives the perceived magnitude of future movement, and it is not directional.
Demand for options tends to increase when the historical volatility of an underlying increases unexpectedly, particularly with large moves to the downside. IV tends to be positively correlated with historical volatility and negatively correlated with price, but it is ultimately based on the
perceived
market risk and not directly on price information.
IV can be used to estimate the expected price range of an instrument. IV gives a one standard deviation
expected
range because it is based on how the market is using options to hedge against future periods of volatility.
Because stock returns are assumed to be normally distributed, theoretically, there is a 68.2% chance the price of an equity lands within its expected range over a given time frame. However, historical data show that prices stay within their expected ranges more often than theoretically estimated. For example, market IV (estimated using the IV for SPY) overstated the realized move 87% of the time between 2016 and 2021.
Options sellers have the long‐term statistical advantage over options buyers, with the trade‐off of exposure to unlikely, potentially significant losses. Because IV is a proxy for the demand for options and the inflation of premium, it can be used to identify opportune times to sell insurance. Premium sellers can also use IV to structure positions so they are likely to expire worthless, the ideal outcome for the short position.
Volatility profiles differ significantly between assets, but all IV signals are assumed to revert back to some long‐term value following significant diversions. Stated differently, IV tends to contract back to a long‐term value following significant expansions from its lull volatility state. This phenomenon ind


rofiles differ significantly between assets, but all IV signals are assumed to revert back to some long‐term value following significant diversions. Stated differently, IV tends to contract back to a long‐term value following significant expansions from its lull volatility state. This phenomenon indicates that there is some degree of statistical validity when making downward directional assumptions about volatility once it's inflated.
Notes
1
Implied volatility (IV), like historical volatility, is a percentage and pertains to log returns. It is common to represent IV as either a decimal (0.X) or percentage (X%). An IV index, which is an instrument that tracks IV and will be introduced later in this chapter, is typically represented using points (X) but should be understood as a percentage (X%).
2
It is possible to get directional expected move information about an underlying by analyzing the IV across various strikes. This will be elaborated on more in the appendix.
3
IV yields a rough approximation for the expected price range, but this is not how the expected range is typically calculated on most trading platforms. Refer to the appendix for more information about how expected range is calculated more precisely. For the time being, we are using this simplified formula since it is most intuitive.
4
When ignoring the risk‐free rate, the expected price range over
T
days for a stock with price
S
and volatility σ can be estimated by
. The formula in
Equation (2.2)
is an approximation because, for small
x
values, e
x
≈ 1 +
x
. This approximation becomes less valid when
x
is large, meaning this expected range calculation is less accurate when IV is high. This will be explored more in the appendix.
5
Note that volatility indices, such as the VIX, will be represented using points but are meant to be understood as a percentage. For example, a VIX of 30 corresponds to an annualized implied volatility of 30%.
6
It's important to note that the threshold for high IV is different for every asset because each instrument is subject to unique risk factors. Evaluating IV can be difficult because there is so much variability between assets, but there will be a more in‐depth discussion of this in the following chapter.
7
The value that the signal reverts back to is roughly the long‐term mode of the distribution, or the volatility that has occurred most often historically.
8
Such underlyings can be used for earnings plays, which will be discussed in a
Chapter 9
.


Chapter 3
Trading Short Premium
Options are highly versatile instruments. They can be used to hedge the directional risk of a stock, or they can be used as a source of profits. As alluded to in the example of hurricane insurance, short premium positions can be used to generate small, consistent profits for those willing to accept the tail risk. The mechanics of short premium trading are subtle, but many of the core concepts can be introduced in an intuitive way with some simple gambling analogies. For example, when using options for profit generation (i.e., not risk mitigation), the long‐term performance of long and short options can be analogized with slot machines.
Buying Options for profit
is like playing the slot machines. Gamblers who play enough times may hit the jackpot and receive a huge payout. However, despite the potential payouts, most players average a loss in the long run because they are taking small losses the majority of the time. Investors who buy options are betting on large, often directional moves in the underlying asset. Those assumptions
may be correct and yield significant profits occasionally, but underlying prices ultimately stay within their expected ranges most of the time. This results in small, frequent losses on unused contracts and an average loss over time.
Selling Options for profit
is like owning the slot machines. Casino owners have the long‐run statistical advantage for every game, an edge particularly high for slots. Owners may occasionally pay out large jackpots, but as long as people play enough and the payouts are manageable, they are compensated for taking on this risk with nearly guaranteed profit in the long term. Similarly, because short options carry tail risk but provide small, consistent profits from implied volatility (IV) overstatement, then they should average a profit in the long run if risk is managed appropriately.
Long premium strategies have a high profit potential but cannot be consistently timed to ensure profit in the long term. This is because outlier underlying moves and IV expansions that benefit long premium positions are strongly linked to external events (such as natural disasters or political conflict), which are relatively difficult to reliably predict. Short premium strategies, on the other hand, profit more often and have the long‐term statistical advantage if investors manage risks appropriately.
Similar to the slot machine owner, a short premium trader must reduce the impact of outlier losses to reach a large number of occurrences (trades) and realize the positive long‐term averages. This is most effectively done by limiting position size and by adjusting portfolio exposure according to current market conditions. This chapter will, therefore, cover the following broader concepts in volatility trading:
Trading in high IV: Identifying favorable conditions for opening short premium trades.
Number of occurrences: Reaching the minimum number of trades required to achieve long‐term averages.
Portfolio allocation and position sizing: Establishing an appropriate level of risk for the given market conditions.
Active management and efficient capital allocation: Understanding the benefits of managing trades prior to expiration.
IV plays a crucial role in trading short premium. Remember that IV is a measure of the
sentiment
of uncertainty in the market. It is a proxy for the amount of
fear
among premium buyers (or
excitement
, depending on your personality) and a measure of
opportunity
for premium sellers. When market uncertainty increases, premium prices also increase, and premium sellers receive more compensation for being exposed to large losses. However, IV is also instrumental when managing exposure to extreme losses and establishing appropriate position sizes.
Background: A Note on Visualizing Option Risk
When discussing the risk‐reward trade‐off of trading short premium, it is helpful to contextualize concepts and statistics with respect to a specific strategy. The next few chapters will focus on a
short strangle
, an options strategy consisting of a short out‐of‐the‐money (OTM) call and a short OTM put:
A short OTM call (the right to buy an asset at a certain price) has a bearish directional assumption. The seller profits when the underlying price stays below the specified strike price.
A short OTM put (the right to sell an asset at a certain price) has a bullish directional assumption. The seller profits when the underlying price stays above the specified strike price.
These two contracts combine to form a strangle. This is an example of an
undefined risk
strategy, where the loss is theoretically unlimited. The short call has undefined risk because stock prices can increase indefinitely, meaning the potential loss to the upside is unknown. Though short puts technically cannot lose more than 100 times the strike price, this potential loss is large enough that they are also considered undefined risk. Defined risk strategies, where the maximum

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se stock prices can increase indefinitely, meaning the potential loss to the upside is unknown. Though short puts technically cannot lose more than 100 times the strike price, this potential loss is large enough that they are also considered undefined risk. Defined risk strategies, where the maximum loss is limited by the construction of the trade, have pros and cons that will be discussed in
Chapter 5
. For simplicity, the strangle is used to formulate most examples in this book.
Strangles have a neutral directional assumption for the contract seller, meaning it is typically profitable when the price of the underlying stays
within the range defined by the short call strike and short put strike. Investors often define the strikes of a strangle according to the expected range of the underlying price (or some multiple of the expected range) over the contract duration. The one standard deviation expected range can be approximated with the current implied volatility of the underlying, as shown in
Chapter 2
.
Figure 3.1
The price of SPY in the last five months of 2019. Included is the 45‐day expected move cone calculated from the IV of SPY in December 2019. The edges of the cones are labeled according to appropriate strikes for an example strangle.
Figure 3.1
shows that SPY was priced at roughly $315 around December 2019, when the current IV for SPY was 12% (corresponding to a VIX level of 12). This means the price for SPY was forecasted to move between –4.2% and +4.2% over the next 45 days with a 68% certainty. This is equivalent to a 45‐day forecast of the price of SPY staying between $302 and $328 approximately. A contract with a strike price corresponding to the
expected move range is approximately a 16
contract. In this scenario, a 45 days to expiration (DTE) short SPY call with a strike price of $328 is a –16
contract roughly, and a 45
DTE short SPY put with a strike price of $302 is approximately a 16
contract. The two positions combined form a delta‐neutral position known as a 45 DTE 16
SPY strangle.
1
The strangle buyer and seller are making different bets:
The strangle buyer assumes that SPY's price will move beyond expectation within the next 45 days, either to the upside or the downside. More specifically, the long strangle yields profit if the price of SPY significantly increases above $328 or decreases below $302 prior to expiration.
The strangle seller profits if the position expires when the underlying price is within or near its expected range or if the position is closed when the contract is trading for a cheaper price than when it was opened (IV contraction).
Because there is a 68% chance the underlying will stay within its expected range, the short position theoretically has a 68% chance of being profitable. However, since the underlying price tends to stay in its expected range more often than theoretically predicted, this results in the probability of profit (POP) of short strangles held to expiration being much higher.
For example, consider the profit and loss (P/L) distributions for the short 45 DTE 16
SPY strangle in
Figures 3.2
(a)–(c). These distributions were generated using historical options data and are useful for visualizing the long‐term risk‐reward profile and likely trade‐by‐trade outcomes for this type of contract. Each occurrence in the histogram corresponds to the final P/L of a short strangle held to expiration.
2
P/L can be represented as a raw dollar amount or as a percentage of initial credit (the fraction of option premium that the seller ultimately kept).
3
Figure 3.2
(a) Historical P/L distribution (% of initial credit) for short 45 DTE 16
SPY strangles, held to expiration from 2005–2021. (b) Historical P/L distribution ($) for short 45 DTE 16
SPY strangles, held to expiration from 2005–2021. (c) The same distribution as in (b) but zoomed in near $0. The percentage of occurrences on either side of $0 have been labeled.
Figure 3.2
(c) shows that 81% of occurrences are
positive
and only 19% are
negative
. This means this strategy has historically profited 81% of the time and only taken losses 19% of the time, significantly higher than the 68% POP that the simplified theory suggests. Over the long run, this strategy was
profitable
and averaged a P/L of $44 (or 28% of the initial credit) per trade. However, notice the P/L distributions for this strategy are highly skewed and carry significant tail risk. As shown in
Figure 3.2
(a), these tail losses are unlikely but could potentially amount
to –1,000% or even –4,000% of the initial credit. In other words, if a trader receives $100 in initial credit for selling a SPY strangle, there is a slim chance of losing upward of $4,000 on that trade according to historical behavior. This is the trade‐off for the high POPs of short premium strategies.
The possibility of outlier losses should not be surprising because placing a short premium trade is betting against large, unexpected price swings. For a relatively stable asset like SPY