o see some cash but with the possibility of trading away aserious profit. Again, it’sall about flexibility. In an example, for astock that’snow trading at $50, you could buy acall with astrike price of $55 and sell acall with astrike price of $60. You’ll likely pay more to buy your call than you gain from selling the second call. Let’ssay it was $0.25 for the $55 contract and $0.60 for the $60, leaving you paying $0.35 in total to set your position. For this to work best, you’re hoping that the stock will end up somewhere between $55 and $60 at the deadline because the second contract will not be exercised and you will make aprofit. If it rises above the $60, you’ll still make aprofit but it will be capped at that exact profit if your buyer exercises their right to purchase the stock. The downside is that your stock could skyrocket to $65 and you won’tsee aprofit above the $60 but that can be acceptable if you’re looking to cut down your costs and still make aprofit. The example above is abull spread. That can also work on abear spread if you reverse the trades and sell your call lower than you buy your call.
Cash Secured Puts:
That can be used as away of purchasing aparticular stock at adiscount. It only works if your account has enough money to actually buy the stock because you will be obliged to do so if the option is exercised. If it isn’t, you’ve made some money because it will expire without forcing you to buy but you’ll still bank the premium in the process. Either way, assuming you really do want that stock, you win. In thats strategy, you’ll set the strike price at the exact price you’re looking to obtain that stock for. The only downside is that it could drop alot lower and at that point you really won’tfeel like you got the best bargain. Out of the money puts have abetter chance of expiring without being exercised. If you’re only looking to make profit on the premium or you’re not desperate to own the underlying stock, that can often be your best bet. If you do end up owning the stock, your usual hope is that it will change direction and you can trade it to make another profit.
Married Puts:
To do that, purchase stock and aput at the same time. That provides an insurance for you and a “floor” to protect you if that stock suddenly plunges. It will make sure you don’tlose the clothes on your back if the stock does plummet but it also has the chance to make alittle money if your timing is good and the stock price rises.
Rolling Your Positions:
We covered that briefly but, just as areminder, rolling your position can help you increase your profit over time. When you do that, you simply set up anew call as soon as the old one expires in the hope that the stock will continue to move in the same direction it has been doing until now. You will be looking to go up in strike price and out in time to an expiration date. It can be risky because there is no guarantee that the stock will continue to do what it has been doing, so, it’sonly worth taking the risk if you think there is areasonable chance the stock will continue to move in the same direction. If you roll aput, on the other hand, you’re going down in strike price and out in time to deadline because you want to avoid actually selling the stock. For both those alternatives, you’ll be entering abuy to close order and initiating anew contract.
In Conclusion
From novice to initiated, you’ve now gained the basics of the knowledge that will help you enter the exciting world of options trading. It certainly isn’teverything there is to know, but you now have enough of agrounding to get started.
From here on out, it’sall about practice and being conservative as you improve your understanding and develop your own strategies. Only you will know what works best for you, how much risk you want to play with, and how your personal ability to predict and determine the stock market can be best put into practice.
As you dip your feet into the water, you’ll start to see profits coming in and you’ll feel that buzz that all options traders enjoy. The more you trade, the more you’ll see all of the fundamental mechanics at play and the more you’ll start to connect the dots and figure out your own personality as atrader.
You’re in for atreat. Options trading is rewarding and exciting when done right. Remember to keep that calendar updated and to stay conservative at least in the beginning and you’ll enjoy the learning curve every step of the way!
Special Thanks
Iwould like to give special thanks to all the readers from around the globe who chose to share their kind and encouraging words with me.
Knowing even just one person found this book helpful means the world to me.
If you've benefited from this book at all, Iwould be honored to have you share your thoughts on it, so that others would get something valuable out of this book too.
Your reviews are the fuel for my writing soul, and I'dbe
forever grateful
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Preface
If the conditions are just right, extraordinary things happen when many individual pieces come together: Water molecules organize and form snowflakes; Cells arrange and create organs; Jet streams combine and cause tornadoes; Grains of sand rally and produce avalanches; Investors panic sell and induce financial crashes. Complex systems are composed of many interacting parts, and
emergence
occurs when these parts organize to create collective phenomena that no one part is capable of creating alone. Complex systems can be found in nearly every discipline, and the mathematics describing emergent properties is not only fascinating but indicates fundamental similarities between seemingly unrelated complex systems. The extinction of aspecies of fly due to an invasive species of frog has really nothing to do with financial markets, yet the dynamics of the fly population undergoing ecological collapse look nearly indistinguishable from that of astock undergoing economic collapse. Many physicists gravitate toward finance because physical systems and financial systems can be analyzed with similar theoretical, statistical, and computational tools. It was my interest in those mathematical connections that drew me to finance initially. However, after placing my first trade at the start of the 2020 crash, Iquickly learned the
importance of financial intuition as well, particularly when trading options.
Atrader'sintuition comes from experience, but atrader can more efficiently build that intuition by supplementing market engagement with some basic trading philosophies. Many of the papers, books, and blogs Iread as anew options trader offered detailed coverage of options theory and its mathematics, but Inever encountered aresource that explicitly laid out the most essential elements of practical strategy development. Without asystem of core trading principles, applying financial theory, interpreting and analyzing data, and cultivating any sense of market intuition was challenging. However, once afoundation of options trading fundamentals was in place, overcoming the options learning curve became aconsiderably more manageable process. In my personal case, this foundation developed from conversations with my coworkers at tastytrade (most of which were debates with Anton Kulikov), watching options markets regularly, using options data and theory to build actionable strategies, and alot of trial and error. My goal in writing this book is to help new traders build their own intuitions more effectively by breaking down the philosophies that formed the basis for my own, beginning with abit of math and market theory and building from there. Nothing substitutes for experience, and investors' first options trade will likely teach them more than any book. However, it'smy hope this framework that Anton and Iorganized will allow new traders to enter the options market with confidence and gain meaningful value from their first trading experiences, in both the monetary and educational senses.
Julia Spina

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Introduction: Why Trade Options?
The house always wins
. This cautionary quote is certainly true, but it does not tell the entire story. From table limits to payout odds, every game in acasino is designed to give the house astatistical edge. The casino may take large, infrequent losses at the slot machines or small, frequent losses at the blackjack table, but as long as patrons play long enough, the house will inevitably turn aprofit. Casinos have long relied on this principle as the foundation of their business model: People can either bet
against
the house and hope that luck lands in their favor or
be
the house and have probability on their side.
Unlike casinos, where the odds are fixed against the players, liquid financial markets offer adynamic, level playing field with more room to strategize. However, similar to casinos, asuccessful trader does not rely on luck. Rather, traders' long‐term success depends on their ability to obtain aconsistent, statistical edge from the tools, strategies, and information available to them. Today'smarkets are becoming increasingly accessible to the average person, as online and commission‐free trading have basically become industry standards. Investors have access to an
almost unlimited selection of strategies, and options play an interesting role in this development. An option is atype of financial contract that gives the holder the right to buy or sell an asset on or before some future date, aconcept that will be explained more in the following chapter. Options have tunable risk‐reward profiles, allowing traders to reliably select the probability of profit, max loss, and max profit of aposition and potentially profit in any type of market (bullish, bearish, or neutral). These highly versatile instruments can be used to hedge risk and diversify aportfolio,
or
options can be structured to give more risk‐tolerant traders aprobabilistic edge.
In addition to being customizable according to specific risk‐reward preferences, options are also tradable with accounts of nearly any size because they are
leveraged
instruments. In the world of options, leverage refers to the ability to gain or lose more than the initial investment of atrade. An investor may pay $100 for an option and make $200 by the end of the trade, or an investor may make $100 by selling an option and lose $200 by the end of the trade. Leverage may seem unappealing because of its association with risk, but it is not inherently dangerous. When
misused
, leverage can easily wreak financial havoc. However, when used responsibly, the capital efficiency of leverage is apowerful tool that enables traders to achieve the same risk‐return exposure as astock position with significantly less capital.
There is no free lunch in the market. Aleveraged instrument that has a 70% chance of profiting must come with some trade‐off of risk, risk which may even be undefined in some cases. This is why the core principle of sustainable options trading is risk management. Just as casinos control the size of jackpot payouts by limiting the maximum amount aplayer can bet, options traders must control their exposure to potential losses from leveraged positions by limiting position size. And just as casinos diversify risk across different games with different odds, strategy diversification is essential to the long‐term success of an options portfolio.
Beyond the potential downside risk of options, other factors can make them unattractive to investors. Unlike equities, which are passive instruments, options require amore active trading approach due to their volatile nature and time sensitivity. Depending on the choice of strategies, options portfolios should be monitored anywhere from
daily to once every two weeks. Options trading also has afairly steep learning curve and requires alarger base of math knowledge compared to equities. Although the mathematics of options can easily become complicated and burdensome, for the type of options trading covered in this book, trading decisions can often be made with aselection of indicators and intuitive, back‐of‐the‐envelope calculations.
The goal of this book is to educate traders to make personalized and informed decisions that best align with their unique profit goals and risk tolerances. Using statistics and historical backtests, this book contextualizes the downside risk of options, explores the strategic capacity of these contracts, and emphasizes the key risk management techniques in building aresilient options portfolio. To introduce these concepts in astraightforward way, this book begins with discussion of the math and finance basics of quantitative options trading (
Chapter 1
), followed by an intuitive explanation of implied volatility (
Chapter 2
) and trading short premium (
Chapter 3
). With these foundational concepts covered, the book then moves onto trading in practice, beginning with buying power reduction and option leverage (
Chapter 4
), followed by trade construction (
Chapte

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Chapter 1
Math and Finance Preliminaries
The purpose of this book is to provide aqualitative
framework for options investing based on aquantitative
analysis of financial data and theory. Mathematics plays acrucial role when developing this framework, but it is predominantly ameans to an end. This chapter therefore includes abrief overview of the prerequisite math and financial concepts required to understand this book. Because this isn'tin‐depth coverage of the following topics, we encourage you to explore the supplemental texts listed in the references section for those mathematically inclined. Formulae and their descriptions are included in several sections for reference, but they are not necessary to follow the remainder of the book.
Stocks, Exchange‐Traded Funds, and Options
From swaptions to non‐fungible tokens (NFTs), new instruments and opportunities frequently emerge as markets evolve. By the time this book
reaches the shelf, the financial landscape and the instruments occupying it may be very different from when it was written. Rather than focus on awide range of instruments, this book discusses fundamental trading concepts using asmall selection of asset classes (stocks, exchange‐traded funds, and options) to formulate examples.
Ashare of
stock
is asecurity that represents afraction of ownership of acorporation. Stock shares are normally issued by the corporation as asource of funding, and these instruments are usually publicly traded on stock exchanges, such as the New York Stock Exchange (NYSE) and the Nasdaq. Shareholders are entitled to afraction of the company'sassets and profits based on the proportion of shares they own relative to the number of outstanding shares.
An
exchange‐traded fund (ETF)
is abasket of securities, such as stocks, bonds, or commodities. Like stocks, shares of ETFs are traded publicly on stock exchanges. Similar to mutual funds, these instruments represent afraction of ownership of adiversified portfolio that is usually managed professionally. These assets track aspects of the market such as an index, sector, industry, or commodity. For example, SPDR S&P 500 (SPY) is amarket index ETF tracking the S&P 500, Energy Select Sector SPDR Fund (XLE) is asector ETF tracking the energy sector, and SPDR Gold Trust (GLD) is acommodity ETF tracking gold. ETFs are typically much cheaper to trade than the individual assets in an ETF portfolio and are inherently diversified. For instance, ashare of stock for an energy company is subject to company‐specific risk factors, while ashare of an energy ETF is diversified over several energy companies.
When assessing the price dynamics of astock or ETF and comparing the dynamics of different assets, it is common to convert price information into returns. The return of astock is the amount the stock price increased or decreased as aproportion of its value rather than adollar amount. Returns can be scaled over any time frame (daily, monthly, annual), with calculations typically calling for daily returns. The two most common types of returns are simple returns, represented as apercentage and calculated using
Equation (1.1)
, and log returns, calculated using
Equation (1.2)
. The logarithm'smathematical definition and properties are covered in the appendix for those interested, but that information is not necessary to know to follow the remainder of the book.
(1.1)
(1.2)
where
is the price of the asset on day
and
is the price of the asset the prior day. For example, an asset priced at $100 on day 1 and $101 on day 2 has asimple daily return of 0.01 (1%) and alog return of 0.00995. Simple and log returns have different mathematical characteristics (e.g., log returns are time‐additive), which impact more advanced quantitative analysis. However, these factors are not relevant for the purposes of this book because the difference between log returns and simple returns is fairly negligible when working on daily timescales. Simple daily returns are used for all returns calculations shown.
An
option
is atype of financial derivative, meaning its price is based on the value of an underlying asset. Options contracts are either traded on public exchanges (exchange‐traded options) or traded privately with little regulatory oversight (over‐the‐counter [OTC] options). As OTC options are nonstandardized and usually inaccessible for retail investors, only exchange‐traded options will be discussed in this book.
An option gives the holder the right (but not the obligation) to buy or sell some amount of an underlying asset, such as astock or ETF, at apredetermined price on or before afuture date. The two most common styles of options are 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 o

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rican 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 acertain price within agiven 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 aone 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'sduration or days to expiration (DTE).
Note that the price of the option is commonly denoted as
Cfor calls,
Pfor puts, and
Vif 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 along premium trade. The seller of the contract receives the option premium to adopt the
short
side of the position, thus placing ashort 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 aposition, 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 aloss if it was exercised immediately and thus has no intrinsic value.
At‐the‐money (ATM): The contract has astrike price equal to the price of the underlying and thus has no intrinsic value.
The intrinsic value of aposition 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 astrike 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 astrike price of $100 on an underlying with acurrent price per share of $105. Suppose that, due to aperiod of recent market turbulence, investors are fearful the underlying price will crash within the next 45 days and create ademand for these OTM put contracts. The surge in demand inflates the price of the put contract to $10


underlying with acurrent price per share of $105. Suppose that, due to aperiod of recent market turbulence, investors are fearful the underlying price will crash within the next 45 days and create ademand 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 amarket 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 asample trade that applies the long call profit formula.
Example trade: Acall 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 aposition nears expiration, the price of an option converges toward its intrinsic value.
Options pricing clearly plays alarge 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 anumber of assumptions prior to placing atrade. Options traders must make directional assumptions about the price of the underlying over agiven 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 apersonal choice, but traders can formulate consistent assumptions by referring to the efficient market hypothesis (EMH). The EMH states that instruments are traded at afair 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 atrader 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 asemi‐strong form of the EMH. Rather than constructing portfolios according to forecasts of future price trends, the purpo


ble 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 asemi‐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 atrade‐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 aprofit 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 arandom 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 aprobabilistic system makes it possible to form expectations about the future, including the uncertainty associated with those expectations.
Let'sbegin with an example of asimple probabilistic system: rolling apair of fair, six‐sided dice. In this case, if
represents the sum of the dice, then
is arandom 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 asum of 7 ([1,6], [2,5], [3,4], [4,3], [5,2], [6,1]) than asum of 10 ([4,6], [5,5], [6,4]), there is ahigher 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 ahistogram. 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 aprobabilistic 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
Ahistogram for 100,000 simulated rolls for apair 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
adistribution. 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
asufficiently large number of occurrences
, the mean corresponds to the expected value of that distribution. The expected value of arandom variable is the weighted average of outcomes and the anticipated average outcome over future trials. The expected value of arandom 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 occurrin


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 ameasure 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 alarge 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 arandom 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 adistribution 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 adistribution, the third moment is used.
Skew
(third moment): This is ameasure of the asymmetry of adistribution. Adistribution'sskew 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 apure number that quantifies the degree of asymmetry according to the following formula:
(1.9)
Figure 1.2
Ahistogram 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 amodification to the dice rolling example. Suppose that the dice are biased rather than fair. Let'sconsider two scenarios: apair of unfair dice with asmall number bias (two and three more likely) and apair of unfair dice with alarge 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 askew of zero. However, the distributions when rolling the unfair dice are skewed, as shown in
Figures 1.3
(a) and (b).
The skew of adistribution 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 alonger 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
Figur


hat 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 alonger 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 alonger 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 adistribution 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 avalid 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 amean of 7.8 and astandard 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 alater chapter, as distributions of financial instruments are commonly skewed, and there is ambiguity in defining risk under those circumstances.
Figure 1.3
(a) Ahistogram 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) Ahistogram 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 agreat deal of study. Assuming the underlying distribution of an experiment resembles awell 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 aresult known as the central limit theorem. This theorem says, roughly, that if arandom 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 abell shape. This is shown in
Figure 1.4
.
The normal distribution is asymmetric, 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 anormal 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
Ahistogram for 100,000 simulated rolls with agroup 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 astock 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 aquantitative 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
Adetailed plot of the normal distribution and


lity estimation provides aquantitative 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
Adetailed 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'simportant 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 atheoretical value for the fair price of an option. In practice, an option'sprice 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 asuperficial grasp of the Black‐Scholes model to understand (1) the foundational assumptions of financial markets and (2) where implied volatility (agauge for the market'sperception
of risk) comes from.
The Black‐Scholes model is based on aset of assumptions related to the dynamics of financial assets and the market as awhole. 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, amacroeconomic 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 atype of stochastic process or asystem that experiences random fluctuations as it evolves with time. Traditionally used to describe the positional fluctuations of aparticle suspended in fluid at thermal equilibrium,
7
astandard 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 aprocess that follows arandom path. Each step in this path is probabilistic and independent of one another. When disjoint steps of equal duration are plotted in ahistogram, that distribution is normal with aconstant 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 aresult of the history of large price moves.
Figure 1.6
(a) The 2D position of aparticle in afluid, moving with Brownian motion. The particle begins at acoordinate of
and drifts to anew 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 displac


n of aparticle in afluid, moving with Brownian motion. The particle begins at acoordinate of
and drifts to anew 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 ahighly 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 astock 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 (aphenomenon known as heteroscedasticity). Stock returns are also not typically independent of one another across time (aphenomenon known as autocorrelation), which is arequirement 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 amean 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 alarger 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 aslight 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 astock that follows ageometric 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 adrift rate, and
is the volatility of the stock. The drift rate and volatility of the stock are assumed to be constant, and it'simportant to reiterate that neither of these variables are directly observable. These constants can be approximated using the average return of astock 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 apredictable 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 abasis for the derivation, assuming ariskless 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 adependence on
and
),
is the price of the stock (with adependence 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 anon‐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
,
Tis the time that the option will expire (
is the duration of the contract),
is the price of the stock at time
t
,
Kis 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'sbecause 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 t


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 astock, 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 ametric, 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 apast‐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 atheoretical
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'sprice from its theoretical value as aresult 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 ameasure of the
sentiment
of risk for agiven underlying according to the supply and demand for options contracts. For an example, suppose anon‐dividend‐paying stock currently trading at $100 per share has ahistorical 45‐day returns volatility of 20%. Suppose its call option with a 45‐day duration and astrike 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 acall 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 anormal distribution.
Stock price movements are independent across time (i.e., future price changes are independent of past price changes, requiring aminimum 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 aset of risk measures, and each describes the sensitivity of an option'sprice 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 afirst‐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
Vis the price of the option (acall or aput) and
Sis 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 along call option with adelta of 0.50 (denoted 50
because that is the total
for aone 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 along stock, along call, and ashort put are all bullish strategies, meaning they will profit when the underlying price increases. Similarly, becau


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 along stock, along call, and ashort 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 aloss when the underlying price increases.
Delta has asign and magnitude, so it is ameasure of the
degree
of
directional risk
of aposition. 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. Acontract with adelta 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'sprice per share. Acontract with adelta 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 ameasure of directional exposure, it plays alarge role when hedging directional risks. For instance, if atrader 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 asecond‐order Greek and ameasure 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 acontract is typically transient, so the gamma of aposition gives abetter indication of the long‐term directional exposure. Suppose traders wanted to construct adelta neutral position by pairing ashort call (negative delta) with ashort 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 alarge enough account to handle the large P/Lswings and loss potential of the trade, the 40
contracts are more suitable.
Theta
is afirst‐order Greek that measures the expected P/Lchanges resulting from the decay of the option'sextrinsic 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
Vis the price of the option (acall or aput) and
tis 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, along call with atheta 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, aresult of the value of the option converging to its intrinsic value as uncertainty dissipates. Because the extrinsic value of acontract 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 atrade‐off between the gamma and theta of aposition. For instance, along call with the benefit of alarge, positive gamma will also be subjected to alarge amount of negative time decay. Consider these examples:
Position 1:
A 4


M options and lower for ITM and OTM positions, all else constant.
There is atrade‐off between the gamma and theta of aposition. For instance, along call with the benefit of alarge, positive gamma will also be subjected to alarge amount of negative time decay. Consider these examples:
Position 1:
A 45 DTE, 16
call with astrike price of $50 is trading on a $45 underlying. The long position has agamma of 5.4 and atheta of –1.3.
Position 2:
A 45 DTE, 44
call with astrike price of $50 is trading on a $49 underlying. The long position has agamma of 7.9 and atheta 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 agrain 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'salso important to supplement that framework with model‐free statistics.
Covariance and Correlation
Up until now we have discussed trading with respect to asingle 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
Xand
Y,
this is equivalent to the following:
15
(1.19)
Simplified, covariance quantifies the tendency of the linear relationship between two variables:
Apositive
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.
Anegative
covariance indicates that the high values of one signal coincide with the low values of the other and vice versa.
Acovariance of zero indicates that no linear trend was observed between the two variables.
Covariance can be best understood with agraphical 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 astrong linear relationship between these variables.
Covariance measures the direction of the linear relationship between two variables, but it does not give aclear 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 anormalized 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 aperfect positive linear relationship, –1 corresponding to aperfect 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 astrong, positive linear relationship. For
Figure 1.9
(b), TLT returns versus SPY returns, the correlation between these assets is –0.43, indicating amoderate, 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 ahuge role in portfolio construction, particularly from arisk managem


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 ahuge role in portfolio construction, particularly from arisk 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 ameasure of systematic risk and specifically quantifies the volatility of the stock relative to that of the overall market, which is typically estimated with areference asset, such as SPY. Given the market'sreturns,
, astock with returns
has the following beta:
(1.22)
The volatility of astock 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 astock 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 astock 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 awhole.
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 aportfolio or position over agiven time frame at aspecific likelihood level based on historical behavior. For example, aposition with adaily VaR of –$100 at the 5% likelihood level can expect to lose $100 (or more) in asingle day at most 5% of the time. This means that the bottom 5% of occurrences on the historical daily P/Ldistribution are –$100 or worse. For avisualization, 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 anumerical estimate for the extreme loss potential according to past tendencies. To place more emphasis on the negative tail of adistribution and determine amore 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 ametric 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 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, no


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 areturns 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. Asimple random walk is adiscrete 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 away 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 afair and intuitive approximation in most cases because the difference between log returns and simple returns is typically negligible on daily timescales.
11
dis asymbol used in calculus to represent amathematical derivative. It equivalently represents an infinitesimal change in the variable it'sapplied to.
dS
(
t
) is merely avery small, incremental movement of the stock price at time
t
. ∂ is the partial derivative, which also represents avery 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 asingle derivative of
Vand is first‐order. Greeks of second‐order are reached by taking aderivative 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 avariable 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 agiven underlying (aresult 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 agiven 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 agiven 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, aproxy 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 anumerical 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 Aand 5% of the stock price for underlying B. This suggests that there is more perceived uncertainty associated with the price of underlying Acompared to underlying B. Equivalently, this indicates that the anticipated magnitude of future moves in the underlying price is larger for underlying Acompared 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 aone 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 arough annualized volatility forecast.
3
Example: An asset has aprice 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 aduration 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 atheoretical estimate for ho

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zed 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 atheoretical 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 astock 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 aprice 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 avariety 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 atech‐sector specific event will have abigger 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'srevisit 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 asimilar risk‐reward trade‐off as sellers make small, consistent profits most of the


h 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 asimilar 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 agiven 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 aproxy 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 aspecific 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 aproxy for the broader market. It is also abaseline 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 aproxy 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 apeak 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 along‐term value following acyclic trend. This is because equities are used to estimate the perceived value of acompany, 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 arelatively low value at or below its average of 18.5. This is known as alull 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 alull 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 aprolonged 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 acontinued decline in IV. Acontraction turns into alull 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 first major spike due to COVID‐19 hap


t 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 ayear, 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 alower 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 avolatility contraction. However, if traders sell premium during alull 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 aloss 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 avolatility 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 arelatively 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 along‐term value following asignificant divergence. Although this concept cannot be empirically proven or disproven, the reversion of IV is acore 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. Acomparison 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 asingle 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 along‐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