Chapter 1
Math and Finance Preliminaries

The purpose of this book is to provide a qualitative framework for options investing based on a quantitative analysis of financial data and theory. Mathematics plays a crucial role when developing this framework, but it is predominantly a means to an end. This chapter therefore includes a brief overview of the prerequisite math and financial concepts required to understand this book. Because this isn't in‐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 a wide range of instruments, this book discusses fundamental trading concepts using a small selection of asset classes (stocks, exchange‐traded funds, and options) to formulate examples.

A share of stock is a security that represents a fraction of ownership of a corporation. Stock shares are normally issued by the corporation as a source 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 a fraction of the company's assets and profits based on the proportion of shares they own relative to the number of outstanding shares.

An exchange‐traded fund (ETF) is a basket 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 a fraction of ownership of a diversified 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 a market index ETF tracking the S&P 500, Energy Select Sector SPDR Fund (XLE) is a sector ETF tracking the energy sector, and SPDR Gold Trust (GLD) is a commodity ETF tracking gold. ETFs are typically much cheaper to trade than the individual assets in an ETF portfolio and are inherently diversified. For instance, a share of stock for an energy company is subject to company‐specific risk factors, while a share of an energy ETF is diversified over several energy companies.

When assessing the price dynamics of a stock or ETF and comparing the dynamics of different assets, it is common to convert price information into returns. The return of a stock is the amount the stock price increased or decreased as a proportion of its value rather than a dollar 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 a percentage and calculated using Equation (1.1), and log returns, calculated using Equation (1.2). The logarithm's mathematical 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.

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 a simple daily return of 0.01 (1%) and a log 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 a type 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 a stock or ETF, at a predetermined price on or before a future 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 options are calls and puts. American calls give the holder the right to buy the underlying asset at a certain price within a given time frame, and American puts give the holder the right to sell the underlying asset. The contract parameters must be specified prior to opening the trade and are listed below:

• The underlying asset trading at the spot price, or the current per share price .
• The number of underlying shares. One option usually covers 100 shares of the underlying, known as a one lot.
• The price at which the underlying shares can be bought or sold prior to expiration. This price is called the strike price .
• The expiration date, after which the contract is worthless. The time between the present day and the expiration date is the contract's duration or days to expiration (DTE).

Note that the price of the option is commonly denoted as C for calls, P for puts, and V if the type of contract is not specified. Options traders may buy or sell these contracts, and the conditions for profitability differ depending on the choice of position. The purchaser of the contract pays the option premium (current market price of the option) to adopt the long side of the position. This is also known as a long premium trade. The seller of the contract receives the option premium to adopt the short side of the position, thus placing a short premium trade. The choice of strategy corresponds to the directional assumption of the trader. For calls and puts, the directional assumption is either bullish, assuming the underlying price will increase, or bearish, assuming the underlying price will decrease. The directional assumptions and scenarios for profitability for these contracts are summarized in the following table.

Table 1.1 The definitions, conditions for profitability, and directional assumptions for long/short calls/puts.

The relationship between the strike price and the current price of the underlying determines the moneyness of the position. This is equivalently the intrinsic value of a position, or the value of the contract if it were exercised immediately. Contracts can be described as one of the following, noting that options cannot have negative intrinsic value:

• In‐the‐money (ITM): The contract would be profitable if it was exercised immediately and thus has intrinsic value.
• Out‐of‐the‐money (OTM): The contract would result in a loss if it was exercised immediately and thus has no intrinsic value.
• At‐the‐money (ATM): The contract has a strike price equal to the price of the underlying and thus has no intrinsic value.

The intrinsic value of a position is based entirely on the type of position and the choice of strike price relative to the price of the underlying:

• Call options
  • Intrinsic Value = Either (stock price – strike price) or 0
  • ITM:
  • OTM:
  • ATM:
• Put options
  • Intrinsic Value = Either or 0
  • ITM:
  • OTM:
  • ATM:

For example, consider a 45 DTE put contract with a strike price of $100:

• Scenario 1 (ITM): The...