name="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, normalized by the number of data points in the set.
5
The skew of the returns distribution is also used to estimate the directional risk of an asset. The fourth moment (kurtosis) quantifies how heavy the tails of a returns distribution are and is commonly used to estimate the outlier risk of an asset.
6
Dividends can be accounted for in variants of the original model.
7
This application of Wiener processes as well as their use in financial mathematics are due to them arising as the scaling limit of simple random walk. A simple random walk is a discrete process that takes independent
8
Note that, unless stated or shown otherwise, the date ranges throughout this book generally end on the first of the final year. For the range shown here, the data begins on January 1, 2010 and ends on January 1, 2015.
9
Displacement along the X‐axis is the difference between the current horizontal location of the particle and the previous horizontal location of the particle for each step.
10
Simple returns will also be approximated as normally distributed throughout this book. Although this is not explicitly implied by the Black‐Scholes model, it is a fair and intuitive approximation in most cases because the difference between log returns and simple returns is typically negligible on daily timescales.
11
12
The log function and log‐normal distribution are both covered in the appendix.
13
Order refers to the number of mathematical derivatives taken on the price of the option. Delta has a single derivative of
14
In practice, the strike and underlying prices for 50Δ contracts tend to differ
15
The covariance of a variable with itself (e.g., Cov(X, X)) is merely the variance of the signal itself.
