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correlation between these assets is 0.00, indicating no measurable linear relationship between these variables. According to the correlation values for the pairs shown, the strongest linear relationship is between SPY and QQQ because the magnitude of the correlation coefficient is largest.

      The correlation coefficient plays a huge role in portfolio construction, particularly from a risk management perspective. Correlation quantifies the relationship between the directional tendencies of two assets. If portfolio assets have highly correlated returns (either positively or negatively), the portfolio is highly exposed to directional risk. To understand how correlation impacts risk, consider the additive property of variance. For two random variables with individual variances and covariance , the combined variance is given by the following:

      (1.21)

      When combining two assets, the overall impact on the uncertainty of the portfolio depends on the uncertainties of the individual assets as well as the covariance between them. Therefore, for every new position that occupies additional portfolio capital, the covariance will increase portfolio uncertainty (high correlation), have little effect on portfolio uncertainty (correlation near zero), or reduce portfolio uncertainty (negative correlation).

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      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

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 steps with probability . The scaling limit is reached by shrinking the size of the steps while speeding up their rate in such a way that the process neither sits at its initial location nor runs off to infinity immediately.

8

Note that, unless stated or shown otherwise, the date ranges throughout this book generally end on the first of the final year. For the range shown here, the data begins on January 1, 2010 and ends on January 1, 2015.

9

Displacement along the X‐axis is the difference between the current horizontal location of the particle and the previous horizontal location of the particle for each step.

10

Simple returns will also be approximated as normally distributed throughout this book. Although this is not explicitly implied by the Black‐Scholes model, it is a fair and intuitive approximation in most cases because the difference between log returns and simple returns is typically negligible on daily timescales.

11

d is a symbol used in calculus to represent a mathematical derivative. It equivalently represents an infinitesimal change in the variable it's applied to. dS(t) is merely a very small, incremental movement of the stock price at time t. ∂ is the partial derivative, which also represents a very small change in one variable with respect to variations in another.

12

The log function and log‐normal distribution are both covered in the appendix.

13

Order refers to the number of mathematical derivatives taken on the price of the option. Delta has a single derivative of V and is first‐order. Greeks of second‐order are reached by taking a derivative of first‐order Greeks.

14

In practice, the strike and underlying prices for 50Δ contracts tend to differ slightly due to strike skew.

15

The covariance of a variable with itself (e.g., Cov(X, X)) is merely the variance of the signal itself.

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