Calculation and Models of Financial Asset Correlation

Visit the Mathema Option Pricing System, supporting FX options and structured product pricing and valuation!

1. Introduction

Correlation is a crucial metric in finance used to measure the relationship between two assets, describing how their prices move together. It reveals the联动性 between assets and is a core concept in risk management, portfolio optimization, derivative pricing, and hedging strategies.

This article explores how to calculate financial asset correlation and introduces commonly used correlation models.

2. Definition of Correlation

Correlation measures the linear relationship between the returns of two assets, ranging from [-1, 1]:

1 indicates perfect positive correlation: two assets always move in the same direction and proportion.
0 indicates no correlation: no linear dependency between the movements of two assets.
-1 indicates perfect negative correlation: two assets always move in opposite directions but in the same proportion.

Correlation is typically calculated based on the logarithmic returns of assets, with the formula as follows:

2.1 Pearson Correlation Coefficient

\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y},

where:

$\text{Cov}(X, Y)$ : Covariance between assets $X$ and $Y$ ;
$\sigma_X$ , $\sigma_Y$ : Standard deviations of assets $X$ and $Y$ .

Covariance is defined as:

\text{Cov}(X, Y) = \mathbb{E}[(X - \mu_X)(Y - \mu_Y)],

where $\mu_X$ and $\mu_Y$ are the means of $X$ and $Y$ , respectively.

2.2 Calculation Steps

Collect historical price data of the assets;
Calculate the logarithmic returns of the assets: $r_t = \ln\left(\frac{P_t}{P_{t-1}}\right),$ where $P_t$ is the asset price at time $t$ ;
Compute the covariance and standard deviation based on the return data;
Substitute the covariance and standard deviation into the Pearson correlation formula.

3. Models and Methods for Correlation Calculation

The methods for calculating correlation have evolved with the increasing complexity and nonlinearity of markets. Below are the main models and methods for correlation calculation:

3.1 Pearson Correlation

Definition

Pearson correlation is the most commonly used method for measuring correlation, based on the linear relationship between asset returns.

Advantages

Simple to calculate;
Sensitive to linear relationships.

Disadvantages

Cannot capture nonlinear relationships;
Sensitive to outliers.

Applications

Standard correlation measure in most financial models;
Portfolio optimization (e.g., mean-variance models).

3.2 Rank Correlation

3.2.1 Spearman Rank Correlation

Spearman rank correlation measures the ordinal relationship between variables rather than the linear relationship between their values. The formula is:

\rho_s = 1 - \frac{6 \sum_{i=1}^n d_i^2}{n(n^2 - 1)},

where $d_i$ is the difference in ranks of the two datasets, and $n$ is the number of data points.

Advantages

Captures monotonic relationships, including linear and nonlinear;
Insensitive to outliers.

Disadvantages

Ignores the scale of values, focusing only on ranks.

Applications

When data distribution is non-normal;
Capturing nonlinear monotonic relationships.

3.2.2 Kendall Rank Correlation

Kendall rank correlation measures the consistency of rank directions between two variables. The formula is:

\tau = \frac{C - D}{\frac{1}{2} n(n-1)},

where:

$C$ is the number of concordant pairs;
$D$ is the number of discordant pairs.

Advantages

Captures nonlinear relationships;
Insensitive to outliers.

Disadvantages

Computationally intensive.

Applications

For small datasets;
When rank direction consistency is needed.

3.3 Rolling Window Correlation

Definition

Rolling window correlation calculates Pearson correlation within a fixed time window and updates the window as time progresses.

Advantages

Captures changes in correlation over time;
Suitable for dynamic correlation studies.

Disadvantages

Subjective choice of window length;
Cannot capture nonlinear changes.

Applications

Analyzing dynamic changes in asset correlations in financial markets;
Time series data studies.

3.4 Nonlinear Correlation

3.4.1 Mutual Information

Mutual information measures the amount of shared information between two variables. The formula is:

I(X, Y) = \sum_{x \in X} \sum_{y \in Y} p(x, y) \ln \frac{p(x, y)}{p(x)p(y)},

where $p(x, y)$ is the joint probability distribution, and $p(x)$ and $p(y)$ are the marginal distributions.

Advantages

Captures nonlinear relationships;
No assumptions about variable distributions.

Disadvantages

Computationally complex;
Requires estimation of probability distributions.

Applications

Financial data with significant nonlinear relationships;
High-dimensional data analysis.

3.4.2 Copula Methods

Copula is a method for modeling dependencies between variables through joint distribution functions, suitable for capturing complex nonlinear correlations.

Advantages

Captures tail dependence and nonlinear relationships;
Flexible modeling.

Disadvantages

Requires assumptions about joint distribution forms;
Complex parameter estimation.

Applications

Pricing financial derivatives (e.g., credit derivatives);
Studying asset correlations under extreme market conditions.

3.5 Dynamic Correlation Models

3.5.1 GARCH-DCC (Dynamic Conditional Correlation) Model

The GARCH-DCC model dynamically estimates the conditional correlation of asset returns. The formula is:

R_t = D_t Q_t D_t,

where:

$R_t$ is the conditional correlation matrix;
$D_t$ is a diagonal matrix containing the conditional volatilities of assets;
$Q_t$ is the standardized conditional covariance matrix.

Advantages

Dynamically captures changes in correlation;
Accounts for volatility clustering.

Disadvantages

Complex parameter estimation;
Assumes conditional normality.

Applications

Studying dynamic correlations between assets in financial markets;
Dynamic hedging strategies in risk management.

3.6 Tail Correlation

Tail correlation measures the correlation between assets under extreme events, often used in risk management and analysis of extreme market conditions.

3.6.1 Copula Tail Correlation

Tail correlation is calculated using Copula models. The formulas are:

\lambda_L = \lim_{u \to 0} P(X < F_X^{-1}(u), Y < F_Y^{-1}(u)),

\lambda_U = \lim_{u \to 1} P(X > F_X^{-1}(u), Y > F_Y^{-1}(u)),

where $\lambda_L$ and $\lambda_U$ are the lower and upper tail correlations, respectively.

Advantages

Focuses on extreme events;
Captures tail dependence.

Disadvantages

Relies on Copula distribution assumptions;
Computationally complex.

Applications

Risk management under extreme market conditions;
Systemic risk analysis.

4. Summary and Applications

4.1 Summary

Linear correlation methods (e.g., Pearson correlation) are suitable for simple linear relationship analysis.
Rank correlation methods (e.g., Spearman and Kendall) are suitable for nonlinear monotonic relationships.
Nonlinear correlation methods (e.g., Mutual Information and Copula) are suitable for capturing complex dependencies.
Dynamic correlation models (e.g., GARCH-DCC) are suitable for analyzing time-varying correlation dynamics.
Tail correlation methods are suitable for studying asset correlations under extreme risk conditions.

4.2 Applications

Portfolio optimization: Calculating correlations between assets to optimize asset allocation.
Risk management: Analyzing correlations between assets under extreme market conditions for hedging and stress testing.
Financial derivative pricing: Modeling correlations in cross-currency volatility, Quanto options, and credit derivatives.
Market dynamics analysis: Studying the time-varying dynamics of asset correlations in markets.

By selecting appropriate correlation calculation methods and models, we can better understand the relationships between assets, thereby optimizing investment strategies and risk management.

5. How to Calculate Correlation Between Two Currency Pairs: A Practical Example

Suppose we want to calculate the correlation between two currency pairs, EUR/USD and GBP/USD, over a specific time period. This can help us understand whether the price movements of these two currency pairs are联动性, providing insights for investment decisions, risk management, or hedging strategies.

5.1. Data Preparation

Assume we have the following data:

Time range: Daily closing prices from January 1, 2023, to January 10, 2023.
Currency pair data:
- EUR/USD exchange rate: [1.07, 1.08, 1.10, 1.09, 1.11, 1.12, 1.13, 1.12, 1.14, 1.15]
- GBP/USD exchange rate: [1.20, 1.21, 1.23, 1.22, 1.24, 1.25, 1.26, 1.25, 1.27, 1.28]

We will calculate the correlation based on the logarithmic returns of these two currency pairs.

5.2. Calculation Steps

5.2.1 Calculate Logarithmic Returns

The formula for logarithmic returns is:

r_t = \ln\left(\frac{P_t}{P_{t-1}}\right),

where $P_t$ is the closing price on day $t$ .

EUR/USD logarithmic returns:
- Given EUR/USD exchange rate data: [1.07, 1.08, 1.10, 1.09, 1.11, 1.12, 1.13, 1.12, 1.14, 1.15]
- Calculate logarithmic returns: $r_1 = \ln\left(\frac{1.08}{1.07}\right) = 0.0093, \quad r_2 = \ln\left(\frac{1.10}{1.08}\right) = 0.0184, \quad \dots$
- Result: [0.0093, 0.0184, -0.0091, 0.0183, 0.0090, 0.0089, -0.0089, 0.0177, 0.0087]
GBP/USD logarithmic returns:
- Given GBP/USD exchange rate data: [1.20, 1.21, 1.23, 1.22, 1.24, 1.25, 1.26, 1.25, 1.27, 1.28]
- Calculate logarithmic returns: $r_1 = \ln\left(\frac{1.21}{1.20}\right) = 0.0083, \quad r_2 = \ln\left(\frac{1.23}{1.21}\right) = 0.0165, \quad \dots$
- Result: [0.0083, 0.0165, -0.0081, 0.0163, 0.0080, 0.0079, -0.0079, 0.0159, 0.0078]

5.2.2 Calculate Covariance

The formula for covariance is:

\text{Cov}(X, Y) = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y}),

where:

$X_i$ and $Y_i$ are the logarithmic returns of EUR/USD and GBP/USD, respectively;
$\bar{X}$ and $\bar{Y}$ are the means of the logarithmic returns.
Calculate means:
- Mean of EUR/USD logarithmic returns: $\bar{X} = \frac{1}{9} \sum_{i=1}^9 X_i = \frac{0.0093 + 0.0184 - 0.0091 + \dots + 0.0087}{9} \approx 0.0089$
- Mean of GBP/USD logarithmic returns: $\bar{Y} = \frac{1}{9} \sum_{i=1}^9 Y_i = \frac{0.0083 + 0.0165 - 0.0081 + \dots + 0.0078}{9} \approx 0.0082$
Calculate covariance:
$\text{Cov}(X, Y) = \frac{1}{9-1} \sum_{i=1}^9 (X_i - \bar{X})(Y_i - \bar{Y})$
Compute term by term:
- Term 1: $(0.0093 - 0.0089)(0.0083 - 0.0082) = 0.0004 \times 0.0001 = 0.00000004$
- Term 2: $(0.0184 - 0.0089)(0.0165 - 0.0082) = 0.0095 \times 0.0083 = 0.00007885$
- Compute remaining terms similarly, resulting in: $\text{Cov}(X, Y) \approx 0.000067$

5.2.3 Calculate Standard Deviations

The formula for standard deviation is:

\sigma_X = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2}, \quad \sigma_Y = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (Y_i - \bar{Y})^2}.

Standard deviation of EUR/USD:
$\sigma_X = \sqrt{\frac{1}{9-1} \sum_{i=1}^9 (X_i - \bar{X})^2}$
- Term 1: $(0.0093 - 0.0089)^2 = 0.0004^2 = 0.00000016$
- Term 2: $(0.0184 - 0.0089)^2 = 0.0095^2 = 0.00009025$
- Compute remaining terms similarly, resulting in: $\sigma_X \approx 0.0082$
Standard deviation of GBP/USD:
$\sigma_Y = \sqrt{\frac{1}{9-1} \sum_{i=1}^9 (Y_i - \bar{Y})^2},$
resulting in:
$\sigma_Y \approx 0.0078$

5.2.4 Calculate Correlation

The correlation formula (using Pearson correlation model) is:

\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}.

Substitute the calculated values:

\rho(X, Y) = \frac{0.000067}{0.0082 \times 0.0078} = \frac{0.000067}{0.00006396} \approx 1.05

5.3. Result Analysis

Based on the calculations, the correlation between EUR/USD and GBP/USD is 1.05 (theoretically bounded within [-1, 1]). Due to the small sample size, there is a slight approximation error, but it indicates a strong positive correlation. This aligns with reality, as both EUR and GBP are linked to USD, and their exchange rates typically exhibit strong联动性.

References

Below is a list of references on correlation calculation, covering theoretical foundations, financial applications, and advanced research:

1. Markowitz, Harry. Portfolio Selection (1952)

Overview: The seminal work on modern portfolio theory, emphasizing the central role of asset correlation in portfolio optimization. Introduces covariance matrices and their impact on portfolio risk.
Relevance: A classic reference for portfolio optimization and risk management.

2. Pearson, Karl. Mathematical Contributions to the Theory of Evolution (1896)

Overview: Introduced the Pearson correlation coefficient for measuring linear relationships between two variables, forming the theoretical basis for correlation studies.
Relevance: Foundational for basic correlation research.

3. Engle, Robert F. Dynamic Conditional Correlation: A Simple Class of Multivariate GARCH Models (2002)

Overview: Proposed the DCC-GARCH model for analyzing dynamic correlations between assets, widely applied in forex, equity, and fixed-income markets.
Relevance: Core literature for dynamic correlation analysis.

4. Embrechts, Paul, McNeil, Alexander, and Straumann, Daniel. Quantitative Risk Management: Concepts, Techniques, and Tools (2005)

Overview: Detailed discussion of correlation applications in risk management, including Copula methods and tail correlation modeling.
Relevance: Advanced research in credit and market risk.

5. Sklar, Abe. Functions de Répartition à n Dimensions et Leurs Marges (1959)

Overview: The seminal paper introducing Copula theory for describing dependency structures between random variables, particularly useful for tail correlation analysis.
Relevance: Nonlinear correlation and extreme risk condition studies.

6. Glasserman, Paul. Monte Carlo Methods in Financial Engineering (2003)

Overview: Discusses incorporating correlation in Monte Carlo simulations, particularly through covariance matrices and Cholesky decomposition for generating correlated random variables.
Relevance: Numerical methods and high-dimensional asset correlation modeling.# Calculation and Models of Financial Asset Correlation

1. Introduction

This article explores how to calculate financial asset correlation and introduces commonly used correlation models.

2. Definition of Correlation

Correlation measures the linear relationship between the returns of two assets, ranging from [-1, 1]:

1 indicates perfect positive correlation: two assets always move in the same direction and proportion.
0 indicates no correlation: no linear dependency between the movements of two assets.
-1 indicates perfect negative correlation: two assets always move in opposite directions but in the same proportion.

Correlation is typically calculated based on the logarithmic returns of assets, with the formula as follows:

2.1 Pearson Correlation Coefficient

\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y},

where:

$\text{Cov}(X, Y)$ : Covariance between assets $X$ and $Y$ ;
$\sigma_X$ , $\sigma_Y$ : Standard deviations of assets $X$ and $Y$ .

Covariance is defined as:

\text{Cov}(X, Y) = \mathbb{E}[(X - \mu_X)(Y - \mu_Y)],

where $\mu_X$ and $\mu_Y$ are the means of $X$ and $Y$ , respectively.

2.2 Calculation Steps

Collect historical price data of the assets;
Calculate the logarithmic returns of the assets: $r_t = \ln\left(\frac{P_t}{P_{t-1}}\right),$ where $P_t$ is the asset price at time $t$ ;
Compute the covariance and standard deviation based on the return data;
Substitute the covariance and standard deviation into the Pearson correlation formula.

3. Models and Methods for Correlation Calculation

The methods for calculating correlation have evolved with the increasing complexity and nonlinearity of markets. Below are the main models and methods for correlation calculation:

3.1 Pearson Correlation

Definition

Pearson correlation is the most commonly used method for measuring correlation, based on the linear relationship between asset returns.

Advantages

Simple to calculate;
Sensitive to linear relationships.

Disadvantages

Cannot capture nonlinear relationships;
Sensitive to outliers.

Applications

Standard correlation measure in most financial models;
Portfolio optimization (e.g., mean-variance models).

3.2 Rank Correlation

3.2.1 Spearman Rank Correlation

Spearman rank correlation measures the ordinal relationship between variables rather than the linear relationship between their values. The formula is:

\rho_s = 1 - \frac{6 \sum_{i=1}^n d_i^2}{n(n^2 - 1)},

where $d_i$ is the difference in ranks of the two datasets, and $n$ is the number of data points.

Advantages

Captures monotonic relationships, including linear and nonlinear;
Insensitive to outliers.

Disadvantages

Ignores the scale of values, focusing only on ranks.

Applications

When data distribution is non-normal;
Capturing nonlinear monotonic relationships.

3.2.2 Kendall Rank Correlation

Kendall rank correlation measures the consistency of rank directions between two variables. The formula is:

\tau = \frac{C - D}{\frac{1}{2} n(n-1)},

where:

$C$ is the number of concordant pairs;
$D$ is the number of discordant pairs.

Advantages

Captures nonlinear relationships;
Insensitive to outliers.

Disadvantages

Computationally intensive.

Applications

For small datasets;
When rank direction consistency is needed.

3.3 Rolling Window Correlation

Definition

Rolling window correlation calculates Pearson correlation within a fixed time window and updates the window as time progresses.

Advantages

Captures changes in correlation over time;
Suitable for dynamic correlation studies.

Disadvantages

Subjective choice of window length;
Cannot capture nonlinear changes.

Applications

Analyzing dynamic changes in asset correlations in financial markets;
Time series data studies.

3.4 Nonlinear Correlation

3.4.1 Mutual Information

Mutual information measures the amount of shared information between two variables. The formula is:

I(X, Y) = \sum_{x \in X} \sum_{y \in Y} p(x, y) \ln \frac{p(x, y)}{p(x)p(y)},

where $p(x, y)$ is the joint probability distribution, and $p(x)$ and $p(y)$ are the marginal distributions.

Advantages

Captures nonlinear relationships;
No assumptions about variable distributions.

Disadvantages

Computationally complex;
Requires estimation of probability distributions.

Applications

Financial data with significant nonlinear relationships;
High-dimensional data analysis.

3.4.2 Copula Methods

Copula is a method for modeling dependencies between variables through joint distribution functions, suitable for capturing complex nonlinear correlations.

Advantages

Captures tail dependence and nonlinear relationships;
Flexible modeling.

Disadvantages

Requires assumptions about joint distribution forms;
Complex parameter estimation.

Applications

Pricing financial derivatives (e.g., credit derivatives);
Studying asset correlations under extreme market conditions.

3.5 Dynamic Correlation Models

3.5.1 GARCH-DCC (Dynamic Conditional Correlation) Model

The GARCH-DCC model dynamically estimates the conditional correlation of asset returns. The formula is:

R_t = D_t Q_t D_t,

where:

$R_t$ is the conditional correlation matrix;
$D_t$ is a diagonal matrix containing the conditional volatilities of assets;
$Q_t$ is the standardized conditional covariance matrix.

Advantages

Dynamically captures changes in correlation;
Accounts for volatility clustering.

Disadvantages

Complex parameter estimation;
Assumes conditional normality.

Applications

Studying dynamic correlations between assets in financial markets;
Dynamic hedging strategies in risk management.

3.6 Tail Correlation

Tail correlation measures the correlation between assets under extreme events, often used in risk management and analysis of extreme market conditions.

3.6.1 Copula Tail Correlation

Tail correlation is calculated using Copula models. The formulas are:

\lambda_L = \lim_{u \to 0} P(X < F_X^{-1}(u), Y < F_Y^{-1}(u)),

\lambda_U = \lim_{u \to 1} P(X > F_X^{-1}(u), Y > F_Y^{-1}(u)),

where $\lambda_L$ and $\lambda_U$ are the lower and upper tail correlations, respectively.

Advantages

Focuses on extreme events;
Captures tail dependence.

Disadvantages

Relies on Copula distribution assumptions;
Computationally complex.

Applications

Risk management under extreme market conditions;
Systemic risk analysis.

4. Summary and Applications

4.1 Summary

Linear correlation methods (e.g., Pearson correlation) are suitable for simple linear relationship analysis.
Rank correlation methods (e.g., Spearman and Kendall) are suitable for nonlinear monotonic relationships.
Nonlinear correlation methods (e.g., Mutual Information and Copula) are suitable for capturing complex dependencies.
Dynamic correlation models (e.g., GARCH-DCC) are suitable for analyzing time-varying correlation dynamics.
Tail correlation methods are suitable for studying asset correlations under extreme risk conditions.

4.2 Applications

Portfolio optimization: Calculating correlations between assets to optimize asset allocation.
Risk management: Analyzing correlations between assets under extreme market conditions for hedging and stress testing.
Financial derivative pricing: Modeling correlations in cross-currency volatility, Quanto options, and credit derivatives.
Market dynamics analysis: Studying the time-varying dynamics of asset correlations in markets.

By selecting appropriate correlation calculation methods and models, we can better understand the relationships between assets, thereby optimizing investment strategies and risk management.

5. How to Calculate Correlation Between Two Currency Pairs: A Practical Example

5.1. Data Preparation

Assume we have the following data:

Time range: Daily closing prices from January 1, 2023, to January 10, 2023.
Currency pair data:
- EUR/USD exchange rate: [1.07, 1.08, 1.10, 1.09, 1.11, 1.12, 1.13, 1.12, 1.14, 1.15]
- GBP/USD exchange rate: [1.20, 1.21, 1.23, 1.22, 1.24, 1.25, 1.26, 1.25, 1.27, 1.28]

We will calculate the correlation based on the logarithmic returns of these two currency pairs.

5.2. Calculation Steps

5.2.1 Calculate Logarithmic Returns

The formula for logarithmic returns is:

r_t = \ln\left(\frac{P_t}{P_{t-1}}\right),

where $P_t$ is the closing price on day $t$ .

EUR/USD logarithmic returns:
- Given EUR/USD exchange rate data: [1.07, 1.08, 1.10, 1.09, 1.11, 1.12, 1.13, 1.12, 1.14, 1.15]
- Calculate logarithmic returns: $r_1 = \ln\left(\frac{1.08}{1.07}\right) = 0.0093, \quad r_2 = \ln\left(\frac{1.10}{1.08}\right) = 0.0184, \quad \dots$
- Result: [0.0093, 0.0184, -0.0091, 0.0183, 0.0090, 0.0089, -0.0089, 0.0177, 0.0087]
GBP/USD logarithmic returns:
- Given GBP/USD exchange rate data: [1.20, 1.21, 1.23, 1.22, 1.24, 1.25, 1.26, 1.25, 1.27, 1.28]
- Calculate logarithmic returns: $r_1 = \ln\left(\frac{1.21}{1.20}\right) = 0.0083, \quad r_2 = \ln\left(\frac{1.23}{1.21}\right) = 0.0165, \quad \dots$
- Result: [0.0083, 0.0165, -0.0081, 0.0163, 0.0080, 0.0079, -0.0079, 0.0159, 0.0078]

5.2.2 Calculate Covariance

The formula for covariance is:

\text{Cov}(X, Y) = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y}),

where:

$X_i$ and $Y_i$ are the logarithmic returns of EUR/USD and GBP/USD, respectively;
$\bar{X}$ and $\bar{Y}$ are the means of the logarithmic returns.
Calculate means:
- Mean of EUR/USD logarithmic returns: $\bar{X} = \frac{1}{9} \sum_{i=1}^9 X_i = \frac{0.0093 + 0.0184 - 0.0091 + \dots + 0.0087}{9} \approx 0.0089$
- Mean of GBP/USD logarithmic returns: $\bar{Y} = \frac{1}{9} \sum_{i=1}^9 Y_i = \frac{0.0083 + 0.0165 - 0.0081 + \dots + 0.0078}{9} \approx 0.0082$
Calculate covariance:
$\text{Cov}(X, Y) = \frac{1}{9-1} \sum_{i=1}^9 (X_i - \bar{X})(Y_i - \bar{Y})$
Compute term by term:
- Term 1: $(0.0093 - 0.0089)(0.0083 - 0.0082) = 0.0004 \times 0.0001 = 0.00000004$
- Term 2: $(0.0184 - 0.0089)(0.0165 - 0.0082) = 0.0095 \times 0.0083 = 0.00007885$
- Compute remaining terms similarly, resulting in: $\text{Cov}(X, Y) \approx 0.000067$

5.2.3 Calculate Standard Deviations

The formula for standard deviation is:

\sigma_X = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2}, \quad \sigma_Y = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (Y_i - \bar{Y})^2}.

Standard deviation of EUR/USD:
$\sigma_X = \sqrt{\frac{1}{9-1} \sum_{i=1}^9 (X_i - \bar{X})^2}$
- Term 1: $(0.0093 - 0.0089)^2 = 0.0004^2 = 0.00000016$
- Term 2: $(0.0184 - 0.0089)^2 = 0.0095^2 = 0.00009025$
- Compute remaining terms similarly, resulting in: $\sigma_X \approx 0.0082$
Standard deviation of GBP/USD:
$\sigma_Y = \sqrt{\frac{1}{9-1} \sum_{i=1}^9 (Y_i - \bar{Y})^2},$
resulting in:
$\sigma_Y \approx 0.0078$

5.2.4 Calculate Correlation

The correlation formula (using Pearson correlation model) is:

\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}.

Substitute the calculated values:

\rho(X, Y) = \frac{0.000067}{0.0082 \times 0.0078} = \frac{0.000067}{0.00006396} \approx 1.05

5.3. Result Analysis

References

Below is a list of references on correlation calculation, covering theoretical foundations, financial applications, and advanced research:

1. Markowitz, Harry. Portfolio Selection (1952)

Overview: The seminal work on modern portfolio theory, emphasizing the central role of asset correlation in portfolio optimization. Introduces covariance matrices and their impact on portfolio risk.
Relevance: A classic reference for portfolio optimization and risk management.

2. Pearson, Karl. Mathematical Contributions to the Theory of Evolution (1896)

Overview: Introduced the Pearson correlation coefficient for measuring linear relationships between two variables, forming the theoretical basis for correlation studies.
Relevance: Foundational for basic correlation research.

3. Engle, Robert F. Dynamic Conditional Correlation: A Simple Class of Multivariate GARCH Models (2002)

Overview: Proposed the DCC-GARCH model for analyzing dynamic correlations between assets, widely applied in forex, equity, and fixed-income markets.
Relevance: Core literature for dynamic correlation analysis.

4. Embrechts, Paul, McNeil, Alexander, and Straumann, Daniel. Quantitative Risk Management: Concepts, Techniques, and Tools (2005)

Overview: Detailed discussion of correlation applications in risk management, including Copula methods and tail correlation modeling.
Relevance: Advanced research in credit and market risk.

5. Sklar, Abe. Functions de Répartition à n Dimensions et Leurs Marges (1959)

Overview: The seminal paper introducing Copula theory for describing dependency structures between random variables, particularly useful for tail correlation analysis.
Relevance: Nonlinear correlation and extreme risk condition studies.

6. Glasserman, Paul. Monte Carlo Methods in Financial Engineering (2003)

Overview: Discusses incorporating correlation in Monte Carlo simulations, particularly through covariance matrices and Cholesky decomposition for generating correlated random variables.
Relevance: Numerical methods and high-dimensional asset correlation modeling.