Quantitative Finance:

Introduction

Probability theory plays a crucial role in quantitative finance, as it provides the foundation for modeling and understanding the complex dynamics of financial markets. In this blog post, we explore some of the key areas of probability theory that are particularly relevant for quantitative finance, including random variables, moments, probability distributions, and their applications in portfolio management.

We will discuss the characteristics and properties of common probability distributions, such as the Uniform, Binomial, Gaussian, Lognormal, and Poisson, which are frequently used in finance. Lastly, we will examine the sums of random variables and their implications for portfolio risk and return characteristics. This post aims to provide a detailed overview of finance related probability topics for those interested in further study of the world of quantitative finance, where probability theory serves as a critical building block.

Random variables and Moments

Random Variables

A random variable, denoted by \(X\), or any variable of your choosing, is a function that maps the outcomes of a random experiment to real numbers. Random variables can be classified into two main categories:

• Discrete Random Variables: These variables can take only a finite or countably infinite number of distinct values.
• Continuous Random Variables: These variables can take any value within a continuous interval of real numbers.

Discrete Random Variables

A discrete random variable \(X\) has a probability mass function (pmf), denoted by \(p_X(x)\), which describes the probability that the variable takes on a specific value \(x\). The pmf satisfies the following conditions:

1. \(p_X(x) \ge 0\) for all \(x\)
2. \(\sum_{x} p_X(x) = 1\)

Continuous Random Variables

A continuous random variable \(X\) has a probability density function (pdf), denoted by \(f_X(x)\), which describes the probability that the variable takes a value in a small interval around \(x\). The pdf satisfies the following conditions:

1. \(f_X(x) \ge 0\) for all \(x\)
2. \(\int_{-\infty}^{\infty} f_X(x) dx = 1\)

Probability Distributions

For a random variable \(X\), its probability distribution is a function that provides the probabilities of different possible outcomes or values of the variable. The distribution can be represented as:

• Probability Mass Function (pmf) for discrete random variables: \(P(X = x_i) = p(x_i)\)
• Probability Density Function (pdf) for continuous random variables: \(f_X(x)\)

This graph is a PDF of the standard normal Gaussian distribution. Note that it is defined over the range of \(-\infty\) to \(+\infty\).

Cumulative Distribution Function

The cumulative distribution function (CDF) of a random variable \(X\) is a function that gives the probability that \(X\) takes a value less than or equal to \(x\). It is denoted by \(F_X(x)\) which is interpreted as the cumulative probability and defined as:

For discrete random variables: \(F_X(x) = P(X \le x) = \sum_{x_i \le x} p(x_i)\)

For continuous random variables: \(F_X(x) = P(X \le x) = \int_{-\infty}^x f_X(t) dt\)

This graph is a CDF of the standard normal Gaussian distribution. The CDF is a monotonically increasing function. The CDF has a one-to-one mapping of \(X\) to \(F_X(x)\). Note that it is defined over the range of \(-\infty\) to \(+\infty\).

Functions of Random Variables and Change of Variable

In probability theory, the change of variable allows us to transform a probability distribution from one variable to another. Let's consider two random variables, \(X\) and \(Y\), where \(Y = g(X)\). We want to find the probability density function of \(Y\), denoted by \(g(y)\), given the probability density function of \(X\), denoted by \(p(x)\).

To achieve this, we can use the change of variable formula, which states:

\[\begin{equation} g(y) dy = p(x) dx \end{equation}\]

From this equation, we can derive the formula for \(g(y)\) as follows:

\[\begin{equation} g(y) = \frac{p(x)}{\left|\frac{dy}{dx}\right|} \end{equation}\]

This formula allows us to find the probability density function of \(Y\), given the probability density function of \(X\) and the relationship between the two random variables.

Expectations and Moments

Mean is the expectation of the random variable

The mean, also known as the expected value, of a random variable \(X\) is denoted by \(\mathbb{E}[X]\) and defined as:

For discrete random variables: \(\mathbb{E}[X] = \sum_{i} x_i p(x_i)\)

For continuous random variables: \(\mathbb{E}[X] = \int_{-\infty}^{\infty} x f_X(x) dx\)

Moments of a distribution are the expectations of powers of the random variable

The \(n\)-th moment of a random variable \(X\) is the expected value of \(X^n\) and is denoted by \(\mathbb{E}[X^n]\). It is defined as:

For discrete random variables: \(\mathbb{E}[X^n] = \sum_{i} x_i^n p(x_i)\)

For continuous random variables: \(\mathbb{E}[X^n] = \int_{-\infty}^{\infty} x^n f_X(x) dx\)

Linearity of Expectations

The linearity of expectations states that for any random variables \(X\) and \(Y\), and constants \(a\) and \(b\), \(\mathbb{E}[aX + bY] = a\mathbb{E}[X] + b\mathbb{E}[Y]\).

Variance and Standard Deviation

The variance of a random variable \(X\) is a measure of the dispersion of the variable around its mean and is denoted by \(\text{Var}(X)\). It is defined as:

\[\begin{align} \text{Var}(X) &= \mathbb{E}[(X - \mathbb{E}[X])^2] \\ &= \mathbb{E}[X^2 - 2\mathbb{E}[X]X + \mathbb{E}[X]^2] \\ &= \mathbb{E}[X^2] - 2\mathbb{E}[X]\mathbb{E}[X] + \mathbb{E}[X]^2 \\ &= \mathbb{E}[X^2] - 2\mathbb{E}[X]^2 + \mathbb{E}[X]^2 \\ &= \mathbb{E}[X^2] - \mathbb{E}[X]^2 \end{align}\]

The standard deviation, denoted by \(\sigma(X)\), is the square root of the variance: \(\sigma(X) = \sqrt{\text{Var}(X)}\).

Skewness

Skewness is a measure of the asymmetry of the probability distribution of a random variable \(X\). It is defined as:

\(\text{Skew}(X) = \frac{\mathbb{E}[(X - \mathbb{E}[X])^3]}{\sigma^3(X)}\)

Kurtosis

Kurtosis is a measure of the "tailedness" of the probability distribution of a random variable \(X\). It is defined as:

\(\text{Kurt}(X) = \frac{\mathbb{E}[(X - \mathbb{E}[X])^4]}{\sigma^4(X)}\)

Covariance and Correlation

The covariance between two random variables \(X\) and \(Y\) is a linear measure of the degree to which they vary together. It is denoted by \(\text{Cov}(X, Y)\) and defined as:

\[\begin{align} \text{Cov}(X, Y) &= \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])] \\ &= \mathbb{E}[XY - X\mathbb{E}[Y] - \mathbb{E}[X]Y + \mathbb{E}[X]\mathbb{E}[Y]] \\ &= \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] - \mathbb{E}[X]\mathbb{E}[Y] + \mathbb{E}[X]\mathbb{E}[Y] \\ &= \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] \end{align}\]

If the covariance is positive, it means that when \(X\) increases, \(Y\) also tends to increase, and vice versa. If the covariance is negative, it means that when \(X\) increases, \(Y\) tends to do the opposite and decreases, and vice versa. If the covariance is close to zero, it indicates that there is no linear relationship between \(X\) and \(Y\).

The correlation coefficient between two random variables \(X\) and \(Y\) is a normalized measure of their linear relationship. It is denoted by \(\rho_{XY}\) and defined as:

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

The correlation coefficient ranges between -1 and 1. A value of -1 indicates a perfect negative linear relationship, a value of 1 indicates a perfect positive linear relationship, and a value of 0 indicates no linear relationship between the variables.

Common Distributions used in Quantitative Finance

Continuous Uniform Distribution

The continuous uniform distribution is defined by a constant probability density function on the interval \([a, b]\). The pdf is given by:

\(f_X(x) = \frac{1}{b - a}\) for \(a \le x \le b\), and \(f_X(x) = 0\) otherwise.

Moments

Mean: \(\mathbb{E}[X] = \frac{a + b}{2}\)

Variance: \(\text{Var}(X) = \frac{(b - a)^2}{12}\)

Continuous Uniform Distribution: Applications in Quantitative Finance

In quantitative finance, the uniform distribution is often used as a starting point or a basic building block for more complex models and simulations. It has several applications in finance, including the following:

• Monte Carlo simulations: The uniform distribution can be used to generate random numbers for Monte Carlo simulations, which are widely used in option pricing, risk management, and portfolio optimization. By generating random uniform numbers and transforming them into other desired distributions, financial analysts can create scenarios for various financial instruments and evaluate their performance under different market conditions.
• Sampling and bootstrapping: The uniform distribution can be used in resampling techniques, such as bootstrapping, where random samples are drawn with replacement from an original dataset. By assigning equal probability to each data point in the dataset, the uniform distribution helps create new samples that preserve the original dataset's characteristics while allowing for the estimation of statistical properties, such as confidence intervals and bias.

Although the uniform distribution is a relatively simple distribution, it provides a useful foundation for various applications in quantitative finance. Its simplicity makes it easy to work with and understand, while its flexibility allows for more sophisticated models and techniques to be built upon it.

Binomial Distribution

The discrete binomial distribution describes the number of successes in \(n\) independent Bernoulli trials, each with probability of success \(p\). The probability mass function is given by:

\(P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}\) for \(k = 0, 1, \dots, n\).

Moments

Mean: \(\mathbb{E}[X] = np\)

Variance: \(\text{Var}(X) = np(1 - p)\)

Binomial Distribution: Applications in Quantitative Finance

The binomial distribution is a discrete probability distribution that is widely used in quantitative finance for modeling situations with a finite number of trials and binary outcomes. It has several applications in finance, including the following:

• Option pricing: The binomial distribution plays a crucial role in the binomial option pricing model, which is a popular method for pricing European and American options. The model uses a binomial tree to represent the possible price paths of the underlying asset over time, with each node representing the probability of the asset price moving up or down. The binomial distribution is implicitly used to calculate these probabilities and to determine the option's value at each node in the tree.
• Modeling credit risk: The binomial distribution can be used to model the credit risk of a portfolio of loans or bonds. For example, if the probability of default for each loan is known, a binomial distribution can be used to estimate the probability of a given number of defaults occurring in the portfolio. This information can be valuable for risk management and assessing the credit quality of the portfolio.
• Deriving probability of success: In quantitative finance, the binomial distribution can be used to determine the probability of achieving a specific number of successful trades or investments over a given period. This can be useful for evaluating the performance of trading strategies or investment portfolios.
• Risk management: The binomial distribution can be applied to model the probability of specific numbers of loss occurrences in a given time period, given the probability of a single loss occurring. This is particularly useful for estimating Value-at-Risk (VaR) or Expected Shortfall (ES) in risk management frameworks.

The binomial distribution's flexibility and simplicity make it a powerful tool for modeling various financial phenomena with binary outcomes and a known number of trials.

Gaussian Distribution

The Gaussian distribution, also known as the normal distribution, is characterized by its mean \(\mu\) and standard deviation \(\sigma\). The probability density function is given by:

\(f_X(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x - \mu)^2}{2 \sigma^2}}\)

Moments

Mean: \(\mathbb{E}[X] = \mu\)

Variance: \(\text{Var}(X) = \sigma^2\)

Gaussian Distribution: Applications in Quantitaive Finance

The Gaussian distribution, also known as the normal distribution, is widely used in quantitative finance due to its well-understood properties and its prevalence in many financial data sets. There are several applications where the Gaussian distribution is preferred over the lognormal distribution:

• Linear regression and portfolio optimization: Gaussian distribution is used to model errors in linear regression and portfolio optimization problems. The normality assumption of errors allows for the use of ordinary least squares (OLS) estimation and Markowitz's mean-variance optimization framework. In these cases, the Gaussian distribution is more suitable than the lognormal distribution because the latter may lead to biased estimations.
• Modeling asset returns: While the lognormal distribution is more suitable for modeling asset prices, the Gaussian distribution can be used to model asset returns. Asset returns often exhibit properties that are more consistent with the Gaussian distribution, such as symmetry around the mean and relatively light tails compared to asset prices.
• Central Limit Theorem: The Gaussian distribution is the natural choice when modeling the sum or average of a large number of random variables due to the Central Limit Theorem, which states that the sum or average of a large number of independent and identically distributed random variables approaches a Gaussian distribution, regardless of their original distribution. This property is particularly useful when modeling portfolio returns or aggregating risk factors in financial models.
• Gaussian Copula: The Gaussian copula is a widely used method for modeling the dependence structure among multiple random variables, especially in fields like credit risk modeling. By capturing the correlation structure among these variables, the Gaussian copula allows for diverse marginal distributions, providing flexibility in cases where traditional distributions like the lognormal are not suitable. However, it does assume the dependencies transform into a multivariate normal structure on the copula scale. While this makes the Gaussian copula versatile, it is not without limitations, particularly in capturing tail dependencies, which may be significant in risk management contexts. Therefore, careful consideration of the copula choice is necessary, especially in applications requiring robust modeling of extreme events.
• Interest rate modeling: In some interest rate models, like the Vasicek model and the Hull-White model, Gaussian distribution is used to describe the changes in interest rates. These models are often based on the assumption that interest rates follow a mean-reverting process with normally distributed innovations.

While the Gaussian distribution has many applications in quantitative finance, it is essential to recognize its limitations, especially in cases where the underlying data exhibit non-normal features such as fat tails or skewness. Or where it is impossible for an asset to have a negative value (limited liability of stock ownership) yet the Gaussian model would allow such occurrence. In such cases, alternative distributions, like the lognormal or t-distribution, might be more appropriate.

Lognormal Distribution

The lognormal distribution is a continuous probability distribution of a random variable whose logarithm follows a Gaussian distribution. It is parameterized by the mean \(\mu\) and standard deviation \(\sigma\) of the underlying Gaussian distribution. The log-normal distribution limits the price to positive values, which is a realistic constraint for many financial assets. However, note that the lognormal distribution as a data generating process will never generate the value zero. Values can be very close to zero, but never equal to zero. There are some circumstances where you might want to take this into consideration. The probability density function is given by:

\(f_X(x) = \frac{1}{x \sigma \sqrt{2 \pi}} e^{-\frac{(\ln x - \mu)^2}{2 \sigma^2}}\) for \(x > 0\).

Moments

Mean: \(\mathbb{E}[X] = e^{\mu + \frac{1}{2} \sigma^2}\)

Variance: \(\text{Var}(X) = (e^{\sigma^2} - 1) e^{2 \mu + \sigma^2}\)

Lognormal Distribution: Applications in Quantitaive Finance

The lognormal distribution is widely used in quantitative finance due to its ability to model positively skewed and non-negative data. Some common applications of the lognormal distribution in quantitative finance include:

• Asset pricing: The lognormal distribution is often used to model the distribution of stock prices, as it ensures non-negative values and can capture the right-skewed characteristics often observed in stock price movements. This makes it a popular choice in option pricing models, such as the Black-Scholes model.
• Portfolio simulation: When simulating the future value of a portfolio, the lognormal distribution can be used to model the growth of individual assets within the portfolio. This allows for more realistic projections that account for the fact that asset prices cannot be negative and typically exhibit skewed returns.
• Commodity prices: Commodity prices, like oil or gold, often exhibit characteristics similar to those of asset prices, with positive skewness and non-negative values. The lognormal distribution can be employed to model the distribution of commodity prices and to estimate the probability of extreme price movements.
• Derivatives pricing: The lognormal distribution is used in the pricing of various financial derivatives, such as options and futures. The Black-Scholes model, for instance, relies on the assumption that the underlying asset price follows a lognormal distribution.
• Modeling exchange rates: Exchange rates often exhibit characteristics that are consistent with a lognormal distribution, such as non-negativity and positive skewness. Therefore, the lognormal distribution can be employed to model and forecast exchange rate movements.
• Modeling insurance claims: In the insurance industry, claim amounts can be modeled using a lognormal distribution due to its ability to capture the positively skewed and non-negative nature of claim sizes.
• Modeling income and wealth distributions: The lognormal distribution is often used to model income and wealth distributions in economics, as it captures the right-skewed nature of these distributions, with a majority of individuals having lower incomes and a small number of individuals having extremely high incomes.

This is not an exaustive list but rather a good sample of examples of how the lognormal distribution is applied in quantitative finance. Its ability to model non-negative and positively skewed data makes it a valuable tool for analyzing various financial phenomena.

Poisson Distribution

The discrete Poisson distribution models the number of events in a fixed interval of time or space, given a constant average rate \(\lambda\). The probability mass function is given by:

\(P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\) for \(k = 0, 1, 2, \dots\).

Limiting case of binomial distribution

The Poisson distribution can be considered as the limiting case of a binomial distribution when the number of trials \(n\) approaches infinity, the probability of success \(p\) approaches zero, and the product \(np = \lambda\) remains constant. In practice, this can often be a good approximation when n is large and p is small. It is a simple exercise to use both distributions for a series of calculations and then observe the error or difference between values calculated using the binomial vs. the Poisson to see if the error is acceptable for your purpose.

The Poisson distribution formula involves fewer and simpler operations: exponentiation, multiplication, and division involving factorials, which are generally less computationally intensive than those required for the binomial coefficient in the context of large \(n\).

When \(n\) is very large (e.g., in the thousands or more) and \(p\) is very small (such that \(np\) is moderate), the Poisson approximation is not only more computationally efficient but also sufficiently accurate for many practical purposes.

Moments

Mean: \(\mathbb{E}[X] = \lambda\)

Variance: \(\text{Var}(X) = \lambda\)

Poisson Distribution: Applications in Quantitaive Finance

The Poisson distribution is used in quantitative finance to model events that occur infrequently but have a significant impact when they do occur. Some common applications of the Poisson distribution in quantitative finance include:

• Operational risk modeling: The Poisson distribution can be used to model the number of operational failures, such as system breakdowns, fraud, or legal disputes, that occur within a given time frame. This allows financial institutions to estimate the probability of such events and allocate resources accordingly to mitigate potential losses.
• Modeling rare events: The Poisson distribution is particularly useful for modeling rare events, like market crashes or extreme price movements, that can have a significant impact on financial markets. By modeling these events with a Poisson distribution, risk managers can estimate the likelihood of such occurrences and develop strategies to manage the associated risks.
• Default modeling: In credit risk management, the Poisson distribution can be employed to model the number of defaults within a loan portfolio over a given period. This allows banks and financial institutions to estimate the probability of default and manage the associated credit risk.
• Arrival of orders in market microstructure: The Poisson distribution can be used to model the arrival of orders in high-frequency trading or market microstructure analysis. By estimating the arrival rate of orders, traders can develop trading strategies to exploit short-term price movements and liquidity dynamics.
• Insurance claim modeling: In the insurance industry, the Poisson distribution can be used to model the number of claims filed within a given time frame. This enables insurers to estimate the likelihood of claims and allocate resources to handle potential payouts.
• Modeling transaction counts: The Poisson distribution can be employed to model the number of financial transactions, such as trades or payments, that occur within a specific period. This information can be useful for risk management, liquidity analysis, and capacity planning purposes.

These are just a handful of examples of how the Poisson distribution is applied in quantitative finance. Its ability to model infrequent but impactful events makes it a valuable tool for analyzing various financial phenomena and managing the associated risks.

Summary of Common Distributions

The Uniform, Binomial, Gaussian, Lognormal and Poisson distributions are commonly used in finance. Each of these distributions has well-defined moments. Other properties of these distributions that make them useful in finance applications include:

• The Uniform distribution is useful for modeling scenarios where all outcomes have equal probability, such as random sampling or simple Monte Carlo simulations.
• The Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, which can be useful for modeling asset prices in the Binomial Tree model.
• The Gaussian distribution, also known as the normal distribution, is widely used in finance due to its nice mathematical properties and the prevalence of the Central Limit Theorem, which states that the sum of many independent random variables converges to a normal distribution. It is often used to model asset returns, portfolio risk, and option pricing.
• The Lognormal distribution is used to model quantities that cannot be negative, such as stock prices or financial indices. The lognormal distribution is particularly useful for modeling multiplicative processes, such as compound interest and exponential growth.
• The Poisson distribution is often used to model the number of events in a fixed interval of time or space, such as the number of trades in a given period or the arrival rate of orders in a market.

Sums of Random Variables

Multiple Random Variables

Often we are interested in the sum of multiple random variables. While it may be desired to have the full probability distribution of such a sum, it may be sufficient for many applications to only have the expectation (mean) and variance of the distribution. This is possible due to the central limit theorem.

Central Limit Theorem (CLT): The Central Limit Theorem states that the sum (or average) of a large number of independent and identically distributed (i.i.d.) random variables will tend to follow a normal distribution, regardless of the original distribution of the variables. This theorem is widely used in quantitative finance to model the distribution of returns, estimate confidence intervals, and conduct hypothesis testing.

Further, the linearity of the expectation operator makes it easy to determine the expectation of the sum since the expectation of the sum is the sum of the expectations. However, determining the variance of a sum can be a bit more complicated.

For example, consider a portfolio of stocks that we want to model, where \(w_i\) is the portfolio weight of stock \(i\), \(R_i\) is the actual measured return on stock \(i\), \(\mu_i\) is the expected return on stock \(i\) and \(\sigma_i^2\) is the variance of stock \(i\). Then we have the following:

\[ R_p = \sum_{i=1}^N w_iR_i = w_1R_1 + w_2R_2 + \dots + w_NR_N \] \[ \mu_p = E[R_p] = \sum_{i=1}^N w_iE[R_i] = \sum_{i=1}^N w_i\mu_i\]

The variance \(\sigma_p^2\) is a bit more work since stocks are often correlated at some level.

\[\begin{aligned} \sigma_p^2 &= Var(R_p) = E[(R_p - \mu_p)^2] = E\left[\left(\sum_{i=1}^Nw_i(R_i-\mu_i)\right)^2\right] \\ &= \sum_{i=1}^N\sum_{j=1}^Nw_iw_jE\left[(R_i-\mu_i)(R_j-\mu_j)\right] \\ &= \sum_{i=1}^Nw_i^2E\left[(R_i-\mu_i)^2\right] + 2\sum_{i>j}^Nw_iw_jE\left[(R_i-\mu_i)(R_j-\mu_j)\right] \\ &= \sum_{i=1}^Nw_i^2Var(R_i) + 2\sum_{i>j}^Nw_iw_jCov(R_i,R_j) \\ &= \sum_{i=1}^Nw_i^2\sigma_i^2 + 2\sum_{i>j}^Nw_iw_j\sigma_i\sigma_j\rho_{ij} \end{aligned}\]

Where \(\sigma_i\) and \(\sigma_j\) are the standard deviation of stocks i and j (recognizing that \(\sigma_i^2 = \sigma_{i}\sigma_{i}\)), and \(\rho_{ij}\) is the correlation between stock i and stock j.

The covariance matrix is revealing when we view the indices of the covariance matrix \(\Sigma_p\) and at the same time study the summation notation above.

\[ \begin{align} \Sigma_p = \begin{vmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \\ \end{vmatrix} = \begin{vmatrix} \sigma_{1}\sigma_{1}\rho_{11} & \sigma_{1}\sigma_{2}\rho_{12} & \sigma_{1}\sigma_{3}\rho_{13} \\ \sigma_{2}\sigma_{1}\rho_{21} & \sigma_{2}\sigma_{2}\rho_{22} & \sigma_{2}\sigma_{3}\rho_{23} \\ \sigma_{3}\sigma_{1}\rho_{31} & \sigma_{3}\sigma_{2}\rho_{32} & \sigma_{3}\sigma_{3}\rho_{33} \\ \end{vmatrix} \end{align} \]

We can see that for this specific 3x3 case and recognizing that \(\sigma_{ij}=\sigma_i\sigma_j\rho_{ij} \) and that \(\sigma_i^2 = \sigma_{i}\sigma_{i} = \sigma_{ii}\) we have:

\[\sigma_p^2 = \sum_{i=1}^3w_i^2\sigma_i^2 + 2\sum_{i>j}^3w_iw_j\sigma_{ij}\]

Due to the symmetry of the covariance matrix \(\Sigma_p\), we only need to work with the upper or lower triangular matrix portion of the covariance matrix to find the variance of the portfolio. By choosing \(i>j\) in the above summation we effectively chose to work with the lower triangular matrix. The term on the left above, (the sum of the three \(w_i^2\sigma_i^2 \) ) is the sum of the diagnol of the matrix weighted by the portfolio weights and the term on the right is the sum of all terms where \(i>j\).

\[ \begin{align} \Sigma_p = \begin{vmatrix} \sigma_{11} & & \\ \sigma_{21} & \sigma_{22} & \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \\ \end{vmatrix} = \begin{vmatrix} \sigma_{1}\sigma_{1}\rho_{11} & & \\ \sigma_{2}\sigma_{1}\rho_{21} & \sigma_{2}\sigma_{2}\rho_{22} & \\ \sigma_{3}\sigma_{1}\rho_{31} & \sigma_{3}\sigma_{2}\rho_{32} & \sigma_{3}\sigma_{3}\rho_{33} \\ \end{vmatrix} \end{align} \]

So in this specific example, and by recognizing that \(\sigma_{ii}=\sigma_i\sigma_i\rho_{ii} = \sigma_i^2\), the variance of the portfolio would be:

\[ \begin{aligned} \sigma_p^2 &= w_1w_1\sigma_1\sigma_1\rho_{11}+w_2w_2\sigma_2\sigma_2\rho_{22}+w_3w_3\sigma_3\sigma_3\rho_{33} \\ &+2\left(w_2w_1\sigma_2\sigma_1\rho_{21}+w_3w_1\sigma_3\sigma_1\rho_{31}+w_3w_2\sigma_3\sigma_2\rho_{32}\right) \\ &= w_1^2\sigma_{11}+w_2^2\sigma_{22}+w_3^2\sigma_{33}+2\left(w_2w_1\sigma_{21}+w_3w_1\sigma_{31}+w_3w_2\sigma_{32}\right) \end{aligned} \]

This setup reveals that there are special cases that can simplify the calculation of the variance of a sum of random variables (in this case the sum of the returns of the stock that make up the portfolio).

1. No Correlation Between Stocks

   If there is no correlation between any of the stocks then the variance of the portfolio is simply the weighted sum of the variances.

   If \(\rho_{ij}=0\) for all \(i\) and \(j\):

   \[\begin{align}\sigma_p^2 &= \sum_{i=1}^Nw_i^2\sigma_i^2 + 2\sum_{i>j}^3w_iw_j\sigma_i\sigma_j\rho_{ij} \\ &= \sum_{i=1}^Nw_i^2\sigma_i^2 \end{align}\]

2. Perfect Correlation Between Stocks

   If there is perfect correlation between all the stocks then the variance of the portfolio is simply the square of the wighted sum of the variances.

   If \(\rho_{ij}=1\) for all \(i\) and \(j\):

   \[\begin{align} \sigma_p^2 &= \sum_{i=1}^Nw_i^2\sigma_i^2 + 2\sum_{i>j}^Nw_iw_j\sigma_i\sigma_j\rho_{ij} \\ &= \sum_{i=1}^Nw_i^2\sigma_i^2 + 2\sum_{i<j}^Nw_iw_j\sigma_i\sigma_j(1) \\ &= \sum_{i=1}^Nw_i^2\sigma_i^2 + 2\sum_{i<j}^Nw_iw_j\sigma_i\sigma_j \\ &= \sum_{i=1}^Nw_i^2\sigma_i^2 + \sum_{i=1}^N\sum_{j=1}^Nw_iw_j\sigma_i\sigma_j - \sum_{i=1}^Nw_i^2\sigma_i^2 \\ &= \sum_{i=1}^N\sum_{j=1}^Nw_iw_j\sigma_i\sigma_j \\ &= \left(\sum_{i=1}^Nw_i\sigma_i\right)\left(\sum_{j=1}^Nw_j\sigma_j\right) \\ &= \left(\sum_{i=1}^Nw_i\sigma_i\right)^2 \end{align} \]

3. No Correlation Between Stocks, Equal Weights, Same Distribution

   If there is no correlation between any of the stocks, the stock returns all come from the same distribution in that they all have the same variance \(\sigma_0\), and the portfolio is equally weighted in that \(w_i = w_j = \frac{1}{N}\) for all \(i\) and \(j\), then we have:

   \[\begin{align}\sigma_p^2 &= \sum_{i=1}^Nw_i^2\sigma_i^2 + 2\sum_{i>j}w_iw_j\sigma_i\sigma_j\rho_{ij} \\ &= \sum_{i=1}^N\frac{\sigma_0^2}{N^2} \\ &= \frac{\sigma_0^2}{N^2}\sum_{i=1}^N1 \\ &= \frac{\sigma_0^2}{N^2}N=\frac{\sigma_0^2}{N} \end{align}\]

Importance of Sums of Random Variables in Quantitative Finance

Sums of random variables play an important role in quantitative finance as they form the foundation for several important concepts and models. Some of the key areas where sums of random variables are fundamental include:

Portfolio theory: In portfolio theory, as we saw above, the return on a portfolio is the sum of the returns of its individual assets, weighted by their respective proportions in the portfolio. By analyzing the sum of these random variables, investors can understand the risk-return trade-off and optimize their portfolio allocations to achieve a desired level of risk and expected return.

Risk management: Quantitative risk management often involves estimating the distribution of the sum of random variables, such as losses from different sources. For instance, Value-at-Risk (VaR) and Expected Shortfall (ES) are widely used risk metrics that rely on understanding the characteristics of the distribution of portfolio returns, which are the sums of individual asset returns.

Option pricing: Option pricing models, such as the Black-Scholes model, assume that the underlying asset's price follows a stochastic process, which can be modeled as a sum of random variables (e.g., a geometric Brownian motion). By understanding the distribution of the sum of these variables, quantitative analysts can derive option prices and hedge ratios.

Time series analysis: Time series models, such as autoregressive (AR) and moving average (MA) models, involve the sum of random variables to capture the dynamic behavior of financial time series data. These models are building blocks used to create more advanced models to forecast asset prices, estimate volatility, and identify trends in financial markets.

Sums of random variables lay the foundation for various topics in quantitative finance, as they are essential for understanding the behavior of financial variables and developing models to manage risk, optimize portfolios, and price derivative instruments. The Central Limit Theorem, in particular, plays a crucial role in many aspects of quantitative finance, as it allows analysts to make inferences and estimates based on the normal distribution.

Conclusion

My hope is that this content will help you have a more clear picture of what probability theory is and how it can be applied in the field of quantitative finance.

We discussed random variables and moments, including expectations, variance, skewness, and kurtosis. We also explored covariance and correlation.

We covered common distributions such as Uniform, Binomial, Gaussian, Lognormal, and Poisson distributions, and their respective forms and moments. We discussed applications of each distribution in quantitative finance.

Finally, we investigated the sums of random variables in the context of portfolio management as well as briefly pointed out their importance in general in quantitative finance. We considered different scenarios, such as when there is no correlation between stocks, perfect correlation, or when the stocks have the same distribution and equal weights in the portfolio. This analysis, while being an oversimplification of reality, is a good starting point for us to begin to understand the impact of diversification of investments on a portfolio's risk and return characteristics.

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