Have you ever wondered how extreme events in financial markets, such as stock market crashes or major price fluctuations, can be modeled and analyzed? Well, one powerful tool that has gained popularity among financial researchers is Extreme Value Theory (EVT). In particular, the concept of block maxima and peak-over-threshold has proven to be valuable when studying extreme outcomes.

When looking at how prices behave in unusual markets, investment or risk analysts often use stress tests and scenario analysis. these methods are good for simulating how a portfolio's value might change in extreme market situations, but they have certain limitations. they can't explore all possible scenarios, and they don't tell us how likely the scenarios they consider are.

there's another way to approach this – using something called Extreme Value Theory (EVT). EVT is a well-established part of probability and statistics that focuses on the tails of a distribution. It helps us figure out the chance of extreme values without assuming anything about the shape of the distribution that led to those extremes. to do this, EVT finds suitable probability distributions for the tails based on past actual outcomes.

EVT is important in managing market risks because it helps answer questions like: "What are the really bad movements our portfolio or risks might face?" and "Have we already seen the worst, or could things get even worse?"

#### What is Extreme Value Theory (EVT)?

Extreme Value Theory is a branch of statistics that deals with the analysis of rare events or extremes. It provides a framework to model and quantify the probabilities of extreme events occurring in a given dataset. EVT is based on the assumption that extreme events, such as stock market crashes, follow a specific distribution known as the Generalized Extreme Value (GEV) distribution.

The Generalized Extreme Value (GEV) distribution is a statistical probability distribution that is commonly used in extreme value theory to model the distribution of extreme values from a set of observations. It is particularly employed to analyze the tail behavior of distributions, focusing on extreme values that lie beyond a certain threshold.

The GEV distribution is defined by three parameters: location parameter (μ), scale parameter (σ), and shape parameter (ξ). these parameters determine the location, spread, and shape of the distribution. The shape parameter is crucial as it characterizes the type of the distribution:

When ξ = 0, the GEV distribution reduces to the Gumbel distribution.

When ξ < 0, it represents a heavy-tailed distribution.

When ξ > 0, it indicates a distribution with a finite upper bound.

The probability density function (PDF) of the GEV distribution is given by the formula:

PDF of GEV Dist. = 1 / σ [ 1 + ξ ((x - μ) / σ)]^(-1/ξ-1) . exp{-[1 + ξ((x - μ) / σ)]^(-1/ξ)}

Here:

x is the variable.

μ is the location parameter.

σ is the scale parameter.

ξ is the shape parameter.

#### What is Block Maxima?

Block maxima refer to the maximum values observed within non-overlapping blocks of data. In the context of equity returns, the idea is to divide the historical returns into fixed-size blocks and identify the absolute value of the maximum return within each block. these block maxima represent extreme events, as they capture the largest gains or losses experienced over a specific time period. these extreme values are then used to estimate the GEV distribution where the parameters of the distribution can be estimated via the maximum likelihood estimator.

### Analyzing Equity Returns Using Block Maxima

By using block maxima, financial researchers can apply EVT to model extreme events in equity returns. The process typically involves the following steps:

Data Preparation: Gather a historical dataset of equity returns over a specific time period.

Block Selection: Choose an appropriate block size, considering the trade-off between capturing extreme events and having enough data points for reliable analysis.

Maxima Identification: Identify the maximum return within each block, which represents an extreme event.

GEV Distribution Estimation: Fit the block maxima data to the GEV distribution using statistical techniques. This estimation allows for understanding the distribution of extreme events and calculating their probabilities.

#### What is Peak-Over-Threshold?

Instead of looking only at the maximum values, the POT (Peak Over Threshold) method focuses on all events above a certain large threshold that we set beforehand.

(the mean excess theorem, which helps us pick the threshold parameter, denoted as "m." The theorem states that if a random variable X follows a distribution with a shape parameter ξ less than 1, then for values less than a certain threshold, the mean excess function eX(u) is given by the formula (σ+ξμ)/(1- ξ), where μ is the mean excess.)

The "Excess" function is straightforward – it's the sum of the excesses over the threshold "m", divided by the number of observations that exceed this threshold.

To illustrate POT modeling, let's consider an example using weekly return data from a hedge fund portfolio. We'll focus on the negative weekly return values, specifically the left tail of the return distribution, and take the absolute value of these losses. The dataset consists of 300 observations. To apply the Picklands–Balkema–de Hann theorem, we must choose an appropriate threshold parameter, "m." Since we aim to model extreme risks, we prefer a high value for "m." However, a higher "m" means fewer observations are available to estimate the parameters.

Determine an appropriate threshold value, often referred to as the "return level" in finance. this threshold represents the minimum value for an event to be considered extreme.

### Benefits and Limitations of Block Maxima and EVT

The use of block maxima and EVT offers several advantages. It provides a rigorous statistical framework to analyze extreme events, allowing for better risk assessment and the estimation of tail probabilities. This information is crucial for financial institutions, investors, and risk managers to make informed decisions and develop appropriate risk management strategies.

However, it's important to note that EVT assumes certain underlying conditions, such as stationarity and independence of the data, which may not always hold true for equity returns. Moreover, the estimation of the GEV distribution parameters requires a sufficient number of extreme events, which can be challenging to obtain, especially in the context of rare financial crises.

## Block Maxima and Extreme Value Theory in Finance

As financial markets continue to evolve and become increasingly complex, the application of EVT and block maxima analysis is likely to gain even more importance. Researchers are exploring modifications and extensions to EVT to address its limitations and make it more suitable for capturing the dynamics of equity returns and other financial assets.

Additionally, advancements in computational power and data availability enable the analysis of larger datasets, providing researchers with more accurate estimates of extreme events and their associated risks.