Extreme Value Theory in Quantitative Finance: Applications and Limitations

Extreme Value Theory in Quantitative Finance: Applications and Limitations

Introduction to Extreme Value Theory (EVT)

Extreme value theory (EVT) is a statistical theory that analyses extreme events that occur infrequently but have significant consequences. EVT is used in many fields, including finance, to model rare events that can profoundly impact asset prices and risk management. EVT is beneficial for modelling tail risk, the risk of extreme events occurring beyond what is usually expected.

EVT in Equity Markets

Equity markets are characterized by many assets, each with its unique risk and return profile. EVT can be used to model tail risk in equity markets, which is essential for portfolio risk management. EVT can also be used in equity portfolio construction, where the goal is to construct a portfolio optimized for risk-adjusted returns.

EVT in Derivative Markets

Derivative markets are characterized by complex financial instruments that are used for hedging and speculation. EVT can be used to model tail risk in derivative markets, which is important for pricing and risk management. EVT is particularly useful in options pricing, where the goal is to accurately price options that are out of the money.

EVT Models and Their Applications in Finance

EVT models come in two main types: univariate EVT models and multivariate EVT models. Univariate EVT models are used for modelling extreme events in a single asset, while multivariate EVT models are used for modelling extreme events in a portfolio of assets. EVT-based risk measures, such as value at risk (VaR) and expected shortfall (ES), can quantify tail risk in financial markets.

Empirical Applications of EVT in Finance

EVT has been used in many empirical studies in finance, including studies on equity market crashes, credit risk, and extreme events in commodity markets. EVT has also been used in risk management and hedging strategies, where the goal is to reduce tail risk exposure in financial portfolios.

Criticisms and Limitations of EVT in Finance

Despite its usefulness, EVT has some limitations in financial applications. EVT models assume that the data is stationary and independent, which may not be accurate in financial markets. EVT models also assume that the distribution of extreme events is independent of the distribution of non-extreme events, which may not be accurate in practice. Alternative models, such as copula and regime-switching, have been proposed to address these limitations.

Future Research

EVT is a powerful tool for modelling tail risk in financial markets, but it has limitations that should be considered when applying it in practice. Future research in EVT and finance should focus on addressing the limitations of EVT models and exploring alternative models and approaches for modelling tail risk in financial markets.

EVT is a valuable tool for modelling rare events in financial markets, but it should be used with caution and understand its limitations. EVT models can help to improve risk management and portfolio construction. Still, they should be used with other models and approaches to understand financial market risk comprehensively.

Python Code to calculate extreme values:

Sample Python code that imports the price data of a portfolio of securities and calculates the extreme value using EVT:

import pandas as p
from scipy.stats import genextreme

# Import price data
prices = pd.read_csv('portfolio_prices.csv', index_col='date')

# Calculate the daily returns
returns = prices.pct_change().dropna()

# Calculate the maximum daily return
max_return = returns.max()

# Fit GEV distribution to the data
params = genextreme.fit(max_return)

# Calculate the extreme value at the desired confidence level
conf_level = 0.01  # 99% confidence level
extreme_value = genextreme.ppf(1 - conf_level, *params)

print(f"The extreme value of the portfolio at {conf_level:.0%} confidence level is {extreme_value:.2%}.")        

In this code, we first import the price data of a portfolio of securities from a CSV file using pandas. We then calculate the daily returns of the portfolio and drop the first row containing NaN values. We then calculate the maximum daily return, which is an extreme value of the portfolio.

Next, we fit a GEV distribution to the maximum daily returns using the genextreme.fit() function from the scipy.stats library. The genextreme distribution is a parametric distribution commonly used in EVT to model the tail distribution of extreme values.

Finally, we calculate the extreme value at a desired confidence level using the genextreme.ppf() function. The ppf() function calculates the inverse of the cumulative distribution function (CDF) at a given probability. In this case, we calculate the extreme value at the 99% confidence level, equivalent to the 1% tail probability. The calculated extreme value is then printed to the console.

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