Technical Summary: Assessing Extreme Risk using Stochastic Simulation of Extremes

Problem Statement

Risk management across sectors such as climatology and finance relies heavily on the assessment of tail risks—events with low probability but high severity. A primary challenge in this domain is data scarcity, particularly in multivariate contexts where extreme events are rare. Traditional parametric approaches based on Extreme Value Theory (EVT) often face limitations in multivariate settings because the limiting distribution of multivariate extremes is not inherently parametric, necessitating complex model selection and restrictive assumptions about dependence structures (e.g., uniform dependence via copulas). Furthermore, empirical estimates of tail risk metrics (TRMs) are often unstable or impossible to compute when the number of observed exceedances above a high threshold is insufficient. Existing non-parametric methods for conditional tail inference are also limited by data availability and dimensionality.

Methodology

The paper proposes two non-parametric simulation algorithms designed to extend samples of multivariate extremes, thereby addressing data scarcity. These algorithms are grounded in the stochastic representation of Multivariate Generalised Pareto (MGP) distributions, specifically utilizing the decomposition derived by Rootzén et al. (2018).

Theoretical Foundation

The approach assumes that the vector of excesses above a high threshold converges to a standard MGP distribution. A standard MGP vector $Z$ can be decomposed as $Z = E + T - \max(T)$, where $E$ is a unit exponential variable and $T$ is a $d$-dimensional random vector independent of $E$. This decomposition allows for the representation of the vector components through differences $\Delta_{j,k} = T_j - T_k$.

Algorithm 1: Non-parametric Joint MGP Simulation

This algorithm generates new independent and identically distributed (i.i.d.) samples of a standard MGP vector to expand the dataset for joint risk assessment.

1. Transformation: Observed data is standardized to exponential margins and transformed into a standard MGP vector $Z$.
2. Difference Calculation: The algorithm computes the vector of differences $\Delta^{(1)}$ based on the first component.
3. Simulation:
   • New unit exponential variables $E_\ell$ are generated.
   • New realizations of the difference vector $\Delta^{(1)}$ are generated via non-parametric bootstrapping (resampling from observed differences).
   • New MGP vectors $\tilde{Z}$ are reconstructed using the stochastic relation $Z_j = E + \sum \Delta_{j,k} \prod \mathbb{1}_{\Delta_{\ell,k}<0}$.
4. Output: The simulated vectors are projected back to the original space to estimate risk metrics.

Algorithm 2: Non-parametric Conditional MGP Simulation

This algorithm generates samples of a specific risk factor $X_j$ conditional on other risk factors $X_{-j}$ being at extreme levels. This facilitates the inference of conditional tail distributions.

1. Conditional Distribution Derivation: The algorithm derives the conditional density of the difference variable $\Delta_{q,j}$ given $Z_{-j} = z_{-j}$. Three cases are distinguished based on the maximum value of the conditioning vector $z^* = \max(z_{-j})$:
   • Case 1: $z^* > 0$ and $z^* = z_q$.
   • Case 2: $z^* > 0$ and $z^* \neq z_q$.
   • Case 3: $z^* \leq 0$.
2. Sampling:
   • In Cases 1 and 3, where the conditional distribution is independent of the specific values of $z_{-j}$, bootstrapping is used directly.
   • In Case 2, where the distribution depends on $z_{-j}$, a rejection sampling method is employed, combining bootstrapping with a rejection algorithm to ensure the correct conditional density is sampled.
3. Output: New realizations of $Z_j$ are generated conditionally on $Z_{-j}$, allowing for the estimation of conditional expectations.

Key Contributions

1. Extension to Multivariate Dimensions: The authors generalize the bivariate simulation algorithms proposed by Legrand et al. (2023) to arbitrary dimensions ($d > 2$), enabling the analysis of complex, high-dimensional risk factors.
2. Non-Parametric Flexibility: By relying on bootstrapping and the stochastic representation of MGP vectors, the method avoids the restrictive parametric assumptions often required in multivariate EVT, such as specific copula families.
3. Dual Simulation Schemes: The paper provides a unified framework for both joint simulation (to estimate unconditional tail risk metrics) and conditional simulation (to infer quantities related to conditional tail distributions).
4. Theoretical Validation: The paper includes rigorous theoretical proofs (Propositions 7.1–7.6) demonstrating that the simulated samples converge to the correct standard MGP distribution and that the rejection sampling method in the conditional algorithm is valid.

Results

The performance of the algorithms was evaluated using both synthetic data (financial returns modeled with Student's $t$-distributions and Gumbel copulas) and real financial data (weekly negative returns of HSBC, Lloyds, and RBS).

• Tail Risk Metric (TRM) Estimation:

  • The joint simulation algorithm significantly improved the estimation of Value at Risk (VaR), Expected Shortfall (ES), Marginal Expected Shortfall (MES), and Dependent Conditional Tail Expectation (DCTE).
  • Compared to using only the original sample, the simulated and extended samples reduced relative approximation errors and dispersion, particularly at high confidence levels ($\alpha \in \{0.9975, 0.999, 0.9997\}$).
  • The method successfully estimated TRMs in scenarios where the original sample contained zero or very few exceedances (e.g., DCTE estimation for certain stocks was impossible with original data but feasible with simulation).

• Conditional Expectation Estimation:

  • The conditional simulation algorithm was tested against linear regression for estimating $\mu(x_{-j}) = E[X_j | X_{-j} = x_{-j}]$.
  • In synthetic data, the conditional simulation approach yielded lower errors than linear regression, with the advantage increasing as conditioning observations became more extreme.
  • In real data applications, the simulation approach provided shock estimates that were comparable to or more robust than linear regression, particularly for assets with heavier tails (Lloyds and RBS) where linear regression is known to be less suitable.

Significance and Claims

The paper claims that the proposed non-parametric simulation schemes effectively address the critical issue of data scarcity in extreme risk management. By expanding the available dataset through stochastic simulation, the authors demonstrate that it is possible to:

• Extrapolate beyond the range of observed historical data.
• Obtain more reliable and stable estimates of tail risk metrics (ES, MMES, DCTE) even when the number of observed extremes is extremely limited.
• Perform inference on conditional tail distributions in high-dimensional settings where traditional regression or parametric methods struggle.

The authors position these algorithms as practical tools for risk managers who need to quantify risks under extraordinary circumstances where historical data is insufficient. They note that while the current implementation uses empirical estimators on the simulated data, future work could integrate these simulation approaches with more sophisticated estimation techniques. The paper concludes that these methods enhance the quantity of observations in extreme regions, leading to more trustworthy risk metrics without imposing restrictive parametric models on the dependence structure.