Distributionally Robust Rare-Event Simulation in Finance

This repository contains the code from my summer research project on distributionally robust rare-event simulation for financial risk.

Project Overview

The research addresses robust risk management problems of the form:

sup_{ν: d(ν,μ) ≤ δ} P_ν(Loss ≥ threshold)

where we seek the worst-case probability over all distributions ν within distance δ of a nominal distribution with mean μ.

Repository Structure

Core Problem Types

• Linear/ - Linear constraint problems (portfolio loss functions)
• Quadratic/ - Quadratic constraint problems
• piecewise/ - Piecewise linear constraints (options portfolios)
• Rebalance Delta Hedging/ - Dynamic hedging with rebalancing costs
• American Options/ - American option boundary analysis
• American option boundary/ - Additional American option studies

Estimation Methods

Each problem type contains subdirectories for different estimation approaches:

• CIS-CIS/ - Two-stage Conditional Importance Sampling
• MC-MC/ - Two-stage Monte Carlo
• CIS-MC/ - Hybrid: CIS for radius estimation, MC for probability
• MC-CIS/ - Hybrid: MC for radius estimation, CIS for probability

Key Algorithms

Two-Stage Estimation Framework

1. Stage 1: Estimate the optimal ambiguity radius u* that satisfies the constraint
2. Stage 2: Estimate the worst-case probability at radius u*

Method Implementations

• Monte Carlo (MC): Standard sampling-based estimation
• Conditional Importance Sampling (CIS): Variance reduction through optimal importance sampling
• Hybrid Methods: Combine MC and CIS for different stages

Main Files

Core Functions (per problem type)

• get_worst_p_[type]_[method1]_[method2].py - Main estimation functions
• experiments_[method]_[type].py - Experimental frameworks and parameter sweeps
• get_radius_[method]_[type].py - Stage 1 radius estimation
• robust_[method]_[type].py - Stage 2 probability estimation

Utilities

• find_xstar.py - Optimization problem solver
• shift_samples.py - Importance sampling transformations
• Various helper functions for each constraint type

Data and Results

Input Data

The experiments use S&P 500 stock data (MSFT, NVDA, AAPL, AMZN, META, GOOGL, BRK.B, TSLA, JPM, UNH):

• Returns from June 2021 to June 2024
• Mean returns μ and correlation matrix Rho defined in experiment files

Output Files

• *_results.csv - Experimental results with performance metrics
• ratios_vs_CIS_linear_results_with_MC.csv - Method comparisons
• plot.ipynb - Visualization and analysis

Running Experiments

Basic Usage

# Example: Linear CIS-CIS experiment
from Linear.CIS_CIS.experiments_cis_linear import run_threshold_sweep

# Set parameters
thresholds = [1, 2, 4, 6, 10]  # Loss thresholds to test
delta = 0.02                    # Ambiguity radius
N_tot = int(1e5)               # Total samples
runs = 5                       # Independent runs

# Run experiment
run_threshold_sweep(thresholds, delta, mu, Sigma, w, N_alpha, N_tot, runs, rng)

Key Parameters

• delta: Ambiguity set radius (Wasserstein distance)
• loss_threshold: Loss level for probability estimation
• mu, Sigma: Mean and covariance of nominal distribution
• w: Portfolio weights (linear case) or constraint matrices (other cases)
• N_alpha, N_tot: Sample allocation between stages
• runs: Number of independent replications

Performance Metrics

The experiments track several key metrics:

• mean_p: Average worst-case probability estimate
• rel_error: Relative error (confidence interval / estimate)
• lsre_mean: Log-scale relative efficiency
• mean_time_sec: Computational time
• var_H, var_p: Variance estimates for each stage

Method Comparison

The ratios_vs_CIS_linear_results_with_MC.csv file contains comparative analysis showing:

• Efficiency ratios between methods (CIS, MC, hybrid approaches)
• Time complexity comparisons
• Accuracy trade-offs

Specialized Applications

American Options

• [american_put_boundary.py](American option boundary/CIS-CIS ( no exercise) /american_put_boundary.py) - Boundary computation
• Analysis of early exercise decisions under ambiguity

VaR Estimation

• VaR_cis_cis.py - Value-at-Risk under distributional ambiguity
• Bootstrap confidence intervals for risk measures

Delta Hedging

• Dynamic rebalancing with transaction costs
• Robust hedging under parameter uncertainty

Dependencies

numpy
scipy
pandas
matplotlib
numba  # For performance optimization

Research Context

This work addresses fundamental challenges in financial risk management:

1. Model Uncertainty: Traditional risk models assume known distributions
2. Computational Efficiency: Robust optimization problems are computationally intensive
3. Practical Implementation: Methods must scale to realistic portfolio sizes

The hybrid CIS-MC approaches often provide the best balance of accuracy and computational efficiency, as demonstrated in the comparative results.