Stochastic Initial Margin for Counterparty Credit Risk (CCR)

A Python framework for computing stochastic Initial Margin (IM) in the context of Counterparty Credit Risk, comparing brute-force Nested Monte Carlo with the Johnson distribution approximation (McWalter et al., 2018).

Overview

This project computes exposure metrics (EE, EEE, EEPE) for a portfolio of European options under bilateral CSA with stochastic IM. It implements two approaches to estimate the conditional 99% VaR of P&L changes over the Margin Period of Risk (MPOR):

1. Nested Monte Carlo (brute-force) — inner simulations at each outer node to build the empirical P&L distribution
2. Johnson approximation — fits a Johnson distribution (SU/SB/SL/SN) using the first four conditional moments, avoiding the costly inner MC loop

The comparison demonstrates that the Johnson method achieves comparable accuracy with a speedup factor proportional to the number of inner scenarios.

Key Features

• Multi-asset correlated GBM diffusion under the risk-neutral measure (exact discretization via Cholesky)
• Black-Scholes closed-form pricing for European calls and puts
• Nested Monte Carlo IM with configurable inner/outer scenario counts
• Johnson distribution fit using moment matching (skewness/kurtosis plane classification)
• Exposure metrics: Expected Exposure (EE), Effective EE, EEPE, and EAD computation
• Centralized configuration — all parameters in a single file (config/parameters.py)
• Fully vectorized NumPy implementation for performance

Portfolio

#	Type	Underlying	S₀	Strike	Maturity	Vol	Position
1	Call	A	100	105	2Y	20%	Long
2	Put	A	100	95	2Y	20%	Long
3	Call	B	150	160	2Y	25%	Long
4	Put	C	80	75	2Y	30%	Short
5	Call	C	80	85	2Y	30%	Long

Correlations: ρ(A,B) = 0.6, ρ(A,C) = 0.4, ρ(B,C) = 0.5

Parameters

Parameter	Value	Description
Risk-free rate	3%	Continuous compounding
MPOR	10 business days (≈ 2/52 yr)	Standard bilateral CSA
EEPE horizon	1 year	Regulatory standard
IM confidence	99%	VaR quantile level
N_outer	500	Outer Monte Carlo scenarios
N_inner	500	Inner MC scenarios (nested only)
Time steps	52	Weekly grid over 1 year

Project Structure

├── config/
│   └── parameters.py          # All configurable parameters
├── lib/
│   ├── black_scholes.py       # BS pricing (call/put)
│   ├── diffusion.py           # Correlated multi-asset GBM simulation
│   ├── portfolio.py           # Portfolio class and MtM computation
│   ├── margin.py              # IM calculation (nested MC + exposure with IM)
│   ├── exposure.py            # EE, EEE, EEPE metrics
│   ├── johnson.py             # Johnson distribution fit and IM approximation
│   └── utils.py               # Helpers, validation, timing
├── notebooks/
│   ├── 01_test_phase1.ipynb   # Validation: BS pricing, GBM, exposure
│   ├── 02_test_phase2.ipynb   # Validation: nested MC IM
│   └── 03_test_phase3.ipynb   # Validation: Johnson approximation
├── tests/                     # Unit tests (pytest)
├── report/                    # Generated figures and theoretical writeup
├── main.py                    # Full pipeline: both methods + comparison
└── CLAUDE.md                  # Project specification

Quick Start

Requirements

• Python 3.10+
• NumPy, SciPy, Matplotlib

Installation

pip install numpy scipy matplotlib

Run

# Full run (N_outer=500, N_inner=500)
python main.py

# Fast mode for quick testing (N_outer=200, N_inner=200)
python main.py --fast

Tests

pytest tests/

Output

The pipeline produces:

• Console output: EEPE values, IM statistics, comparison table, timing
• Figures saved in report/:
  • fig1 — Sample GBM paths for each underlying
  • fig2 — EE/EEE profiles without IM
  • fig3 — EE/EEE comparison: no IM vs nested MC vs Johnson
  • fig4 — Scatter plots: nested MC IM vs Johnson IM
  • fig5 — Mean IM over time + MAE between methods
  • fig6 — IM distribution at t = 0.5Y

Methodology

IM Computation

The Initial Margin at time t is defined as:

IM(t) = Q_99%(ΔPV | F_t)    where ΔPV = PV(t) - PV(t + δ)

A positive ΔPV represents a portfolio loss over the MPOR period δ.

Nested Monte Carlo (Deliverable 1)

For each outer scenario at each time step:

1. Simulate N_inner sub-scenarios over the MPOR
2. Price the portfolio at t + δ using Black-Scholes
3. Compute P&L changes: ΔPV = PV(t) - PV(t + δ)
4. IM = empirical 99th percentile of ΔPV

Johnson Approximation (Deliverable 2)

Replaces the inner MC loop by:

1. Estimating the first four conditional moments of ΔPV via polynomial regression
2. Fitting a Johnson distribution (SU/SB/SL/SN) based on the (skewness², kurtosis) pair
3. Computing the 99% quantile analytically

Exposure with IM (Deliverable 3)

Residual exposure after IM collateralization:

E_IM(t) = max(0, ΔPV(t, t+δ) - IM(t))

Then EE → EEE → EEPE are computed as usual.

References

• McWalter, T., Kienitz, J., Nowaczyk, N., Rudd, R., & Acar, S. (2018). Dynamic Initial Margin Estimation Based on Quantiles of Johnson Distributions
• CRR Art. 284 — EEPE computation under the risk-neutral measure
• BCBS/IOSCO — Margin requirements for non-centrally cleared derivatives

License

Academic project — EXIOM Partners × CentraleSupélec (M2 Quantitative Finance).