This module provides institutional-grade quantitative tools used by hedge funds and proprietary trading desks.

Value at Risk (VaR)

Maximum expected loss at a given confidence level over a specific time horizon.

• Parametric: Assumes Normal distribution
• Historical: Uses actual past returns (no distribution assumption)
• Monte Carlo: 10,000 random simulations
• CVaR/ES: Average loss beyond the VaR threshold

Sharpe & Sortino Ratios

Risk-adjusted return metrics. Sortino penalizes only downside volatility.

GARCH(1,1)

Models time-varying volatility with clustering effects. Used by banks and quant funds for VaR and option pricing.

Z-Score & Mean Reversion

Identifies when a currency is statistically over/under-valued vs its historical mean. Basis of statistical arbitrage strategies.

Hurst Exponent

Detects the market regime: trending vs mean-reverting vs random walk. Determines which strategy type to apply.

For informational purposes only. Not financial advice.

VaR Formula (Parametric): VaR = μ − z × σ

• 95% VaR: z = 1.645 → 1-in-20 daily loss
• 99% VaR: z = 2.326 → 1-in-100 daily loss

CVaR (Expected Shortfall): Average of all losses beyond the VaR threshold. More conservative and preferred by Basel III regulators.

Regulatory Note: Basel III requires banks to use CVaR (ES) instead of VaR for market risk capital requirements since 2019.

σ²(t) = ω + α × ε²(t-1) + β × σ²(t-1)

• ω (omega): Long-run variance weight
• α (alpha): Reaction to market shocks (0.10)
• β (beta): Volatility persistence (0.85)
• α + β < 1: Stationarity condition

Usage: Options pricing (Black-Scholes inputs), risk capital allocation, volatility regime detection.

GARCH parameters (α=0.10, β=0.85) are calibrated to Forex markets. Full MLE estimation requires specialized software.

R/S Analysis (Rescaled Range):

• H < 0.4: Anti-persistent — mean-reverting. Pairs trading, stat arb.
• H ≈ 0.5: Random Walk (Efficient Market Hypothesis) — no edge.
• H > 0.6: Persistent — trending. Trend-following, momentum strategies.

Practical use: Hedge funds use Hurst to select strategy type before entering a trade. If H < 0.45, use mean-reversion. If H > 0.55, use trend-following.