To What Extent Does a Student-t GARCH(1,1) Model Capture Observable Phenomena of Crypto Markets Better Than EWMA Black-Scholes Assumptions?

• Compare t-GARCH(1,1) against EWMA-anchored BSM for BTC derivative pricing
• Strict out-of-sample evaluation (2023–2026) across Calm / Normal / Panic regimes
• Metrics: volatility smile fit, VaR exceedances, straddle backtest P&L

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File Hierarchy

paper/
├── __init__.py                              # Abstract + file map
├── 1_introduction/
│   ├── 1_1_financial_markets.py             # Crypto market structure
│   ├── 1_2_options_mechanics.py             # Options pricing fundamentals
│   └── 1_3_research_gap.py                  # EWMA backward-looking bias
├── 2_literature_review/
│   ├── 2_1_bsm_paradigm.py                  # BSM, volatility smile, Newton-Raphson IV
│   ├── 2_2_conditional_heteroskedasticity.py # GARCH, CRIX, Duan risk-neutral
│   ├── 2_3_jumps_heavy_tails.py             # ARJI-GARCH, SETAR-GARCH, dist shifts
│   └── 2_4_computational_frontiers.py       # MC, PINNs, LSTM-GARCH, HyperIV
├── 3_theoretical_framework/
│   ├── 3_1_bsm_model.py                     # GBM → closed-form BSM + EWMA feed
│   └── 3_2_tgarch_model.py                  # Conditional variance + Student-t errors
├── 4_methodology/
│   ├── 4_1_data_acquisition.py              # yfinance + Deribit, OTM/ATM filter
│   ├── 4_2_oos_testing.py                   # In-sample fit 2020–2023, freeze params
│   ├── 4_3_regime_classification.py         # 30-day rolling vol percentile buckets
│   └── 4_4_monte_carlo_pricing.py           # Implied yield + beta dampening (VRP)
├── 5_results/
│   ├── 5_1_volatility_forecasting.py        # IGARCH hangover, leading vs. trailing
│   ├── 5_2_volatility_smile.py              # Smile curvature, smirk asymmetry
│   ├── 5_3_trading_backtest.py              # Signal-driven straddle, +20% OOS P&L
│   ├── 5_4_pricing_error_distribution.py    # BS smoothness vs. GARCH symmetry fails
│   └── 5_5_var_exceedances.py               # 95% VaR by regime (MC vs. EWMA-Gaussian)
├── 6_discussion.py                          # Limitations: speed, smirk, IGARCH
├── 7_conclusion.py                          # Summary + future directions
└── 8_references.py                          # Full bibliography + REFERENCES dict

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1. Introduction

• 1.1 Financial Markets — Crypto: continuous, unique macro drivers, non-normal returns
• 1.2 Options Mechanics — Call/Put, strike K, early exercise, fat-tail pricing need
• 1.3 Research Gap — EWMA trails shocks; this paper quantifies forward-looking GARCH advantage OOS

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2. Literature Review

• 2.1 BSM Paradigm — Constant vol, volatility smile, Newton-Raphson implied vol
• 2.2 Conditional Heteroskedasticity — GARCH(1,1), Duan LRNVR, BTC/CRIX vol index
• 2.3 Jumps & Heavy Tails — ARJI-GARCH time-varying jumps, SETAR-GARCH regime switching
• 2.4 Computational Frontiers — Monte Carlo cost, PINNs, LSTM-GARCH, HyperIV smoothing
• Gap — No rigorous OOS backtest of t-GARCH straddle vs. EWMA-BSM in panic regimes

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3. Theoretical Framework

• 3.1 BSM Model
  • dS_t = μS_t dt + σS_t dW_t
  • Closed form: C = S0·N(d1) − K·e^{−rT}·N(d2)
  • EWMA benchmark: 30-day rolling vol fed into BSM instead of constant σ
• 3.2 t-GARCH(1,1)
  • σ_t² = ω + α·ε_{t−1}² + β·σ_{t−1}²
  • Innovations ε_t = σ_t·z_t, z_t ~ Student-t(ν)
  • Heavy tails built into variance mechanism via ν degrees of freedom

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4. Methodology

• 4.1 Data — BTC-USD + ^IRX via yfinance; Deribit options API; drop ITM + illiquid
• 4.2 OOS Protocol — Fit params on 2020–2023 only; freeze; evaluate 2023–2026
• 4.3 Regime Classification — 30-day rolling vol: Calm (<p25) / Normal (p25–p90) / Panic (>p90)
• 4.4 Monte Carlo — 2,000 paths; implied yield replaces risk-free rate; beta dampened to 0.95

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5. Empirical Results

• 5.1 Volatility Forecasting — α+β≈1 (IGARCH); ν=2.95; GARCH ≈ EWMA over long horizons

• 5.2 Volatility Smile — GARCH reproduces smile curvature; fails to capture downside smirk

• 5.3 Trading Backtest — GARCH signal fires 27/39 periods; +20% OOS P&L vs. −50% naive

• 5.4 Pricing Errors — BS fails at low TTM near-the-money; GARCH fails on OTM calls

• 5.5 VaR Exceedances

  Regime	Target	GARCH	BS EWMA
  Calm	5.0%	5.8%	10.7%
  Normal	5.0%	0.7%	2.7%
  Panic	5.0%	0.0%	0.0%

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6. Discussion & Limitations

• GARCH MC too slow for HFT → future: PINNs to solve SDEs (Lu et al., 2021)
• Symmetric variance → cannot capture smirk → fix: GJR-GARCH or EGARCH
• IGARCH hangover from static fit → fix: rolling-window re-fitting

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7. Conclusion

• EWMA-BSM: fast but inadequate for BTC tail risk
• t-GARCH(1,1) wins on: smile fit, calm-regime VaR, straddle signal quality
• GARCH caveats: IGARCH persistence, symmetric smile, volatility hangover in Normal regime
• Next steps: asymmetric extensions + dynamic re-fitting to replace BS EWMA across all regimes

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References

See paper/8_references.py for full bibliography.