systems.builtin.stochastic.discrete.DiscreteGARCH11

systems.builtin.stochastic.discrete.DiscreteGARCH11(*args, **kwargs)

GARCH(1,1) - time-varying volatility model (Nobel Prize 2003).

The industry standard for modeling financial volatility, capturing volatility clustering, persistence, and mean reversion with just three parameters. This represents one of the most important models in financial econometrics.

Model Equations

Return process: r[k] = μ + ε[k] ε[k] = σ[k]·w[k]

Variance process (GARCH): σ²[k] = ω + α·ε²[k-1] + β·σ²[k-1]

where: - r[k]: Return (observable) - μ: Mean return (typically ≈ 0 for daily) - ε[k]: Innovation (zero-mean shock) - σ²[k]: Conditional variance (time-varying, latent) - w[k] ~ N(0,1): Standardized innovation (iid) - ω > 0: Constant term - α ≥ 0: ARCH coefficient (shock effect) - β ≥ 0: GARCH coefficient (persistence)

State-Space Form:

Augmented state: Z = [r, σ²] (return and variance)

Dynamics (nonlinear!): r[k] = μ + σ[k]·w[k] σ²[k+1] = ω + α·(r[k] - μ)² + β·σ²[k]

Physical Interpretation

What GARCH Models:

Financial markets exhibit volatility clustering: - Large price movements followed by large movements - Calm periods followed by calm periods - “Volatility begets volatility”

Why This Happens:

Information Flow:
- News arrives in clusters
- Uncertainty propagates
Market Microstructure:
- Feedback: Vol → wider spreads → more vol
- Herding behavior
Risk Preferences:
- Risk-off → sell → higher vol → more risk-off
- Feedback loops

GARCH Equation Interpretation:

σ²[k] = ω + α·ε²[k-1] + β·σ²[k-1]

Components: 1. ω: Baseline variance (floor) 2. α·ε²[k-1]: Recent shock (ARCH effect) - Large |ε[k-1]| → high σ²[k] - “News impact curve” 3. β·σ²[k-1]: Past variance (persistence) - High σ²[k-1] → high σ²[k] - Volatility clustering

Key Features

Time-Varying Variance: Unlike constant-variance models, σ²[k] changes every period.

Volatility Clustering: High volatility tends to persist (α + β near 1).

Mean Reversion: Variance reverts to ω/(1-α-β) over time.

Conditional vs Unconditional: - Conditional: Var[r[k]|past] = σ²[k] (time-varying) - Unconditional: Var[r[k]] = ω/(1-α-β) (constant)

Heavy Tails: Even with Gaussian w[k], unconditional distribution has excess kurtosis.

Predictable Volatility: Can forecast σ²[k+h] from past (even if returns unpredictable).

Mathematical Properties

Stationarity Condition: α + β < 1

If α + β = 1: IGARCH (integrated, non-stationary variance) If α + β > 1: Explosive (unstable)

Unconditional Moments:

Mean: E[r] = μ

Variance: Var[r] = ω/(1 - α - β)

Kurtosis: Excess kurtosis > 0 (heavy tails) Kurt = 3·(1 - (α+β)²)/(1 - (α+β)² - 2α²)

Autocorrelation: - Returns r[k]: ρ(h) ≈ 0 (unpredictable) - Squared returns ε²[k]: ρ(h) = (α+β)^h (geometric decay)

Persistence: Measured by α + β (sum of ARCH and GARCH): - α + β ≈ 0.99: Very persistent (typical stocks) - α + β ≈ 0.95: Persistent - α + β ≈ 0.80: Moderate persistence

Volatility Forecast: h-step ahead: σ²[k+h|k] = σ̄² + (α+β)^h·(σ²[k] - σ̄²)

where σ̄² = ω/(1-α-β) is long-run variance.

Exponential decay to long-run average.

Physical Interpretation

Parameter ω (Baseline): - Units: [return]² - Long-run variance: σ̄² = ω/(1-α-β) - Typical: Relates to annual volatility

Parameter α (ARCH Effect): - Dimensionless - Weight on recent shock - Typical: 0.05-0.15 - Higher α: More reactive to news

Parameter β (GARCH Effect): - Dimensionless - Weight on past variance - Typical: 0.80-0.92 - Higher β: More persistent

Sum α + β (Total Persistence): - Typical: 0.95-0.99 (very high!) - Near 1: “Volatility never dies” - Financial data: Usually >0.90

Typical Values (Daily Stock Returns): - ω ≈ 0.000001-0.00001 (depends on scaling) - α ≈ 0.05-0.10 - β ≈ 0.85-0.92 - α + β ≈ 0.95-0.98

State Space

Augmented state: Z = [r, σ²] ∈ ℝ² - r: Observable return - σ²: Latent conditional variance

Observable: r ∈ ℝ - What we actually measure (price changes)

Latent: σ² ∈ ℝ₊ - Unobserved (must be estimated/filtered)

Control: u (typically none for GARCH) - Could add for interventions

Parameters

Name	Type	Description	Default
omega	float	Constant term (must be positive) - Sets baseline variance - Typical: 1e-6 to 1e-4 (daily returns)	`0.00001`
alpha	float	ARCH coefficient (must be non-negative) - News impact (shock sensitivity) - Typical: 0.05-0.15	`0.1`
beta	float	GARCH coefficient (must be non-negative) - Persistence (volatility memory) - Typical: 0.80-0.92	`0.85`
mu	float	Mean return - Often ≈ 0 for daily returns - Can estimate or fix	`0.0`
dt	float	Sampling period (1 = daily typical)	`1.0`

Stochastic Properties

System Type: NONLINEAR (σ² equation)
Conditional: Variance time-varying
Stationary: If α + β < 1
Heavy Tails: Yes (excess kurtosis)
Volatility Clustering: Yes (by design)
Leverage Effect: No (symmetric)

Applications

1. Risk Management: - VaR calculation - Expected shortfall (CVaR) - Stress testing

2. Option Pricing: - Volatility forecasting - Monte Carlo with GARCH - Implied vs realized vol

3. Portfolio: - Time-varying covariance (MGARCH) - Dynamic hedging - Risk parity

4. Trading: - Volatility strategies - Position sizing (risk targeting) - Options trading signals

5. Regulation: - Basel capital requirements - Internal VaR models - Stress testing

Numerical Simulation

Recursion:

Initialize: σ²[0] = ω/(1-α-β)

For k = 0, 1, 2, …: 1. Draw w[k] ~ N(0,1) 2. Compute r[k] = μ + σ[k]·w[k] 3. Update σ²[k+1] = ω + α·(r[k]-μ)² + β·σ²[k]

State-Space: State Z = [r, σ²] - r observable - σ² latent (filtered in practice)

Comparison with Constant Variance

Constant Variance: r[k] = μ + σ·w[k]

Problems: - Misses volatility clustering - Underestimates tail risk - Poor VaR forecasts

GARCH: r[k] = μ + σ[k]·w[k] σ²[k] = f(past)

Advantages: - Captures clustering - Better tail risk - Accurate VaR

Limitations

Symmetric (leverage effect ignored)
Gaussian (can extend to Student-t)
Low frequency (daily typically, not intraday)
Univariate (for MGARCH extension)

Extensions

EGARCH: Asymmetry
GJR-GARCH: Threshold effects
FIGARCH: Long memory
DCC-GARCH: Dynamic conditional correlation
GARCH-M: Risk premium in mean

Methods

Name	Description
define_system	Define GARCH(1,1) dynamics.
forecast_variance	Forecast variance h steps ahead.
get_half_life	Get volatility shock half-life (periods to decay by 50%).
get_persistence	Get volatility persistence α + β.
get_unconditional_variance	Get long-run (unconditional) variance σ̄² = ω/(1-α-β).

define_system

systems.builtin.stochastic.discrete.DiscreteGARCH11.define_system(
    omega=1e-05,
    alpha=0.1,
    beta=0.85,
    mu=0.0,
    dt=1.0,
)

Define GARCH(1,1) dynamics.

Parameters

Name	Type	Description	Default
omega	float	Constant term in variance equation (must be positive) - Sets baseline variance - Typical daily: 1e-6 to 1e-4	`0.00001`
alpha	float	ARCH coefficient (must be non-negative) - Weight on past squared shock - Typical: 0.05-0.15 - Higher: More reactive to news	`0.1`
beta	float	GARCH coefficient (must be non-negative) - Weight on past variance - Typical: 0.80-0.92 - Higher: More persistent	`0.85`
mu	float	Mean return - Typically ≈ 0 for daily - Can be non-zero for longer horizons	`0.0`
dt	float	Sampling period (1 = daily typical)	`1.0`

Raises

Name	Type	Description
	ValueError	If ω ≤ 0, α < 0, β < 0, or α + β ≥ 1

Notes

Parameter Constraints:

Positivity:
- ω > 0: Ensures σ²[k] > 0
- α ≥ 0, β ≥ 0: Non-negativity
Stationarity:
- α + β < 1: Variance mean-reverting
- α + β = 1: IGARCH (non-stationary)
- α + β > 1: Explosive (invalid)

Long-Run Variance: σ̄² = ω/(1 - α - β)

Must be finite (requires α + β < 1).

Persistence: α + β

Typical values: - Stocks: 0.95-0.99 (very persistent!) - Bonds: 0.90-0.95 - FX: 0.93-0.97

Half-Life: Time for variance shock to decay by 50%: h_{1/2} ≈ ln(0.5)/ln(α+β)

Example: α + β = 0.95 → h ≈ 13.5 days

Typical Configurations:

S&P 500 (daily): - ω ≈ 0.000001 - α ≈ 0.10 - β ≈ 0.88 - α + β = 0.98 (very persistent)

Emerging Market (more volatile): - ω ≈ 0.00005 - α ≈ 0.15 - β ≈ 0.80 - α + β = 0.95

Low Volatility Stock: - ω ≈ 0.0000005 - α ≈ 0.05 - β ≈ 0.92 - α + β = 0.97

Initialization:

For σ²[0], common choices: 1. Unconditional: ω/(1-α-β) 2. Sample variance from data 3. Estimated as parameter

State-Space Structure:

This is NONLINEAR state-space: - r[k] depends on σ[k] (multiplicatively) - σ²[k+1] depends on ε²[k] (quadratically)

Not amenable to standard Kalman filter. Need extended/unscented/particle filter.

Examples

>>> # Standard stock GARCH
>>> stock = DiscreteGARCH11(
...     omega=0.000002,
...     alpha=0.08,
...     beta=0.90,
...     mu=0.0005
... )
>>>
>>> # Check constraints
>>> persistence = 0.08 + 0.90
>>> print(f"Stationary: {persistence < 1}")
>>>
>>> # Long-run volatility
>>> sigma_lr = np.sqrt(0.000002 / (1 - persistence))
>>> print(f"Long-run daily vol: {sigma_lr:.4f}")
>>> annual_vol = sigma_lr * np.sqrt(252)
>>> print(f"Annualized: {annual_vol:.2%}")

forecast_variance

systems.builtin.stochastic.discrete.DiscreteGARCH11.forecast_variance(
    current_variance,
    horizon=10,
)

Forecast variance h steps ahead.

σ²[k+h|k] = σ̄² + (α+β)^h·(σ²[k] - σ̄²)

Parameters

Name	Type	Description	Default
current_variance	float	Current conditional variance σ²[k]	required
horizon	int	Forecast horizon	`10`

Returns

Name	Type	Description
	np.ndarray	Variance forecasts for h=1,…,horizon

Examples

>>> garch = DiscreteGARCH11(omega=0.00001, alpha=0.1, beta=0.85)
>>> forecasts = garch.forecast_variance(current_variance=0.0001, horizon=20)
>>> print(f"1-day: {np.sqrt(forecasts[0]):.4f}")
>>> print(f"20-day: {np.sqrt(forecasts[-1]):.4f}")

get_half_life

systems.builtin.stochastic.discrete.DiscreteGARCH11.get_half_life()

Get volatility shock half-life (periods to decay by 50%).

Returns

Name	Type	Description
	float	Half-life [periods]

Examples

>>> garch = DiscreteGARCH11(alpha=0.1, beta=0.85)
>>> hl = garch.get_half_life()
>>> print(f"Volatility half-life: {hl:.1f} days")

get_persistence

systems.builtin.stochastic.discrete.DiscreteGARCH11.get_persistence()

Get volatility persistence α + β.

get_unconditional_variance

systems.builtin.stochastic.discrete.DiscreteGARCH11.get_unconditional_variance()

Get long-run (unconditional) variance σ̄² = ω/(1-α-β).

Returns

Name	Type	Description
	float	Unconditional variance

Raises

Name	Type	Description
	ValueError	If α + β ≥ 1 (non-stationary)

Examples

>>> garch = DiscreteGARCH11(omega=0.00001, alpha=0.1, beta=0.85)
>>> var_uncond = garch.get_unconditional_variance()
>>> vol_uncond = np.sqrt(var_uncond)
>>> print(f"Long-run daily vol: {vol_uncond:.4f}")

# systems.builtin.stochastic.discrete.DiscreteGARCH11 { #cdesym.systems.builtin.stochastic.discrete.DiscreteGARCH11 } ```python systems.builtin.stochastic.discrete.DiscreteGARCH11(*args, **kwargs) ``` GARCH(1,1) - time-varying volatility model (Nobel Prize 2003). The industry standard for modeling financial volatility, capturing volatility clustering, persistence, and mean reversion with just three parameters. This represents one of the most important models in financial econometrics. ## Model Equations {.doc-section .doc-section-model-equations} Return process: r[k] = μ + ε[k] ε[k] = σ[k]·w[k] Variance process (GARCH): σ²[k] = ω + α·ε²[k-1] + β·σ²[k-1] where: - r[k]: Return (observable) - μ: Mean return (typically ≈ 0 for daily) - ε[k]: Innovation (zero-mean shock) - σ²[k]: Conditional variance (time-varying, latent) - w[k] ~ N(0,1): Standardized innovation (iid) - ω > 0: Constant term - α ≥ 0: ARCH coefficient (shock effect) - β ≥ 0: GARCH coefficient (persistence) **State-Space Form:** Augmented state: Z = [r, σ²] (return and variance) Dynamics (nonlinear!): r[k] = μ + σ[k]·w[k] σ²[k+1] = ω + α·(r[k] - μ)² + β·σ²[k] ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **What GARCH Models:** Financial markets exhibit **volatility clustering**: - Large price movements followed by large movements - Calm periods followed by calm periods - "Volatility begets volatility" **Why This Happens:** 1. **Information Flow:** - News arrives in clusters - Uncertainty propagates 2. **Market Microstructure:** - Feedback: Vol → wider spreads → more vol - Herding behavior 3. **Risk Preferences:** - Risk-off → sell → higher vol → more risk-off - Feedback loops **GARCH Equation Interpretation:** σ²[k] = ω + α·ε²[k-1] + β·σ²[k-1] Components: 1. **ω:** Baseline variance (floor) 2. **α·ε²[k-1]:** Recent shock (ARCH effect) - Large |ε[k-1]| → high σ²[k] - "News impact curve" 3. **β·σ²[k-1]:** Past variance (persistence) - High σ²[k-1] → high σ²[k] - Volatility clustering ## Key Features {.doc-section .doc-section-key-features} **Time-Varying Variance:** Unlike constant-variance models, σ²[k] changes every period. **Volatility Clustering:** High volatility tends to persist (α + β near 1). **Mean Reversion:** Variance reverts to ω/(1-α-β) over time. **Conditional vs Unconditional:** - Conditional: Var[r[k]|past] = σ²[k] (time-varying) - Unconditional: Var[r[k]] = ω/(1-α-β) (constant) **Heavy Tails:** Even with Gaussian w[k], unconditional distribution has excess kurtosis. **Predictable Volatility:** Can forecast σ²[k+h] from past (even if returns unpredictable). ## Mathematical Properties {.doc-section .doc-section-mathematical-properties} **Stationarity Condition:** α + β < 1 If α + β = 1: IGARCH (integrated, non-stationary variance) If α + β > 1: Explosive (unstable) **Unconditional Moments:** Mean: E[r] = μ Variance: Var[r] = ω/(1 - α - β) Kurtosis: Excess kurtosis > 0 (heavy tails) Kurt = 3·(1 - (α+β)²)/(1 - (α+β)² - 2α²) **Autocorrelation:** - Returns r[k]: ρ(h) ≈ 0 (unpredictable) - Squared returns ε²[k]: ρ(h) = (α+β)^h (geometric decay) **Persistence:** Measured by α + β (sum of ARCH and GARCH): - α + β ≈ 0.99: Very persistent (typical stocks) - α + β ≈ 0.95: Persistent - α + β ≈ 0.80: Moderate persistence **Volatility Forecast:** h-step ahead: σ²[k+h|k] = σ̄² + (α+β)^h·(σ²[k] - σ̄²) where σ̄² = ω/(1-α-β) is long-run variance. Exponential decay to long-run average. ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **Parameter ω (Baseline):** - Units: [return]² - Long-run variance: σ̄² = ω/(1-α-β) - Typical: Relates to annual volatility **Parameter α (ARCH Effect):** - Dimensionless - Weight on recent shock - Typical: 0.05-0.15 - Higher α: More reactive to news **Parameter β (GARCH Effect):** - Dimensionless - Weight on past variance - Typical: 0.80-0.92 - Higher β: More persistent **Sum α + β (Total Persistence):** - Typical: 0.95-0.99 (very high!) - Near 1: "Volatility never dies" - Financial data: Usually >0.90 **Typical Values (Daily Stock Returns):** - ω ≈ 0.000001-0.00001 (depends on scaling) - α ≈ 0.05-0.10 - β ≈ 0.85-0.92 - α + β ≈ 0.95-0.98 ## State Space {.doc-section .doc-section-state-space} Augmented state: Z = [r, σ²] ∈ ℝ² - r: Observable return - σ²: Latent conditional variance Observable: r ∈ ℝ - What we actually measure (price changes) Latent: σ² ∈ ℝ₊ - Unobserved (must be estimated/filtered) Control: u (typically none for GARCH) - Could add for interventions ## Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|---------------------------------------------------------------------------------------------------|-----------| | omega | float | Constant term (must be positive) - Sets baseline variance - Typical: 1e-6 to 1e-4 (daily returns) | `0.00001` | | alpha | float | ARCH coefficient (must be non-negative) - News impact (shock sensitivity) - Typical: 0.05-0.15 | `0.1` | | beta | float | GARCH coefficient (must be non-negative) - Persistence (volatility memory) - Typical: 0.80-0.92 | `0.85` | | mu | float | Mean return - Often ≈ 0 for daily returns - Can estimate or fix | `0.0` | | dt | float | Sampling period (1 = daily typical) | `1.0` | ## Stochastic Properties {.doc-section .doc-section-stochastic-properties} - System Type: NONLINEAR (σ² equation) - Conditional: Variance time-varying - Stationary: If α + β < 1 - Heavy Tails: Yes (excess kurtosis) - Volatility Clustering: Yes (by design) - Leverage Effect: No (symmetric) ## Applications {.doc-section .doc-section-applications} **1. Risk Management:** - VaR calculation - Expected shortfall (CVaR) - Stress testing **2. Option Pricing:** - Volatility forecasting - Monte Carlo with GARCH - Implied vs realized vol **3. Portfolio:** - Time-varying covariance (MGARCH) - Dynamic hedging - Risk parity **4. Trading:** - Volatility strategies - Position sizing (risk targeting) - Options trading signals **5. Regulation:** - Basel capital requirements - Internal VaR models - Stress testing ## Numerical Simulation {.doc-section .doc-section-numerical-simulation} **Recursion:** Initialize: σ²[0] = ω/(1-α-β) For k = 0, 1, 2, ...: 1. Draw w[k] ~ N(0,1) 2. Compute r[k] = μ + σ[k]·w[k] 3. Update σ²[k+1] = ω + α·(r[k]-μ)² + β·σ²[k] **State-Space:** State Z = [r, σ²] - r observable - σ² latent (filtered in practice) ## Comparison with Constant Variance {.doc-section .doc-section-comparison-with-constant-variance} **Constant Variance:** r[k] = μ + σ·w[k] Problems: - Misses volatility clustering - Underestimates tail risk - Poor VaR forecasts **GARCH:** r[k] = μ + σ[k]·w[k] σ²[k] = f(past) Advantages: - Captures clustering - Better tail risk - Accurate VaR ## Limitations {.doc-section .doc-section-limitations} - Symmetric (leverage effect ignored) - Gaussian (can extend to Student-t) - Low frequency (daily typically, not intraday) - Univariate (for MGARCH extension) ## Extensions {.doc-section .doc-section-extensions} - EGARCH: Asymmetry - GJR-GARCH: Threshold effects - FIGARCH: Long memory - DCC-GARCH: Dynamic conditional correlation - GARCH-M: Risk premium in mean ## Methods | Name | Description | | --- | --- | | [define_system](#cdesym.systems.builtin.stochastic.discrete.DiscreteGARCH11.define_system) | Define GARCH(1,1) dynamics. | | [forecast_variance](#cdesym.systems.builtin.stochastic.discrete.DiscreteGARCH11.forecast_variance) | Forecast variance h steps ahead. | | [get_half_life](#cdesym.systems.builtin.stochastic.discrete.DiscreteGARCH11.get_half_life) | Get volatility shock half-life (periods to decay by 50%). | | [get_persistence](#cdesym.systems.builtin.stochastic.discrete.DiscreteGARCH11.get_persistence) | Get volatility persistence α + β. | | [get_unconditional_variance](#cdesym.systems.builtin.stochastic.discrete.DiscreteGARCH11.get_unconditional_variance) | Get long-run (unconditional) variance σ̄² = ω/(1-α-β). | ### define_system { #cdesym.systems.builtin.stochastic.discrete.DiscreteGARCH11.define_system } ```python systems.builtin.stochastic.discrete.DiscreteGARCH11.define_system( omega=1e-05, alpha=0.1, beta=0.85, mu=0.0, dt=1.0, ) ``` Define GARCH(1,1) dynamics. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|-----------------------------------------------------------------------------------------------------------------------------|-----------| | omega | float | Constant term in variance equation (must be positive) - Sets baseline variance - Typical daily: 1e-6 to 1e-4 | `0.00001` | | alpha | float | ARCH coefficient (must be non-negative) - Weight on past squared shock - Typical: 0.05-0.15 - Higher: More reactive to news | `0.1` | | beta | float | GARCH coefficient (must be non-negative) - Weight on past variance - Typical: 0.80-0.92 - Higher: More persistent | `0.85` | | mu | float | Mean return - Typically ≈ 0 for daily - Can be non-zero for longer horizons | `0.0` | | dt | float | Sampling period (1 = daily typical) | `1.0` | #### Raises {.doc-section .doc-section-raises} | Name | Type | Description | |--------|------------|--------------------------------------| | | ValueError | If ω ≤ 0, α < 0, β < 0, or α + β ≥ 1 | #### Notes {.doc-section .doc-section-notes} **Parameter Constraints:** 1. **Positivity:** - ω > 0: Ensures σ²[k] > 0 - α ≥ 0, β ≥ 0: Non-negativity 2. **Stationarity:** - α + β < 1: Variance mean-reverting - α + β = 1: IGARCH (non-stationary) - α + β > 1: Explosive (invalid) **Long-Run Variance:** σ̄² = ω/(1 - α - β) Must be finite (requires α + β < 1). **Persistence:** α + β Typical values: - Stocks: 0.95-0.99 (very persistent!) - Bonds: 0.90-0.95 - FX: 0.93-0.97 **Half-Life:** Time for variance shock to decay by 50%: h_{1/2} ≈ ln(0.5)/ln(α+β) Example: α + β = 0.95 → h ≈ 13.5 days **Typical Configurations:** **S&P 500 (daily):** - ω ≈ 0.000001 - α ≈ 0.10 - β ≈ 0.88 - α + β = 0.98 (very persistent) **Emerging Market (more volatile):** - ω ≈ 0.00005 - α ≈ 0.15 - β ≈ 0.80 - α + β = 0.95 **Low Volatility Stock:** - ω ≈ 0.0000005 - α ≈ 0.05 - β ≈ 0.92 - α + β = 0.97 **Initialization:** For σ²[0], common choices: 1. Unconditional: ω/(1-α-β) 2. Sample variance from data 3. Estimated as parameter **State-Space Structure:** This is NONLINEAR state-space: - r[k] depends on σ[k] (multiplicatively) - σ²[k+1] depends on ε²[k] (quadratically) Not amenable to standard Kalman filter. Need extended/unscented/particle filter. #### Examples {.doc-section .doc-section-examples} ```python >>> # Standard stock GARCH >>> stock = DiscreteGARCH11( ... omega=0.000002, ... alpha=0.08, ... beta=0.90, ... mu=0.0005 ... ) >>> >>> # Check constraints >>> persistence = 0.08 + 0.90 >>> print(f"Stationary: {persistence < 1}") >>> >>> # Long-run volatility >>> sigma_lr = np.sqrt(0.000002 / (1 - persistence)) >>> print(f"Long-run daily vol: {sigma_lr:.4f}") >>> annual_vol = sigma_lr * np.sqrt(252) >>> print(f"Annualized: {annual_vol:.2%}") ``` ### forecast_variance { #cdesym.systems.builtin.stochastic.discrete.DiscreteGARCH11.forecast_variance } ```python systems.builtin.stochastic.discrete.DiscreteGARCH11.forecast_variance( current_variance, horizon=10, ) ``` Forecast variance h steps ahead. σ²[k+h|k] = σ̄² + (α+β)^h·(σ²[k] - σ̄²) #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |------------------|--------|------------------------------------|------------| | current_variance | float | Current conditional variance σ²[k] | _required_ | | horizon | int | Forecast horizon | `10` | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|------------|----------------------------------------| | | np.ndarray | Variance forecasts for h=1,...,horizon | #### Examples {.doc-section .doc-section-examples} ```python >>> garch = DiscreteGARCH11(omega=0.00001, alpha=0.1, beta=0.85) >>> forecasts = garch.forecast_variance(current_variance=0.0001, horizon=20) >>> print(f"1-day: {np.sqrt(forecasts[0]):.4f}") >>> print(f"20-day: {np.sqrt(forecasts[-1]):.4f}") ``` ### get_half_life { #cdesym.systems.builtin.stochastic.discrete.DiscreteGARCH11.get_half_life } ```python systems.builtin.stochastic.discrete.DiscreteGARCH11.get_half_life() ``` Get volatility shock half-life (periods to decay by 50%). #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|---------------------| | | float | Half-life [periods] | #### Examples {.doc-section .doc-section-examples} ```python >>> garch = DiscreteGARCH11(alpha=0.1, beta=0.85) >>> hl = garch.get_half_life() >>> print(f"Volatility half-life: {hl:.1f} days") ``` ### get_persistence { #cdesym.systems.builtin.stochastic.discrete.DiscreteGARCH11.get_persistence } ```python systems.builtin.stochastic.discrete.DiscreteGARCH11.get_persistence() ``` Get volatility persistence α + β. ### get_unconditional_variance { #cdesym.systems.builtin.stochastic.discrete.DiscreteGARCH11.get_unconditional_variance } ```python systems.builtin.stochastic.discrete.DiscreteGARCH11.get_unconditional_variance() ``` Get long-run (unconditional) variance σ̄² = ω/(1-α-β). #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|------------------------| | | float | Unconditional variance | #### Raises {.doc-section .doc-section-raises} | Name | Type | Description | |--------|------------|-------------------------------| | | ValueError | If α + β ≥ 1 (non-stationary) | #### Examples {.doc-section .doc-section-examples} ```python >>> garch = DiscreteGARCH11(omega=0.00001, alpha=0.1, beta=0.85) >>> var_uncond = garch.get_unconditional_variance() >>> vol_uncond = np.sqrt(var_uncond) >>> print(f"Long-run daily vol: {vol_uncond:.4f}") ```