Vasicek Base¶

Background¶

Vasicek Base

Educational script demonstrating vasicek base concepts.

Code¶

```python """ Vasicek Base

Educational script demonstrating vasicek base concepts. """

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vasicek/vasicek_base.py¶

============================================================================¶

from dataclasses import dataclass from enum import Enum from typing import Dict

class VasicekScheme(Enum): """Vasicek discretization schemes.""" EULER_MARUYAMA = "euler_maruyama" EXACT = "exact" # Vasicek has exact simulation MILSTEIN = "milstein" # Same as Euler for linear drift

class VasicekConfig: """Configuration class for Vasicek model settings."""

# Default model parameters
DEFAULT_R0 = 0.03
DEFAULT_B = 0.05  # long-term mean (b)
DEFAULT_A = 0.1   # mean reversion speed (a)
DEFAULT_SIGMA = 0.03
DEFAULT_MATURITY = 10.0

# Numerical constraints
MIN_RATE = -0.1  # Vasicek can go negative
MAX_RATE = 1.0
MIN_TIME = 1e-6
MAX_TIME = 100.0

# Simulation defaults
DEFAULT_NUM_PATHS = 1000

# Validation tolerances
VALIDATION_TOLERANCE = 0.05

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Custom Exceptions¶

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class VasicekError(Exception): """Base exception for Vasicek model errors.""" pass

class VasicekParameterError(VasicekError): """Exception for invalid Vasicek model parameters.""" pass

class VasicekSimulationError(VasicekError): """Exception for simulation-related errors.""" pass

class VasicekValidationError(VasicekError): """Exception for validation failures.""" pass

class VasicekNumericalError(VasicekError): """Exception for numerical computation errors.""" pass

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Parameter Classes¶

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@dataclass(frozen=True) class VasicekParameters: """Immutable container for Vasicek model parameters.""" r0: float # Initial rate b: float # Long-term mean (θ in some notation) a: float # Mean reversion speed (κ in some notation) sigma: float # Constant volatility maturity_time: float

def __post_init__(self):
    """Validate parameters after initialization."""
    self._validate_parameters()

def _validate_parameters(self) -> None:
    """Validate all Vasicek parameters."""
    if not (VasicekConfig.MIN_RATE < self.r0 < VasicekConfig.MAX_RATE):
        raise VasicekParameterError(
            f"Initial rate r0={self.r0} must be in "
            f"({VasicekConfig.MIN_RATE}, {VasicekConfig.MAX_RATE})"
        )

    if not (VasicekConfig.MIN_RATE < self.b < VasicekConfig.MAX_RATE):
        raise VasicekParameterError(
            f"Long-term mean b={self.b} must be in "
            f"({VasicekConfig.MIN_RATE}, {VasicekConfig.MAX_RATE})"
        )

    if self.a <= 0:
        raise VasicekParameterError(
            f"Mean reversion speed a={self.a} must be positive"
        )

    if self.sigma <= 0:
        raise VasicekParameterError(
            f"Volatility sigma={self.sigma} must be positive"
        )

    if not (VasicekConfig.MIN_TIME < self.maturity_time < VasicekConfig.MAX_TIME):
        raise VasicekParameterError(
            f"Maturity time={self.maturity_time} must be in "
            f"({VasicekConfig.MIN_TIME}, {VasicekConfig.MAX_TIME})"
        )

def to_dict(self) -> Dict[str, float]:
    """Convert parameters to dictionary."""
    return {
        'r0': self.r0,
        'b': self.b,
        'a': self.a,
        'sigma': self.sigma,
        'maturity_time': self.maturity_time
    }

if name == "main": pass ```

Exercises¶

Exercise 1. The Vasicek SDE is \(dr = a(b - r)\,dt + \sigma\,dW\). Explain why the MIN_RATE in VasicekConfig is \(-0.1\) rather than a small positive number as in the CIR model.

Exercise 2. Create a VasicekParameters instance with \(r_0 = 0.02\), \(b = 0.04\), \(a = 0.15\), \(\sigma = 0.01\), and maturity_time = 20.0. Convert it to a dictionary and verify all fields.

Exercise 3. What exception is raised if you try to create VasicekParameters(r0=0.03, b=0.05, a=-0.1, sigma=0.02, maturity_time=10.0)? Explain why this constraint exists.

Exercise 4. Compare the VasicekParameters dataclass to CIRParameters. List two key structural differences and explain how they reflect the mathematical differences between the models.