Vasicek Formula¶

Background¶

Vasicek Formula

Educational script demonstrating vasicek formula concepts.

Code¶

```python """ Vasicek Formula

Educational script demonstrating vasicek formula concepts. """

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

vasicek/vasicek_formula.py¶

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

import numpy as np from .vasicek_base import VasicekParameters, VasicekNumericalError

class VasicekAnalytical: """Analytical formulas for Vasicek model."""

@staticmethod
def mean(params: VasicekParameters, t: float) -> float:
    """Analytical mean of Vasicek process at time t."""
    if t <= 0:
        return params.r0
    return params.b + (params.r0 - params.b) * np.exp(-params.a * t)

@staticmethod
def variance(params: VasicekParameters, t: float) -> float:
    """Analytical variance of Vasicek process at time t."""
    if t <= 0:
        return 0.0
    return (params.sigma**2 / (2 * params.a)) * (1 - np.exp(-2 * params.a * t))

@staticmethod
def standard_deviation(params: VasicekParameters, t: float) -> float:
    """Analytical standard deviation at time t."""
    return np.sqrt(VasicekAnalytical.variance(params, t))

class VasicekBondPricer: """Bond pricing utilities for Vasicek model."""

@staticmethod
def zero_coupon_bond_price(
    params: VasicekParameters,
    current_rate: float,
    time_to_maturity: float
) -> float:
    """
    Calculate analytical zero-coupon bond price using Vasicek formula.

    P(r,t,T) = A(t,T) * exp(-B(t,T) * r)
    """
    if time_to_maturity <= 0:
        return 1.0

    try:
        # B(t,T) coefficient
        if params.a == 0:
            B_T = time_to_maturity
        else:
            B_T = (1 - np.exp(-params.a * time_to_maturity)) / params.a

        # A(t,T) coefficient
        if params.a == 0:
            A_T = np.exp(-params.b * time_to_maturity + 
                        (params.sigma**2 * time_to_maturity**3) / 6)
        else:
            term1 = (params.b - params.sigma**2 / (2 * params.a**2)) * (B_T - time_to_maturity)
            term2 = (params.sigma**2 / (4 * params.a)) * B_T**2
            A_T = np.exp(term1 - term2)

        # Bond price
        bond_price = A_T * np.exp(-B_T * current_rate)

        return float(np.clip(bond_price, 1e-10, 1.0))

    except (OverflowError, ZeroDivisionError, ValueError) as e:
        raise VasicekNumericalError(f"Bond pricing computation failed: {e}")

@staticmethod
def yield_to_maturity(
    params: VasicekParameters,
    current_rate: float,
    time_to_maturity: float
) -> float:
    """Calculate yield to maturity from bond price."""
    if time_to_maturity <= 0:
        return current_rate

    bond_price = VasicekBondPricer.zero_coupon_bond_price(
        params, current_rate, time_to_maturity
    )
    return -np.log(bond_price) / time_to_maturity

@staticmethod
def yield_curve(
    params: VasicekParameters,
    current_rate: float,
    maturities: np.ndarray
) -> np.ndarray:
    """Calculate yield curve for given maturities."""
    yields = np.zeros_like(maturities)

    for i, T in enumerate(maturities):
        yields[i] = VasicekBondPricer.yield_to_maturity(params, current_rate, T)

    return yields

@staticmethod
def forward_rate(
    params: VasicekParameters,
    current_rate: float,
    t1: float,
    t2: float
) -> float:
    """Calculate forward rate between times t1 and t2."""
    if t1 >= t2:
        raise ValueError("t1 must be less than t2")

    P_t1 = VasicekBondPricer.zero_coupon_bond_price(params, current_rate, t1)
    P_t2 = VasicekBondPricer.zero_coupon_bond_price(params, current_rate, t2)

    return np.log(P_t1 / P_t2) / (t2 - t1)

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

Exercises¶

Exercise 1. Using the Vasicek analytical formulas, compute the mean and variance of \(r(10)\) given \(r_0 = 0.03\), \(b = 0.05\), \(a = 0.1\), and \(\sigma = 0.02\).

Exercise 2. Derive the \(B(t,T)\) coefficient for the Vasicek bond pricing formula \(P = A \cdot e^{-B \cdot r}\) and explain why \(B(t,T) \to T - t\) as \(a \to 0\).

Exercise 3. Compute the Vasicek yield curve for maturities \(T \in \{1, 5, 10, 30\}\) given \(r = 0.03\), \(b = 0.05\), \(a = 0.1\), \(\sigma = 0.02\). Is the curve upward-sloping?

Exercise 4. Compute the forward rate \(f(0, 5, 10)\) if \(P(0,5) = 0.8450\) and \(P(0,10) = 0.6900\).