Cox-Ingersoll-Ross (CIR) model

Short rate dynamics: mean and variance

The short rate under the CIR model has the dynamics:

\[dr_t = \kappa (\theta - r_t)dt + \sigma \sqrt{r_t}dB_t\]

For a moment, if we drop the stochastic term, and merely consider the first order linear ODE $\frac{dr_t}{dt} + \kappa r_t = \kappa \theta$, the integrating factor for this differential equation is $e^{\int \kappa dt} = e^{\kappa t}$. Multiplying both sides by the integrating factor, we have:

\[\begin{align*} e^{\kappa t} dr_t &= \kappa(\theta - r_t) e^{\kappa t}dt + \sigma e^{\kappa t}\sqrt{r_t} dB_t \\ e^{\kappa t} dr_t + r_t e^{\kappa t}dt &= \kappa e^{\kappa t}\theta dt + \sigma e^{\kappa t}\sqrt{r_t} dB_t \\ d(e^{\kappa t} r_t) &= \kappa e^{\kappa t}\theta dt + \sigma e^{\kappa t}\sqrt{r_t} dB_t \\ \int_{0}^{t} d(e^{\kappa s} r_s) &= \theta \kappa\int_{0}^{t} e^{\kappa s} ds + \sigma \int_{0}^{t} e^{\kappa s}\sqrt{r_s} dB_s \\ [e^{\kappa s} r_s]_{0}^{t} &= \kappa \theta \left[\frac{e^{\kappa s}}{\kappa}\right]_{0}^{t} + \sigma \int_{0}^{t} e^{\kappa s}\sqrt{r_s} dB_s\\ e^{\kappa t}r_t - r_0 &= \theta (e^{\kappa t} - 1) + \sigma \int_{0}^{t} e^{\kappa s}\sqrt{r_s} dB_s \\ e^{\kappa t} r_t &= r_0 + \theta (e^{\kappa t} - 1) + \sigma \int_{0}^{t} e^{\kappa s}\sqrt{r_s} dB_s \\ r_t &= r_0 e^{-\kappa t} + \theta (1 - e^{-\kappa t}) + \sigma \int_{0}^{t} e^{-\kappa (t-s)}\sqrt{r_s} dB_s \end{align*}\]

The mean is given by:

\[\begin{align*} \mathbf{E}[r_t] &= r_0 e^{-\kappa t} + \theta (1 - e^{-\kappa t}) \end{align*}\]

The random variable $\sigma \int_{0}^{t} e^{-\kappa (t-s)}\sqrt{r_s} dB_s$ has mean $0$ and variance:

\[\begin{align*} \mathbf{E}\left[\left(\sigma \int_{0}^{t} e^{-\kappa (t-s)}\sqrt{r_s} dB_s\right)^2\right] &= \sigma^2 \int_{0}^{t}e^{-2\kappa(t-s)} \mathbf{E}[r_s] ds \\ &= \sigma^2 e^{-2\kappa t}\int_{0}^{t}e^{2\kappa s} \left(r_0 e^{-\kappa s} + \theta(1-e^{-\kappa s})\right) ds\\ &= \sigma^2 r_0 e^{-2\kappa t} \int_{0}^{t} e^{\kappa s} ds + \sigma^2 \theta e^{-2\kappa t} \int_{0}^{t}(e^{2\kappa s}-e^{\kappa s}) ds \\ &= \sigma^2 r_0 e^{-2\kappa t} \left[\frac{e^{\kappa s}}{\kappa} \right]_{0}^{t} +\sigma^2 \theta e^{-2\kappa t} \left[\frac{e^{2\kappa s}}{2\kappa} - \frac{e^{\kappa s}}{\kappa}\right]_{0}^{t}\\ &= \frac{\sigma^2 r_0}{\kappa} e^{-2\kappa t} (e^{\kappa t} - 1)+\sigma^2 \theta e^{-2\kappa t} \left[\frac{e^{2\kappa s}}{2\kappa} - \frac{2e^{\kappa s}}{2\kappa}\right]_{0}^{t}\\ &= \frac{\sigma^2 r_0}{\kappa} e^{-2\kappa t}(e^{\kappa t} - 1)+\frac{\sigma^2 \theta}{2\kappa} e^{-2\kappa t}(e^{2\kappa t} - 2e^{\kappa t} - (1 - 2))\\ &= \frac{\sigma^2 r_0}{\kappa} e^{-2\kappa t}(e^{\kappa t} - 1)+\frac{\sigma^2 \theta}{2\kappa}e^{-2\kappa t} (1 + e^{2\kappa t} - 2e^{\kappa t})\\ &= \frac{\sigma^2 r_0}{\kappa} (e^{-\kappa t} - e^{-2\kappa t})+\frac{\sigma^2 \theta}{2\kappa} (1 - e^{-\kappa t})^2 \end{align*}\]

Naive python implementation

`CIRProcess` class

The class CIRProcess is designed as an engine to generate sample paths of the CIR process.

import math
from dataclasses import dataclass

import joypy
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from matplotlib import cm
from tqdm import tqdm


@dataclass
class CIRProcess:
    """An engine for generating sample paths of the Cox-Ingersoll-Ross process"""

    kappa: float
    theta: float
    sigma: float
    step_size: float
    total_time: float
    r_0: float

    def generate_paths(self, paths: int):
        """Generate sample paths"""
        num_steps = int(self.total_time / self.step_size)
        dz = np.random.standard_normal((paths, num_steps))
        r_t = np.zeros((paths, num_steps))
        zero_vector = np.full(paths, self.r_0)
        prev_r = zero_vector
        for i in range(num_steps):
            r_t[:, i] = (
                prev_r
                + self.kappa * np.subtract(self.theta, prev_r) * self.step_size
                + self.sigma
                * np.sqrt(np.abs(prev_r))
                * math.sqrt(self.step_size)
                * dz[:, i]
            )

            prev_r = r_t[:, i]

        return r_t

Sample Paths

We generate $N=10$ paths of the CIR process.

Show the code

cir_process = CIRProcess(
    kappa=3,
    r_0=9,
    sigma=0.5,
    step_size=10e-3,
    theta=3,
    total_time=1.0,
)

num_paths = 10

paths = cir_process.generate_paths(num_paths)

t = np.linspace(0.01, 1.0, 100)

plt.grid(True)
plt.xlabel(r"Time $t$")
plt.ylabel(r"$R(t)$")
plt.title(r"$N=10$ paths of the Cox-Ingersoll-Ross process")
for path in paths:
    plt.plot(t, path)

plt.show()

Evolution of the distribution.

The evolution of the distribution with time can be visualized.

Show the code

# TODO: - this is where slowness lies, generating paths is a brezze

# Wrap the paths 2d-array in a dataframe
paths_tr = paths.transpose()
# Take 20 samples at times t=0.05, 0.10, 0.15, ..., 1.0 along each path
samples = paths_tr[4::5]
# Reshape in a 1d column-vector
samples_arr = samples.reshape(num_paths * 20)
samples_df = pd.DataFrame(samples_arr, columns=["values"])
samples_df["time"] = [
    "t=" + str((int(i / num_paths) + 1) / 20) for i in range(num_paths * 20)
]

# TODO: end

fig, ax = joypy.joyplot(
    samples_df,
    by="time",
    colormap=cm.autumn_r,
    column="values",
    grid="y",
    kind="kde",
    range_style="own",
    tails=10e-3,
)
plt.vlines(
    [cir_process.theta, cir_process.r_0],
    -0.2,
    1,
    color="k",
    linestyles="dashed",
)
plt.show()

--- title: "Cox-Ingersoll-Ross (CIR) model" author: "Quasar" date: "2024-05-08" categories: [Rates Modelling] image: image.png toc: true toc-depth: 3 --- ## Short rate dynamics: mean and variance The short rate under the CIR model has the dynamics: $$dr_t = \kappa (\theta - r_t)dt + \sigma \sqrt{r_t}dB_t$$ For a moment, if we drop the stochastic term, and merely consider the first order linear ODE $\frac{dr_t}{dt} + \kappa r_t = \kappa \theta$, the integrating factor for this differential equation is $e^{\int \kappa dt} = e^{\kappa t}$. Multiplying both sides by the integrating factor, we have: \begin{align*} e^{\kappa t} dr_t &= \kappa(\theta - r_t) e^{\kappa t}dt + \sigma e^{\kappa t}\sqrt{r_t} dB_t \\ e^{\kappa t} dr_t + r_t e^{\kappa t}dt &= \kappa e^{\kappa t}\theta dt + \sigma e^{\kappa t}\sqrt{r_t} dB_t \\ d(e^{\kappa t} r_t) &= \kappa e^{\kappa t}\theta dt + \sigma e^{\kappa t}\sqrt{r_t} dB_t \\ \int_{0}^{t} d(e^{\kappa s} r_s) &= \theta \kappa\int_{0}^{t} e^{\kappa s} ds + \sigma \int_{0}^{t} e^{\kappa s}\sqrt{r_s} dB_s \\ [e^{\kappa s} r_s]_{0}^{t} &= \kappa \theta \left[\frac{e^{\kappa s}}{\kappa}\right]_{0}^{t} + \sigma \int_{0}^{t} e^{\kappa s}\sqrt{r_s} dB_s\\ e^{\kappa t}r_t - r_0 &= \theta (e^{\kappa t} - 1) + \sigma \int_{0}^{t} e^{\kappa s}\sqrt{r_s} dB_s \\ e^{\kappa t} r_t &= r_0 + \theta (e^{\kappa t} - 1) + \sigma \int_{0}^{t} e^{\kappa s}\sqrt{r_s} dB_s \\ r_t &= r_0 e^{-\kappa t} + \theta (1 - e^{-\kappa t}) + \sigma \int_{0}^{t} e^{-\kappa (t-s)}\sqrt{r_s} dB_s \end{align*} The mean is given by: \begin{align*} \mathbf{E}[r_t] &= r_0 e^{-\kappa t} + \theta (1 - e^{-\kappa t}) \end{align*} The random variable $\sigma \int_{0}^{t} e^{-\kappa (t-s)}\sqrt{r_s} dB_s$ has mean $0$ and variance: \begin{align*} \mathbf{E}\left[\left(\sigma \int_{0}^{t} e^{-\kappa (t-s)}\sqrt{r_s} dB_s\right)^2\right] &= \sigma^2 \int_{0}^{t}e^{-2\kappa(t-s)} \mathbf{E}[r_s] ds \\ &= \sigma^2 e^{-2\kappa t}\int_{0}^{t}e^{2\kappa s} \left(r_0 e^{-\kappa s} + \theta(1-e^{-\kappa s})\right) ds\\ &= \sigma^2 r_0 e^{-2\kappa t} \int_{0}^{t} e^{\kappa s} ds + \sigma^2 \theta e^{-2\kappa t} \int_{0}^{t}(e^{2\kappa s}-e^{\kappa s}) ds \\ &= \sigma^2 r_0 e^{-2\kappa t} \left[\frac{e^{\kappa s}}{\kappa} \right]_{0}^{t} +\sigma^2 \theta e^{-2\kappa t} \left[\frac{e^{2\kappa s}}{2\kappa} - \frac{e^{\kappa s}}{\kappa}\right]_{0}^{t}\\ &= \frac{\sigma^2 r_0}{\kappa} e^{-2\kappa t} (e^{\kappa t} - 1)+\sigma^2 \theta e^{-2\kappa t} \left[\frac{e^{2\kappa s}}{2\kappa} - \frac{2e^{\kappa s}}{2\kappa}\right]_{0}^{t}\\ &= \frac{\sigma^2 r_0}{\kappa} e^{-2\kappa t}(e^{\kappa t} - 1)+\frac{\sigma^2 \theta}{2\kappa} e^{-2\kappa t}(e^{2\kappa t} - 2e^{\kappa t} - (1 - 2))\\ &= \frac{\sigma^2 r_0}{\kappa} e^{-2\kappa t}(e^{\kappa t} - 1)+\frac{\sigma^2 \theta}{2\kappa}e^{-2\kappa t} (1 + e^{2\kappa t} - 2e^{\kappa t})\\ &= \frac{\sigma^2 r_0}{\kappa} (e^{-\kappa t} - e^{-2\kappa t})+\frac{\sigma^2 \theta}{2\kappa} (1 - e^{-\kappa t})^2 \end{align*} ## Naive python implementation ### `CIRProcess` class The class `CIRProcess` is designed as an engine to generate sample paths of the CIR process. ```{python} import math from dataclasses import dataclass import joypy import matplotlib.pyplot as plt import numpy as np import pandas as pd from matplotlib import cm from tqdm import tqdm @dataclass class CIRProcess: """An engine for generating sample paths of the Cox-Ingersoll-Ross process""" kappa: float theta: float sigma: float step_size: float total_time: float r_0: float def generate_paths(self, paths: int): """Generate sample paths""" num_steps = int(self.total_time / self.step_size) dz = np.random.standard_normal((paths, num_steps)) r_t = np.zeros((paths, num_steps)) zero_vector = np.full(paths, self.r_0) prev_r = zero_vector for i in range(num_steps): r_t[:, i] = ( prev_r + self.kappa * np.subtract(self.theta, prev_r) * self.step_size + self.sigma * np.sqrt(np.abs(prev_r)) * math.sqrt(self.step_size) * dz[:, i] ) prev_r = r_t[:, i] return r_t ``` ### Sample Paths We generate $N=10$ paths of the CIR process. ```{python} # | code-fold: true # | code-summary: "Show the code" cir_process = CIRProcess( kappa=3, r_0=9, sigma=0.5, step_size=10e-3, theta=3, total_time=1.0, ) num_paths = 10 paths = cir_process.generate_paths(num_paths) t = np.linspace(0.01, 1.0, 100) plt.grid(True) plt.xlabel(r"Time $t$") plt.ylabel(r"$R(t)$") plt.title(r"$N=10$ paths of the Cox-Ingersoll-Ross process") for path in paths: plt.plot(t, path) plt.show() ``` ### Evolution of the distribution. The evolution of the distribution with time can be visualized. ```{python} # | code-fold: true # | code-summary: "Show the code" # TODO: - this is where slowness lies, generating paths is a brezze # Wrap the paths 2d-array in a dataframe paths_tr = paths.transpose() # Take 20 samples at times t=0.05, 0.10, 0.15, ..., 1.0 along each path samples = paths_tr[4::5] # Reshape in a 1d column-vector samples_arr = samples.reshape(num_paths * 20) samples_df = pd.DataFrame(samples_arr, columns=["values"]) samples_df["time"] = [ "t=" + str((int(i / num_paths) + 1) / 20) for i in range(num_paths * 20) ] # TODO: end fig, ax = joypy.joyplot( samples_df, by="time", colormap=cm.autumn_r, column="values", grid="y", kind="kde", range_style="own", tails=10e-3, ) plt.vlines( [cir_process.theta, cir_process.r_0], -0.2, 1, color="k", linestyles="dashed", ) plt.show() ```

Short rate dynamics: mean and variance

Naive python implementation

CIRProcess class

Sample Paths

Evolution of the distribution.

`CIRProcess` class