Interest Rate Simulation

Cox-Ingersoll-Ross model

This Shiny app simulates interest rate paths for one-factor equilibrium models using various stochastic differential equations (SDEs). Users can specify different types of drift $b(r, t) dt$ and diffusion terms $\sigma(r, t) dW(t)$ to observe how interest rates might evolve over time.

Stochastic Differential Equations

Constant Long-term Equilibrium and CEV Volatility

For a constant long-term equilibrium rate $\bar{r}$ and a constant elasticity of variance (CEV) model, the interest rate $r(t)$ follows the SDE:

$$ dr = -\alpha (r - \bar{r}) \ dt + \sigma r^{\gamma} \ dW(t) $$

where:

• $\alpha$ is the speed of mean reversion.
• $\sigma$ is the volatility parameter.
• $\gamma$ is the elasticity of volatility.
• $dW(t)$ represents the increments of a Wiener process (Brownian motion).

Special Cases

• $\gamma = 0$: Vasicek model (normal volatility model)
• $\gamma = 0.5$: Cox-Ingersoll-Ross model (square root volatility model)
• $\gamma = 1$: Dothan model (proportional volatility model)

Constant Long-term Equilibrium and Dynamic Volatility

For a constant long-term equilibrium rate $\bar{r}$ and a dynamic volatility $\sigma(t)$, the interest rate $r(t)$ follows the SDE:

$$ dr = -\alpha (r - \bar{r}) \ dt + \sigma(t) \ dW(t) $$

Dynamic Long-term Equilibrium and CEV Volatility

For a dynamic long-term equilibrium rate $\theta(t)$ and a CEV volatility model, the interest rate $r(t)$ follows the SDE:

$$ dr = -\alpha (r - \theta(t)) \ dt + \sigma r^{\gamma} \ dW(t) $$

Dynamic Long-term Equilibrium and Dynamic Volatility

For a dynamic long-term equilibrium rate $\theta(t)$ and a dynamic volatility $\sigma(t)$, the interest rate $r(t)$ follows the SDE:

$$ dr = -\alpha (r - \theta(t)) \ dt + \sigma(t) \ dW(t) $$

Parameters and Inputs

• Equilibrium Type:

  • Constant: User specifies $\bar{r}$.
  • Dynamic: User specifies $\theta(t)$ as a function of time $t$.

• Volatility Term:

  • CEV: User specifies $\sigma$ and $\gamma$.
  • Dynamic: User specifies $\sigma(t)$ as a function of time $t$.

• $\alpha$: Speed of mean reversion.

• $\sigma$: Volatility parameter (used if volatilityType is CEV).

• $\gamma$: Elasticity of volatility (used if volatilityType is CEV).

• $r_0$: Initial interest rate.

• $T$: Time horizon for the simulation.

• steps: Number of discrete time steps.

• nPaths: Number of simulated paths.

• confInterval: Confidence interval for the summary statistics.

Using the App

1. Equilibrium Type: Select either Constant or Dynamic equilibrium.
2. Volatility Type: Select either CEV or Dynamic volatility.
3. Input Parameters: Provide values for $\alpha$, $\sigma$, $\gamma$, $r_0$, $T$, steps, nPaths, and confInterval.
4. Simulate: Click the Simulate button to generate interest rate paths.
5. Download: Use the Download Simulated Paths button to download the simulated data.

Outputs

• Interest Rate Plot: Displays the simulated interest rate paths.
• Summary Plot: Shows the median and confidence interval of the simulated paths.

Feature Pipeline

• Dynamic equilibrium and volatility also a function of current rate $r$
• no-arbitrage and multi-factor models
• paramater estimation based on empirical data