📂Stochastic Differential Equations

CKLS Mean Reverting Gamma Stochastic Differential Equation

Model ¹

$$d X_{t} = \left( \alpha - \beta X_{t} \right) dt + \sigma X_{t}^{\gamma} d W_{t} \qquad , X_{0} > 0$$ Let’s assume $\alpha, \beta, \sigma, \gamma > 0$. This stochastic differential equation is called the CKLS Mean Reverting Gamma Stochastic Differential Equation.

Variables

• $X_{t}$: Represents the Interest Rate or the Gene Frequency.

Parameters

• $\alpha / \beta$: The Mean Reversion, towards which $X_{t}$ tends to revert over the long term.
• $\alpha > 0$: The Speed of Adjustment, where a higher value means a faster return to the mean.
• $\sigma > 0$: Represents the Volatility.
• $\gamma > 0$: Represents the nonlinear relationship between $X_{t}$ and volatility.

Explanation

The CKLS equation proposed by Chan, Károlyi, Longstaff, Sanders is a stochastic differential equation that can be seen as a generalization of several well-known models in financial mathematics.

• $\gamma = 0$: Becomes the Ornstein-Uhlenbeck Equation.
• $\gamma = {{ 1 } \over { 2 }}$: Becomes the CIR Model.
• $\gamma = 1$: Becomes the Geometric Brownian Motion (GBM).