justification of square root process

In finance, many stochastic processes $X(t)$ are defined via $$dX = \text{(some drift term)} dt + \sigma X^\gamma dW_t$$ with $\gamma = 1/2$ (for instance the Heston model or the CIR process). Generally, this is called a square-root process. My question is: How does one justify the choice of $\gamma = 1/2$. I am aware that it is convenient to chose $0 < \gamma < 1$ since for $\gamma > 1$, no unique Martingale measure exists. But why exactly $\gamma = 1/2$ and not, say $\gamma = 6/7$. (I have found one related question here Why square root of volatility in Heston model? but no satisfying answer has been given.)

1 Answer

1. C.I.R Process belongs the class of affine diffusion processes.For processes within this class, a closed form solution of the characteristic function exists(Duffie,et al). For more details, Suppose we have given a scalar SDEs, i.e., $$dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dW_t$$ this process ($\{X_t\}_{0\leq t\leq T}$) is said to be of the affine form if \begin{align} &&\mu(X_t,t)=\alpha_0+\alpha_1X_t\\ &&\sigma^2(X_t,t)=\beta_0+\beta_1X_t \end{align} where $\alpha_j,\beta_j\in R$. We claim C.I.R Process belongs the class of affine diffusion processes,because

\begin{align} & \mu (t,{{r}_{t}})\,\,=\kappa (\theta -{{r}_{t}})=\underbrace{\kappa \theta }_{{{\alpha }_{0}}}+\underbrace{(-\kappa )}_{{{\alpha }_{1}}}\,{{r}_{t}} \\ & {{\sigma }^{2}}(t,{{r}_{t}})=\sigma^2r_t=\underbrace{0}_{\beta_0}+\underbrace{{{\sigma }^{2}}\,}_{\beta {{}_{1}}}\,{{r}_{t}} \\ \end{align} Now, if $\gamma\ne\frac{1}{2}$,then C.I.R Process doesn't belong the class of affine diffusion processes(please check yourself) and the discounted characteristic function is not of the following form $$\Phi(\phi,r_t,t,T)=e^{A(\phi,\tau)+B(\phi,\tau)r_t}$$ Consequently, conditional distribution of on $r_t$ doesn't follow a non-central chi-square distribution.

1. If $\gamma>\frac{1}{2}$ e.g $\gamma=\frac{6}{7}$ then Feller's Condition holds for any value of $\kappa$ and $\theta$ (we know $\kappa,\theta>0$)

$$\underset{{{r}_{t}}\to 0}{\mathop{\lim }}\,\,\left( \kappa (\theta -{{r}_{t}})-\frac{1}{2}\frac{\partial }{\partial r}(\sigma\,r_{t}^{\,\gamma})^2 \right)=\kappa \theta>0$$ In other words, $r_t$ is always positive and this is inconsistent with financial Modeling. Also,if $\gamma<\frac{1}{2}$ then

$$\underset{{{r}_{t}}\to 0}{\mathop{\lim }}\,\,\left( \kappa (\theta -{{r}_{t}})-\frac{1}{2}\frac{\partial }{\partial r}(\sigma\,r_{t}^{\,\gamma})^2 \right)\rightarrow-\infty$$ In other words, $r_t$ is always negative and this is inconsistent with reality.