justification of square root process - Quantitative Finance Stack Exchange most recent 30 from quant.stackexchange.com 2019-08-22T22:43:29Z https://quant.stackexchange.com/feeds/question/18922 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://quant.stackexchange.com/q/18922 4 justification of square root process user56643 https://quant.stackexchange.com/users/16950 2015-07-18T09:23:38Z 2015-07-18T12:52:05Z <p>In finance, many stochastic processes $X(t)$ are defined via \begin{equation} dX = \text{(some drift term)} dt + \sigma X^\gamma dW_t \end{equation} 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 $0 &lt; \gamma &lt; 1$ since for $\gamma &gt; 1$, no unique Martingale measure exists. But why exactly $\gamma = 1/2$ and not, say $\gamma = 6/7$. (I have found one related question here <a href="https://quant.stackexchange.com/questions/18904/why-square-root-of-volatility-in-heston-model">Why square root of volatility in Heston model?</a> but no satisfying answer has been given.)</p> https://quant.stackexchange.com/questions/18922/-/18924#18924 3 Answer by user16891 for justification of square root process user16891 https://quant.stackexchange.com/users/0 2015-07-18T12:52:05Z 2015-07-18T12:52:05Z <ol> <li>C.I.R Process belongs the class of affine diffusion processes.For processes within this class, a closed form solution of the <strong>characteristic function</strong> exists(<a href="http://pages.stern.nyu.edu/~dbackus/Disasters/DuffiePanSingleton%20jumps%20Econometrica%2000.PDF" rel="nofollow noreferrer">Duffie,et al</a>). For more details, Suppose we have given a <strong>scalar SDEs</strong>, 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} &amp;&amp;\mu(X_t,t)=\alpha_0+\alpha_1X_t\\ &amp;&amp;\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</li> </ol> <p>\begin{align} &amp; \mu (t,{{r}_{t}})\,\,=\kappa (\theta -{{r}_{t}})=\underbrace{\kappa \theta }_{{{\alpha }_{0}}}+\underbrace{(-\kappa )}_{{{\alpha }_{1}}}\,{{r}_{t}} \\ &amp; {{\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 <strong>is not of the following form</strong> $$\Phi(\phi,r_t,t,T)=e^{A(\phi,\tau)+B(\phi,\tau)r_t}$$ Consequently, conditional distribution of on $r_t$ doesn't follow a non-central chi-square distribution.</p> <ol start="2"> <li>If $\gamma&gt;\frac{1}{2}$ e.g $\gamma=\frac{6}{7}$ then <a href="https://quant.stackexchange.com/questions/18904">Feller's Condition</a> holds for any value of $\kappa$ and $\theta$ (we know $\kappa,\theta&gt;0$)</li> </ol> <p>$$\underset{{{r}_{t}}\to 0}{\mathop{\lim }}\,\,\left( \kappa (\theta -{{r}_{t}})-\frac{1}{2}\frac{\partial }{\partial r}(\sigma\,r_{t}^{\,\gamma})^2 \right)=\kappa \theta&gt;0$$ In other words, $r_t$ is always positive and this is inconsistent with financial Modeling. Also,if $\gamma&lt;\frac{1}{2}$ then</p> <p>$$\underset{{{r}_{t}}\to 0}{\mathop{\lim }}\,\,\left( \kappa (\theta -{{r}_{t}})-\frac{1}{2}\frac{\partial }{\partial r}(\sigma\,r_{t}^{\,\gamma})^2 \right)\rightarrow-\infty$$ In other words, $r_t$ is always negative and this is inconsistent with reality.</p>