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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 ...


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V(t) is the variance process of the stock price, not volatility process. Cox-Ingersoll-Ross demonstrated that that specific process can be non-negative under certain conditions, which is what you want for variance.


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I have approximate the integrals by Monte Carlo Method but you can use several method such as Newton-Cotes formulas and Gaussian quadrature. Function Example Solutions Call = 34.0976 Put = 4.8941 Parameters were extracted from Jianwei Zhu(2008),Page 10,Table 4


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Since $v_0$ and $\theta$ are responsible for the initial and long-term level of the variance,Zhu (2010) recommends basing vega on those two parameters. Both parameters represent variance, so to create measures of sensitivity to volatility, Zhu (2010) defines two vegas, one based on $\upsilon=\sqrt v_0$ and the other based on $\omega=\sqrt \theta$ for the ...


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In this paper http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2626552 the authors compare the Heston model with volatility given by $ dV_t = \kappa_V(\bar{V}-V_t)dt+\sigma_V\sqrt{V_t}dW_t $ with the a model where the volatiltiy is given by $ dV_t = \kappa_V(\bar{V}-V_t)dt+\sigma_VV_tdW_t $. They show that the latter is inverse gamma distributed and ...


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The reason is that Heston managed to solve the case with square root. The log-normal vol process leads to nasty properties. The 3/2 model is another case that have been solved.


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Edit we assume $X_t$ follows the differential stochastic process $$d X(t)=\mu (t,{{X}_{t}})dt+\sigma (t,{{X}_{t}}) dW(t)$$ if $$\underset{{{X}_{t}}\to 0}{\mathop{\lim }}\,\,\mu (t,{{X}_{t}} )-\frac{1}{2}\frac{\partial }{\partial x}{{\sigma }^{2}}(t,{{X}_{t}})\geq 0$$ then $$P(\{\,t\in [0\,,\infty )|\,X(t\,,x )\leq 0\})=0$$ in the C.I.R Model ,we have ...



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