CIR Variance Characteristic Functions from the Noncentral Chi-Square Law and Affine ODEs
Summary
The note gives two ways to obtain the characteristic function of the Cox–Ingersoll–Ross (CIR) variance process, whose mean-reverting diffusion has volatility proportional to the square root of the state. One route uses the known transition law: the variance at a future time is a scaled noncentral chi-square random variable. Substituting the corresponding argument into that distribution’s characteristic function yields the expression for the CIR state.
A second route uses the affine-process structure and Feynman–Kac: write the conditional transform as an exponential affine in the state, then solve coupled ordinary differential equations for its coefficients. The note identifies the initial conditions and equations, but leaves their solution as an exercise. It also distinguishes the variance or short-rate transform from the characteristic function of the Heston model’s log stock price, which is the transform commonly needed for option pricing. The result is conditional on the specified CIR dynamics and does not by itself provide the Heston stock-price transform.
Key ideas
- The CIR state at a future time has a scaled noncentral chi-square distribution.
- Its characteristic function follows by rescaling the known transform of that distribution.
- The affine approach represents the conditional transform with two time-varying coefficients.
- Feynman–Kac reduces those coefficients to coupled ordinary differential equations.
- The characteristic function of CIR variance differs from that of Heston log stock price.
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Full text
# CIR process characteristic function
# CIR process characteristic function
what is the characteristic function of the CIR process given by $dv_t = \kappa (\theta - v_t)dt + \sigma \sqrt{v_t}dW_t$ Unfortunately, I could not find the answer in the literature. I know it is in the class of affine diffusion processes, but how can we find the characteristic function?
## Answer by Kevin (score 5)
https://quant.stackexchange.com/a/64209
The distribution of the variance $v_t$ is known, see here. We have $$v_t=\frac{1}{2c_t}Y,$$ where $$c_t=\frac{2\kappa}{(1-e^{-\kappa t})\sigma^2}$$ and $Y$ follows a non-central $\chi^2$ distribution with $k=\frac{4\kappa\theta}{\sigma^2}$ degrees of freedom and non-centrality parameter $\lambda_t=2c_tv_0e^{-\kappa t}$. The characteristic function of $Y$ is known to be $$\varphi_Y(u)=\frac{1}{(1-2iu)^{k/2}}\exp\left(\frac{iu\lambda_t}{1-2iu}\right).$$ The characteristic function of $v_t$ is thus $$\varphi_{v_t}(u)=\varphi_{Y/2c_t}(u)=\varphi_Y\left(\frac{u}{2c_t}\right)=\frac{1}{(1-iu/c_t)^{k/2}}\exp\left(\frac{iu\lambda_t/2}{c_t-iu}\right).$$
This is the characteristic function of $v_t$, i.e. the variance in the Heston (1993) model or the short rate $r_t$ in the Cox et al. (1985) model. This is very different to the characteristic function of the (log-)stock price in the Heston (1993) model that you need for usual option pricing.
## Answer by Kermittfrog (score 3)
https://quant.stackexchange.com/a/64224
In addition to @Kevin ‘s splendid answer, you can make use of the affine properties you have just described. The relevant machinery is found in DPS2000, equations (2.4) thru (2.6).
We are looking for
$$ \phi(u,t) \equiv E\left(e^{iuv_t}\right)$$
where $E(\cdot)\equiv E(\cdot|\mathcal{F_0})$. For affine processes, this expectation can be solved thru a cleverly chosen set of differential equations, i.e. Feynman-Kac (see paper):
$$ \phi(u,s) = E\left(e^{\alpha_s + \beta_s v_s}\right) \mathrm{s.t.} \alpha_t=0, \beta_t=ui$$
Where the functions $\alpha(t), \beta(t)$ satisfy ordinary differential equations in case of a CIR process:
$$\frac{\partial \beta}{\partial \tau} = -\kappa\beta_\tau + \frac{1}{2}\sigma^2\beta_\tau^2$$ subject to $\beta_0 = ui$ and $$ \frac{\partial \alpha}{\partial \tau}=\kappa\theta\beta_\tau$$ subject to $\alpha_0=0$
NB: I have flipped the time dimension from $s:0\to t$ to $\tau:t\to 0$, hence you find different signs when comparing to the paper
You can now solve the ODEs, starting with $\beta$ and plugging the result back into the ODE for $\alpha$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.