Chi-Squared and Related Distributions

Motivation: why this matters in quant finance

The chi-squared distribution is what appears when Gaussian shocks are squared and added. Squaring removes sign and measures magnitude, so chi-squared variables naturally arise in variance estimation, residual diagnostics, goodness-of-fit tests, and positive stochastic state variables.

In finance, this gives two roles: statistical testing and positive diffusion modelling. Sample variance theory uses chi-squared variables. CIR short rates and Heston variance transitions use non-central chi-squared laws.

Definition

If $Z_1,\ldots,Z_k$ are independent standard normals, then

V=\sum_{i=1}^{k}Z_i^2\sim\chi^2_k.

Its density is

f(v)=\frac{1}{2^{k/2}\Gamma(k/2)}v^{k/2-1}e^{-v/2}, \qquad v>0.

A non-central chi-squared variable arises from squaring normals with nonzero means:

\sum_iY_i^2\sim\chi^2_k(\delta), \qquad \delta=\sum_i\mu_i^2.

Key Properties

For $V\sim\chi^2_k$ ,

\mathbb{E}[V]=k, \qquad \operatorname{Var}(V)=2k.

If independent $V_i\sim\chi^2_{k_i}$ , then $\sum_iV_i\sim\chi^2_{\sum_i k_i}$ . The MGF is

M_V(t)=(1-2t)^{-k/2}, \qquad t<1/2.

Related distributions are built from chi-squared variables:

Distribution	Construction
Exponential	$\chi^2_2\sim\operatorname{Exp}(1/2)$
Student's $t$	$Z/\sqrt{V/\nu}$
$F$	$(V_1/k_1)/(V_2/k_2)$
Gamma	$\chi^2_k=\operatorname{Gamma}(k/2,2)$

In Quant Finance

If $X_1,\ldots,X_n\sim\mathcal{N}(\mu,\sigma^2)$ and $S^2$ is the sample variance, then

\frac{(n-1)S^2}{\sigma^2}\sim\chi^2_{n-1}.

This is the exact Gaussian theory behind volatility confidence intervals.

For goodness-of-fit, Pearson's statistic

\chi^2=\sum_i\frac{(O_i-E_i)^2}{E_i}

measures whether observed bin counts match model probabilities. VaR exception backtests and distributional diagnostics often use this logic.

For the CIR/Heston variance process,

dv_t=\kappa(\theta-v_t)dt+\xi\sqrt{v_t}dW_t,

the transition law is scaled non-central chi-squared. The Feller condition is tied to the degrees-of-freedom parameter $4\kappa\theta/\xi^2$ .

Worked Example: Variance Uncertainty

With $n=252$ Gaussian daily returns and sample volatility $\widehat{\sigma}$ ,

\frac{251\widehat{\sigma}^2}{\sigma^2}\sim\chi^2_{251}.

Even if the Gaussian return model were correct, volatility estimated from one year of data would still have sampling uncertainty.

Common Confusions and Pitfalls

Chi-squared distributions are symmetric. They are right-skewed for small degrees of freedom.

Non-central means shifted. Non-centrality changes the whole shape, not merely the location.

A goodness-of-fit test proves the model is right. It can reject visible misfit; it cannot prove truth.

Where This Goes Next

Student's t-Distribution: divides a normal by chi-squared scale uncertainty.
Exponential Distribution: the $\chi^2_2$ special case.
Moment-Generating Functions: prove additivity and gamma-family relationships.

References

Bertsekas and Tsitsiklis, Introduction to Probability, 2nd ed., Ch. 3 Sec. 3.3 and Ch. 4 Sec. 4.4.
Mosteller, Fifty Challenging Problems in Probability, Problems 27-30 and 45-46, for count and matching-problem intuition behind goodness-of-fit thinking.
James, Witten, Hastie, and Tibshirani, An Introduction to Statistical Learning, 2nd ed., Ch. 3 and Ch. 7, for standard errors, $t$ and $F$ tests, and ANOVA output.