Deriving the Distribution of the Square of Arithmetic Brownian Motion
Summary
The document considers an arithmetic Brownian motion with constant drift and volatility, then defines an asset price as the square of that process. Applying Ito’s lemma gives the stochastic differential equation for the squared process, with drift depending on the current state and a diffusion term proportional to that state. The question focuses on how to find the resulting price distribution.
At a fixed time, the underlying process is normally distributed with mean equal to its initial value plus drift times elapsed time, and variance equal to volatility squared times elapsed time. Squaring a scaled version therefore gives a noncentral chi-squared variable with one degree of freedom. For the cumulative distribution, the answer uses the event that the absolute value of the normal variable is no greater than the square root of the threshold, expressing the result as a difference of two normal CDF values; differentiating yields the density. This transformation also works without naming the chi-squared family. The result assumes the stated normal process and applies to nonnegative thresholds; it does not discuss model fit or parameter estimation.
Key ideas
- Ito’s lemma gives the dynamics of the squared process from the arithmetic Brownian motion.
- At a fixed time, the underlying process is normal with drift-adjusted mean and variance proportional to time.
- A scaled square of that normal variable follows a noncentral chi-squared distribution with one degree of freedom.
- The CDF of a squared variable can be found from the probability that the original variable lies between the positive and negative square roots of the threshold.
- Differentiating the CDF relationship gives the density of the squared process.
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Full text
# Square of arithmetic brownian motion process
# Square of arithmetic brownian motion process
We have an arithmetic Brownian motion process $X_t$ that follows $dX_t=\mu dt + \sigma dZ_t$ and we define the asset price $S_t=X_t^2$ and we are asked to find the stochastic differential equation that $S_t$ satisfies, as well as the density and distribution functions of $S_t$. The first part I accomplished through a straightforward application of Ito's lemma: \begin{align*} dS_t&=\left(\frac{\partial S_t}{\partial t}+\mu\frac{\partial S_t}{\partial X_t}+\frac{1}{2}\sigma^2\frac{\partial^2 S_t}{\partial X_t^2}\right)dt+\sigma\frac{\partial S_t}{\partial X_t}dZ_t\\ &=\left(2\mu X_t+\sigma^2\right)dt+2\sigma X_tdZ_t\\ \end{align*}
but the second part I am having more trouble on. We know that $X_t$ is normally distributed in that $X_t-X_0\sim N(\mu t,\sigma\sqrt{t})$, so I know from other classes that $S_t=X_t^2$ will follow some sort of $\chi^2$ distribution, however because $X_t$ has neither zero mean nor unit variance, $S_t$ ends up being a scaled noncentral $\chi^2$ distribution and the CDF and PDF are ugly at best (not to mention I'm pretty sure that the knowledge that the square of normal is $\chi^2$ is beyond the scope of this class).
Is there something wrong with my logic in going from $X_t\sim$ Normal to the distribution of $S_t$?
## Answer by Quantuple (score 5)
https://quant.stackexchange.com/a/25127
Your logic is fine $$ X_t \sim \mathcal {N}(X_0+\mu t, \sigma^2 t) $$
Thus, $\left (\frac {X_t}{\sigma\sqrt {t}}\right)^2 $ indeed exhibits a non central chi-squared distribution
$$ \left (\frac {X_t}{\sigma\sqrt {t}}\right)^2 \sim \chi^2\left(k=1,\lambda=\left (\frac {X_0+\mu t}{\sigma\sqrt {t}}\right)^2\right) $$
whence the law of $S_t := X_t^2$.
As regards the pdf/cdf of $S_t$, the key here is that there is only one degree of freedom, so no need to know the non central $\chi^2$ pdf/cdf by heart.
Actually, there is no need to know that its a chi-squared distribution to begin with.
Indeed, for any variable $X^2$ with support on $[0,\infty [$, the cumulative distribution function $F_{X^2}$ writes:
$$ F_{X^2}(x) := P [X^2 \leq x] $$ which is equivalent to writing \begin{align} F_{X^2}(x) &= P [\vert X \vert \leq \sqrt {x}] \\ &= P [-\sqrt {x} \leq X \leq +\sqrt {x}] \\ &= F_X (\sqrt {x}) - F_X (-\sqrt {x}) \end{align}
Hence the relationship between the cumulative distribution function of $X^2$ and that of $X $.
Now differentiate the above equation with respect to $x$ to obtain the relationship between the probability density functions.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.