Model for this price dynamic

Question

I would like to know if someone has idea on how to simulate the corresponding price dynamic :

The price moves x% hourly on either direction. The maximum the price can move up in a day is y%, and the minimum the price can move down in a day is z%. That's all there is to it.

I had consider a simple model where $X_{n} = X_{0} * Product ( 1 + Z * x)$ with Z a Bernouilli and wanted to fix p such that for n = 24, the probability of $X_{n} > (1+y) * X_{0}$ and $X_{n} < (1 -z) X_{0}$ are zero, but it wont work.

Is there an easy way to model the dynamic and later to simulate it ? ( i dont feel like simulating 100 random walks and then filtering those who dont feet the criteria after 24 moves, really qualify for this dynamic simulation).

Answer 1

A bit cursorily, I'd set it up as follows.

Let the upper return limit be $u$, the lower return limit be $l$. Given a fixed return size $r$, we have the constraints

$$ \begin{align} nr&\leq u\\ l&\leq -nr\\ \end{align} $$ Thus

$$r=\frac{1}{n}\min(|u|,|l|)$$

Then, simply simulate the sign of each return event thru $n$ Bernoulli trials as in your original post.

If you want non symmetrical boundaries, simply choose up/down returns $r_u$ and $r_d$ as

$$ \begin{align} r_u&=\frac{u}{n}\\ r_d&=\frac{d}{n} \end{align} $$

and simulate draws from either $r_u$ or $r_d$ as Bernoulli.

HTH?