Simulating Bounded Daily Price Moves with Fixed Return Steps

Summary

The answer proposes simulating a price process with a fixed number of hourly steps and bounded daily returns. It sets the size of each step from the upper and lower daily limits and the number of steps, then uses Bernoulli draws to choose the direction of each move. For asymmetric limits, it gives separate step sizes for upward and downward moves, scaled to the corresponding boundary.

This construction makes the path’s cumulative movement respect the stated bounds under the answer’s additive return convention. It is a simple discrete model for the question’s fixed-magnitude moves, rather than a general model of market returns. The post does not specify the Bernoulli probability, calibrate it to data, or address whether percentage returns should compound multiplicatively; those choices affect the resulting distribution and realism. The proposed setup therefore matches the stated constraints but leaves the directional behavior to be specified separately.

Key ideas

• Set the fixed return step using the daily boundary and the number of intraday steps.
• Use Bernoulli trials to choose the direction of each step.
• Use separate upward and downward step sizes when the daily bounds are asymmetric.
• The method leaves the probability of each direction unspecified.
• The simple setup does not model variable step sizes or calibrate its assumptions to observed returns.

Tags

# Model for this price dynamic


# Model for this price dynamic












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 by Kermittfrog (score 1)

https://quant.stackexchange.com/a/80239

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?

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