Skip to content
All library documents

Estimating Band-Crossing Counts for an Arithmetic Brownian Motion

Article Quant Q&A · Author: rudinable
Read the original

Summary

The document considers an arithmetic Brownian motion with constant volatility and asks how often it is expected to move a fixed distance from its current level, resetting the reference band after each crossing. The question is framed in terms of repeated barrier events, with a possible application to successive double-barrier knock-in option fills.

The answer uses the diffusion scaling relation that a typical move over time grows with volatility times the square root of time. Equating this scale to the band width gives a characteristic crossing time; dividing the interval by that time yields a heuristic expected count proportional to volatility squared and elapsed time, and inversely proportional to the squared band width. This is an order-of-magnitude heuristic, not a rigorous derivation of the expected first-passage renewal count. It assumes a simple constant-volatility Brownian model and does not address drift, discrete monitoring, or execution effects.

Key ideas

  • For an arithmetic Brownian motion, the typical displacement over time scales with volatility times the square root of time.
  • Equating typical displacement to the band width gives a characteristic crossing timescale.
  • The proposed heuristic count increases with squared volatility and elapsed time.
  • The heuristic count decreases with the square of the band width.
  • The estimate omits drift, discrete monitoring, and execution details.

Tags

StatisticsVolatilityDerivatives pricingRisk management
Full text
# ABM Crossing Times


# ABM Crossing Times












Suppose I have a process that follows an arithmetic brownian motion

$dX_t = \sigma dW_t$

How do I calculate, within a certain interval $\Delta t$ , the expected number of times that the process will "leave" a certain band $\delta$ from the starting time.

I.e, suppose I have a starting point $X_0$ and $t_0$ . Suppose that by $t_1,X_1>X_0+\delta$ or $X_1<X_0−\delta $ . This should add one to the count. Then I want to reset the band to $X_1±\delta$ etc. What are the expected number of times that the process $X_t$ will leave these bands within $\Delta t$?

Otherwise said, if I'm repetitively buying double barrier knock-in options on successive fills on an ABM, how many times should I get hit within a certain time interval.

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/80170

Heuristically, the time $t$ taken to travel an absolute distance $\delta$ is given by $$\sigma \sqrt{t} = \delta$$. So this gives $t= (\delta/\sigma)^2$ so the number of times this happens in a time period $T$ should be $$\sigma^2 T/\delta^2$$

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.