On joint distribution of range and terminal value of a Brownian motion
Oleg Svirschi
TL;DR
This paper derives a closed-form joint distribution for the range and terminal value of a Brownian motion, introduces a scaled distribution, and empirically compares it with currency pair data.
Contribution
It provides the first closed-form solution for the joint distribution of range and terminal value of Brownian motion and applies it to currency data.
Findings
Closed-form joint distribution derived
Range scaled terminal value distribution constructed
Empirical comparison with currency data conducted
Abstract
In this note, we present the closed form solution for the joint distribution of the range and terminal value of a Brownian motion. Based on this distribution we build a range scaled terminal value distribution and show the derivation steps of its density, further s-density. Finally, we sample the s-density from different groups of currency pairs and compare them with theoretical result.
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Taxonomy
TopicsComplex Systems and Time Series Analysis · Stochastic processes and financial applications · Financial Risk and Volatility Modeling