Truncated Variation, Upward Truncated Variation and Downward Truncated Variation of Brownian Motion with Drift - their Characteristics and Applications

TL;DR

This paper introduces and analyzes truncated variations of Brownian motion with drift, including upward and downward variants, providing formulas for their moment-generating functions and applications in financial mathematics.

Contribution

It defines truncated variation and related quantities for Brownian motion with drift, deriving their moment-generating functions and applications in finance.

Findings

Truncated variation has finite exponential moments.

Explicit formulas for Laplace transforms of truncated variations.

Applications in modeling maximal returns in finance.

Abstract

In the paper "On Truncated Variation of Brownian Motion with Drift" (Bull. Pol. Acad. Sci. Math. 56 (2008), no.4, 267 - 281) we defined truncated variation of Brownian motion with drift, $W_t = B_t + \mu t, t\geq 0,$ where $(B_t)$ is a standard Brownian motion. Truncated variation differs from regular variation by neglecting jumps smaller than some fixed $c > 0$. We prove that truncated variation is a random variable with finite moment-generating function for any complex argument. We also define two closely related quantities - upward truncated variation and downward truncated variation. The defined quantities may have some interpretation in financial mathematics. Exponential moment of upward truncated variation may be interpreted as the maximal possible return from trading a financial asset in the presence of flat commission when the dynamics of the prices of the asset follows a geometric Brownian motion process. We calculate the Laplace transform with respect to time parameter of the moment-generating functions of the upward and downward truncated variations. As an application of the obtained formula we give an exact formula for expected value of upward and downward truncated variations. We give also exact (up to universal constants) estimates of the expected values of the mentioned quantities.