On the correlation of the supremum and the infimum and of maximum gain and maximum loss of Brownian motion with drift


Vardar-Acar C. , Zirbel C. L. , Szekely G. J.

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, vol.248, pp.61-75, 2013 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 248
  • Publication Date: 2013
  • Doi Number: 10.1016/j.cam.2013.01.010
  • Title of Journal : JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
  • Page Numbers: pp.61-75
  • Keywords: Brownian motion with drift, Scaling property, Uniform continuity, Markov property, Correlation coefficient

Abstract

Investors are naturally interested in the supremum and the infimum of stock prices, also in the maximum gain and the maximum loss over a time period. To shed light on these relatively complicated aspects of sample paths, we consider Brownian motion with and without drift. We provide explicit calculations of the correlation between the supremum and the infimum of Brownian motion with drift. We establish a number of results concerning the distributions of maximum gain and maximum loss. We present simulation studies of maximum gain and of maximum loss of Brownian motion with a range of values for the drift. We conjecture that the correlation between maximum gain and maximum loss has a minimum value of -0.5 at drift 2. (C) 2013 Elsevier B.V. All rights reserved.