Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T02:07:02.719Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Frequency of Drawdowns for Brownian Motion Processes

Part of: Markov processes Stochastic processes Mathematical economics

Published online by Cambridge University Press:  30 January 2018

David Landriault ,
Bin Li  and
Hongzhong Zhang
David Landriault*
Affiliation:
University of Waterloo
Bin Li*
Affiliation:
University of Waterloo
Hongzhong Zhang*
Affiliation:
Columbia University
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada.
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada.
∗∗∗∗ Postal address: Department of Statistics, Columbia University, New York, NY 10027, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Drawdowns measuring the decline in value from the historical running maxima over a given period of time are considered as extremal events from the standpoint of risk management. To date, research on the topic has mainly focused on the side of severity by studying the first drawdown over a certain prespecified size. In this paper we extend the discussion by investigating the frequency of drawdowns and some of their inherent characteristics. We consider two types of drawdown time sequences depending on whether a historical running maximum is reset or not. For each type we study the frequency rate of drawdowns, the Laplace transform of the nth drawdown time, the distribution of the running maximum, and the value process at the nth drawdown time, as well as some other quantities of interest. Interesting relationships between these two drawdown time sequences are also established. Finally, insurance policies protecting against the risk of frequent drawdowns are also proposed and priced.

Keywords

DrawdownfrequencyBrownian motion

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J65: Brownian motion 91B24: Price theory and market structure
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 191 - 208
Copyright
© Applied Probability Trust 

References

Abate, J. and Whitt, W. (2006). A unified framework for numerically inverting Laplace transforms. INFORMS J. Comput. 18, 408–421.Google Scholar
Andersen, E. S. (1957). On the collective theory of risk in the case of contagion between the claims. In Transactions of the XVth International Congress of Actuaries, Vol. II, New York, pp. 104–125.Google Scholar
Borodin, A. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.Google Scholar
Burghardt, G., Duncan, R. and Liu, L. (2003). Deciphering drawdown. Risk magazine 5, S16–S20.Google Scholar
Carr, P., Zhang, H. and Hadjiliadis, O. (2011). Maximum drawdown insurance. Internat. J. Theoret. Appl. Finance 14, 1195–1230.Google Scholar
Chekhlov, A., Uryasev, S. and Zabarankin, M. (2005). Drawdown measure in portfolio optimization. Internat. J. Theoret. Appl. Finance 8, 13–58.Google Scholar
Consul, P. C. and Famoye, F. (2006). Lagrangian Probability Distributions. Birkhäuser, Boston, MA.Google Scholar
Douady, R., Shiryaev, A. N. and Yor, M. (2000). On probability characteristics of ‘downfalls’ in a standard Brownian motion. Theory Prob. Appl. 44, 29–38.Google Scholar
Grossman, S. J. and Zhou, Z. (1993). Optimal investment strategies for controlling drawdowns. Math. Finance 3, 241–276.Google Scholar
Hadjiliadis, O. and Večeř, J. (2006). Drawdowns preceding rallies in a Brownian motion model. Quant. Finance 6, 403–409.Google Scholar
Hamelink, F. and Hoesli, M. (2004). Maximum drawdown and the allocation to real estate. J. Property Res. 21, 5–29.CrossRefGoogle Scholar
Jarrow, R., Kchia, Y. and Protter, P. (2011). How to detect an asset bubble. SIAM J. Financial Math. 2, 839–865.Google Scholar
Lehoczky, J. P. (1977). Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Prob. 5, 601–607.Google Scholar
Magdon-Ismail, M., Atiya, A. F., Pratap, A. and Abu-Mostafa, Y. S. (2004). On the maximum drawdown of a Brownian motion. J. Appl. Prob. 41, 147–161.CrossRefGoogle Scholar
Mijatović, A. and Pistorius, M. R. (2012). On the drawdown of completely asymmetric Lévy processes. Stoch. Process. Appl. 122, 3812–3836.CrossRefGoogle Scholar
Pfeffer, D. (2001). Sequential barrier options. Algo Res. Quart. 4, 65–74.Google Scholar
Pospisil, L. and Večeř, J. (2010). Portfolio sensitivity to changes in the maximum and the maximum drawdown. Quant. Finance 10, 617–627.Google Scholar
Pospisil, L., Večeř, J. and Hadjiliadis, O. (2009). Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups. Stoch. Process. Appl. 119, 2563–2578.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.CrossRefGoogle Scholar
Taylor, H. M. (1975). A stopped Brownian motion formula. Ann. Prob. 3, 234–246.Google Scholar
Večeř, J. (2006). Maximum drawdown and directional trading. Risk 19, 88–92.Google Scholar
Willmot, G. E. and Lin, X. S. (2011). Risk modelling with the mixed Erlang distribution. Appl. Stoch. Models Business Industry 27, 8–22.Google Scholar
Zhang, H. (2015). Occupation times, drawdowns, and drawups for one-dimensional regular diffusions. Adv. Appl. Prob. 1, 209–229.Google Scholar
Zhang, H. and Hadjiliadis, O. (2010). Drawdowns and rallies in a finite time-horizon. Drawdowns and rallies. Methodol. Comput. Appl. Prob. 12, 293–308.Google Scholar
Zhang, H. and Hadjiliadis, O. (2012). Drawdowns and the speed of market crash. Methodol. Comput. Appl. Prob. 14, 739–752.Google Scholar
Zhang, H., Leung, T. and Hadjiliadis, O. (2013). Stochastic modeling and fair valuation of drawdown insurance. Insurance Math. Econom. 53, 840–850.Google Scholar