Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T17:58:22.634Z Has data issue: false hasContentIssue false
  • English
  • Français

Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  04 January 2016

Hongzhong Zhang
Hongzhong Zhang*
Affiliation:
Columbia University
*
∗ Postal address: Department of Statistics, 1255 Amsterdam Avenue, Columbia University, New York, NY 10027, USA. Email address: hzhang@stat.columbia.edu
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.

The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

Keywords

Drawdowndrawupdiffusion processoccupation time

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60G17: Sample path properties
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 1 , March 2015 , pp. 210 - 230
Copyright
© Applied Probability Trust 

References

Albrecher, H., Gerber, H. U. and Shiu, E. S. W. (2011). “The optimal dividend barrier in the gamma-omega model.” Europ. Actuarial J.. 1, 43–55.Google Scholar
Borodin, A. N. and Salminen, P. (2002). “Handbook of Brownian Motion—Facts and Formulae, 2nd edn. “Birkhauser, Basel.Google Scholar
Cai, N., Chen, N. and Wan, X. (2010). “Occupation times of Jump-diffusion processes with double exponential Jumps and the pricing of options.” Math. Operat. Res. 35, 412–437.Google Scholar
Carr, P., Zhang, H. and Hadjiliadis, O. (2011). “Maximum drawdown insurance.” Internat. J. Theoret. Appl. Finance 14, 1195–1230.Google Scholar
Cheridito, P., Nikeghbali, A. and Platen, E. (2012). “Processes of class sigma, last passage times, and drawdowns.” SIAM J. Financial Math. 3, 280–303.Google Scholar
Chesney, M., Jeanblanc-Picque, M. and Yor, M. (1997). “Brownian excursions and Parisian barrier options.” Adv. Appl. prob. 29, 165–184.Google Scholar
Forde, M., Pogudin, A. and Zhang, H. (2013). “Hitting times, occupation times, trivariate laws and the forward Kolmogorov equation for a one-dimensional diffusion with memory.” Adv. Appl. Prob. 45, 860–875.Google Scholar
Gerber, H. U., Shiu, E. S. W. and Yang, H. (2012). “The Omega model: from bankruptcy to occupation times in the red.” Europ. Actuarial J.. 2, 259–272.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 the Brownian motion model.” Quant. Finance 6, 403–409.Google Scholar
Hadjiliadis, O., Zhang, H. and Poor, H. V. (2009). “One shot schemes for decentralized quickest change detection.” IEEE Trans. Inf. Theory 55, 3346–3359.Google Scholar
Kyprianou, A. E., Pardo, J. C. and Pérez, J. L. (2013). Occupation times of refracted Lévy processes. J. Theoret. Prob. 10.1007/s10959-013-0501-4.Google Scholar
Landriault, D., Renaud, J.-F. and Zhou, X. (2011). “Occupation times of spectrally negative Lévy processes with applications.” Stoch. Process. Appl. 121, 2629–2641.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
Li, B. and Zhou, X. (2013). “The Joint Laplace transforms for diffusion occupation times.” Adv. Appl. Prob. 45, 1049–1067.Google Scholar
Magdon-Ismail, M. and Atiya, A. (2004). “Maximum drawdown.” Risk 17, 99–102.Google Scholar
Meilijson, I. (2003). “The time to a given drawdown in Brownian motion.“In Séminaire de Probabilités XXXVII (Lecture Notes Math. 1832), Springer, Berlin, pp. 94–108.Google Scholar
Mijatović, A. and Pistorius, M. R. (2012). “On the drawdown of completely asymmetric Lévy processes.” Stoch. Process. Appl. 122, 3812–3836.Google Scholar
Miura, R. (1992). “A note on look-back options based on order statistics.” Hitotsubashi J. Commerce Manag. 27, 15–28.Google Scholar
Miura, R. (2007). “Rank process, stochastic corridor and applications to finance.“In Advances in Statistical Modeling and Inference (Ser. Biostat. 3), World Scientific, Hackensack, NJ, pp. 529–542.Google Scholar
Peskir, G. (1998). “Optimal stopping of the maximum process: the maximality principle.” Ann. Prob. 26, 1614–1640.Google Scholar
Pitman, J. and Yor, M. (1999). “Laplace transforms related to excursions of a one-dimensional diffusion.” Bernoulli 5, 249–255.Google Scholar
Pitman, J. and Yor, M. (2003). “Hitting, occupation and inverse local times of one-dimensional diffusions: martingale and excursion approaches.” Bernoulli 9, 1–24.CrossRefGoogle Scholar
Poor, H. V. and Hadjiliadis, O. (2009). “Quickest Detection.” Cambridge University Press.Google Scholar
Pospisil, L. and Vecer, J. (2008). “PDE methods for the maximum drawdown.” J. Comput. Finance 12, 59–76.Google Scholar
Pospisil, L. and Vecer, J. (2010). “Portfolio sensitivity to changes in the maximum and the maximum drawdown.” Quant. Finance 10, 617–627.Google Scholar
Pospisil, L., Vecer, 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
Protter, P. E. (2004). “Stochastic Integration and Differential Equations, 2nd edn. “Springer, Berlin.Google Scholar
Salminen, P. and Vallois, P. (2007). “On maximum increase and decrease of Brownian motion.” Ann. Inst. H. Poincaré B Prob. Statis. 43, 655–676.Google Scholar
Shepp, L. and Shiryaev, A. N. (1993). “The Russian option: reduced regret.” Ann. Appl. Prob. 3, 631–640.Google Scholar
Shiryaev, A. N. (1996). “Minimax optimality of the method of cumulative sums (CUSUM) in the continuous time case.” Russian Math. Surveys 51, 750–751.Google Scholar
Vecer, J. (2006). “Maximum draw-down and directional trading.” Risk 19, 88–92.Google Scholar
Vecer, J. (2007). “Preventing portfolio losses by hedging maximum drawdown.” Wilmott 5, 1–8.Google Scholar
Yamamoto, K., Sato, S. and Takahashi, A. (2010). “Probability distribution and option pricing for drawdown in a stochastic volatility environment.” Internat. J. Theoret. Appl. Finance 13, 335–354.Google Scholar
Zhang, H. (2010). “Drawdowns, drawups, and their applications.” , City University of New York.Google Scholar
Zhang, H. and Hadjiliadis, O. (2010). “Drawdowns and rallies in a finite time-horizon.” 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. and Hadjiliadis, O. (2012). “Quickest detection in a system with correlated noise.“In Proc. 51st IEEE Conf. Decision Control, IEEE, New York, pp. 4757–4763.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
Zhang, H., Hadjiliadis, O., Schäfer, T. and Poor, H. V. (2014). “Quickest detection in coupled systems.“SIAM J. Control Optimization 52, 1567–1596.Google Scholar