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

Zhang, Hongzhong

doi:10.1017/s0001867800007771

Cited by 26 publications

(47 citation statements)

References 35 publications

(33 reference statements)

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“…Drawdowns of spectrally negative Lévy processes were analyzed in Mijatovic and Pistorius (2012). The notion of drawup, which measures the maximum cumulative gain relative to a running minimum, has also been investigated probabilistically, particularly in terms of its relationship to drawdown; see for example Hadjiliadis and Vecer (2006), Pospisil et al (2009), andZhang andHadjiliadis (2010).…”

Section: Introductionmentioning

confidence: 99%

“…Grossman and Zhou (1993) considered an asset allocation problem subject to drawdown constraints; Cvitanic and Karatzas (1995) extended the same optimization problem to the multi-variate framework; Chekhlov et al (2003Chekhlov et al ( , 2005) developed a linear programming algorithm for a sample optimization of portfolio expected return subject to constraints on drawdown, which, in Krokhmal et al (2003), was numerically compared to shortfall optimzation with applications to hedge funds in mind; Carr et al (2011) introduced a new European style drawdown insurance contract and derivative-based drawdown hedging strategies; and most recently Cherney and Obloj (2013), Sekine (2013), Zhang et al (2013) and Zhang (2015) studied drawdown optimization and drawdown insurance under various stochastic modeling assumptions. reformulated the necessary optimality conditions for a portfolio optimization problem with drawdown in the form of the Capital Asset Pricing Model (CAPM), which is used to derive a notion of drawdown beta.…”

Section: Introductionmentioning

confidence: 99%

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Drawdown: from practice to theory and back again

Goldberg

Mahmoud

2016

Math Finan Econ

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Abstract. Maximum drawdown, the largest cumulative loss from peak to trough, is one of the most widely used indicators of risk in the fund management industry, but one of the least developed in the context of measures of risk. We formalize drawdown risk as Conditional Expected Drawdown (CED), which is the tail mean of maximum drawdown distributions. We show that CED is a degree one positive homogenous risk measure, so that it can be linearly attributed to factors; and convex, so that it can be used in quantitative optimization. We empirically explore the differences in risk attributions based on CED, Expected Shortfall (ES) and volatility. An important feature of CED is its sensitivity to serial correlation. In an empirical study that fits AR(1) models to US Equity and US Bonds, we find substantially higher correlation between the autoregressive parameter and CED than with ES or with volatility. We are grateful to Robert Anderson for insightful comments on the material discussed in this article; to Alexei Chekhlov, Stan Uryasev, and Michael Zabarankin for their feedback on a previous draft of this work; to Vladislav Dubikovsky, Michael Hayes, and Márk Horváth for their contributions to an earlier version of this article; to Carlo Acerbi for providing detailed comments on a previous draft; and to the referees and editors of Mathematics and Financial Economics for their valuable feedback.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Drawdown: from practice to theory and back again

Goldberg

Mahmoud

2016

Math Finan Econ

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“…The probability that a drawdown precedes a drawup in a finite time-horizon was studied under drifted Brownian motions and simple random walks in [24]. More recently, Zhang [23] and Zhang and Hadjiliadis [25] studied Laplace transforms of the drawdown time, the so-called speed of market crash, and various occupation times at the first exit and the drawdown time for a general time-homogeneous diffusion process.…”

Section: Literature Reviewmentioning

confidence: 99%

On the Frequency of Drawdowns for Brownian Motion Processes

Landriault

Zhang

2015

Journal of Applied Probability

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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.

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“…Zhang, Leung, and Hadjiliadis () introduced a drawdown insurance contract whereby the protection buyer pays a constant premium over time to insure against a drawdown of a prespecified amount, with features such as early cancellation and drawup contingencies. Zhang () studied the law of the occupation times of the drawdown and drawup processes and used the results to price cumulative Parisian options and α‐quantile options on the drawdown process. American options on maximum drawdown were also considered in the recent paper by Gapeev and Rodosthenous ().…”

Section: Introductionmentioning

confidence: 99%

A variation of the Azéma martingale and drawdown options

Dassios

Lim

2019

Mathematical Finance

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In this paper, we derive a variation of the Azéma martingale using two approaches—a direct probabilistic method and another by projecting the Kennedy martingale onto the filtration generated by the drawdown duration. This martingale links the time elapsed since the last maximum of the Brownian motion with the maximum process itself. We derive explicit formulas for the joint densities of (τ,Wτ,Mτ), which are the first time the drawdown period hits a prespecified duration, the value of the Brownian motion, and the maximum up to this time. We use the results to price a new type of drawdown option, which takes into account both dimensions of drawdown risk—the magnitude and the duration.

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

Cited by 26 publications

References 35 publications

Drawdown: from practice to theory and back again

Drawdown: from practice to theory and back again

On the Frequency of Drawdowns for Brownian Motion Processes

A variation of the Azéma martingale and drawdown options

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