A variation of the Azéma martingale and drawdown options

Dassios, Angelos; Lim, Jia Wei

doi:10.1111/mafi.12202

Cited by 2 publications

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“…Azéma martingales for the Brownian motion were discovered by Azéma (1985), and can be obtained by projecting martingales onto the slow filtration. A variation of the Azéma martingale are used to price Parisian options by Chesney, Jeanblanc‐Picqué, and Yor (1997), and an extension of it involving the local time is derived by Dassios and Lim (2016). Here, we use excursion theory to prove a variation of the Azéma martingale for the CIR and Bessel processes.…”

Section: Introductionmentioning

confidence: 99%

Azéma martingales for Bessel and CIR processes and the pricing of Parisian zero‐coupon bonds

Dassios

Lim

2020

Mathematical Finance

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In this paper, we study the excursions of Bessel and Cox–Ingersoll–Ross (CIR) processes with dimensions 0<δ<2. We obtain densities for the last passage times and meanders of the processes. Using these results, we prove a variation of the Azéma martingale for the Bessel and CIR processes based on excursion theory. Furthermore, we study their Parisian excursions, and generalize previous results on the Parisian stopping time of Brownian motion to that of the Bessel and CIR processes. We obtain explicit formulas and asymptotic results for the densities of the Parisian stopping times, and develop exact simulation algorithms to sample the Parisian stopping times of Bessel and CIR processes. We introduce a new type of bond, the zero‐coupon Parisian bond. The buyer of such a bond is betting against zero interest rates, while the seller is effectively hedging against a period where interest rates fluctuate around 0. Using our results, we propose two methods for pricing these bonds and provide numerical examples.

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Section: Introductionmentioning

confidence: 99%

Azéma martingales for Bessel and CIR processes and the pricing of Parisian zero‐coupon bonds

Dassios

Lim

2020

Mathematical Finance

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“…(2017). Solutions related to the valuation of drawdown options and insurance contracts can be found in Dassios and Lim (2019), Palmowski and Tumilewicz (2020), Yamamoto et al. (2010) and Zhang et al.…”

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A review on drawdown risk measures and their implications for risk management

Geboers

Depaire

Annaert

2022

Journal of Economic Surveys

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As highlighted by the recent market turmoil following COVID‐19, markets can experience significant retracements or drawdowns. While these recent market moves have definitely been large, significant drawdowns have been around since the start of financial markets. Various risk metrics such as Value at Risk and volatility are used to describe risk. The intuitive drawdown risk measure, which is often used in practice alongside the above metrics, is receiving more and more academic attention. In this article we provide a systematic review of the literature on the drawdown risk measures. We describe two different methodologies for calculating drawdowns and analyze drawdown based risk measures used in risk management, portfolio construction and optimization. Finally we discuss the statistical properties related to drawdowns. Based on the research done so far, we identify several areas for further research.

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A variation of the Azéma martingale and drawdown options

Cited by 2 publications

References 39 publications

Azéma martingales for Bessel and CIR processes and the pricing of Parisian zero‐coupon bonds

Azéma martingales for Bessel and CIR processes and the pricing of Parisian zero‐coupon bonds

A review on drawdown risk measures and their implications for risk management

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