An Analytical Solution for the Two‐sided Parisian Stopping Time, Its Asymptotics, and the Pricing of Parisian Options

Dassios, Angelos; Lim, Jia Wei

doi:10.1111/mafi.12091

Cited by 14 publications

(13 citation statements)

References 14 publications

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“…1, we compare the estimated density based on 100,000 samples generated from the algorithm, to the theoretical density of M τ , and the results are very close. For τ , we compare the estimated density from the simulations to the density computed using the recursive formula derived in Dassios and Lim (2015).…”

Section: Numerical Studiesmentioning

confidence: 99%

“…These options pay off an amount proportional to the running maximum every time the drawdown duration reaches length 1, until the option expires. Pricing using the recursive formulas in Dassios and Lim (2015) would become too cumbersome when pricing multiple drawdown options, as an additional integral would need to be evaluated for each drawdown time. This simulation algorithm, however, eliminates the need to evaluate multiple integrals.…”

Section: Multiple Drawdown Optionmentioning

confidence: 99%

See 1 more Smart Citation

An Efficient Algorithm for Simulating the Drawdown Stopping Time and the Running Maximum of a Brownian Motion

Dassios

Lim

2017

Methodol Comput Appl Probab

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We define the drawdown stopping time of a Brownian motion as the first time its drawdown reaches a duration of length 1. In this paper, we propose an efficient algorithm to efficiently simulate the drawdown stopping time and the associated maximum at this time. The method is straightforward and fast to implement, and avoids simulating sample paths thus eliminating discretisation bias. We show how the simulation algorithm is useful for pricing more complicated derivatives such as multiple drawdown options.

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Section: Numerical Studiesmentioning

confidence: 99%

Section: Multiple Drawdown Optionmentioning

confidence: 99%

An Efficient Algorithm for Simulating the Drawdown Stopping Time and the Running Maximum of a Brownian Motion

Dassios

Lim

2017

Methodol Comput Appl Probab

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“…Hence, the optional stopping theorem applies. As

U_{τ} = 1

(we have let

D = 1

), we have

E (e^{- β τ} e^{- γ M_{τ}}) = \frac{1}{1 + γ \sqrt{\frac{π}{2}} e^{β} + β e^{β} \int_{0}^{1} e^{- β w} \frac{1}{\sqrt{w}} dw} .

We use the same method of Laplace inversion which was used in Dassios and Lim (, ) to obtain the densities of the one‐ and two‐sided Parisian stopping times. Rearranging the expression, we have

\begin{matrix} true \\ right E ((), e^{- β τ}, e^{- γ M_{τ}}) & = & left \frac{e^{- β}}{γ \sqrt{\frac{π}{2}} + 2 \sqrt{π β} \int_{0}^{\sqrt{2 β}} \frac{1}{\sqrt{2 π}} e^{- \frac{x^{2}}{2}} d x + e^{- β}} \\ = & left \frac{e^{- β}}{γ \sqrt{\frac{π}{2}} + \sqrt{π β} + \int_{1}^{\infty} \frac{e^{- β s}}{2 s^{3 / 2}} ds} \\ = & left \frac{e^{- β}}{(\sqrt{π β} + γ \sqrt{\frac{π}{2}}) (1 + \frac{1}{\sqrt{π β} + γ \sqrt{\frac{π}{2}}} \int_{1}^{\infty} \frac{e^{β s}}{2 s^{3 / 2}} ds)} \\ = & left e^{- β} \frac{1}{\sqrt{π β} + γ \sqrt{\frac{π}{2}}} \sum_{k = 0}^{\infty} false(- 1 false) \end{matrix}

…”

Section: Joint Density Of τ Wτ and Mτmentioning

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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“…The two-sided Parisian options have been introduced in Dassios and Wu (2010). The fully symmetric case has been studied in Dassios and Lim (2017). Let us remark that in this latter case the Laplace transform of the two-sided Parisian time is given in Getoor (1979), Proposition (9.2).…”

Section: The Geometric Brownian Motionmentioning

confidence: 99%

On the first positive and negative excursion exceeding a given length

Sirovich

Testa

2019

Statistics & Probability Letters

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For a one-dimensional diffusion process X, we derive the Laplace transform and the moments of the first time at which the age of an excursion above (or below) the level x is longer than u. The result is then illustrated for diffusion processes that are found relevant in applications. In the context of pricing Parisian options, the Brownian motion and the Geometric Brownian motion are considered and the Laplace transform can be made explicit and explicit expression for the moments can be derived. In the context of neuronal modeling, the Ornstein-Uhlenbeck process and the Cox-Ingersoll-Ross process are considered and the Laplace transform and the moments must be approximated by numerical inversion.

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An Analytical Solution for the Two‐sided Parisian Stopping Time, Its Asymptotics, and the Pricing of Parisian Options

Cited by 14 publications

References 14 publications

An Efficient Algorithm for Simulating the Drawdown Stopping Time and the Running Maximum of a Brownian Motion

An Efficient Algorithm for Simulating the Drawdown Stopping Time and the Running Maximum of a Brownian Motion

A variation of the Azéma martingale and drawdown options

On the first positive and negative excursion exceeding a given length

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