Parisian Option Pricing: A Recursive Solution for the Density of the Parisian Stopping Time

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.

“…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

2019

Mathematical Finance

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“…Now we refer to Dassios and Lim [Dassios and Lim (2013)] for the derivation of the following equality…”

Section: Formal Proofmentioning

confidence: 99%

“…In this paper, we derive a recursive formula for the density of the double barrier Parisian stopping time. In [Dassios and Lim (2013)], an explicit solution for the density of the Parisian stopping time with a single barrier was obtained. But here, we consider excursions both above the upper barrier and below the lower barrier.…”

Section: Introductionmentioning

confidence: 99%

Recursive formula for the double-barrier Parisian stopping time

2018

J. Appl. Probab.

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“…Other methods of pricing Parisian options include the PDE method, studied by Haber, Schönbucher, and Wilmott (1999), the simulation method, as in Anderluh (2008) and Bernard and Boyle (2011), and the combinatorial approach in Costabile (2002). In Dassios and Lim (2013), a recursive solution for the density of the one-sided Parisian stopping time was found and a procedure for pricing Parisian options was proposed, which does not require any numerical inversion of Laplace transforms.…”

Section: Introductionmentioning

confidence: 99%

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

2015

Mathematical Finance

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In this paper, we obtain a recursive formula for the density of the two-sided Parisian stopping time. This formula does not require any numerical inversion of Laplace transforms, and is similar to the formula obtained for the one-sided Parisian stopping time derived in Dassios and Lim [6]. However, when we study the tails of the two distributions, we find that the two-sided stopping time has an exponential tail, while the one-sided stopping time has a heavier tail. We derive an asymptotic result for the tail of the two-sided stopping time distribution and propose an alternative method of approximating the price of the two-sided Parisian option.