Pricing Double Barrier Parisian Options Using Laplace Transforms

Labart, Céline; Lelong, Jérôme

doi:10.1142/s0219024909005154

Cited by 27 publications

(21 citation statements)

References 11 publications

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“…Table 2 shows the prices of Parisian down-and-in calls, valued using parameters σ = 0.2, r = 0.05, T = 1 year, K = 95, and L = 90, and at different window lengths D and initial stock price S 0 . Table 3 gives a comparison of the CPU times for our algorithm and that using the Laplace inversion technique in [9], computed using the above parameters and S 0 = 90. Due to the increasing number of recursions required, the computation times increase rapidly as the window length decreases.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Avellandea and Wu [3] used a lattice method. Labart and Lelong [9] used an inversion formula based on the Abate and Whitt [1] method, while Bernard, Courtois, and Quittard-Pinon [4] obtained numerical prices by approximating the Laplace transforms using a linear combination of fractional functions. In this paper, we used a different method to obtain the option price without numerically inverting its Laplace transform.…”

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confidence: 99%

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

Dassios¹,

Lim²

2013

SIAM J. Finan. Math.

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Abstract. In this paper, we obtain the density function of the single barrier one-sided Parisian stopping time.The problem reduces to that of solving a Volterra integral equation of the first kind, where a recursive solution is consequently obtained. 1. Introduction. Parisian options were first introduced by Chesney, Jeanblanc-Picque, and Yor [5]. They are path-dependent options whose payoff depends not only on the final value of the underlying asset, but also on the path trajectory of the underlying asset above or below a predetermined barrier L. The owner of a Parisian down-and-out call loses the option when the underlying asset price S reaches the level L and remains constantly below this level for a time interval longer than D, while for a Parisian down-and-in call, the same event gives the owner the right to exercise the option. Parisian options are a kind of barrier option. However, it has the advantage of not being as easily manipulated by an influential agent as a simple barrier option, and thus is a guarantee against easy arbitrage.No explicit pricing formula is known for this type of option. Previous literature has largely focused on using Laplace transforms to price Parisian options. In [5,7,10], the problem is reduced to finding the Laplace transform of the Parisian stopping time, which is the first time the length of the excursion reaches level D. In [5], the Laplace transform of the stopping time was obtained using the Brownian meander and Azema martingale, while Dassios and Wu [7] introduced a perturbed Brownian motion and a semi-Markov model to obtain the Laplace transform. In both of these, an explicit form of the Laplace transform of the distribution of the Parisian stopping time and consequently that of the option price is found. Other methods of pricing Parisian options include the PDE method, studied by Haber, Schonbucher, and Wilmott [8]. There exist also other types of Parisian options. Cumulative Parisian options,

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Section: Numerical Resultsmentioning

confidence: 99%

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confidence: 99%

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

Dassios¹,

Lim²

2013

SIAM J. Finan. Math.

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“…We have established pricing formulae for MinParisianHit and MaxParisianHit options. These fair prices contain single Laplace transforms which need to be inverted numerically using techniques as in Labart and Lelong [17], Abate and Whitt [1] and Bernard et al [4].…”

Section: Option Triggered At Maximum Of Parisian and Hitting Timesmentioning

confidence: 99%

“…Here, we study excursion and hitting time using a three state semi-Markov model indicating whether the process is in a positive or negative excursion and above or below a fixed barrier. This will allow us to compute the double Laplace transform of these two stopping times, which can be inverted numerically using techniques as in Labart and Lelong [17]. Gauthier [14], [15] studies the first instant when a standard Brownian motion either spends consecutively more than a certain time above a certain level, or reaches another level, i.e.…”

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The joint distribution of Parisian and hitting times of Brownian motion with application to Parisian option pricing

Dassios

Zhang

2016

Finance Stoch

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We study the joint law of Parisian time and hitting time of a drifted Brownian motion by using a three-state semi-Markov model, obtained through perturbation. We obtain a martingale, to which we can apply the optional sampling theorem and derive the double Laplace transform. This general result is applied to address problems in option pricing. We introduce a new option related to Parisian options, being triggered when the age of an excursion exceeds a certain time or/and a barrier is hit. We obtain an explicit expression for the Laplace transform of its fair price.

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“…The concluding numerical examples will show the reader how the various Parisian option types behave that can be constructed from the doublesided Parisian knock-in call. The double-sided Parisian option treated by [17] differs from the one treated here as will be pointed out in the next section.…”

Section: Introductionmentioning

confidence: 97%

Double-sided Parisian option pricing

Anderluh

Weide

2009

Finance Stoch

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In this paper we derive Fourier transforms for double-sided Parisian option contracts. The double-sided Parisian option contract is triggered by the stock price process spending some time above an upper level or below some lower level. The double-sided Parisian knock-in call contract is the general type of Parisian contract from which also the single-sided contract types follow. The paper gives an overview of the different types of contracts that can be derived from the double-sided Parisian knock-in calls, and, after discussing the Fourier inversion, it concludes with various numerical examples, explaining the, sometimes peculiar, behavior of the Parisian option.The paper also yields a nice result on standard Brownian motion. The Fourier transform for the double-sided Parisian option is derived from the Laplace transform of the double-sided Parisian stopping time. The probability that a standard Brownian motion makes an excursion of a given length above zero before it makes an excursion of another length below zero follows from this Laplace transform and is not very well known in the literature. In order to arrive at the Laplace transform, a very careful application of the strong Markov property is needed, together with a non-intuitive lemma that gives a bound on the value of Brownian motion in the excursion.

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Pricing Double Barrier Parisian Options Using Laplace Transforms

Cited by 27 publications

References 11 publications

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

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

The joint distribution of Parisian and hitting times of Brownian motion with application to Parisian option pricing

Double-sided Parisian option pricing

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