Pricing double barrier Parisian options using Laplace transforms.

*(English)*Zbl 1190.91143Parisian options were introduced by M. Chesney, M. Jeanblanc-Picqué and M. Yor [Adv. Appl. Probab. 29, No. 1, 165–184 (1997; Zbl 0882.60042)] and are defined as barrier options where the condition involves the time spent above or below a certain level. If the condition of the asset involves the time spent out of the range of the two barriers \([L_1,L_2]\), then the option is known as a double barrier Parisian option. Monte Carlo simulations, lattices, Laplace transforms and partial differential equations are all methods used to price Parisian options. The authors present a comprehensive review, including citations, for the pricing of Parisian options. The valuation of double barrier options has not been studied to the extent of the single barrier options. P. Baldi, L. Caramellino and M. G. Iovino [in: Monte Carlo and quasi-Monte Carlo methods 1998. Proceedings of a conference held at the Claremont Graduate Univ., Claremont, CA, USA, June 22-26, 1998. Berlin: Springer. 149-162 (2000; Zbl 0937.91062)] proposed a method of pricing double barrier options based upon Monte Carlo simulation.

In this article, the authors price double barrier Parisian options using Laplace transforms. The authors price double Parisian out calls and double Parisian in calls using a Laplace transform with a term \(\widehat{A}\) whose calculation differs significantly from that in the single Parisian option case. The calculation of this term is similar to the manner in which H. Geman and M. Yor [Math. Finance 6, No. 4, 365–378 (1996; Zbl 0915.90016)] calculated the equivalent term for standard double barrier options.

Suppose the double barrier Parisian option is activated or canceled after spending more than a given time \(D\) in an excursion. The authors define Parisian times \(T_b^-\) (and \(T_b^+\)) to be the time the process makes an excursion longer than \(D\) below (resp. above) the level \(b\). The authors then present an upper bound on the error between the approximated price and the exact price that relies upon the regularity of the Parisian option with respect to maturity times. They determine in theorem 5.1 that the price of a double barrier Parisian option is of class \(\mathcal{C}^{\infty}\) when the lower barrier for the Brownian motion corresponding to \(L_1\) is less than zero and the upper barrier for the Brownian motion corresponding to \(L_2\) is greater than zero. The price is discontinuous when the lower barrier of the Brownian motion is greater than zero or the upper barrier of the Brownian motion is less than zero. The authors also consider the special case when one of the barriers of the Brownian motion is zero (with the result that the price of the double barrier Parisian option is continuous and may be \(\mathcal{C}^1\)). In section 5.2, to assist in the proof of the regularity of single Parisian option prices, the authors determine the regularity of the density of the Parisian times and determine when this density exists. The authors conclude by examining the numerical inversion of Laplace transforms using contour integrals.

In this article, the authors price double barrier Parisian options using Laplace transforms. The authors price double Parisian out calls and double Parisian in calls using a Laplace transform with a term \(\widehat{A}\) whose calculation differs significantly from that in the single Parisian option case. The calculation of this term is similar to the manner in which H. Geman and M. Yor [Math. Finance 6, No. 4, 365–378 (1996; Zbl 0915.90016)] calculated the equivalent term for standard double barrier options.

Suppose the double barrier Parisian option is activated or canceled after spending more than a given time \(D\) in an excursion. The authors define Parisian times \(T_b^-\) (and \(T_b^+\)) to be the time the process makes an excursion longer than \(D\) below (resp. above) the level \(b\). The authors then present an upper bound on the error between the approximated price and the exact price that relies upon the regularity of the Parisian option with respect to maturity times. They determine in theorem 5.1 that the price of a double barrier Parisian option is of class \(\mathcal{C}^{\infty}\) when the lower barrier for the Brownian motion corresponding to \(L_1\) is less than zero and the upper barrier for the Brownian motion corresponding to \(L_2\) is greater than zero. The price is discontinuous when the lower barrier of the Brownian motion is greater than zero or the upper barrier of the Brownian motion is less than zero. The authors also consider the special case when one of the barriers of the Brownian motion is zero (with the result that the price of the double barrier Parisian option is continuous and may be \(\mathcal{C}^1\)). In section 5.2, to assist in the proof of the regularity of single Parisian option prices, the authors determine the regularity of the density of the Parisian times and determine when this density exists. The authors conclude by examining the numerical inversion of Laplace transforms using contour integrals.

Reviewer: Thomas Wakefield (Youngstown)

##### MSC:

91G20 | Derivative securities (option pricing, hedging, etc.) |

91B25 | Asset pricing models (MSC2010) |

44A10 | Laplace transform |

91G60 | Numerical methods (including Monte Carlo methods) |

##### Keywords:

double barrier option; Parisian option; option price regularity; numerical inversion; Laplace transform; Brownian excursions; Euler summations
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\textit{C. Labart} and \textit{J. Lelong}, Int. J. Theor. Appl. Finance 12, No. 1, 19--44 (2009; Zbl 1190.91143)

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