Parisian options are essentially a crossover between barrier options and Asian options. They have predominant barrier option features in that they can be knocked in or out depending on hitting a barrier from under or above; they differ from standard barrier options in that extreme outlier asset movements will not trigger the Parisian, and for the trigger to be activated or extinguished, the asset must lie outside or inside the barrier for a predetermined time period t.

For example, an up-and-out Parisian call option becomes extinguished if the underlying asset remains above a predetermined barrier level for a prescribed amount of time. Compared to standard barrier options, it can be more beneficial for the holder of a Parisian option on a volatile asset.

Because of its highly path-dependent nature, Parisian options cannot be solved via a closed form method and common methods involve the consideration of the PDE and the use of a Laplace transform and solving via finite difference methods. The classic paper on pricing Parisian options under continuous monitoring is that of Chesney, Jeanblanc-Picque, Yor (1995) who show the evaluation of the Laplace transform followed by a rapid inversion via the Euler method - also see Hartley (2000).

Cornwall & Kentwell (1995) extended Chesney, Jeanblanc-Picque & Yor's (1995) approach to a quasi-analytical model and also incorporating discrete time monitoring, which is actually more often the case in actual markets for Parisian options.

1) The Partial Differential Equation

Like many other path dependent exotics, solving the PDE on a lattice using finite differences is generally the most reliable method, and in the case of Parisians and barriers, this is even more so the case.

For these options, there are 16 differential equations governing the various cases in which a Parisian can take. Considering all the combinations of down, up, in & out Parisian calls and puts, and then assessing whether the asset price is below or above the strike price, we are able to determine the equations.

By considering a three dimensional problem (time, price and duration of out/in), one can solve the PDE via finite differences on a lattice - see e.g. Haber, Schonbucher, Wilmott (1999), Vetzal & Forsyth (1999) and Zhu & Stokes (1999).

Zhu & Stokes use finite element schemes to numerically evaluate the Parisian option and shows that convergence and accuracy of this method can be relied upon when pricing these options.

2) Binomial Trees

As we touched on earlier in the text we can solve Parisians via the use of lattice methods such as a binomial, trinomial or adaptive mesh tree and solving the PDE using finite methods. For more on finite methods, Tavella & Randall (2000) extensively consider the use of finite differences to solve PDEs within option pricing.

By starting off with our basic Cox, Ross, Rubinstein (1979) framework, a binomial tree can incorporate Parisian options by counting the number of nodes in which the Parisian has satisfied its duration condition and determining the price at the node. For example, an up-and-out Parisian call option which has a duration of 5 periods (note that the term 'duration' merely represents the window it must remain above the barrier for the Parisian to become extinguished. Given the up-and-out Parisian having a strike price of 100, a barrier level of 105 and a window period of 10 days, as long as the underlying does not remain above a level of 105 for a period of 10 days, the option remains active.

3) Trinomial Trees

Avellaneda & Wu (1999) consider the use of pricing Parisian options using the more readily convergent trinomial lattice, extending on from Boyle's (1986) approach to pricing standard options using a trinomial tree.

By considering the inputs of the trinomial model, one can price via backwards induction, similar to that of Boyle, and even of the CRR binomial tree.

By noting the possibilities that the underlying asset remains outside or inside the barrier for the window period, Avellaneda & Wu can solve the PDE governing the Parisian under this condition by specifying the magnitudes of the movements and the probabilities:

	m = 0

4) Other Lattice Methods

Similarly, other lattice methods such as an adaptive mesh or forward shooting grid (FSG) can be used for faster convergence than standard lattice methods such as binomial or trinomial lattice trees. An adaptive mesh transposes further grids on existing lattices to enable more accurate pricing near the barriers in a lattice which is similar to:

Where the black grid is the original trinomial lattice and the orange grid is another trinomial lattice transposed onto the original. Figlewski & Gao (1999) consider this treatment for standard barrier options, but it can be extended for Parisians.

Kwok & Lau (2001) assess the use of a FSG method first proposed by Hull & White (1993) for pricing Parisian style options. They then extend the approach to price the cumulative style Parisians known as Parasian options.

5) Cumulative Parisian Options

Increasing amounts of literature have been given to a hybrid type of Parisian options known as Parasian options. We consider these options on a separate page which is set to appear shortly.

6) American Parisians

In the case of American Parisians, one might imagine them to be more computationally intensive than their European siblings. This is so for most American style Parisians, but we put an emphasis on the case when closed form analytical solutions can actually be found - when considering the case of perpetual American Parisians.

Other Known Names / Variants

Barrier Options
Edokko Options
Cumulative Parisian Options
Moving Window Options
Parasian Options
Tokyo Options
Window Options

Additional/Useful List of resources

Papers:

Avellaneda, M., Wu, L., "Pricing Parisian-Style Options with a Lattice Method", International Journal of Theoretical & Applied Finance, 2, 1, pp. 1-16, 1999
Black, F. & Scholes, M. "The Pricing of Options & Corporate Liabilities", The Journal of Political Economy (May '73)
Brennan, M. & Schwartz, E., "Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims". Journal of Finance and Quantitative Analysis, 13:462--474 (1978)
Boyle, P., "Option Valuation Using a Three-Jump Process", International Options Journal 3, pages 7-12 (1986)
Chesney, M., Jeanblanc-Picque, M., Yor, M., "Parisian Options & Excursions Theory", 5th Annual Derivatives Conference, Cornell University (1995)
Chesney, M., Cornwall, J., Jeanblanc-Picque, M., Kentwell, G., Yor, M., "Parisian Pricing", 10, 1, 77-79 (1997)
Cornwall, J,. Kentwell, G., "A Quasi-Analytical Approach to Occupation Time Barrier", Bankers Trust Working Paper (1995)
Cox, J., Ross, S., & Rubinstein M., “Option Pricing: A Simplified Approach." Journal of Financial Economics, 7. (Sept '79)
Figlewski S., & Gao, B., "The Adaptive Mesh Model: A New Approach to Efficient Option Pricing", Journal of Financial Economics, 53, pp. 313-51, 1999
Fujita, T., & Miura, R., "Edokko Options: A New Framework of Barrier Options'', Asia-Pacific Financial Markets 9(2) pp. 141-15 (2003)
Grau, A., "Moving Windows", University of Waterloo Working Paper, June 2003
Haber, R.J., Schonbucker, P.J., Wilmott, P., "Pricing Parisian Options", Journal of Derivatives, 6, pp. 71-79, 1999
Hartley, P., "Pricing Parisian Options by Laplace Inversion", Decisions in Economics & Finance, 2002
Hull, J., & A. White "Efficient Procedures for Valuing European & American Path-Dependent Options", Journal of Derivatives, 1, 21-31 (1993)
Kwok, Y.K., Lau, K.W., "Pricing Algorithms for Options with Exotic Path-Dependence", Journal of Derivatives, 2001
Moraux, F., "On Pricing Cumulative Parisian Options", Universite de Rennes I Working Paper (2002)
Schroder, M., "On the Valuation of Paris Options: Foundational Results", Universitat Mannheim Working Paper, (2000)
Tavella, D., & Randall, R., "Pricing Financial Instruments: the Finite Difference Method" (2000)
Vetzal, K.R., Forsyth, P.A., "Discrete Parisian & Delayed Barrier Options: A General Numerical Approach", Advances in Futures & Options Research, 10, pp. 1-15, 1999
Vorst, A., "Parisian Option Valuation", Derivatives Week, 8, 8, 6-7 (1999)
Zhu, Z., Stokes, N., "A Finite Element Platform for Pricing Path-Dependent Exotic Options", CSIRO Mathematical & Information Sciences Working Paper, 1999
Zvan, R., Forsyth, P.A., Vetzal, K.R., "Discrete Asian Barrier Options", Working Paper, 1998