Parisian options with jumps: a maturity–excursion randomization approach

Abstract

This paper introduces an analytically tractable method for the pricing of European and American Parisian options in a flexible jump–diffusion model. Our contribution is threefold. First, using a double Laplace–Carson transform with respect to the option maturity and the Parisian (excursion) time, we obtain closed-form solutions for different types of Parisian contracts. Our approach allows us also to analytically disentangle contributions of the jump and diffusion components for Parisian options in the excursion region. Second, we provide numerical examples and quantify the impact of jumps on the option price and the Greeks. Finally, we study the non-monotonic effects of volatility and jump intensity close to the excursion barrier, which are important for shareholders’ investment policy decisions in a levered firm.

Keywords:

• Parisian options
• Maturity–excursion randomization
• Hyper-exponential jump–diffusion model
• Excursion disentanglement

AMS Subject Classifications:

• G13
• C02
• C63

Acknowledgements

We would like to thank Jérôme Detemple, João Pedro Vidal Nunes, Paola Pederzoli and Felix Stang and two anonymous reviewers for their valuable comments and suggestions. We are also grateful to the participants at Gerzensee Research Days 2014, and at the Bachelier Finance Conference 2014 in Brussels, Belgium. We gratefully acknowledge support from the Swiss Finance Institute (SFI) and the Department for Banking and Finance (DBF) at the University of Zurich (UZH). The previous version of this paper was entitled ‘European and American Parisian options in a jump-diffusion model’.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 We note that Carr (1998) relied on the Post–Widder inversion, whereas our implementation rests on the Gaver–Stehfest inversion. For further discussion and technical details regarding the two approaches, we refer an interested reader to Kimura (2010) and Leippold and Vasiljević (2017).

2 E.g. see Albrecher et al. (2012), Anderluh (2008), Anderluh and van der Weide (2009), Avellaneda and Wu (1999), Bernard et al. (2005), Bernard and Boyle (2011), Chesney et al. (1997), Chesney and Gauthier (2006), Costabile (2002), Czarna and Palmowski (2011), Dassios and Wu (2010), Dassios and Wu (2011), Dassios and Lim (2013), Dassios and Lim (2017), Dassios and Zhang (2016), Fujita and Miura (2002), Haber et al. (1999), Hugonnier (1999), Labart and Lelong (2009), Landriault et al. (2011), Li and Zhao (2009), Loeffen et al. (2014), Schröder (2003), Vetzal and Forsyth (1999), Zhu and Stokes (1998), Zhu and Chen (2013).

3 Some other types of occupation–time derivatives are analysed in the presence of jump risk (Ait Aoudia and Renaud 2016, Cai et al. 2010, Wu and Zhou 2016).

4 In the case of foreign exchange options, r and represent a domestic and a foreign risk-free interest rate, respectively.

5 Naturally, the probabilities of positive and negative jumps satisfy the condition .

6 For example, a European Parisian up-and-out option is a European-style derivative contract with a knock-out feature which is activated when the underlying process consecutively spends a prespecified amount of time above a certain threshold level.

7 We arbitrarily choose the barrier level to be above the strike price, i.e. .

8 The associated boundary conditions for EPOUP and APOUP options are given in equations (21(21) ) and (43(43) ), respectively.

9 Both modifications of the original pricing domain are thoroughly discussed in Zhu and Chen (2013), pp. 876–879.

10 For example, see Detemple (2005), pp. 55–59, and Jeanblanc et al. (2009), pp. 191–198.

11 We conducted extensive numerical tests and concluded that sensitivities of EPUOP and APUOP options exhibit similar behaviour. Therefore, to facilitate the presentation of our numerical results, we focus on European-style Parisian options in figure 2. To allow for future comparisons, additional numerical results are provided in table E1.

12 We only consider EPUOP options, however similar conclusions can be derived for their American counterparts.

14 This result is intuitive. If the Parisian window shrinks, then it becomes more likely that an APUOP option might be knocked out. As a consequence, the early exercise boundary is pushed higher, i.e. an option holder has an incentive to exercise earlier due to the increased likelihood of the option being cancelled at some point in the future.