Fluctuation Identities with Continuous Monitoring and Their Application to Price Barrier Options

Abstract

We present a numerical scheme to calculate fluctuation identities for exponential Lévy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-Laplace domain and require numerical inverse transforms. Thus we cover a gap in the literature that has mainly studied the discrete monitoring case; indeed, there are no existing numerical methods that deal with the continuous case. As a motivating application we price continuously monitored barrier options with the underlying asset modelled by an exponential Lévy process. We perform a detailed error analysis of the method and develop error bounds to show how the performance is limited by the truncation error of the sinc-based fast Hilbert transform used for the Wiener-Hopf factorisation. By comparing the results for our new technique with those for the discretely monitored case (which is in the Fourier-z domain) as the monitoring time step approaches 1 arXiv:1712.00077v1 [q-fin.CP] 30 Nov 2017 zero, we show that the error convergence with continuous monitoring represents a limit for the discretely monitored scheme.

American step options

“…See also Grant, Vora, & Weeks (1997), Davidov & Linetsky (2001) and Kaishev & Dimitrova (2009), among others, for the family of path-dependent options. The pricing of exotic options such as barrier options is an important topic that is examined in the operations research literature; see e.g., Kou (2007), Feng & Linetsky (2008), Cai, Chen, & Wan (2009), Dingec & Hörmann (2012), Giesecke & Smelov (2013), Jin, Li, Tan, & Wu (2013), Wang & Tan (2013), Sesana, Marazzina, & Fusai (2014), Date & Islyaev (2015), Fusai, Germano, & Marazzina (2016), Phelan, Marazzina, Fusai, & Germano (2018). 2 Step options belong to the class of occupation time derivatives.…”

Pricing methods for α-quantile and perpetual early exercise options based on Spitzer identities

“…Similarly to the discrete time case, the inverse transform is not generally available in closed form and so we use the numerical inverse Laplace transform by Abate and Whitt (1995), used for option pricing by Phelan et al (2018a). The relationship between the forward Laplace and z transforms is…”

“…This relationship can be exploited to derive versions of the fluctuation identities with continuous monitoring (Baxter and Donsker, 1957;Green et al, 2010;Fusai et al, 2016;Phelan et al, 2018a).…”

“…As described by Green et al (2010), and implemented in Phelan et al (2018a), we can exploit the relationship between Laplace and z-transforms described in Eq. ( 9) to extend the pricing methods from discrete to continuous monitoring (i.e.…”

Pricing discretely-monitored double barrier options with small probabilities of execution

“…The main advantage of MCS over other pricing methods is its model-free property and its non-dependence on the dimension N of the approximated equation. The latter is an important property since as N → ∞ (∆t → 0), the price of a discretely monitored barrier option converges to that of a continuously monitored one (Broadie et al, 1997;Phelan et al, 2018a). However, MCS has a serious drawback, as high volatility makes it difficult for the asset to remain within barriers -especially when the gap between them is small-which in turn, makes a positive payoff a rare event (Glasserman et al, 1999).…”