A truncated Lévy subordinator is a Lévy subordinator in R + with Lévy measure restricted from above by a certain level b. In this paper, we study the path and distribution properties of this type of processes in detail and set up an exact simulation framework based on a marked renewal process. In particular, we focus on a typical specification of truncated Lévy subordinator, namely the truncated stable process. We establish an exact simulation algorithm for the truncated stable process, which is very accurate and efficient. Compared to the existing algorithm suggested in Chi (2012), our algorithm outperforms over all parameter settings. Using a distribution decomposition technique, we also develop an exact simulation algorithm for the truncated tempered stable process and other related processes. We illustrate an application of our algorithm as a valuation tool for stochastic hyperbolic discounting, and numerical analysis are provided to demonstrate the accuracy and effectiveness of our methods. We also show that variations of the result can also be used to sample two-sided truncated Lévy processes, two-sided Lévy processes via subordinating Brownian motions, and truncated Lévy driven Ornstein-Uhlenbeck processes.
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,
In this paper, we obtain a recursive formula for the density of the two-sided Parisian stopping time. This formula does not require any numerical inversion of Laplace transforms, and is similar to the formula obtained for the one-sided Parisian stopping time derived in Dassios and Lim [6]. However, when we study the tails of the two distributions, we find that the two-sided stopping time has an exponential tail, while the one-sided stopping time has a heavier tail. We derive an asymptotic result for the tail of the two-sided stopping time distribution and propose an alternative method of approximating the price of the two-sided Parisian option.
We define the drawdown stopping time of a Brownian motion as the first time its drawdown reaches a duration of length 1. In this paper, we propose an efficient algorithm to efficiently simulate the drawdown stopping time and the associated maximum at this time. The method is straightforward and fast to implement, and avoids simulating sample paths thus eliminating discretisation bias. We show how the simulation algorithm is useful for pricing more complicated derivatives such as multiple drawdown options.
In this paper, we propose a continuous-time stochastic intensity model, namely, two-phase dynamic contagion process (2P-DCP), for modelling the epidemic contagion of COVID-19 and investigating the lockdown effect based on the dynamic contagion model introduced by Dassios and Zhao (2011). It allows randomness to the infectivity of individuals rather than a constant reproduction number as assumed by standard models. Key epidemiological quantities, such as the distribution of final epidemic size and expected epidemic duration, are derived and estimated based on real data for various regions and countries. The associated time lag of the effect of intervention in each country or region is estimated. Our results are consistent with the incubation time of COVID-19 found by recent medical study. We demonstrate that our model could potentially be a valuable tool in the modeling of COVID-19. More importantly, the proposed model of 2P-DCP could also be used as an important tool in epidemiological modelling as this type of contagion models with very simple structures is adequate to describe the evolution of regional epidemic and worldwide pandemic.
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