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In this paper we obtain a recursive formula for the density of the double-barrier Parisian stopping time. We present a probabilistic proof of the formula for the first few steps of the recursion, and then a formal proof using explicit Laplace inversions. These results provide an efficient computational method for pricing double-barrier Parisian options.

We consider a generalised Vervaat perpetuity of the form X = Y1W1 +Y2W1W2 + · · ·, where $W_i \sim {\cal U}^{1/t}$ and (Yi)i≥0 is an independent and identically distributed sequence of random variables independent from (Wi)i≥0. Based on a distributional decomposition technique, we propose a novel method for exactly simulating the generalised Vervaat perpetuity. The general framework relies on the exact simulation of the truncated gamma process, which we develop using a marked renewal representation for its paths. Furthermore, a special case arises when Yi = 1, and X has the generalised Dickman distribution, for which we present an exact simulation algorithm using the marked renewal approach. In particular, this new algorithm is much faster than existing algorithms illustrated in Chi (2012), Cloud and Huber (2017), Devroye and Fawzi (2010), and Fill and Huber (2010), as well as being applicable to the general payments case. Examples and numerical analysis are provided to demonstrate the accuracy and effectiveness of our method.

In this paper, we derive a variation of the Azéma martingale using two approaches—a direct probabilistic method and another by projecting the Kennedy martingale onto the filtration generated by the drawdown duration. This martingale links the time elapsed since the last maximum of the Brownian motion with the maximum process itself. We derive explicit formulas for the joint densities of (τ,W τ , M τ ), which are the first time the drawdown period hits a prespecified duration, the value of the Brownian motion, and the maximum up to this time. We use the results to price a new type of drawdown option, which takes into account both dimensions of drawdown risk—the magnitude and the duration.

A truncated Lévy subordinator is a Lévy subordinator in R + with Lévy measure restricted from above by a certain level b . In this article, we study the path and distribution properties of this type of process 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, our algorithm outperforms over all parameter settings. Using the distributional 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 is 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.

In this paper, we study the excursions of Bessel and CIR processes with dimensions 0 < δ < 2. We obtain densities for the last passage times and meanders of the processes. Using these results, we prove a variation of the Azéma martingale for the Bessel and CIR processes based on excursion theory. Furthermore, we study their Parisian excursions, and generalise previous results on the Parisian stopping time of Brownian motion to that of the Bessel and CIR processes. We obtain explicit formulas and asymptotic results for the densities of the Parisian stopping times, and develop exact simulation algorithms to sample the Parisian stopping times of Bessel and CIR processes. We introduce a new type of bond, the zero coupon Parisian bond. The buyer of such a bond is betting against zero interest rates, while the seller is effectively hedging against a period where interest rates fluctuate around 0. Using our results, we propose two methods for pricing these bonds and provide numerical examples.

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 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. The advantage of this new method as compared to that in previous literature is that the recursions are easy to program as the resulting formula involves only a finite sum and does not require a numerical inversion of the Laplace transform. For long window periods, an explicit formula for the density of the stopping time can be obtained. For shorter window lengths, we derive a recursive equation from which numerical results are computed. From these results, we compute the prices of one-sided Parisian options.