Recursive formula for the double-barrier Parisian stopping time

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

Parisian excursions of Brownian motion and their applications in mathematical finance

In this thesis, we study Parisian excursions, which are defined as excursions of Brownian motion above or below a pre-determined barrier, exceeding a certain time length. Employing a new method, a recursion formula for the densities of single barrier and double barrier Parisian stopping times are computed. This new approach allows us to obtain a semi-closed form solution for the density of the one-sided stopping times, and does not require any numerical inversions of Laplace transforms. Further, it is backed by an intuitive argument which is premised on the recursive nature of the excursions and the strong Markov property of the Brownian motion. The same method is also employed in our computation of the two-sided and the double barrier Parisian stopping times. In turn, the resultant densities are used to price Parisian options. In particular, we provide numerical expressions for down-and-in Parisian calls. Additionally, we study the tail of the distribution of the two-sided Parisian stopping time. Based on the asymptotic properties of its distribution, we propose an approximation for the option prices, alleviating the heavy computational load arising from the recursions. Finally, we use the infinitesimal generator to obtain several results on other variations of Parisian excursions. Specifically, apart from the length, we are interested in the number of excursions and the maximum height achieved during an excursion. Using the same generator, we derive the joint Laplace transform of the occupation times of the Brownian motion above and below zero, but only starting the clock each time after a certain length

Parisian Option Pricing: A Recursive Solution for the Density of the Parisian Stopping Time

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

Parisian times, Bessel processes and Poisson-Dirichlet random variables

In this thesis, we study the first hitting time and Parisian time of Brownian motion and squared Bessel process, as well as the exact simulation algorithm of the two-parameter Poisson-Dirichlet distribution. Let the underlying process be a reflected Brownian motion with drift moving on a finite collection of rays. Using a recursive method, we derive the Laplace transform of the first hitting time of the underlying process. This generalises the well-known result about the first hitting time of a Walsh Brownian motion on spider. We also invert the Laplace transform explicitly using two different methods, and obtain the density and distribution functions of the first hitting time. Then we consider the Parisian time of the underlying process, which is defined as the first exceeding time of the excursion time length. Using the same recursive method, we derive the Laplace transform of the Parisian time. The exact simulation algorithm for the Parisian time is also proposed. Next, we extend the result to the Parisian time of a squared Bessel process with a linear excursion boundary. Based on a variation of the Azéma martingale, we obtain the distributional properties of the Parisian time, and design the algorithm for sampling from the Parisian time. Finally, as an extension of the simulation of the Parisian time, we propose two exact simulation algorithms for sampling from the two-parameter Poisson-Dirichlet distribution

Pricing Step Options under the CEV and other Solvable Diffusion Models

We consider a special family of occupation-time derivatives, namely proportional step options introduced by Linetsky in [Math. Finance, 9, 55--96 (1999)]. We develop new closed-form spectral expansions for pricing such options under a class of nonlinear volatility diffusion processes which includes the constant-elasticity-of-variance (CEV) model as an example. In particular, we derive a general analytically exact expression for the resolvent kernel (i.e. Green's function) of such processes with killing at an exponential stopping time (independent of the process) of occupation above or below a fixed level. Moreover, we succeed in Laplace inverting the resolvent kernel and thereby derive newly closed-form spectral expansion formulae for the transition probability density of such processes with killing. The spectral expansion formulae are rapidly convergent and easy-to-implement as they are based simply on knowledge of a pair of fundamental solutions for an underlying solvable diffusion process. We apply the spectral expansion formulae to the pricing of proportional step options for four specific families of solvable nonlinear diffusion asset price models that include the CEV diffusion model and three other multi-parameter state-dependent local volatility confluent hypergeometric diffusion processes.Comment: 30 pages, 16 figures, submitted to IJTA

Brownian excursions in mathematical finance

The Brownian excursion is defined as a standard Brownian motion conditioned on starting and ending at zero and staying positive in between. The first part of the thesis deals with functionals of the Brownian excursion, including first hitting time, last passage time, maximum and the time it is achieved. Our original contribution to knowledge is the derivation of the joint probability of the maximum and the time it is achieved. We include a financial application of our probabilistic results on Parisian default risk of zero-coupon bonds. In the second part of the thesis the Parisian, occupation and local time of a drifted Brownian motion is considered, using a two-state semi-Markov process. New versions of Parisian options are introduced based on the probabilistic results and explicit formulae for their prices are presented in form of Laplace transforms. The main focus in the last part of the thesis is on the joint probability of Parisian and hitting time of Brownian motion. The difficulty here lies in distinguishing between different scenarios of the sample path. Results are achieved by the use of infinitesimal generators on perturbed Brownian motion and applied to innovative equity exotics as generalizations of the Barrier and Parisian option with the advantage of being highly adaptable to investors’ beliefs in the market

On the Dual Risk Models

Abstract This thesis focuses on developing and computing ruin-related quantities that are potentially measurements for the dual risk models which was proposed to describe the annuity-type businesses from the perspective of the collective risk theory in 1950’s. In recent years, the dual risk models are revisited by many researchers to quantify the risk of the similar businesses as the annuity-type businesses. The major extensions included in this thesis consist of two aspects: the first is to search for new ruin-related quantities that are potentially indices of the risk for well-established dual models; the other aspect is to generalize the settings of the dual models instead of the ruin quantities. There are four separate articles in this thesis, in which the first (Chapter 2) and the last (Chapter 5) belong to the first type of extensions while the others (Chapter 3 and Chapter 4) belong to the generalizations of the dual models. The first article (Chapter 2) studies the discounted moments of the surplus at the time of the last jump before ruin for the compound Poisson dual risk model. The idea comes from that the ruin of the compound Poisson dual models is caused by absence of positive jumps within a period with length being propotional to the surplus at the time of the last jump. As a quantity related to a non-stopping time, the explicit expression of the target quantity is obtained through integro-differential equations. The second article (Chapter 3) investigate the Sparre-Andersen dual risk models in which the epochs are independently, identically distributed generalized Erlang-n random variables. An important difference between this model and some other models such as the Erlang-n dual risk models is that the roots to the generalized Lundberg’s equation are not necessarily distinct. By taking the multiple roots into account, the explicit expressions of the Laplace transform of the time to ruin and expected discounted aggregate dividends under the threshold strategy and exponential distributed revenues are derived. The third article (Chapter 4) revisits the the dual Lévy risk model. The target ruin quantity is the expected discounted aggregate dividends paid up to ruin under the threshold dividend strategy. The explicit expression is obtained in terms of the q-scale functions through constructing a new dividend strategy having the target ruin quantity converging to that under the threshold strategy. Also, the optimality of the threshold strategy among all the absolutely continuous stategies when evaluating the target quantity as a value function is discussed. The fourth article (Chapter 5) initiate the study of the Parisian ruin problem for the general dual Lévy risk models. Unlike the regular ruin for the dual models, the deficit at Parisian ruin is not necessarily equal to zero. Hence we introduce the Gerber-Shiu expected discounted penalty function (EPDF) at the Parisian ruin and obtain an explicit expression for this function. Keywords: Sparre-Andersen dual models, expected discounted aggregate dividends, dual Levy risk models, Parisian ruin, Gerber-Shiu function ii