A General Approach for Parisian Stopping Times under Markov Processes

Consider a stochastic process $X$ and define

for a given level $L$. The ages of excursion of $X$ below and above $L$ are given by $t-g^{-}_{L,t}$ and $t-g^{+}_{L,t}$, respectively. Consider the following two stopping times:

which are called Parisian stopping times. $\tau_{L,D}^{+}$ ($\tau_{L,D}^{-}$) is the first time that the age of excursion of $X$ above (below) $L$ reaches $D$. In these definitions, we use the convention that $\sup\emptyset=0$ and $\inf\emptyset=\infty$.

Parisian stopping times appear in important finance and insurance applications. Parisian options are proposed in Chesney et al. (1997) as an alternative to barrier options and they have the advantage of being less prone to market manipulation (Labart (2010)). These options are activated or cancelled if the process stays long enough below or above a prespecified level. In the insurance literature, the Parisian stopping time for the excursion below the solvency level is used to define the ruin event (e.g., Dassios and Wu (2008), Czarna and Palmowski (2011), Loeffen et al. (2013)).

There is an extensive literature on the study of Parisian stopping times and their applications in finance and insurance with many papers focused on specific models. For the Brownian motion and the geometric Brownian motion model, see Chesney et al. (1997), Bernard et al. (2005), Dassios and Wu (2010), Dassios and Wu (2011), Zhu and Chen (2013), Dassios and Lim (2013), Dassios and Zhang (2016), Le et al. (2016), Dassios and Lim (2017) and Dassios and Lim (2018). For models with jumps, methods for Parisian option pricing were developed in Albrecher et al. (2012) for Kou’s double-exponential jump-diffusion model and in Chesney and Vasiljević (2018) for the hyper-exponential jump-diffusion model. Czarna and Palmowski (2011) and Loeffen et al. (2013) derived the ruin probability for spectrally negative Lévy processes. In addition, Chesney and Gauthier (2006), Le et al. (2016) and Lu et al. (2018) studied American-style Parisian option pricing under the Black-Scholes model while Chesney and Vasiljević (2018) dealt with the hyper-exponential jump-diffusion model. As for the hedging of Parisian options, Kim and Lim (2016) developed a quasi-static strategy and tested it under the double-exponential jump-diffusion model.

Several numerical methods have also been applied to the Parisian option pricing problem. Baldi et al. (2000) developed a Monte Carlo simulation method based on large deviation estimates for general diffusions. For the Black-Scholes model, Avellaneda and Wu (1999) proposed a trinomial tree approach while Haber et al. (1999) developed an explicit finite difference scheme to numerically solve the pricing PDE, which is not unconditionally stable. These two methods can also be applied to general diffusions, but they need to incorporate the current length of excursion as an additional dimension. Furthermore, they may require a very small time step to achieve a target accuracy level, making them inefficient. If the model contains jumps, the lattice method and the finite difference approach will face more challenges for accuracy and efficiency.

In this paper, we aim to develop a computationally efficient approach for Parisian stopping time problems that is applicable to general Markov models. We consider a one-dimensional (1D) Markov process $X$ living on an interval $I\subseteq\mathbb{R}$ with its infinitesimal generator $\mathcal{G}$ given by

for $f\in C_{c}^{2}(I)$ (the space of twice continuously differentiable functions on $I$ with compact support) with $\int_{\mathbb{R}}(z^{2}\wedge 1)\nu(x,dz)<\infty$. The quantities $\mu(x)$, $\sigma(x)$, $\nu(x,dz)$ are known as the drift, volatility and jump measure of $X$, respectively. We would like to obtain the distribution of $\tau_{L,D}^{-}$ and $\tau_{L,D}^{+}$, i.e., $\mathbb{P}_{x}[\tau_{L,D}^{\pm}\leq t]$. The Laplace transform of the probability as a function of $t$ is given by

For a general Markov process, there is no analytical solution for this transform. Our idea is approximating $X$ by a continuous time Markov chain (CTMC). We will show later that the Laplace transform of $\tau_{L,D}^{\pm}$ under a CTMC model is the solution to a linear system which involves the distribution and the Laplace transform of a first passage time of the CTMC, and this solution can be obtained in closed form. To recover the distribution of $\tau_{L,D}^{\pm}$, we simply invert its Laplace transform numerically using the efficient algorithm of Abate and Whitt (1992). Using the result for $\tau_{L,D}^{\pm}$, we can compute the Laplace transfrom of a Parisian option price as a function of maturity in closed form for a CTMC model. We can further solve a range of Parisian problems under a CTMC model such as the distribution of the multi-sided Parisian stopping time, the joint distribution of a Parisian stopping time and a first passage time and the pricing of Parisian bonds. For more complex models like regime-switching models and stochastic volatility models, we can also approximate them by a CTMC and solve the Parisian stopping time problems. Thus, our approach is very general and versatile. Numerical experiments for various popular models further demonstrate the computational efficiency of our method, which can outpeform traditional numerical methods significantly.

Our paper fits into a burgeoning literature on CTMC approximation for option pricing. Mijatović and Pistorius (2013) used it for pricing European and barrier options under general 1D Markov models. They showed that the barrier option price can be obtained in closed form using matrix exponentials under a CTMC model. Sharp converegnce rate analysis of their algorithm for realistic option payoffs is a challenging problem. Answers are provided for diffusion models first in Li and Zhang (2018) for uniform grids and then in Zhang and Li (2021a)) for general non-uniform grids while the problem still remains open for general Markov models with jumps. For diffusions, Zhang and Li (2021a) also developed principles for designing grids to achieve second order convergence whereas the algorithm only converges at first order in general. The idea of CTMC approximation has also been exploited to develop efficient methods for pricing various types of options for 1D Markov models: Eriksson and Pistorius (2015) for American options, Cai et al. (2015), Song et al. (2018), Cui et al. (2018) for Asian options, and Zhang and Li (2021b) and Zhang et al. (2021) for maximum drawdown options. Furthermore, CTMC approximation can be extended to deal with more complex models where the asset price follows a regime-switching process (Cai et al. (2019)) or exhibits stochastic volatility (Cui et al. (2017, 2018, 2019)).

One can view the present paper as a generalization of Mijatović and Pistorius (2013) and Zhang and Li (2019) from barrier options to Parisian options. However, this generalization is by no means trivial. First, the Parisian problem is much more complex than the first passage problem as it involves the excursion of the process. Second, there is considerable challenge in sharp convergence rate analysis. Even for barrier options, the sharp convergence rate of CTMC approximation is only known for diffusion models, and the difficulty mainly stems from the lack of smoothness in the option payoffs. In this paper, we focus on analyzing the convergence rate of our algorithm for Parisian problems under diffusion models. We use the spectral analysis framework in Zhang and Li (2021a), but there is an important difference which makes the analysis for Parisian problems much harder. Here, we must carefully analyze the dependence of solutions to Sturm-Liouville and elliptic boundary value problems on variable boundary levels whereas in the barrier option problem the boundaries are fixed. Furthermore, we need to establish some new identities that enable us to obtain the sharp rate. Our analysis not only yields the sharp estimate of the convergence rate, but also unveils how the grid of the CTMC should be designed to achieve fast convergence.

For the sake of simplicity, our discussions in the main text will be focused on the algorithms for single-barrier Parisian stopping times and the down-and-in Parisian option as well as their convergence analysis. Extensions for other cases are provided in the appendix. The rest of this paper is organized as follows. In Section 2, we derive the Laplace transforms of Parisian stopping times and Parisian option prices under a general CTMC model with finite number of states. Section 3 reviews how to construct a CTMC to approximate a general jump-diffusion and establishes the convergence of this approximation for the Parisian problems. Section 4 analyzes the convergence rate for diffusion models and proposes grid design principles to ensure second order convergence. Section 5 shows numerical results for various models and Section 6 concludes the paper. In Appendix A, we extend our method to deal with multi-sided Parisian stopping times, the joint distribution of Parisian stopping times and first passage times, Parisian bonds and the Parisian problem for regime-switching and stochastic volatility models. Proofs for the convergence rate analysis are collected in Appendix B.

Consider a CTMC denoted by $Y$ which lives in the state space $\mathbb{S}_{n}=\{y_{0},y_{1},\cdots,y_{n}\}$. The transition rate matrix of $Y$ is denoted by ${\boldsymbol{G}}$ with ${\boldsymbol{G}}(x,y)$ indicating the transition rate from state $x$ to $y$, $x\neq y$ and ${\boldsymbol{G}}(x,x)=-\sum_{y\in\mathbb{S}_{n}\backslash\{x\}}{\boldsymbol{G}}(x,y)$. The action of $\boldsymbol{G}$ on a function $g:\mathbb{S}_{n}\to\mathbb{R}$ is defined as

For any $\xi$, we also introduce the notations

which are the CTMC states that are the right and left neighbor of $\xi$, respectively. We also define $L^{+}=\min\{z\geq L:z\in\mathbb{S}_{n}\}$. Note that $L^{+}$ is different from $\xi^{+}$ in that $L^{+}=L$ if $L\in\mathbb{S}_{n}$, but we always have $\xi^{+}>\xi$. We slightly abuse the notation here to avoid introducing a new one.

In this section, we first derive the Laplace transform of Parisian stopping times and then the Laplace transform of the price of a down-and-in Parisian option.