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This chapter reviews recent developments in the density discontinuity approach. It is well known that agents having perfect control of the forcing variable will invalidate the popular regression discontinuity designs (RDDs). To detect the manipulation of the forcing variable, McCrary (2008) developed a test based on the discontinuity in the density around the threshold. Recent papers have noted that the sorting patterns around the threshold are often either the researcher’s object of interest or may relate to structural parameters such as tax elasticities through known functions. This, in turn, implies that the behavior of the distribution around the threshold is not only informative of the validity of a standard RDD; it can also be used to recover policy-relevant parameters and perform counterfactual exercises.

The purpose of this paper is to carry out a finite element simulation of a physically non-linear phase change problem in a two-dimensional space without adaptive remeshing or moving-mesh algorithms. The extended finite element method (XFEM) and the level set method (LSM) were used to capture the transient solution and motion of phase boundaries. It was crucial to consider the effects of unequal densities of the solid and liquid phases and the flow in the liquid region.

The XFEM and the LSM are applied to solve non-linear transient problems with a phase change in a two-dimensional space. The model assumes thermo-dependent properties of the material and unequal densities of the phases; it also allows for convection in the liquid phase. A non-linear system of equations is derived and a numerical solution is proposed. The Newton-Raphson method is used to solve the problem and the LSM is applied to track the interface.

The robustness and utility of the method are demonstrated on several two-dimensional benchmark problems.

The novel procedure based on the XFEM and the LSM was developed to solve physically non-linear phase change problems with unequal densities of phases in a two-dimensional space.

This paper aims to extend the hybrid atomistic-continuum multiscale method developed by Vu et al. (2016) to study the gas flow problems in long microchannels involving density variations.

The simulation domain is decomposed into three regions: the bulk where the continuous Navier–Stokes and energy equations are solved, the neighbourhood of the wall simulated by molecular dynamics and the overlap region which connects the macroscopic variables (density, velocity and temperature) between the two former regions. For the simulation of long micro/nanochannels, a strategy with multiple molecular blocks all along the fluid/solid interface is adopted to capture accurately the macroscopic velocity and temperature variations.

The validity of the hybrid method is shown by comparisons with a simplified analytical model in the molecular region. Applications to compressible and condensation problems are also presented, and the results are discussed.

The hybrid method proposed in this paper allows cost-effective computer simulations of large-scale problems with an accurate modelling of the transfers at small scales (velocity slip, temperature jump, thin condensation films, etc.).

Multiphase flow computations involve coupled momentum, mass and energy transfer between moving and irregularly shaped boundaries, large property jumps between materials, and computational stiffness. In this study, we focus on the immersed boundary technique, which is a combined Eulerian‐Lagrangian method, to investigate the performance improvement using the multigrid technique in the context of the projection method. The main emphasis is on the interplay between the multigrid computation and the effect of the density and viscosity ratios between phases. Two problems, namely, a rising bubble in a liquid medium and impact dynamics between a liquid drop and a solid surface are adopted. As the density ratio increases, the single grid computation becomes substantially more time‐consuming; with the present problems, an increase of factor 10 in density ratio results in approximately a three‐fold increase in CPU time. Overall, the multigrid technique speeds up the computation and furthermore, the impact of the density ratio on the CPU time required is substantially reduced. On the other hand, the impact of the viscosity ratio does not play a major role on the convergence rates.

The purpose of this paper is to propose a model for ruin‐contingent life annuity (RCLA) contracts under a jump diffusion model, where the dynamics of volatility is governed by the Heston stochastic volatility framework. The paper aims to illustrate that the proposed jump diffusion process, for both asset price and stochastic volatility, will provide a more realistic pricing model for RCLA contracts in comparison to existing models.

Under the assumption of the deterministic withdrawals, the authors use a partial integro differential equation (PIDE) approach to develop the pricing scheme for the fair value of the lump sum charges of RCLA contracts. Consequently, the authors employ an elegant numerical scheme, finite difference method, for solving the PIDEs for the reference portfolio, as well as the volatility. The findings show that a different pricing behaviour of the RCLA contracts under the authors' model parameters is obtained compared to that in the existing literature.

RCLA pricing in the complete market often underestimates the jump risk and the persistent factor in the volatility process. The authors' generalized model shows how these two random sources of risks can be precisely characterized.

The parameter values used in the numerical analysis require more empirical evidence. Hence, in order for more precise pricing practice, the calibration from real data is needed.

The model, as adopted in this study, for pricing of RCLA contracts should provide a general guideline for the commercialization of this product by insurance companies.

The demand for RCLA contracts as an alternative to the commonly‐practised annuitization option has recently increased, rapidly, among the soon‐to‐retire baby boomers. This paper investigates the fair value of this particular product, which could be beneficial to researchers for a better understanding of the product design.

This is the first research paper which prices the RCLA contracts in the incomplete market. The gap between RCLA contract pricing and studies of jump diffusion models for derivative pricing, in the literature, is therefore filled.

This paper empirically investigates the usefulness of extreme events implied into the non-complete option market in which return generating process of underlying asset is different from that of options. The empirical results find that the information about the extreme events implied in the option market prices has more accurate forecasting power within the tail than near the first moment of realized distribution. So, we expect that the implied information of extreme jump can help to improve the back-testing performance of value at risk where it is primarily important to take account of low-probability events. Regardless of whether calibration function for density transformation is the beta-distribution or non-parametric kernel density, extreme jump provides consistently satisfactory predictions.

The purpose of the present article is to obtain the similarity solution for the shock wave generated by a piston propagating in a self-gravitating nonideal gas under the impact of azimuthal magnetic field for adiabatic and isothermal flows.

The Lie group theoretic method given by Sophus Lie is used to obtain the similarity solution in the present article.

Similarity solution with exponential law shock path is obtained for both ideal and nonideal gas cases. The effects on the flow variables, density ratio at the shock front and shock strength by the variation of the shock Cowling number, adiabatic index of the gas, gravitational parameter and nonidealness parameter are investigated. The shock strength decreases with an increase in the shock Cowling number, nonidealness parameter and adiabatic index, whereas the strength of the shock wave increases with an increase in gravitational parameter.

Propagation of shock wave with spherical geometry in a self-gravitating nonideal gas under the impact of azimuthal magnetic field for adiabatic and isothermal flows has not been studied by any author using the Lie group theoretic method.

Due to disequilibrium between supply and demand in the option market, the option market‐maker is under exposure to certain risks because of their net option positions. This paper aims to pay attention to whether the risk award affects the option price and the shape of implied volatility in the market‐maker system.

The paper first eliminates the part of implied volatility explained by underlying asset's stochastic volatility‐jump price process, and second sorts out market investors' net demand data from TAIEX Options tick by tick deal data and then finally considers three market maker's risks – unhedgeable risk, capital constrain risk and asymmetric information risk, and how they affect implied volatility's level and slope.

Through the research in the TAIEX Option market, the paper finds that, under unhedgeable risk, net demand pressure has a significant impact on implied volatility. Especially, unhedgeable risk due to underlying asset's stochastic volatility has the best explanation for implied volatility level, and unhedgeable risk due to underlying asset's jump can explain implied volatility slope to some extent. Capital constrain risk and asymmetric information risk have an insignificant impact on implied volatility.

The findings in this study suggest that the risk award affects the option price and the shape of implied volatility in the market‐maker system and different risks have different effects on the level and slope of option implied volatility.

This paper finds the influence factors of the option price in the market‐maker system. It's useful for China's financial government and investors to learn the price tendency and regular pattern in the future China option market.

This is the first time that a net demand pressure based option pricing model is used, which is derived by Garleanu, Pedersen and Poteshman, to study the TAIEX Options' implied volatility. And the paper improves the methods eliminating the part of implied volatility explained by underlying asset's stochastic volatility‐jump price process.

The purpose of this paper is to solve the optimal dynamic portfolio problem under the double-exponential jump diffusion (DEJD) distribution, which can allow the asset returns to jump asymmetrically.

The authors solve the problem by solving the HJB equation. Meanwhile, in the presence of jump component in the asset returns, the investor may suffer a large loss due to high leveraged position, so the authors impose the short-sale and borrowing constraints when solving the optimization problem.

The authors provide sufficient conditions such that the optimal solution exists and show theoretically that the optimal risky asset weight is an increasing function of jump-up probability and average jump-up size and a decreasing function of average jump-down size.

In this study, the authors assume that the jump-up and jump-down intensities are constant. In the future, the authors will relax the assumption and allows the jump intensities to be time varying.

Empirical studies based on Chinese Shanghai stock index data show that the jump distribution of Shanghai index returns is asymmetric, and the DEJD model can fit the data better than the log-normal jump-diffusion model. The numerical results are consistent with the theoretical prediction, and the authors find that the less risk-averse investor will suffer more economic cost if ignoring asymmetric jump distribution.

This study first examines how asymmetric jumps affect the investor’s portfolio allocation.