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In this paper, we propose an integral equation approach for pricing an American-style Parisian up-and-out call option under the Black-Scholes framework. The main difficulty of pricing this option lies in the determination of its optimal exercise price, which is a three-dimensional surface, instead of a two-dimensional (2-D) curve as is the case for a "one-touch" barrier option. In our approach, we first reduce the 3-D pricing problem to a 2-D one by using the "moving window" technique developed by Zhu and Chen (2013, Pricing Parisian and Parasian options analytically. Journal of Economic Dynamics and Control, 37(4): 875-896), then apply the Fourier sine transform to the 2-D problem to obtain two coupled integral equations in terms of two unknown quantities: the option price at the asset barrier and the optimal exercise price. Once the integral equations are solved numerically by using an iterative procedure, the calculation of the option price and the hedging parameters is straightforward from their integral representations. Our approach is validated by a comparison between our results and those of the trusted finite difference method. Numerical results are also provided to show some interesting features of the prices of American-style Parisian up-and-out call options and the behaviour of the associated optimal exercise boundaries.

In this paper, we propose an integral equation approach for pricing an American-style Parisian up-and-out call option under the Black-Scholes framework. The main difficulty of pricing this option lies in the determination of its optimal exercise price, which is a three-dimensional surface, instead of a two-dimensional (2-D) curve as is the case for a "one-touch" barrier option. In our approach, we first reduce the 3-D pricing problem to a 2-D one by using the "moving window" technique developed by Zhu and Chen (2013, Pricing Parisian and Parasian options analytically. Journal of Economic Dynamics and Control, 37(4): 875-896), then apply the Fourier sine transform to the 2-D problem to obtain two coupled integral equations in terms of two unknown quantities: the option price at the asset barrier and the optimal exercise price. Once the integral equations are solved numerically by using an iterative procedure, the calculation of the option price and the hedging parameters is straightforward from their integral representations. Our approach is validated by a comparison between our results and those of the trusted finite difference method. Numerical results are also provided to show some interesting features of the prices of American-style Parisian up-and-out call options and the behaviour of the associated optimal exercise boundaries.

In this paper, we propose an integral equation approach for pricing an American-style Parisian up-and-out call option under the Black-Scholes framework. The main difficulty of pricing this option lies in the determination of its optimal exercise price, which is a three-dimensional surface, instead of a two-dimensional (2-D) curve as is the case for a "one-touch" barrier option. In our approach, we first reduce the 3-D pricing problem to a 2-D one by using the "moving window" technique developed by Zhu and Chen (2013, Pricing Parisian and Parasian options analytically. Journal of Economic Dynamics and Control, 37(4): 875-896), then apply the Fourier sine transform to the 2-D problem to obtain two coupled integral equations in terms of two unknown quantities: the option price at the asset barrier and the optimal exercise price. Once the integral equations are solved numerically by using an iterative procedure, the calculation of the option price and the hedging parameters is straightforward from their integral representations. Our approach is validated by a comparison between our results and those of the trusted finite difference method. Numerical results are also provided to show some interesting features of the prices of American-style Parisian up-and-out call options and the behaviour of the associated optimal exercise boundaries.

In this paper, we propose an integral equation approach for pricing an American-style Parisian up-and-out call option under the Black-Scholes framework. The main difficulty of pricing this option lies in the determination of its optimal exercise price, which is a three-dimensional surface, instead of a two-dimensional (2-D) curve as is the case for a "one-touch" barrier option. In our approach, we first reduce the 3-D pricing problem to a 2-D one by using the "moving window" technique developed by Zhu and Chen (2013, Pricing Parisian and Parasian options analytically. Journal of Economic Dynamics and Control, 37(4): 875-896), then apply the Fourier sine transform to the 2-D problem to obtain two coupled integral equations in terms of two unknown quantities: the option price at the asset barrier and the optimal exercise price. Once the integral equations are solved numerically by using an iterative procedure, the calculation of the option price and the hedging parameters is straightforward from their integral representations. Our approach is validated by a comparison between our results and those of the trusted finite difference method. Numerical results are also provided to show some interesting features of the prices of American-style Parisian up-and-out call options and the behaviour of the associated optimal exercise boundaries.

We imaged Comet 252P/2000 G1 (LINEAR) (hereafter 252P) with the Hubble Space Telescope and both 252P and P/2016 BA$_{14}$ (PanSTARRS) (hereafter BA$_{14}$) with the Discovery Channel Telescope in March and April 2016, surrounding its close encounter to Earth. The r'-band $Af\rho$ of 252P in a 0.2"-radius aperture were $16.8\pm0.3$ and $57\pm1$ cm on March 14 and April 4, respectively, and its gas production rates were: $Q$(OH) = $(5.8\pm0.1)\times10^{27}$ s$^{-1}$, and $Q$(CN) = $(1.25\pm0.01)\times10^{25}$ s$^{-1}$ on April 17. The r'-band upper limit $Af\rho$ of BA1$_{14}$ was $0.19\pm0.01$ cm in a 19.2"-radius aperture, and $Q$(CN) = $(1.4\pm0.1)10^{22}$ s$^{-1}$ on April 17, 2017. 252P shows a bright and narrow jet of a few hundred kilometers long in the sunward direction, changing its projected position angle in the sky with a periodicity consistent with 7.24 hours. However, its photometric lightcurve is consistent with a periodicity of 5.41 hours. We suggest that the nucleus of 252P is likely in a non-principal axis rotation. The nucleus radius of 252P is estimated to be about $0.3\pm0.03$ km, indicating an active fraction of 40% to >100% in its 2016 apparition. Evidence implies a possible cloud of slow-moving grains surrounding the nucleus. The activity level of 252P in the 2016 apparition increased by two orders of magnitude from its previous apparitions, making this apparition unusual. On the other hand, the activity level of BA14 appears to be at least three orders of magnitude lower than that of 252P, despite its ten times or larger surface area., Comment: 31 pages, 15 figures, 4 tables, accepted by AJ

We imaged Comet 252P/2000 G1 (LINEAR) (hereafter 252P) with the Hubble Space Telescope and both 252P and P/2016 BA$_{14}$ (PanSTARRS) (hereafter BA$_{14}$) with the Discovery Channel Telescope in March and April 2016, surrounding its close encounter to Earth. The r'-band $Af\rho$ of 252P in a 0.2"-radius aperture were $16.8\pm0.3$ and $57\pm1$ cm on March 14 and April 4, respectively, and its gas production rates were: $Q$(OH) = $(5.8\pm0.1)\times10^{27}$ s$^{-1}$, and $Q$(CN) = $(1.25\pm0.01)\times10^{25}$ s$^{-1}$ on April 17. The r'-band upper limit $Af\rho$ of BA1$_{14}$ was $0.19\pm0.01$ cm in a 19.2"-radius aperture, and $Q$(CN) = $(1.4\pm0.1)10^{22}$ s$^{-1}$ on April 17, 2017. 252P shows a bright and narrow jet of a few hundred kilometers long in the sunward direction, changing its projected position angle in the sky with a periodicity consistent with 7.24 hours. However, its photometric lightcurve is consistent with a periodicity of 5.41 hours. We suggest that the nucleus of 252P is likely in a non-principal axis rotation. The nucleus radius of 252P is estimated to be about $0.3\pm0.03$ km, indicating an active fraction of 40% to >100% in its 2016 apparition. Evidence implies a possible cloud of slow-moving grains surrounding the nucleus. The activity level of 252P in the 2016 apparition increased by two orders of magnitude from its previous apparitions, making this apparition unusual. On the other hand, the activity level of BA14 appears to be at least three orders of magnitude lower than that of 252P, despite its ten times or larger surface area., Comment: 31 pages, 15 figures, 4 tables, accepted by AJ

In this paper, a completely new integral equation for the price of an American put option as well as its optimal exercise price is successfully derived. Compared to existing integral equations for pricing American options, the new integral formulation has two distinguishable advantages: (i) it is in a form of one-dimensional integral, and (ii) it is in a form that is free from any discontinuity and singularities associated with the optimal exercise boundary at the expiry time. These rather unique features have led to a significant enhancement of the computational accuracy and efficiency as shown in the examples.

We derive an analytical solution for the value of Parisian up-and-in calls by using the "moving window" technique for pricing European-style Parisian up-and-out calls. Our pricing formula can be applied to both European-style and American-style Parisian up-and-in calls, due to the fact that with an "in" barrier, the option holder cannot do or decide on anything before the option is activated, and once the option is activated it is just a plain vanilla call, which could be of American style or European style.

In this paper, we formulate margin call stock loans in finite maturity as American down-and-out calls with rebate and time-dependent strike. The option problem is solved semi-analytically based on the approach in Zhu (2006). An explicit equation for optimal exit price and a pricing formula for loan value are obtained in Laplace space. Final results are obtained by numerical inversion. Examples are provided to show the dependency of the optimal exit price and margin call stock loan value on various parameters.

In this paper, an integral equation approach is adopted to price American-style down-and-out calls. Instead of using the probability theory as used in the literature, we use the continuous Fourier sine transform to solve the partial differential equation system governing the option prices. As a way of validating our approach, we show that the "early exercise premium representation" for American-style down-and-out calls without rebate can be re-derived by using our approach. We then examine the case that time-dependent rebates are included in the contract of American-style down-and-out calls. As a result, a more general integral representation for the price of an American-style down-and-out call, with the presence of an extra term associated with the rebate, is obtained. Our numerical method based on the newly-derived integral representation appears to be efficient in computing the price and the hedging parameters for American-style down-and-out calls with rebates. In addition, significant effects of rebates on the option prices and the optimal exercise boundaries are illustrated through selected numerical results.

In this paper we study stock loans of finite maturity with different dividend distributions semi-analytically using the analytical approximation method in Zhu (2006). Stock loan partial differential equations (PDEs) are established under Black-Scholes framework. Laplace transform method is used to solve the PDEs. Optimal exit price and stock loan value are obtained in Laplace space. Values in the original time space are recovered by numerical Laplace inversion. To demonstrate the efficiency and accuracy of our semi-analytic method several examples are presented, the results are compared with those calculated using existing methods. We also present a calculation of fair service fee charged by the lender for different loan parameters.

In this paper, we price American-style Parisian down-and-in call options under the Black-Scholes framework. Usually, pricing an American-style option is much more difficult than pricing its European-style counterpart because of the appearance of the optimal exercise boundary in the former. Fortunately, the optimal exercise boundary associated with an American-style Parisian knock-in option only appears implicitly in its pricing partial differential equation (PDE) systems, instead of explicitly as in the case of an American-style Parisian knock-out option. We also recognize that the "moving window" technique developed by Zhu and Chen (2013) for pricing European-style Parisian up-and-out call options can be adopted to price American-style Parisian knock-in options as well. In particular, we obtain a simple analytical solution for American-style Parisian down-and-in call options and our new formula is written in terms of four double integrals, which can be easily computed numerically.

In this paper we study stock loans of finite maturity with different dividend distributions semi-analytically using the analytical approximation method in Zhu (2006). Stock loan partial differential equations (PDEs) are established under Black-Scholes framework. Laplace transform method is used to solve the PDEs. Optimal exit price and stock loan value are obtained in Laplace space. Values in the original time space are recovered by numerical Laplace inversion. To demonstrate the efficiency and accuracy of our semi-analytic method several examples are presented, the results are compared with those calculated using existing methods. We also present a calculation of fair service fee charged by the lender for different loan parameters.

In this paper, we price American-style Parisian down-and-in call options under the Black-Scholes framework. Usually, pricing an American-style option is much more difficult than pricing its European-style counterpart because of the appearance of the optimal exercise boundary in the former. Fortunately, the optimal exercise boundary associated with an American-style Parisian knock-in option only appears implicitly in its pricing partial differential equation (PDE) systems, instead of explicitly as in the case of an American-style Parisian knock-out option. We also recognize that the "moving window" technique developed by Zhu and Chen (2013) for pricing European-style Parisian up-and-out call options can be adopted to price American-style Parisian knock-in options as well. In particular, we obtain a simple analytical solution for American-style Parisian down-and-in call options and our new formula is written in terms of four double integrals, which can be easily computed numerically.

In this paper we study stock loans of finite maturity with different dividend distributions semi-analytically using the analytical approximation method in Zhu (2006). Stock loan partial differential equations (PDEs) are established under Black-Scholes framework. Laplace transform method is used to solve the PDEs. Optimal exit price and stock loan value are obtained in Laplace space. Values in the original time space are recovered by numerical Laplace inversion. To demonstrate the efficiency and accuracy of our semi-analytic method several examples are presented, the results are compared with those calculated using existing methods. We also present a calculation of fair service fee charged by the lender for different loan parameters.

In this paper, we price American-style Parisian down-and-in call options under the Black-Scholes framework. Usually, pricing an American-style option is much more difficult than pricing its European-style counterpart because of the appearance of the optimal exercise boundary in the former. Fortunately, the optimal exercise boundary associated with an American-style Parisian knock-in option only appears implicitly in its pricing partial differential equation (PDE) systems, instead of explicitly as in the case of an American-style Parisian knock-out option. We also recognize that the "moving window" technique developed by Zhu and Chen (2013) for pricing European-style Parisian up-and-out call options can be adopted to price American-style Parisian knock-in options as well. In particular, we obtain a simple analytical solution for American-style Parisian down-and-in call options and our new formula is written in terms of four double integrals, which can be easily computed numerically.

In this paper we study stock loans of finite maturity with different dividend distributions semi-analytically using the analytical approximation method in Zhu (2006). Stock loan partial differential equations (PDEs) are established under Black-Scholes framework. Laplace transform method is used to solve the PDEs. Optimal exit price and stock loan value are obtained in Laplace space. Values in the original time space are recovered by numerical Laplace inversion. To demonstrate the efficiency and accuracy of our semi-analytic method several examples are presented, the results are compared with those calculated using existing methods. We also present a calculation of fair service fee charged by the lender for different loan parameters.

In this paper, we price American-style Parisian down-and-in call options under the Black-Scholes framework. Usually, pricing an American-style option is much more difficult than pricing its European-style counterpart because of the appearance of the optimal exercise boundary in the former. Fortunately, the optimal exercise boundary associated with an American-style Parisian knock-in option only appears implicitly in its pricing partial differential equation (PDE) systems, instead of explicitly as in the case of an American-style Parisian knock-out option. We also recognize that the "moving window" technique developed by Zhu and Chen (2013) for pricing European-style Parisian up-and-out call options can be adopted to price American-style Parisian knock-in options as well. In particular, we obtain a simple analytical solution for American-style Parisian down-and-in call options and our new formula is written in terms of four double integrals, which can be easily computed numerically.

The research about asteroids attracts more and more attention recently, especially focusing on their physical structures, such as the spin axis, the rotation period and the shape. The long distance between Earth observers and asteroids makes it impossible to get the shape and other parameters of asteroids directly with the exception of the NEAs (Near Earth Asteroids) and others passed by some spacecrafts. Generally photometric measurement is still the main way to obtain the research data for asteroids now, i.e. the lightcurves recording the brightness and positions of asteroids. Supposing that the shape of the asteroid is a triaxial ellipsoid with a stable spinning status, a new method is present in this article to reconstruct the shape models of asteroids from the lightcurves, with the other physical parameters together. By applying a special curvature function, the method calculates the brightness integration on a unit sphere and Lebedev Quadrature is employed for the discretization. At last the method searches the optimal solution by Levenberg-Marquardt algorithm to minimize the residual of the brightness. By adopting this method not only related physical parameters of asteroids can be obtained at a reasonable accuracy, but also a simple shape model of Ellipsoid can be generated for reconstructing more sophisticated shape model further., Comment: 8 pages, 4 figures

We compare various approaches to find the most efficient method for the practical computation of the lightcurves (integrated brightnesses) of irregularly shaped bodies such as asteroids at arbitrary viewing and illumination geometries. For convex models, this reduces to the problem of the numerical computation of an integral over a simply defined part of the unit sphere. We introduce a fast method, based on Lebedev quadratures, which is optimal for both lightcurve simulation and inversion in the sense that it is the simplest and fastest widely applicable procedure for accuracy levels corresponding to typical data noise. The method requires no tessellation of the surface into a polyhedral approximation. At the accuracy level of 0.01 mag, it is up to an order of magnitude faster than polyhedral sums that are usually applied to this problem, and even faster at higher accuracies. This approach can also be used in other similar cases that can be modelled on the unit sphere. The method is easily implemented in lightcurve inversion by a simple alteration of the standard algorithm/software., Comment: Astronomy and Astrophysics, in press

We compare various approaches to find the most efficient method for the practical computation of the lightcurves (integrated brightnesses) of irregularly shaped bodies such as asteroids at arbitrary viewing and illumination geometries. For convex models, this reduces to the problem of the numerical computation of an integral over a simply defined part of the unit sphere. We introduce a fast method, based on Lebedev quadratures, which is optimal for both lightcurve simulation and inversion in the sense that it is the simplest and fastest widely applicable procedure for accuracy levels corresponding to typical data noise. The method requires no tessellation of the surface into a polyhedral approximation. At the accuracy level of 0.01 mag, it is up to an order of magnitude faster than polyhedral sums that are usually applied to this problem, and even faster at higher accuracies. This approach can also be used in other similar cases that can be modelled on the unit sphere. The method is easily implemented in lightcurve inversion by a simple alteration of the standard algorithm/software., Comment: Astronomy and Astrophysics, in press

The research about asteroids attracts more and more attention recently, especially focusing on their physical structures, such as the spin axis, the rotation period and the shape. The long distance between Earth observers and asteroids makes it impossible to get the shape and other parameters of asteroids directly with the exception of the NEAs (Near Earth Asteroids) and others passed by some spacecrafts. Generally photometric measurement is still the main way to obtain the research data for asteroids now, i.e. the lightcurves recording the brightness and positions of asteroids. Supposing that the shape of the asteroid is a triaxial ellipsoid with a stable spinning status, a new method is present in this article to reconstruct the shape models of asteroids from the lightcurves, with the other physical parameters together. By applying a special curvature function, the method calculates the brightness integration on a unit sphere and Lebedev Quadrature is employed for the discretization. At last the method searches the optimal solution by Levenberg-Marquardt algorithm to minimize the residual of the brightness. By adopting this method not only related physical parameters of asteroids can be obtained at a reasonable accuracy, but also a simple shape model of Ellipsoid can be generated for reconstructing more sophisticated shape model further., Comment: 8 pages, 4 figures

A periodic array of interface cracks was considered earlier as a simple model to study the effect of crack interaction on the elastic fields of a solid composed of two dissimilar isotropic elastic materials. This paper focuses on the perturbation of the elastic compliances of the solid due to the interacting interface cracks. The problem is formulated using Betti's reciprocal theorem. The analysis provides the equivalent elastic compliances in terms of the geometric parameters of the interface cracks for various levels of loadings and material combinations. The results show that the interaction between cracks is significant only when the crack spacing is less than four times the crack length. Also, contact between crack faces introduces coupling between applied shear stress and normal displacement, but it is significant only when the ratio of the magnitudes of the shear and normal stresses exceeds two.