On multigrid for anisotropic equations and variational inequalities: pricing multi-dimensional European and American options

Partial differential operators in finance often originate in bounded linear stochastic processes. As a consequence, diffusion over these boundaries is zero and the corresponding coefficients vanish. The choice of parameters and stretched grids lead to additional anisotropies in the discrete equations or inequalities. In this study various block smoothers are tested in numerical experiments for equations of Black–Scholes-type (European options) in several dimensions. For linear complementarity problems, as they arise from optimal stopping time problems (American options), the choice of grid transfer is also crucial to preserve complementarity conditions on all grid levels. We adapt the transfer operators at the free boundary in a suitable way and compare with other strategies including cascadic approaches and full approximation schemes

On cross-currency models with stochastic volatility and correlated interest rates

We construct multi-currency models with stochastic volatility and correlated stochastic interest rates with a full matrix of correlations. We frst deal with a foreign exchange (FX) model of Heston-type, in which the domestic and foreign interest rates are generated by the short-rate process of Hull-White [HW96]. We then extend the framework by modeling the interest rate by a stochastic volatility displaced-diffusion Libor Market Model [AA02], which can model an interest rate smile. We provide semi-closed form approximations which lead to effcient calibration of the multi-currency models. Finally, we add a correlated stock to the framework and discuss the construction, model calibration and pricing of equity- FX-interest rate hybrid payoffs.Foreign-exchange (FX); stochastic volatility; Heston model; stochastic interest rates; interest rate smile; forward characteristic function; hybrids; affne diffusion; effcient calibration.

Pricing Early-Exercise and Discrete Barrier Options by Fourier-Cosine Series Expansions

We present a pricing method based on Fourier-cosine expansions for early-exercise and discretely-monitored barrier options. The method works well for exponential Levy asset price models. The error convergence is exponential for processes characterized by very smooth transitional probability density functions. The computational complexity is $O((M-1) N \log{N})$ with $N$ a (small) number of terms from the series expansion, and $M$, the number of early-exercise/monitoring dates.

Green walls and building energy consumption

A green wall, where foliage covers the external wall of a building, is considered to be a sustainable solution as it has many benefits for the direct environment. However, the advantages for the consumer (owner or building occupant) remain limited. One of these advantages is the potential energy savings that come from a reduction in the building's cooling load. The covering foliage has three properties that will negatively influence the amount of heat transfer taking place between the indoor and the outdoor climate. The first property of the foliage cover is the reduction of wind speed, or convective heat transfer, along the wall. This influence mainly causes the heat resistance of the external air film to rise from 0.04 to a maximum of 0.17 m2 K W-1, dependent on the density of the foliage. The second property of the foliage is the reduction of solar radiation dissipating in the external wall. The leaf area index determines the size of this influence. The reduced temperature of the external wall causes less heat transfer to the indoor climate. Finally, the third property of the foliage is the evaporation of water inside the leafs. This process converts sensible heat into latent heat, which is beneficial for the Urban Heat Island effect. The energy needed for this process is mainly taken from incoming solar radiation, but on rare occasions with strong wind and hot and dry air, the sensible heat can be taken from the air, causing a small reduction in air temperature. This property is however not quantified here. Living wall systems could also contribute to energy savings through their insulating capacity. The Rc value of three systems has been computed using THERM 6.3. The Rc value of the Greenwave system based on soil varies between the 0.1 and 0.2 m2 K W-1 dependent on the wetness of the growing medium. For the Wonderwall system based on fabric this ranges between the 0.2 and 0.3 m2 K W-1. The insulating capacity of the LivePanel system (rock wool) was found to be more significant: 0.9 - 1.5 m2 K W-1. The results of the quantification of the first two foliage properties and the Rc value of the LivePanel are subjected in a HAMBASE energy consumption simulation. In this study a small Dutch office building is subjected to quantify the potential energy savings aspect of a foliage covering the building's external walls (40% glass). Other investigated variables are the thermal mass of the building, the amount of thermal insulation and the application of a green roof. The results show that applying a foliage cover led to a decrease in annual cooling load, but this was balanced by an increase in annual heating load. In every examined situation, the annual energy consumption decreased by a mere 1%. Changing the model's structure from heavyweight to lightweight, while maintaining the Rl at 3.5 m2 K W-1, induced to a 41% increase in energy consumption. The increase was lower (24%) when the Rl of the heavyweight model was changed to 2.5 m2 K W-1. Applying a green roof on the heavyweight model led to a 2% decrease. Attaching the LivePanel system to the poorly insulated heavyweight model (Rl ? 2.5 m2 K W-1) caused the annual energy consumption to drop from 18,350 kWh to 14,650 kWh. Despite the annual savings of about 815.-, the application's annual maintenance costs (estimated between 2,520.- and 8,400.-) are not relieved. A green wall, where foliage covers the external wall of a building, is considered to be a sustainable solution as it has many benefits for the direct environment. However, the advantages for the consumer (owner or building occupant) remain limited. One of these advantages is the potential energy savings that come from a reduction in the building's cooling load. The covering foliage has three properties that will negatively influence the amount of heat transfer taking place between the indoor and the outdoor climate. The first property of the foliage cover is the reduction of wind speed, or convective heat transfer, along the wall. This influence mainly causes the heat resistance of the external air film to rise from 0.04 to a maximum of 0.17 m2 K W-1, dependent on the density of the foliage. The second property of the foliage is the reduction of solar radiation dissipating in the external wall. The leaf area index determines the size of this influence. The reduced temperature of the external wall causes less heat transfer to the indoor climate. Finally, the third property of the foliage is the evaporation of water inside the leafs. This process converts sensible heat into latent heat, which is beneficial for the Urban Heat Island effect. The energy needed for this process is mainly taken from incoming solar radiation, but on rare occasions with strong wind and hot and dry air, the sensible heat can be taken from the air, causing a small reduction in air temperature. This property is however not quantified here. Living wall systems could also contribute to energy savings through their insulating capacity. The Rc value of three systems has been computed using THERM 6.3. The Rc value of the Greenwave system based on soil varies between the 0.1 and 0.2 m2 K W-1 dependent on the wetness of the growing medium. For the Wonderwall system based on fabric this ranges between the 0.2 and 0.3 m2 K W-1. The insulating capacity of the LivePanel system (rock wool) was found to be more significant: 0.9 - 1.5 m2 K W-1. The results of the quantification of the first two foliage properties and the Rc value of the LivePanel are subjected in a HAMBASE energy consumption simulation. In this study a small Dutch office building is subjected to quantify the potential energy savings aspect of a foliage covering the building's external walls (40% glass). Other investigated variables are the thermal mass of the building, the amount of thermal insulation and the application of a green roof. The results show that applying a foliage cover led to a decrease in annual cooling load, but this was balanced by an increase in annual heating load. In every examined situation, the annual energy consumption decreased by a mere 1%. Changing the model's structure from heavyweight to lightweight, while maintaining the Rl at 3.5 m2 K W-1, induced to a 41% increase in energy consumption. The increase was lower (24%) when the Rl of the heavyweight model was changed to 2.5 m2 K W-1. Applying a green roof on the heavyweight model led to a 2% decrease. Attaching the LivePanel system to the poorly insulated heavyweight model (Rl ? 2.5 m2 K W-1) caused the annual energy consumption to drop from 18,350 kWh to 14,650 kWh. Despite the annual savings of about 815.-, the application's annual maintenance costs (estimated between 2,520.- and 8,400.-) are not relieved