We present a flexible and scalable method to compute global solutions of high-dimensional stochastic dynamic models. Within a time-iteration setup, we interpolate policy functions using an adaptive sparse grid algorithm with piecewise multi-linear (hierarchical) basis functions. As the dimensionality increases, sparse grids grow considerably slower than standard tensor product grids. In addition, the grid scheme we use is automatically refined locally and can thus capture steep gradients or even non-differentiabilities. To further increase the maximum problem size we can handle, our implementation is fully hybrid parallel, i.e. using a combination of distributed and shared memory parallelization schemes. This parallelization enables us to efficiently use high-performance computing architectures. Our algorithm scales up nicely to more than one thousand parallel processes. To demonstrate the performance of our method, we apply it to high-dimensional international real business cycle models with capital adjustment costs and irreversible investment.Keywords: Adaptive Sparse Grids, High-Performance Computing, International Real Business Cycles, Occasionally Binding Constraints JEL Classification: C63, C68, F41 * We are very grateful to Felix Kübler for helpful discussions and support. We thank Ken Judd, Karl Schmedders and seminar participants at University of Zürich, Stanford University, University of Chicago, Argonne National Laboratory, and CEF 2013 in Vancouver for valuable comments. Moreover, we thank Xiang Ma for very instructive email discussions regarding the workings of adaptive sparse grids. We are grateful for the support of Olaf Schenk, Antonio Messina and Riccardo Murri concerning HPC related issues. We acknowledge CPU time granted on the University of Zürich's 'Schrödinger' HPC cluster. Johannes Brumm gratefully acknowledges financial support from the ERC.

We propose a method to compute equilibria in dynamic models with several continuous state variables and occasionally binding constraints. These constraints induce non-differentiabilities in policy functions. We develop an interpolation technique that addresses this problem directly: It locates the non-differentiabilities and adds interpolation nodes there. To handle this flexible grid, it uses Delaunay interpolation, a simplicial interpolation technique. Hence, we call this method Adaptive Simplicial Interpolation (ASI). We embed ASI into a time iteration algorithm to compute recursive equilibria in an infinite horizon endowment economy where heterogeneous agents trade in a bond and a stock subject to various trading constraints. We show that this method computes equilibria accurately and outperforms other grid schemes by far. AbstractWe propose a method to compute equilibria in dynamic models with several continuous state variables and occasionally binding constraints. These constraints induce non-differentiabilities in policy functions. We develop an interpolation technique that addresses this problem directly: It locates the non-differentiabilities and adds interpolation nodes there. To handle this flexible grid, it uses Delaunay interpolation, a simplicial interpolation technique. Hence, we call this method Adaptive Simplicial Interpolation (ASI). We embed ASI into a time iteration algorithm to compute recursive equilibria in an infinite horizon endowment economy where heterogeneous agents trade in a bond and a stock subject to various trading constraints. We show that this method computes equilibria accurately and outperforms other grid schemes by far.

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Reproduction permitted only if source is stated. ISBN Non-technical summary Many financial securities derive their value not only from future cash flows but also from their ability to serve as collateral. This second source of value varies with macroeconomic condi-tions. In this paper, we investigate this collateral premium and its impact on security returns.We examine a model with two agents facing collateral constraints for borrowing. The agents can borrow against positions in assets which differ only by their "collateralizability". The collateralizability of an asset determines the fraction of the asset that can be confiscated in case of default and depends on its physical and legal properties. We document that borrowing against collateral contributes substantially to the return volatility of the assets. In our calibration of the model there are two types of agents who differ with respect to their risk aversion. The agent with the low risk aversion is the natural buyer of risky assets, and leverages to finance these investments. The agent with the high risk aversion has a strong desire to insure against bad shocks and thus is a natural buyer of safe bonds. When the economy is hit by a negative shock, the collateral constraint forces the leveraged agent to reduce consumption and to sell risky assets to the risk-averse agent, leading to substantial changes in the wealth distribution, which in turn affect agents' portfolios and asset prices.We further show that assets with different degrees of collateralizability which are otherwise identical exhibit substantially different return dynamics. In particular, the more collateralizable asset has both a smaller excess return and a smaller return volatility. However, the main economic mechanism leading to this result is straightforward; in response to negative shocks, the less risk-averse agent, holding both infinitely-lived assets and a large negative bond position against the collateralizable asset, must deleverage. She first sells the less col-lateralizable asset since it does not provide collateral value to her. In contrast, the less risk-averse agent holds on to the collateralizable asset as long as possible which leads only to a small drop in its price. As a consequence of this trading pattern, the less collateralizable asset has both a higher excess return and a higher return volatility than the more collateralizable asset.Finally, we document that the prices of collateralizable assets contain a sizable collateral premium which depends strongly on the difference in the collateralizability between the assets and much less so on an asset's expected future cash flows. In fact, assets that never pay dividends can still have a positive price in equilibrium if they are much more collateralizable than other assets in the economy. In der vorliegenden Arbeit wird darüber hinaus gezeigt, dass sich die Renditen unterschiedlich besicherungsfähiger, aber ansonsten gleicher Aktiva sehr verschieden entwickeln. So erzielen Aktiva mit höherer Besicherungsfähigkeit eine niedrig...

Reproduction permitted only if source is stated. ISBN Non-technical summary Many financial securities derive their value not only from future cash flows but also from their ability to serve as collateral. This second source of value varies with macroeconomic condi-tions. In this paper, we investigate this collateral premium and its impact on security returns.We examine a model with two agents facing collateral constraints for borrowing. The agents can borrow against positions in assets which differ only by their "collateralizability". The collateralizability of an asset determines the fraction of the asset that can be confiscated in case of default and depends on its physical and legal properties. We document that borrowing against collateral contributes substantially to the return volatility of the assets. In our calibration of the model there are two types of agents who differ with respect to their risk aversion. The agent with the low risk aversion is the natural buyer of risky assets, and leverages to finance these investments. The agent with the high risk aversion has a strong desire to insure against bad shocks and thus is a natural buyer of safe bonds. When the economy is hit by a negative shock, the collateral constraint forces the leveraged agent to reduce consumption and to sell risky assets to the risk-averse agent, leading to substantial changes in the wealth distribution, which in turn affect agents' portfolios and asset prices.We further show that assets with different degrees of collateralizability which are otherwise identical exhibit substantially different return dynamics. In particular, the more collateralizable asset has both a smaller excess return and a smaller return volatility. However, the main economic mechanism leading to this result is straightforward; in response to negative shocks, the less risk-averse agent, holding both infinitely-lived assets and a large negative bond position against the collateralizable asset, must deleverage. She first sells the less col-lateralizable asset since it does not provide collateral value to her. In contrast, the less risk-averse agent holds on to the collateralizable asset as long as possible which leads only to a small drop in its price. As a consequence of this trading pattern, the less collateralizable asset has both a higher excess return and a higher return volatility than the more collateralizable asset.Finally, we document that the prices of collateralizable assets contain a sizable collateral premium which depends strongly on the difference in the collateralizability between the assets and much less so on an asset's expected future cash flows. In fact, assets that never pay dividends can still have a positive price in equilibrium if they are much more collateralizable than other assets in the economy. In der vorliegenden Arbeit wird darüber hinaus gezeigt, dass sich die Renditen unterschiedlich besicherungsfähiger, aber ansonsten gleicher Aktiva sehr verschieden entwickeln. So erzielen Aktiva mit höherer Besicherungsfähigkeit eine niedrig...