Chebyshev polynomial approximation to approximate partial differential equations

This paper suggests a simple method based on Chebyshev approximation at Chebyshev nodes to approximate partial differential equations. The methodology simply consists in determining the value function by using a set of nodes and basis functions. We provide two examples. Pricing an European option and determining the best policy for chatting down a machinery. The suggested method is flexible, easy to program and efficient. It is also applicable in other fields, providing efficient solutions to complex systems of partial differential equations.

On the Relationship between Market Power and Bank Risk Taking

We analyse risk-taking behaviour of banks in the context of spatial competition. Banks mobilise unsecured deposits by offering deposit rates, which they invest either in a prudent or a gambling asset. Limited liability along with high return of a successful gamble induce moral hazard at the bank level. We show that when the market power is low, banks invest in the gambling asset. On the other hand, for sufficiently high levels of market power, all banks choose the prudent asset to invest in. We further show that a merger of two neighboring banks increases the likelihood of prudent behaviour. Finally, introduction of a deposit insurance scheme exacerbates banks’ moral hazard problem.