When to expect decreasing implied volatilities

Abstract: In this report the goal is to investigate general properties of implied volatility such as the relationship between implied volatility and local volatility as well as the relationship between implied volatility and the sign of the jumps in jump-diffusion models. More specific the question to investigate is whether the implied volatility is decreasing as the strike price increases under the assumption that the market is pricing under a monotonically decreasing local volatility model. To investigate the previously mentioned problem some derivations will have to be done. The core of the derivations are Dupire's equation and Black-Scholes formula. Not only will the result from the derivation show what is tried to be proven, the derived results will also be implemented to visualise the results. This means that the result both is backed by both theory and derivations but also computations which is plotted in the report. An additional goal of this report is to shed light of the relationship between the concepts of implied volatility and local volatility, as well as on the computational aspects on how to determine the implied volatility.