Asymptotics and numerics in rough and local stochastic volatility models

In this dissertation, we investigate various questions arising from the modelling of asset prices. All these models have in common, that they permit the use of rough volatility processes, meaning that the fluctuations of stock prices are modelled via a very irregular process. Besides this underlying structure, statements about asymptotic behaviour play a major role, both in the description of price developments over short time intervals and in the analysis of numerical schemes, that approximate the solutions of the theoretical models. Part I already addresses this issue. Here, we study a certain class of financial models, that can loosely be described as rough Bergomi models. More precisely, we study the behaviour of the weak error between the theoretical model and the actual numerical implementation. It turns out, that this error, up to a certain degree, does not depend on the (ir)regularity of the underlying volatility process. Remarkably, also the correlation between the volatility and the asset has a significant influence on the convergence speed. We finish the discussion with an example, that provides a lower bound for the convergence speed, implying that our result is optimal in a certain sense. In the second part, we are concerned with baskets, financial products composed of several different assets. More precisely, we study these objects in a small noise regime. This regime can, via scaling arguments, also be interpreted in such a way, that the baskets are only observed over a short time period. An important theoretical framework for this is the theory of large deviations, which helps making statements about the asymptotic behaviour of probabilities of rare events. As we work in the context of rough volatility, this gives rise to a non-trivial rate function. Part III thematically returns to the field of numerics. Here we study so-called local stochastic volatility models, which are especially popular among practitioners. Despite their popularity, one quickly encounters difficulties in the theoretical investigation of these models: the occurrence of certain conditional expectations complicates general statements about existence of solutions. Even all one-dimensional marginals of this process would be known. The main idea of our work is a relatively simple observation: when applying the Euler-scheme, a standard method for implementing (stochastic) differential equations, all expressions are well-defined. Thus we studied the weak error of this scheme, compared to the original model. During this analysis, we introduce a novel half-step scheme, which has many practical properties in dealing with conditional expectations. Finally, we use this method to introduce an interacting particle system. The main result then quantifies the error rate, depending on all parameters used. Additionally we provide numerical simulations supporting our theoretical findings.