We adopt deep learning models to directly optimise the portfolio Sharpe ratio. The framework we present circumvents the requirements for forecasting expected returns and allows us to directly optimise portfolio weights by updating model parameters. Instead of selecting individual assets, we trade Exchange-Traded Funds (ETFs) of market indices to form a portfolio. Indices of different asset classes show robust correlations and trading them substantially reduces the spectrum of available assets to choose from. We compare our method with a wide range of algorithms with results showing that our model obtains the best performance over the testing period, from 2011 to the end of April 2020, including the financial instabilities of the first quarter of 2020. A sensitivity analysis is included to understand the relevance of input features and we further study the performance of our approach under different cost rates and different risk levels via volatility scaling.

Geometric mean market makers (G3Ms), such as Uniswap and Balancer, comprise a popular class of automated market makers (AMMs) defined by the following rule: the reserves of the AMM before and after each trade must have the same (weighted) geometric mean. This paper extends several results known for constant-weight G3Ms to the general case of G3Ms with time-varying and potentially stochastic weights. These results include the returns and no-arbitrage prices of liquidity pool (LP) shares that investors receive for supplying liquidity to G3Ms. Using these expressions, we show how to create G3Ms whose LP shares replicate the payoffs of financial derivatives. The resulting hedges are model-independent and exact for derivative contracts whose payoff functions satisfy an elasticity constraint. These strategies allow LP shares to replicate various trading strategies and financial contracts, including standard options. G3Ms are thus shown to be capable of recreating a variety of active trading strategies through passive positions in LP shares.

Markets are subjected to both endogenous and exogenous risks that have caused disruptions to financial and economic markets around the globe, leading eventually to fast stock market declines. In the past, markets have recovered after any economic disruption. On this basis, we focus on the outbreak of COVID-19 as a case study of an exogenous risk and analyze its impact on the Standard and Poor's 500 (S\&P500) index. We assumed that the S\&P500 index reaches a minimum before rising again in the not-too-distant future. Here we present two cases to forecast the S\&P500 index. The first case uses an estimation of expected deaths released on 02/04/2020 by the University of Washington. For the second case, it is assumed that the peak number of deaths will occur 2-months since the first confirmed case occurred in the USA. The decline and recovery in the index were estimated for the following three months after the initial point of the predicted trend. The forecast is a projection of a prediction with stochastic fluctuations described by $q$-gaussian diffusion process with three spatio-temporal regimes. Our forecast was made on the premise that any market response can be decomposed into an overall deterministic trend and a stochastic term. The prediction was based on the deterministic part and for this case study is approximated by the extrapolation of the S\&P500 data trend in the initial stages of the outbreak. The stochastic fluctuations have the same structure as the one derived from the past 24 years. A reasonable forecast was achieved with 85\% of accuracy.

A major drawback of the Standard Heston model is that its implied volatility surface does not produce a steep enough smile when looking at short maturities. For that reason, we introduce the Stationary Heston model where we replace the deterministic initial condition of the volatility by its invariant measure and show, based on calibrated parameters, that this model produce a steeper smile for short maturities than the Standard Heston model. We also present numerical solution based on Product Recursive Quantization for the evaluation of exotic options (Bermudan and Barrier options).