Generalizing earlier works of Delbaen & Haezendonck [5] as well as of [18] and [16] for given compound mixed renewal process S under a probability measure P, we characterize all those probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound mixed renewal process under Q with improved properties. As a consequence, we prove that any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures. Applications related to the ruin problem and to the computation of premium calculation principles in an insurance market without arbitrage opportunities are discussed in [26] and [27], respectively.

We propose an adaptive and explicit fourth-order Runge-Kutta-Fehlberg method coupled with a fourth-order compact scheme to solve the American put options problem. First, the free boundary problem is converted into a system of partial differential equations with a fixed domain by using logarithm transformation and taking additional derivatives. With the addition of an intermediate function with a fixed free boundary, a quadratic formula is derived to compute the velocity of the optimal exercise boundary analytically. As such, it enables us to employ fourth-order spatial and temporal discretization with Dirichlet boundary conditions for obtaining the numerical solution of the asset option, option Greeks, and the optimal exercise boundary. The advantage of the Runge-Kutta-Fehlberg method is based on error control and the adjustment of the time step to maintain the error at a certain threshold. By comparing with some existing methods in numerical experiments, it shows that the present method has a better performance in terms of computational speed and provides a more accurate solution.

We consider the problem of computing the Value Adjustment of European contingent claims when default of either party is considered, possibly including also funding and collateralization requirements. As shown in Brigo et al. (\cite{BLPS}, \cite{BFP}), this leads to a more articulate variety of Value Adjustments ({XVA}) that introduce some nonlinear features. When exploiting a reduced-form approach for the default times, the adjusted price can be characterized as the solution to a possibly nonlinear Backward Stochastic Differential Equation (BSDE). The expectation representing the solution of the BSDE is usually quite hard to compute even in a Markovian setting, and one might resort either to the discretization of the Partial Differential Equation characterizing it or to Monte Carlo Simulations. Both choices are computationally very expensive and in this paper we suggest an approximation method based on an appropriate change of numeraire and on a Taylor's polynomial expansion when intensities are represented by means of affine processes correlated with the asset's price. The numerical discussion at the end of this work shows that, at least in the case of the CIR intensity model, even the simple first-order approximation has a remarkable computational efficiency.

The purpose of this study is to investigate the effects of the COVID-19 pandemic on economic policy uncertainty in the US and the UK. The impact of the increase in COVID-19 cases and deaths in the country, and the increase in the number of cases and deaths outside the country may vary. To examine this, the study employs bootstrap ARDL cointegration approach from March 8, 2020 to May 24, 2020. According to the bootstrap ARDL results, a long-run equilibrium relationship is confirmed for five out of the 10 models. The long-term coefficients obtained from the ARDL models suggest that an increase in COVID-19 cases and deaths outside of the UK and the US has a significant effect on economic policy uncertainty. The US is more affected by the increase in the number of COVID-19 cases. The UK, on the other hand, is more negatively affected by the increase in the number of COVID-19 deaths outside the country than the increase in the number of cases. Moreover, another important finding from the study demonstrates that COVID-19 is a factor of great uncertainty for both countries in the short-term.

Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the 'instantaneous arbitrage capability' generated by the market itself. The cashflow bundle is the vector bundle associated to this stochastic principal fibre bundle for the natural choice of the vector space fibre. The cashflow bundle carries a stochastic covariant differentiation induced by the connection on the principal fibre bundle. The link between arbitrage theory and spectral theory of the connection Laplacian on the vector bundle is given by the zero eigenspace resulting in a parametrization of all risk neutral measures equivalent to the statistical one. This indicates that a market satisfies the (NFLVR) condition if and only if $0$ is in the discrete spectrum of the connection Laplacian on the cash flow bundle or of the Dirac Laplacian of the twisted cash flow bundle with the exterior algebra bundle. We apply this result by extending Jarrow-Protter-Shimbo theory of asset bubbles for complete arbitrage free markets to markets not satisfying the (NFLVR). Moreover, by means of the Atiyah-Singer index theorem, we prove that the Euler characteristic of the asset nominal space is a topological obstruction to the the (NFLVR) condition, and, by means of the Bochner-Weitzenb\"ock formula, the non vanishing of the homology group of the cash flow bundle is revealed to be a topological obstruction to (NFLVR), too. Asset bubbles are defined, classified and decomposed for markets allowing arbitrage.

In this paper, we investigate the mesoscale structure of the World Trade Network. In this framework, a specific role is assumed by short and long-range interactions, and hence by the distance, between countries. Therefore, we identify clusters through a new procedure that exploits Estrada communicability distance and the vibrational communicability distance, which turn out to be particularly suitable for catching the inner structure of the economic network. The proposed methodology aims at finding the distance threshold that maximizes a specific modularity function defined for general metric spaces. Main advantages regard the computational efficiency of the procedure as well as the possibility to inspect intercluster and intracluster properties of the resulting communities. The numerical analysis highlights peculiar relationships between countries and provides a rich set of information that can hardly be achieved within alternative clustering approaches.

The present work addresses theoretical and practical questions in the domain of Deep Learning for High Frequency Trading, with a thorough review and analysis of the literature and state-of-the-art models. Random models, Logistic Regressions, LSTMs, LSTMs equipped with an Attention mask, CNN-LSTMs and MLPs are compared on the same tasks, feature space, and dataset and clustered according to pairwise similarity and performance metrics. The underlying dimensions of the modeling techniques are hence investigated to understand whether these are intrinsic to the Limit Order Book's dynamics. It is possible to observe that the Multilayer Perceptron performs comparably to or better than state-of-the-art CNN-LSTM architectures indicating that dynamic spatial and temporal dimensions are a good approximation of the LOB's dynamics, but not necessarily the true underlying dimensions.

Using the carefully selected industry classification standard, we divide 102 industry securities indices in China's stock market into four demand-oriented sector groups and identify demand-oriented industry-specific volatility spillover networks. The "deman-oriented" is a new idea of reconstructing the structure of the networks considering the relationship between industry sectors and the economic demand their outputs meeting. Networks with the new structure help us improve the understanding of the economic demand change, especially when the macroeconomic is dramatically influenced by exogenous shocks like the outbreak of COVID-19. At the beginning of the outbreak of COVID-19, in China's stock market, spillover effects from industry indices of sectors meeting the investment demand to those meeting the consumption demands rose significantly. However, these spillover effects fell after the outbreak containment in China appeared to be effective. Besides, some services sectors including utility, transportation and information services have played increasingly important roles in the networks of industry-specific volatility spillovers as of the COVID-19 out broke. By implication, firstly, being led by Chinese government, the COVID-19 is successfully contained and the work resumption is organized with a high efficiency in China. The risk of the investment demand therefore was controlled and eliminated relatively fast. Secondly, the intensive using of non-pharmaceutical interventions (NPIs) led to supply restriction in services in China. It will still be a potential threat for the Chinese economic recovery in the next stage.

The objective of this work is twofold: to expand the depression models proposed by Tobin and analyse a supply shock, such as the Covid-19 pandemic, in this Keynesian conceptual environment. The expansion allows us to propose the evolution of all endogenous macroeconomic variables. The result obtained is relevant due to its theoretical and practical implications. A quantity or Keynesian adjustment to the shock produces a depression through the effect on aggregate demand. This depression worsens in the medium/long-term. It is accompanied by increases in inflation, inflation expectations and the real interest rate. A stimulus tax policy is also recommended, as well as an active monetary policy to reduce real interest rates. On the other hand, the pricing or Marshallian adjustment foresees a more severe and rapid depression in the short-term. There would be a reduction in inflation and inflation expectations, and an increase in the real interest rates. The tax or monetary stimulus measures would only impact inflation. This result makes it possible to clarify and assess the resulting depression, as well as propose policies. Finally, it offers conflicting predictions that allow one of the two models to be falsified.

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.

We apply Geometric Arbitrage Theory to obtain results in Mathematical Finance, which do not need stochastic differential geometry in their formulation. First, for a generic market dynamics given by a multidimensional It\^o's process we specify and prove the equivalence between (NFLVR) and expected utility maximization. As a by-product we provide a geometric characterization of the (NUPBR) condition given by the zero curvature (ZC) condition. Finally, we extend the Black-Scholes PDE to markets allowing arbitrage.

On a periodic basis, publicly traded companies report fundamentals, financial data including revenue, earnings, debt, among others. Quantitative finance research has identified several factors, functions of the reported data that historically correlate with stock market performance. In this paper, we first show through simulation that if we could select stocks via factors calculated on future fundamentals (via oracle), that our portfolios would far outperform standard factor models. Motivated by this insight, we train deep nets to forecast future fundamentals from a trailing 5-year history. We propose lookahead factor models which plug these predicted future fundamentals into traditional factors. Finally, we incorporate uncertainty estimates from both neural heteroscedastic regression and a dropout-based heuristic, improving performance by adjusting our portfolios to avert risk. In retrospective analysis, we leverage an industry-grade portfolio simulator (backtester) to show simultaneous improvement in annualized return and Sharpe ratio. Specifically, the simulated annualized return for the uncertainty-aware model is 17.7% (vs 14.0% for a standard factor model) and the Sharpe ratio is 0.84 (vs 0.52).