We consider a system of coupled free boundary problems for pricing American put options with regime-switching. To solve this system, we first employ the logarithmic transformation to map the free boundary for each regime to multi-fixed intervals and then eliminate the first-order derivative in the transformed model by taking derivatives to obtain a system of partial differential equations which we call the asset-delta-gamma-speed equations. As such, the fourth-order compact finite difference scheme can be used for solving this system. The influence of other asset, delta, gamma, and speed options in the present regime is estimated based on Hermite interpolations. Finally, the numerical method is tested with several examples. Our results show that the scheme provides an accurate solution that is fast in computation as compared with other existing numerical methods.

In the recent years money laundering schemes have grown in complexity and speed of realization, affecting financial institutions and millions of customers globally. Strengthened privacy policies, along with in-country regulations, make it hard for banks to inner- and cross-share, and report suspicious activities for the AML (Anti-Money Laundering) measures. Existing topologies and models for AML analysis and information sharing are subject to major limitations, such as compliance with regulatory constraints, extended infrastructure to run high-computation algorithms, data quality and span, proving cumbersome and costly to execute, federate, and interpret. This paper proposes a new topology for exploring multi-banking customer social relations in AML context -- customer-to-customer, customer-to-transaction, and transaction-to-transaction -- using a 3D modeling topological algebra formulated through Poincar\'e embeddings.

In mathematical finance, a process of calibrating stochastic volatility (SV) option pricing models to real market data involves a numerical calculation of integrals that depend on several model parameters. This optimization task consists of large number of integral evaluations with high precision and low computational time requirements. However, for some model parameters, many numerical quadrature algorithms fail to meet these requirements. We can observe an enormous increase in function evaluations, serious precision problems and a significant increase of computational time. In this paper we numerically analyse these problems and show that they are especially caused by inaccurately evaluated integrands. We propose a fast regime switching algorithm that tells if it is sufficient to evaluate the integrand in standard double arithmetic or if a higher precision arithmetic has to be used. We compare and recommend numerical quadratures for typical SV models and different parameter values, especially for problematic cases.

We consider the problem of optimal hedging in an incomplete market with an established pricing kernel. In such a market, prices are uniquely determined, but perfect hedges are usually not available. We work in the rather general setting of a L\'evy-Ito market, where assets are driven jointly by an $n$-dimensional Brownian motion and an independent Poisson random measure on an $n$-dimensional state space. Given a position in need of hedging and the instruments available as hedges, we demonstrate the existence of an optimal hedge portfolio, where optimality is defined by use of an expected least squared-error criterion over a specified time frame, and where the numeraire with respect to which the hedge is optimized is taken to be the benchmark process associated with the designated pricing kernel.

We introduce a novel framework to account for sensitivity to rewards uncertainty in sequential decision-making problems. While risk-sensitive formulations for Markov decision processes studied so far focus on the distribution of the cumulative reward as a whole, we aim at learning policies sensitive to the uncertain/stochastic nature of the rewards, which has the advantage of being conceptually more meaningful in some cases. To this end, we present a new decomposition of the randomness contained in the cumulative reward based on the Doob decomposition of a stochastic process, and introduce a new conceptual tool - the \textit{chaotic variation} - which can rigorously be interpreted as the risk measure of the martingale component associated to the cumulative reward process. We innovate on the reinforcement learning side by incorporating this new risk-sensitive approach into model-free algorithms, both policy gradient and value function based, and illustrate its relevance on grid world and portfolio optimization problems.

The paper develops multiplicative compensation for complex-valued semimartingales and studies some of its consequences. It is shown that the stochastic exponential of any complex-valued semimartingale with independent increments becomes a true martingale after multiplicative compensation, where such compensation is meaningful. This generalization of the L\'evy-Khintchin formula fills an existing gap in the literature. We further report Girsanov-type results based on non-negative multiplicatively compensated semimartingales. In particular, we obtain a simplified expression for the multiplicative compensator under the new measure.

In this paper we study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial diferential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre polynomials. We compare obtained solutions to existing semi-closed pricing formulas. Special attention is paid to the solution of Heston model at the boundary with vanishing volatility.

We introduce the social welfare implications of the Zenga index, a recently proposed index of inequality. Our proposal is derived by following the seminal book by Son (2011) and the recent working paper by Kakwani and Son (2019). We compare the Zenga based approach with the classical one, based on the Lorenz curve and the Gini coefficient, as well as the Bonferroni index. We show that the social welfare specification based on the Zenga uniformity curve presents some peculiarities that distinguish it from the other considered indexes. The social welfare specification presented here provides a deeper understanding of how the Zenga index evaluates the inequality in a distribution.

Structural models with no solution are incoherent, and those with multiple solutions are incomplete. We develop a method to study coherency and completeness conditions for linear dynamic forward-looking rational expectations models under an occasionally binding constraint. In the context of the simple New Keynesian model with a zero lower bound, this method shows that the coherency and completeness condition generally violates the Taylor principle. Rational expectations require time-varying and correlated support restrictions on the distribution of the structural shocks. With appropriate restrictions, a very large number of equilibria can be supported.

We discuss the impact of a Covid-like shock on a simple toy economy, described by the Mark-0 Agent-Based Model that we developed and discussed in a series of previous papers. We consider a mixed supply and demand shock, and show that depending on the shock parameters (amplitude and duration), our toy economy can display V-shaped, U-shaped or W-shaped recoveries, and even an L-shaped output curve with permanent output loss. This is due to the existence of a self-sustained "bad" state of the economy. We then discuss two policies that attempt to moderate the impact of the shock: giving easy credit to firms, and the so-called helicopter money, i.e. injecting new money into the households savings. We find that both policies are effective if strong enough, and we highlight the potential danger of terminating these policies too early. While we only discuss a limited number of scenarios, our model is flexible and versatile enough to allow for a much wider exploration, thus serving as a useful tool for the qualitative understanding of post-Covid recovery. We provide an on-line version of the code at https://gitlab.com/sharma.dhruv/markovid .

Lack of skills is arguably one of the most important determinants of high levels of unemployment and poverty. In response, policymakers often initiate vocational training programs in effort to enhance skill formation among the youth. Using a regression-discontinuity design, we examine a large youth training intervention in Nepal. We find, twelve months after the start of the training program, that the intervention generated an increase in non-farm employment of 10 percentage points (ITT estimates) and up to 31 percentage points for program compliers (LATE estimates). We also detect sizeable gains in monthly earnings. Women who start self-employment activities inside their homes largely drive these impacts. We argue that low baseline educational levels and non-farm employment levels and Nepal's social and cultural norms towards women drive our large program impacts. Our results suggest that the program enables otherwise underemployed women to earn an income while staying at home - close to household errands and in line with the socio-cultural norms that prevent them from taking up employment outside the house.