In this paper, we propose a neural network-based method for approximating expected exposures and potential future exposures of Bermudan options. In a first phase, the method relies on the Deep Optimal Stopping algorithm (DOS) proposed in \cite{DOS}, which learns the optimal stopping rule from Monte-Carlo samples of the underlying risk factors. Cashflow paths are then created by applying the learned stopping strategy on a new set of realizations of the risk factors. Furthermore, in a second phase the risk factors are regressed against the cashflow-paths to obtain approximations of pathwise option values. The regression step is carried out by ordinary least squares as well as neural networks, and it is shown that the latter performs more accurate approximations.

The expected exposure is formulated, both in terms of the cashflow-paths and in terms of the pathwise option values and it is shown that a simple Monte-Carlo average yields accurate approximations in both cases. The potential future exposure is estimated by the empirical $\alpha$-percentile.

Finally, it is shown that the expected exposures, as well as the potential future exposures can be computed under either, the risk neutral measure, or the real world measure, without having to re-train the neural networks.

While standard estimation assumes that all datapoints are from probability distribution of the same fixed parameters $\theta$, we will focus on maximum likelihood (ML) adaptive estimation for nonstationary time series: separately estimating parameters $\theta_T$ for each time $T$ based on the earlier values $(x_t)_{t<T}$ using (exponential) moving ML estimator $\theta_T=\arg\max_\theta l_T$ for $l_T=\sum_{t<T} \eta^{T-t} \ln(\rho_\theta (x_t))$ and some $\eta\in(0,1]$. Computational cost of such moving estimator is generally much higher as we need to optimize log-likelihood multiple times, however, in many cases it can be made inexpensive thanks to dependencies. We focus on such example: exponential power distribution (EPD) $\rho(x)\propto \exp(-|(x-\mu)/\sigma|^\kappa/\kappa)$ family, which covers wide range of tail behavior like Gaussian ($\kappa=2$) or Laplace ($\kappa=1$) distribution. It is also convenient for such adaptive estimation of scale parameter $\sigma$ as its standard ML estimation is $\sigma^\kappa$ being average $\|x-\mu\|^\kappa$. By just replacing average with exponential moving average: $(\sigma_{T+1})^\kappa=\eta(\sigma_T)^\kappa +(1-\eta)|x_T-\mu|^\kappa$ we can inexpensively make it adaptive. It is tested on daily log-return series for DJIA companies, leading to essentially better log-likelihoods than standard (static) estimation, surprisingly with optimal $\kappa$ tails types varying between companies. Presented general alternative estimation philosophy provides tools which might be useful for building better models for analysis of nonstationary time-series.

Algorithmic trading systems are often completely automated, and deep learning is increasingly receiving attention in this domain. Nonetheless, little is known about the robustness properties of these models. We study valuation models for algorithmic trading from the perspective of adversarial machine learning. We introduce new attacks specific to this domain with size constraints that minimize attack costs. We further discuss how these attacks can be used as an analysis tool to study and evaluate the robustness properties of financial models. Finally, we investigate the feasibility of realistic adversarial attacks in which an adversarial trader fools automated trading systems into making inaccurate predictions.

Stock market prediction has been a classical yet challenging problem, with the attention from both economists and computer scientists. With the purpose of building an effective prediction model, both linear and machine learning tools have been explored for the past couple of decades. Lately, deep learning models have been introduced as new frontiers for this topic and the rapid development is too fast to catch up. Hence, our motivation for this survey is to give a latest review of recent works on deep learning models for stock market prediction. We not only category the different data sources, various neural network structures, and common used evaluation metrics, but also the implementation and reproducibility. Our goal is to help the interested researchers to synchronize with the latest progress and also help them to easily reproduce the previous studies as baselines. Base on the summary, we also highlight some future research directions in this topic.

In the presence of monotone information, stochastic Thiele equations describing the dynamics of state-wise prospective reserves are closely related to the classic martingale representation theorem. When the information utilized by the insurer is non-monotone, classic martingale theory does not apply. By taking an infinitesimal approach, we derive generalized stochastic Thiele equations that allow for information discarding. The results and their implication in practice are illustrated via examples where information is discarded upon and after stochastic retirement.

EBIs/ESOs substantially change the traditional production/service function because ESOs/EBIs can have different psychological effects(motivation or de-motivation), and can create intangible capital and different economic payoffs. Although Game Theory is flawed, it can be helpful in describing interactions in ESO/EBIs transactions. ESOs/EBIs involve two-stage games and there are no perfect Nash Equilibria for the two sub-games. The large number of actual and potential participants in these games significantly complicates resolution of equilibria and increases the dynamism of the games given that players are more sensitive to other peoples moves in such games. This article: a) analyzes how ESOs/EBIs affect traditional assumptions of production functions (in both the manufacturing and service sectors), b) analyzes ESOs/EBIs transactions using game theory concepts, c) illustrates some of the limitations of game theory.

We find economically and statistically significant gains from using machine learning to dynamically allocate between the market index and the risk-free asset. We model the market price of risk to determine the optimal weights in the portfolio: reward-risk market timing. This involves forecasting the direction of next month's excess return, which gives the reward, and constructing a dynamic volatility estimator that is optimized with a machine learning model, which gives the risk. Reward-risk timing with machine learning provides substantial improvements in investor utility, alphas, Sharpe ratios, and maximum drawdowns, after accounting for transaction costs, leverage constraints, and on a new out-of-sample test set. This paper provides a unifying framework for machine learning applied to both return- and volatility-timing.

This paper investigates optimal consumption, investment, and healthcare spending under Epstein-Zin preferences. Given consumption and healthcare spending plans, Epstein-Zin utilities are defined over an agent's random lifetime, partially controllable by the agent as healthcare reduces Gompertz' natural growth rate of mortality. In a Black-Scholes market, the stochastic optimization problem is solved through the associated Hamilton-Jacobi-Bellman (HJB) equation. Compared with classical Epstein-Zin utility maximization, the additional controlled mortality process complicates the uniqueness of Epstein-Zin utilities and verification arguments. A combination of probabilistic arguments and analysis of the HJB equation are required to resolve the challenges. In contrast to prior work under time-separable utilities, Epstein-Zin preferences largely facilitate calibration. In five different countries we examined, the model-generated mortality closely approximates actual mortality data; moreover, the calibrated efficacy of healthcare is in close agreement with empirical studies on healthcare across countries.

We apply numerical dynamic programming techniques to solve discrete-time multi-asset dynamic portfolio optimization problems with proportional transaction costs and shorting/borrowing constraints. Examples include problems with multiple assets, and many trading periods in a finite horizon problem. We also solve dynamic stochastic problems, with a portfolio including one risk-free asset, an option, and its underlying risky asset, under the existence of transaction costs and constraints. These examples show that it is now tractable to solve such problems.

In this paper we propose a generalization of the Deep Galerking Method (DGM) of \cite{dgm} to deal with Path-Dependent Partial Differential Equations (PPDEs). These equations firstly appeared in the seminal work of \cite{fito_dupire}, where the functional It\^o calculus was developed to deal with path-dependent financial derivatives contracts. The method, which we call Path-Dependent DGM (PDGM), consists of using a combination of feed-forward and Long Short-Term Memory architectures to model the solution of the PPDE. We then analyze several numerical examples, many from the Financial Mathematics literature, that show the capabilities of the method under very different situations.

A commonly used stochastic model for derivative and commodity market analysis is the Barndorff-Nielsen and Shephard (BN-S) model. Though this model is very efficient and analytically tractable, it suffers from the absence of long range dependence and many other issues. For this paper, the analysis is restricted to crude oil price dynamics. A simple way of improving the BN-S model with the implementation of various machine learning algorithms is proposed. This refined BN-S model is more efficient and has fewer parameters than other models which are used in practice as improvements of the BN-S model. The procedure and the model show the application of data science for extracting a "deterministic component" out of processes that are usually considered to be completely stochastic. Empirical applications validate the efficacy of the proposed model for long range dependence.

We consider dynamic risk measures induced by Backward Stochastic Differential Equations (BSDEs) in enlargement of filtration setting. On a fixed probability space, we are given a standard Brownian motion and a pair of random variables $(\tau, \zeta) \in (0,+\infty) \times E$, with $E \subset \mathbb{R}^m$, that enlarge the reference filtration, i.e., the one generated by the Brownian motion. These random variables can be interpreted financially as a default time and an associated mark. After introducing a BSDE driven by the Brownian motion and the random measure associated to $(\tau, \zeta)$, we define the dynamic risk measure $(\rho_t)_{t \in [0,T]}$, for a fixed time $T > 0$, induced by its solution. We prove that $(\rho_t)_{t \in [0,T]}$ can be decomposed in a pair of risk measures, acting before and after $\tau$ and we characterize its properties giving suitable assumptions on the driver of the BSDE. Furthermore, we prove an inequality satisfied by the penalty term associated to the robust representation of $(\rho_t)_{t \in [0,T]}$ and we discuss the dynamic entropic risk measure case, providing examples where it is possible to write explicitly its decomposition and simulate it numerically.

We show that adversarial reinforcement learning (ARL) can be used to produce market marking agents that are robust to adversarial and adaptively chosen market conditions. To apply ARL, we turn the well-studied single-agent model of Avellaneda and Stoikov [2008] into a discrete-time zero-sum game between a market maker and adversary, a proxy for other market participants who would like to profit at the market maker's expense. We empirically compare two conventional single-agent RL agents with ARL, and show that our ARL approach leads to: 1) the emergence of naturally risk-averse behaviour without constraints or domain-specific penalties; 2) significant improvements in performance across a set of standard metrics, evaluated with or without an adversary in the test environment, and; 3) improved robustness to model uncertainty. We empirically demonstrate that our ARL method consistently converges, and we prove for several special cases that the profiles that we converge to are Nash equilibria in a corresponding simplified single-stage game.