In an economy which could not accommodate the full employment of its labor force, it employs some labor but does not employ others. The bipartition of the labor force is random, and we characterize it by an axiom of equal employment opportunity. We value each employed individual by his or her marginal contribution to the production function; we also value each unemployed individual by the potential marginal contribution the person would make if the market hired the individual. We then use the aggregate individual value to distribute the net production to the unemployment welfare and the employment benefits. Using real-time balanced-budget rule as a constraint and policy stability as an objective, we derive a scientific formula which describes a fair, debt-free, and asymptotic risk-free tax rate for any given unemployment rate and national spending level. The tax rate minimizes the asymptotic mean, variance, semi-variance, and mean absolute deviation of the underlying posterior unemployment rate. The allocation rule stimulates employment and boosts productivity. Under some symmetry assumptions, we even find that an unemployed person should enjoy equivalent employment benefits, and the tax rate goes with this welfare equality. The tool employed is the cooperative game theory in which we assume many players. The players are randomly bipartitioned, and the payoff varies with the partition. One could apply the fair distribution rule and valuation approach to other profit-sharing or cost-sharing situations with these characteristics. This framework is open to alternative identification strategies and other forms of equal opportunity axiom.

We propose a new framework to value employee stock options (ESOs) that captures multiple exercises of different quantities over time. We also model the ESO holder's job termination risk and incorporate its impact on the payoffs of both vested and unvested ESOs. Numerical methods based on Fourier transform and finite differences are developed and implemented to solve the associated systems of PDEs. In addition, we introduce a new valuation method based on maturity randomization that yields analytic formulae for vested and unvested ESO costs. We examine the cost impact of job termination risk, exercise intensity, and various contractual features.

This paper discusses the sensitivity of the long-term expected utility of optimal portfolios for an investor with constant relative risk aversion. Under an incomplete market given by a factor model, we consider the utility maximization problem with long-time horizon. The main purpose is to find the long-term sensitivity, that is, the extent how much the optimal expected utility is affected in the long run for small changes of the underlying factor model. The factor model induces a specific eigenpair of an operator, and this eigenpair does not only characterize the long-term behavior of the optimal expected utility but also provides an explicit representation of the expected utility on a finite time horizon. We conclude that this eigenpair therefore determines the long-term sensitivity. As examples, explicit results for several market models such as the Kim--Omberg model for stochastic excess returns and the Heston stochastic volatility model are presented.

We study a single-period optimal transport problem on $\mathbb{R}^2$ with a covariance-type cost function $c(x,y) = (x_1-y_1)(x_2-y_2)$ and a backward martingale constraint. We show that a transport plan $\gamma$ is optimal if and only if there is a maximal monotone set $G$ that supports the $x$-marginal of $\gamma$ and such that $c(x,y) = \min_{z\in G}c(z,y)$ for every $(x,y)$ in the support of $\gamma$. We obtain sharp regularity conditions for the uniqueness of an optimal plan and for its representation in terms of a map. Our study is motivated by a variant of the classical Kyle model of insider trading from Rochet and Vila (1994).

The multilevel Monte Carlo path simulation method introduced by Giles ({\it Operations Research}, 56(3):607-617, 2008) exploits strong convergence properties to improve the computational complexity by combining simulations with different levels of resolution. In this paper we analyse its efficiency when using the Milstein discretisation; this has an improved order of strong convergence compared to the standard Euler-Maruyama method, and it is proved that this leads to an improved order of convergence of the variance of the multilevel estimator. Numerical results are also given for basket options to illustrate the relevance of the analysis.

We study risk-sharing equilibria with general convex costs on the agents' trading rates. For an infinite-horizon model with linear state dynamics and exogenous volatilities, the equilibrium returns mean-revert around their frictionless counterparts -- the deviation has Ornstein-Uhlenbeck dynamics for quadratic costs whereas it follows a doubly-reflected Brownian motion if costs are proportional. More general models with arbitrary state dynamics and endogenous volatilities lead to multidimensional systems of nonlinear, fully-coupled forward-backward SDEs. These fall outside the scope of known wellposedness results, but can be solved numerically using the simulation-based deep-learning approach of \cite{han.al.17}. In a calibration to time series of returns, bid-ask spreads, and trading volume, transaction costs substantially affect equilibrium asset prices. In contrast, the effects of different cost specifications are rather similar, justifying the use of quadratic costs as a proxy for other less tractable specifications.

Many existing jobs are prone to automation, but since new technologies also create new jobs it is crucial to understand job transitions. Based on empirical data we construct an occupational mobility network where nodes are occupations and edges represent the likelihood of job transitions. To study the effects of automation we develop a labour market model. At the macro level our model reproduces the Beveridge curve. At the micro level we analyze occupation-specific unemployment in response to an automation-related reallocation of labour demand. The network structure plays an important role: workers in occupations with a similar automation level often face different outcomes, both in the short term and in the long term, due to the fact that some occupations offer little opportunity for transition. Our work underscores the importance of directing retraining schemes towards workers in occupations with limited transition possibilities.

For the degree corrected stochastic block model in the presence of arbitrary or even adversarial outliers, we develop a convex-optimization-based clustering algorithm that includes a penalization term depending on the positive deviation of a node from the expected number of edges to other inliers. We prove that under mild conditions, this method achieves exact recovery of the underlying clusters. Our synthetic experiments show that our algorithm performs well on heterogeneous networks, and in particular those with Pareto degree distributions, for which outliers have a broad range of possible degrees that may enhance their adversarial power. We also demonstrate that our method allows for recovery with significantly lower error rates compared to existing algorithms.

We apply the procedure of Lee et al. to the problem of performing inference on the signal noise ratio of the asset which displays maximum sample Sharpe ratio over a set of possibly correlated assets. We find a multivariate analogue of the commonly used approximate standard error of the Sharpe ratio to use in this conditional estimation procedure. We also consider the simple Bonferroni correction for multiple hypothesis testing, fixing it for the case of positive common correlation among assets.

Testing indicates the conditional inference procedure achieves nominal type I rate, and does not appear to suffer from non-normality of returns. The conditional estimation test has low power under the alternative where there is little spread in the signal noise ratios of the assets, and high power under the alternative where a single asset has high signal noise ratio.

Recent progress in the field of artificial intelligence, machine learning and also in computer industry resulted in the ongoing boom of using these techniques as applied to solving complex tasks in both science and industry. Same is, of course, true for the financial industry and mathematical finance. In this paper we consider a classical problem of mathematical finance - calibration of option pricing models to market data, as it was recently drawn some attention of the financial society in the context of deep learning and artificial neural networks. We highlight some pitfalls in the existing approaches and propose resolutions that improve both performance and accuracy of calibration. We also address a problem of no-arbitrage pricing when using a trained neural net, that is currently ignored in the literature.

We provide a framework for determining the centralities of agents in a broad family of random networks. Current understanding of network centrality is largely restricted to deterministic settings, but practitioners frequently use random network models to accommodate data limitations or prove asymptotic results. Our main theorems show that on large random networks, centrality measures are close to their expected values with high probability. We illustrate the economic consequences of these results by presenting three applications: (1) In network formation models based on community structure (called stochastic block models), we show network segregation and differences in community size produce inequality. Benefits from peer effects tend to accrue disproportionately to bigger and better-connected communities. (2) When link probabilities depend on geography, we can compute and compare the centralities of agents in different locations. (3) In models where connections depend on several independent characteristics, we give a formula that determines centralities 'characteristic-by-characteristic'. The basic techniques from these applications, which use the main theorems to reduce questions about random networks to deterministic calculations, extend to many network games.

Mid-price movement prediction based on limit order book (LOB) data is a challenging task due to the complexity and dynamics of the LOB. So far, there have been very limited attempts for extracting relevant features based on LOB data. In this paper, we address this problem by designing a new set of handcrafted features and performing an extensive experimental evaluation on both liquid and illiquid stocks. More specifically, we implement a new set of econometrical features that capture statistical properties of the underlying securities for the task of mid-price prediction. Moreover, we develop a new experimental protocol for online learning that treats the task as a multi-objective optimization problem and predicts i) the direction of the next price movement and ii) the number of order book events that occur until the change takes place. In order to predict the mid-price movement, the features are fed into nine different deep learning models based on multi-layer perceptrons (MLP), convolutional neural networks (CNN) and long short-term memory (LSTM) neural networks. The performance of the proposed method is then evaluated on liquid and illiquid stocks, which are based on TotalView-ITCH US and Nordic stocks, respectively. For some stocks, results suggest that the correct choice of a feature set and a model can lead to the successful prediction of how long it takes to have a stock price movement.

We consider a two-person trading game in continuous time whereby each player chooses a constant rebalancing rule $b$ that he must adhere to over $[0,t]$. If $V_t(b)$ denotes the final wealth of the rebalancing rule $b$, then Player 1 (the `numerator player') picks $b$ so as to maximize $\mathbb{E}[V_t(b)/V_t(c)]$, while Player 2 (the `denominator player') picks $c$ so as to minimize it. In the unique Nash equilibrium, both players use the continuous-time Kelly rule $b^*=c^*=\Sigma^{-1}(\mu-r\textbf{1})$, where $\Sigma$ is the covariance of instantaneous returns per unit time, $\mu$ is the drift vector of the stock market, and $\textbf{1}$ is a vector of ones. Thus, even over very short intervals of time $[0,t]$, the desire to perform well relative to other traders leads one to adopt the Kelly rule, which is ordinarily derived by maximizing the asymptotic exponential growth rate of wealth. Hence, we find agreement with Bell and Cover's (1988) result in discrete time.

Supply chains lend themselves to blockchain technology, but certain challenges remain, especially around invoice financing. For example, the further a supplier is removed from the final consumer product, the more difficult it is to get their invoices financed. Moreover, for competitive reasons, retailers and manufacturers do not want to disclose their supply chains. However, upstream suppliers need to prove that they are part of a `stable' supply chain to get their invoices financed, which presents the upstream suppliers with huge, and often unsurmountable, obstacles to get the necessary finance to fulfil the next order, or to expand their business. Using a fictitious supply chain use case, which is based on a real world use case, we demonstrate how these challenges have the potential to be solved by combining more advanced and specialised blockchain technologies with other technologies such as Artificial Intelligence. We describe how atomic crosschain functionality can be utilised across private blockchains to retrieve the information required for an invoice financier to make informed decisions under uncertainty, and consider the effect this decision has on the overall stability of the supply chain.

Market sectors play a key role in the efficient flow of capital through the modern Global economy. We analyze existing sectorization heuristics, and observe that the most popular - the GICS (which informs the S&P 500), and the NAICS (published by the U.S. Government) - are not entirely quantitatively driven, but rather appear to be highly subjective and rooted in dogma. Building on inferences from analysis of the capital structure irrelevance principle and the Modigliani-Miller theoretic universe conditions, we postulate that corporation fundamentals - particularly those components specific to the Modigliani-Miller universe conditions - would be optimal descriptors of the true economic domain of operation of a company. We generate a set of potential candidate learned sector universes by varying the linkage method of a hierarchical clustering algorithm, and the number of resulting sectors derived from the model (ranging from 5 to 19), resulting in a total of 60 candidate learned sector universes. We then introduce reIndexer, a backtest-driven sector universe evaluation research tool, to rank the candidate sector universes produced by our learned sector classification heuristic. This rank was utilized to identify the risk-adjusted return optimal learned sector universe as being the universe generated under CLINK (i.e. complete linkage), with 17 sectors. The optimal learned sector universe was tested against the benchmark GICS classification universe with reIndexer, outperforming on both absolute portfolio value, and risk-adjusted return over the backtest period. We conclude that our fundamentals-driven Learned Sector classification heuristic provides a superior risk-diversification profile than the status quo classification heuristic.

We introduce a computational framework for dynamic portfolio valuation and risk management building on machine learning with kernels. We learn the replicating martingale of a portfolio from a finite sample of its terminal cumulative cash flow. The learned replicating martingale is given in closed form thanks to a suitable choice of the kernel. We develop an asymptotic theory and prove convergence and a central limit theorem. We also derive finite sample error bounds and concentration inequalities. Numerical examples show good results for a relatively small training sample size.

We study discretizations of polynomial processes using finite state Markov processes satisfying suitable moment matching conditions. The states of these Markov processes together with their transition probabilities can be interpreted as Markov cubature rules. The polynomial property allows us to study such rules using algebraic techniques. Markov cubature rules aid the tractability of path-dependent tasks such as American option pricing in models where the underlying factors are polynomial processes.

We study dynamic optimal portfolio allocation for monotone mean--variance preferences in a general semimartingale model. Armed with new results in this area we revisit the work of Cui, Li, Wang and Zhu (2012, MAFI) and fully characterize the circumstances under which one can set aside a non-negative cash flow while simultaneously improving the mean--variance efficiency of the left-over wealth. The paper analyzes, for the first time, the monotone hull of the Sharpe ratio and highlights its relevance to the problem at hand.

This paper investigates the effects of the launch of Bitcoin futures on the intraday volatility of Bitcoin. Based on one-minute price data collected from four cryptocurrency exchanges, we first examine the change in realized volatility after the introduction of Bitcoin futures to investigate their aggregate effects on the intraday volatility of Bitcoin. We then analyze the effects in more detail utilizing the discrete Fourier transform. We show that although the Bitcoin market became more volatile immediately after the introduction of Bitcoin futures, over time it has become more stable than it was before the introduction.

In this paper, which is the third installment of the author's trilogy on margin loan pricing, we analyze $1,367$ monthly observations of the U.S. broker call money rate, which is the interest rate at which stock brokers can borrow to fund their margin loans to retail clients. We describe the basic features and mean-reverting behavior of this series and juxtapose the empirically-derived laws of motion with the author's prior theories of margin loan pricing (Garivaltis 2019a-b). This allows us to derive stochastic differential equations that govern the evolution of the margin loan interest rate and the leverage ratios of sophisticated brokerage clients (namely, continuous time Kelly gamblers). Finally, we apply Merton's (1974) arbitrage theory of corporate liability pricing to study theoretical constraints on the risk premia that could be generated in the market for call money. Apparently, if there is no arbitrage in the U.S. financial markets, the implication is that the total volume of call loans must constitute north of $70\%$ of the value of all leveraged portfolios.

This paper supplies two possible resolutions of Fortune's (2000) margin-loan pricing puzzle. Fortune (2000) noted that the margin loan interest rates charged by stock brokers are very high in relation to the actual (low) credit risk and the cost of funds. If we live in the Black-Scholes world, the brokers are presumably making arbitrage profits by shorting dynamically precise amounts of their clients' portfolios. First, we extend Fortune's (2000) application of Merton's (1974) no-arbitrage approach to allow for brokers that can only revise their hedges finitely many times during the term of the loan. We show that extremely small differences in the revision frequency can easily explain the observed variation in margin loan pricing. In fact, four additional revisions per three-day period serve to explain all of the currently observed heterogeneity. Second, we study monopolistic (or oligopolistic) margin loan pricing by brokers whose clients are continuous-time Kelly gamblers. The broker solves a general stochastic control problem that yields simple and pleasant formulas for the optimal interest rate and the net interest margin. If the author owned a brokerage, he would charge an interest rate of $(r+\nu)/2-\sigma^2/4$, where $r$ is the cost of funds, $\nu$ is the compound-annual growth rate of the S&P 500 index, and $\sigma$ is the volatility.