We estimate a general non-linear asset pricing model with deep neural networks applied to all U.S. equity data combined with a substantial set of macroeconomic and firm-specific information. Our crucial innovation is the use of the no-arbitrage condition as part of the neural network algorithm. We estimate the stochastic discount factor (SDF or pricing kernel) that explains all asset prices from the conditional moment constraints implied by no-arbitrage. For this purpose, we combine three different deep neural network structures in a novel way: a feedforward network to capture non-linearities, a recurrent Long-Short-Term-Memory network to find a small set of economic state processes and a generative adversarial network to identify the portfolio strategies with the least explained pricing information. Our model allows us to understand what are the key factors that drive asset prices to identify mispricing of stocks and to generate the mean-variance efficient portfolio. Empirically, our approach outperforms out-of-sample all other benchmark approaches: Our optimal portfolio has an annual Sharpe Ratio of 2.6, we explain 8% of the variation in individual stock returns and explain over 90% of average returns for all anomaly sorted portfolios.

We introduce the notion of rough local stochastic volatility models, extending the classical concept to the case where volatility is driven by some Volterra process. In this setting, we show that the pricing function is the solution to a path-dependent PDE, for which we develop a numerical scheme based on Deep Learning techniques. Numerical simulations suggest that the latter is extremely efficient, and provides a good alternative to classical Monte Carlo simulations.

Valuation adjustments are nowadays a common practice to include credit and liquidity effects in option pricing. Funding costs arising from collateral procedures, hedging strategies and taxes are added to option prices to take into account the production cost of financial contracts so that a profitability analysis can be reliably assessed. In particular, when dealing with linear products, we need a precise evaluation of such contributions since bid-ask spreads may be very tight. In this paper we start from a general pricing framework inclusive of valuation adjustments to derive simple evaluation formulae for the relevant case of total return equity swaps when stock lending and borrowing is adopted as hedging strategy.

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.

The rough Bergomi (rBergomi) model, introduced recently in [4], is a promising rough volatility model in quantitative finance. This new model exhibits consistent results with the empirical fact of implied volatility surfaces being essentially time-invariant. This model also has the ability to capture the term structure of skew observed in equity markets. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a time-consuming task. To overcome this issue, we design a novel, alternative, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ) as in [22], and ii) Quasi Monte Carlo (QMC). Both techniques are coupled with Brownian bridge construction and Richardson extrapolation. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e. to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.

We study distributions of realized variance (squared realized volatility) and squared implied volatility, as represented by VIX and VXO indices. We find that Generalized Beta distribution provide the best fits. These fits are much more accurate for realized variance than for squared VIX and VXO -- possibly another indicator that the latter have deficiencies in predicting the former. We also show that there are noticeable differences between the distributions of the 1970-2017 realized variance and its 1990-2017 portion, for which VIX and VXO became available. This may be indicative of a feedback effect that implied volatility has on realized volatility. We also discuss the distribution of the difference between squared implied volatility and realized variance and show that, at the basic level, it is consistent with Pearson's correlations obtained from linear regression.

This chapter examines the geographical meaning of the Sahel, its fluid boundaries, and its spatial dynamics. Unlike other approaches that define the Sahel as a bioclimatic zone or as an ungoverned area, it shows that the Sahel is primarily a space of circulation in which uncertainty has historically been overcome by mobility. The first part of the paper discusses how pre-colonial empires relied on a network of markets and cities that facilitated trade and social relationships across the region and beyond. The second part explores changing regional mobility patterns precipitated by colonial powers and the new approach they developed to control networks and flows. The third part discusses the contradiction between the mobile strategies adopted by local herders, farmers and traders in the Sahel and the territorial development initiatives of modern states and international donors. Particular attention is paid in the last section to how the Sahel was progressively redefined through a security lens.

We derive new formulas for the price of the European call and put options in the Black-Scholes model, under the form of uniformly convergent series generalizing previously known approximations. We also provide precise boundaries for the convergence speed and apply the results to the calculation of hedge parameters (Greeks).

This paper proposes a theory of pricing consistent with two well-documented patterns: customers care about fairness, and firms take these concerns into account when they set prices. The theory assumes that customers find a price unfair when it carries a high markup over cost, and that customers dislike unfair prices. Since markups are not observable, customers must extract them from prices. The theory assumes that customers infer less than rationally: when a price rises after an increase in marginal cost, customers partially misattribute the higher price to a higher markup---which they find unfair. Firms anticipate this response and trim their price increases, which reduces the passthrough of marginal costs into prices below one: prices are somewhat rigid. Embedded in a New Keynesian model---as a replacement of Calvo pricing---our theory produces monetary nonneutrality. When monetary policy loosens and inflation rises, customers misperceive markups as higher and feel unfairly treated; firms mitigate the perceived unfairness of prices by reducing their markups, which in general equilibrium leads to higher output.

We demonstrate an application of risk-sensitive reinforcement learning to optimizing execution in limit order book markets. We represent taking order execution decisions based on limit order book knowledge by a Markov Decision Process; and train a trading agent in a market simulator, which emulates multi-agent interaction by synthesizing market response to our agent's execution decisions from historical data. Due to market impact, executing high volume orders can incur significant cost. We learn trading signals from market microstructure in presence of simulated market response and derive explainable decision-tree-based execution policies using risk-sensitive Q-learning to minimize execution cost subject to constraints on cost variance.

Thanks to the recent availability of comprehensive and detailed online databases of startup companies, it has become possible to more directly investigate startup ecosystems i.e. startup populations in specific regions. In this paper, we analyze the emergence of 20+ such ecosystems in Europe and the USA, with a specific focus on their sectoral diversity. Analyzing the sectoral landscapes of these ecosystems using a new visualization tool indeed highlights marked differences in terms of diversity, which we characterize using metrics derived from ecological sciences. Numerical simulations suggest that the emerging diversity of startup ecosystems can be explained using a simple preferential attachment model based on sectoral funding.

We study a learning dynamic model of routing (congestion) games to explore how an increase in the total demand influences system performance. We focus on non-atomic routing games with two parallel edges of linear cost, where all agents evolve using Multiplicative Weights Updates with a fixed learning rate. Previous game-theoretic equilibrium analysis suggests that system performance is close to optimal in the large population limit, as seen by the Price of Anarchy reduction. In this work, however, we reveal a rather undesirable consequence of non-equilibrium phenomena driven by population increase. As the total demand rises, we prove that the learning dynamics unavoidably become non-equilibrating, typically chaotic. The Price of Anarchy predictions of near-optimal performance no longer apply. To the contrary, the time-average social cost may converge to its worst possible value in the large population limit.

Every system has a carrying capacity, above which the dynamics is non-equilibrating. If the equilibrium flow is a symmetric $50-50\%$ split, the system exhibits one period-doubling bifurcation. A single periodic attractor of period two replaces the attracting fixed point when the demand exceeds the carrying capacity. In general, for asymmetric equilibrium flows, increasing the demand destabilizes the system, so that the system eventually becomes Li-Yorke chaotic with positive topological entropy. This demand-driven instability emerges from any pair of linear cost functions. Remarkably, in any non-equilibrating regime, the time-average flows on the edges converge {\it exactly} to the equilibrium flows, a property akin to no-regret learning in zero-sum games. Our results extend to any sequence of shrinking learning rates, e.g., $1/\sqrt{T}$, by allowing for a dynamically increasing population size.