AbstractThis paper studies the well‐posedness of a class of nonlocal fully nonlinear parabolic systems, which nest the equilibrium Hamilton–Jacobi–Bellman (HJB) systems that characterize the time‐consistent Nash equilibrium point of a stochastic differential game (SDG) with time‐inconsistent (TIC) preferences. The nonlocality of the parabolic systems stems from the flow feature (controlled by an external temporal parameter) of the systems. This paper proves the existence and uniqueness results as well as the stability analysis for the solutions to such systems. We first obtain the results for the linear cases for an arbitrary time horizon and then extend them to the quasilinear and fully nonlinear cases under some suitable conditions. Two examples of TIC SDG are provided to illustrate financial applications with global solvability. Moreover, with the well‐posedness results, we establish a general multidimensional Feynman–Kac formula in the presence of nonlocality (time inconsistency).

AbstractWe study a game between liquidity provider (LP) and liquidity taker agents interacting in an over‐the‐counter market, for which the typical example is foreign exchange. We show how a suitable design of parameterized families of reward functions coupled with shared policy learning constitutes an efficient solution to this problem. By playing against each other, our deep‐reinforcement‐learning‐driven agents learn emergent behaviors relative to a wide spectrum of objectives encompassing profit‐and‐loss, optimal execution, and market share. In particular, we find that LPs naturally learn to balance hedging and skewing, where skewing refers to setting their buy and sell prices asymmetrically as a function of their inventory. We further introduce a novel RL‐based calibration algorithm, which we found performed well at imposing constraints on the game equilibrium. On the theoretical side, we are able to show convergence rates for our multi‐agent policy gradient algorithm under a transitivity assumption, closely related to generalized ordinal potential games.

AbstractThis paper studies an equity market of stochastic dimension, where the number of assets fluctuates over time. In such a market, we develop the fundamental theorem of asset pricing, which provides the equivalence of the following statements: (i) there exists a supermartingale numéraire portfolio; (ii) each dissected market, which is of a fixed dimension between dimensional jumps, has locally finite growth; (iii) there is no arbitrage of the first kind; (iv) there exists a local martingale deflator; (v) the market is viable. We also present the optional decomposition theorem, which characterizes a given nonnegative process as the wealth process of some investment‐consumption strategy. Furthermore, similar results still hold in an open market embedded in the entire market of stochastic dimension, where investors can only invest in a fixed number of large capitalization stocks. These results are developed in an equity market model where the price process is given by a piecewise continuous semimartingale of stochastic dimension. Without the continuity assumption on the price process, we present similar results but without explicit characterization of the numéraire portfolio.

AbstractExpected shortfall (ES, also known as CVaR) is the most important coherent risk measure in finance, insurance, risk management, and engineering. Recently, Wang and Zitikis (2021) put forward four economic axioms for portfolio risk assessment and provide the first economic axiomatic foundation for the family of ES. In particular, the axiom of no reward for concentration (NRC) is arguably quite strong, which imposes an additive form of the risk measure on portfolios with a certain dependence structure. We move away from the axiom of NRC by introducing the notion of concentration aversion, which does not impose any specific form of the risk measure. It turns out that risk measures with concentration aversion are functions of ES and the expectation. Together with the other three standard axioms of monotonicity, translation invariance and lower semicontinuity, concentration aversion uniquely characterizes the family of ES. In addition, we establish an axiomatic foundation for the problem of mean‐ES portfolio selection and new explicit formulas for convex and consistent risk measures. Finally, we provide an economic justification for concentration aversion via a few axioms on the attitude of a regulator towards dependence structures.

AbstractOvernight rates, such as the Secured Overnight Financing Rate (SOFR) in the United States, are central to the current reform of interest rate benchmarks. A striking feature of overnight rates is the presence of jumps and spikes occurring at predetermined dates due to monetary policy interventions and liquidity constraints. This corresponds to stochastic discontinuities (i.e., discontinuities occurring at ex ante known points in time) in their dynamics. In this work, we propose a term structure modeling framework based on overnight rates and characterize absence of arbitrage in a generalized Heath–Jarrow–Morton (HJM) setting. We extend the classical short‐rate approach to accommodate stochastic discontinuities, developing a tractable setup driven by affine semimartingales. In this context, we show that simple specifications allow to capture stylized facts of the jump behavior of overnight rates. In a Gaussian setting, we provide explicit valuation formulas for bonds and caplets. Furthermore, we investigate hedging in the sense of local risk‐minimization when the underlying term structures feature stochastic discontinuities.

AbstractIn financial engineering, prices of financial products are computed approximately many times each trading day with (slightly) different parameters in each calculation. In many financial models such prices can be approximated by means of Monte Carlo (MC) simulations. To obtain a good approximation the MC sample size usually needs to be considerably large resulting in a long computing time to obtain a single approximation. A natural deep learning approach to reduce the computation time when new prices need to be calculated as quickly as possible would be to train an artificial neural network (ANN) to learn the function which maps parameters of the model and of the financial product to the price of the financial product. However, empirically it turns out that this approach leads to approximations with unacceptably high errors, in particular when the error is measured in the ‐norm, and it seems that ANNs are not capable to closely approximate prices of financial products in dependence on the model and product parameters in real life applications. This is not entirely surprising given the high‐dimensional nature of the problem and the fact that it has recently been proved for a large class of algorithms, including the deep learning approach outlined above, that such methods are in general not capable to overcome the curse of dimensionality for such approximation problems in the ‐norm. In this article we introduce a new numerical approximation strategy for parametric approximation problems including the parametric financial pricing problems described above and we illustrate by means of several numerical experiments that the introduced approximation strategy achieves a very high accuracy for a variety of high‐dimensional parametric approximation problems, even in the ‐norm. A central aspect of the approximation strategy proposed in this article is to combine MC algorithms with machine learning techniques to, roughly speaking, learn the random variables (LRV) in MC simulations. In other words, we employ stochastic gradient descent (SGD) optimization methods not to train parameters of standard ANNs but instead to learn random variables appearing in MC approximations. In that sense, the proposed LRV strategy has strong links to Quasi‐Monte Carlo (QMC) methods as well as to the field of algorithm learning. Our numerical simulations strongly indicate that the LRV strategy might indeed be capable to overcome the curse of dimensionality in the ‐norm in several cases where the standard deep learning approach has been proven not to be able to do so. This is not a contradiction to the established lower bounds mentioned above because this new LRV strategy is outside of the class of algorithms for which lower bounds have been established in the scientific literature. The proposed LRV strategy is of general nature and not only restricted to the parametric financial pricing problems described above, but applicable to a large class of approximation problems. In this article we numerically test the LRV strategy in the case of the pricing of European call options in the Black‐Scholes model with one underlying asset, in the case of the pricing of European worst‐of basket put options in the Black‐Scholes model with three underlying assets, in the case of the pricing of European average put options in the Black‐Scholes model with three underlying assets and knock‐in barriers, as well as in the case of stochastic Lorentz equations. For these examples the LRV strategy produces highly convincing numerical results when compared with standard MC simulations, QMC simulations using Sobol sequences, SGD‐trained shallow ANNs, and SGD‐trained deep ANNs.

AbstractThe robustness of risk measures to changes in underlying loss distributions (distributional uncertainty) is of crucial importance in making well‐informed decisions. In this paper, we quantify, for the class of distortion risk measures with an absolutely continuous distortion function, its robustness to distributional uncertainty by deriving its largest (smallest) value when the underlying loss distribution has a known mean and variance and, furthermore, lies within a ball—specified through the Wasserstein distance—around a reference distribution. We employ the technique of isotonic projections to provide for these distortion risk measures a complete characterization of sharp bounds on their value, and we obtain quasi‐explicit bounds in the case of Value‐at‐Risk and Range‐Value‐at‐Risk. We extend our results to account for uncertainty in the first two moments and provide applications to portfolio optimization and to model risk assessment.

AbstractMost insurance contracts are inherently linked to financial markets, be it via interest rates, or—as hybrid products like equity‐linked life insurance and variable annuities—directly to stocks or indices. However, insurance contracts are not for trade except sometimes as surrender to the selling office. This excludes the situation of arbitrage by buying and selling insurance contracts at different prices. Furthermore, the insurer uses private information on top of the publicly available one about financial markets. This paper provides a study of the consistency of insurance contracts in connection with trades in the financial market with explicit mention of the information involved.By defining strategies on an insurance portfolio and combining them with financial trading strategies, we arrive at the notion of insurance–finance arbitrage (IFA). In analogy to the classical fundamental theorem of asset pricing, we give a fundamental theorem on the absence of IFA, leading to the existence of an insurance–finance‐consistent probability. In addition, we study when this probability gives the expected discounted cash‐flows required by the EIOPA best estimate.The generality of our approach allows to incorporate many important aspects, like mortality risk or general levels of dependence between mortality and stock markets. Utilizing the theory of enlargements of filtrations, we construct a tractable framework for insurance–finance consistent valuation.

AbstractWe employ deep learning in forecasting high‐frequency returns at multiple horizons for 115 stocks traded on Nasdaq using order book information at the most granular level. While raw order book states can be used as input to the forecasting models, we achieve state‐of‐the‐art predictive accuracy by training simpler “off‐the‐shelf” artificial neural networks on stationary inputs derived from the order book. Specifically, models trained on order flow significantly outperform most models trained directly on order books. Using cross‐sectional regressions, we link the forecasting performance of a long short‐term memory network to stock characteristics at the market microstructure level, suggesting that “information‐rich” stocks can be predicted more accurately. Finally, we demonstrate that the effective horizon of stock specific forecasts is approximately two average price changes.

AbstractWe study the mean–variance hedging of an American‐type contingent claim that is exercised at a random time in a Markovian setting. This problem is motivated by applications in the areas of employee stock option valuation, credit risk, or equity‐linked life insurance policies with an underlying risky asset value guarantee. Our analysis is based on dynamic programming and uses PDE techniques. In particular, we prove that the complete solution to the problem can be expressed in terms of the solution to a system of one quasi‐linear parabolic PDE and two linear parabolic PDEs. Using a suitable iterative scheme involving linear parabolic PDEs and Schauder's interior estimates for parabolic PDEs, we show that each of these PDEs has a classical C1, 2 solution. Using these results, we express the claim's mean–variance hedging value that we derive as its expected discounted payoff with respect to an equivalent martingale measure that does not coincide with the minimal martingale measure, which, in the context that we consider, identifies with the minimum entropy martingale measure as well as the variance‐optimal martingale measure. Furthermore, we present a numerical study that illustrates aspects of our theoretical results.