# 2022-23

By *Johannes Muhle-Karbe, Xiaofei Shi, and Chen Yang*

Mathematics of Operations Research | 2023 | 48(3), 1423-1453.

We study a risk-sharing economy where an arbitrary number of heterogeneous agents trades an arbitrary number of risky assets subject to quadratic transaction costs. For linear state dynamics, the forward–backward stochastic differential equations characterizing equilibrium asset prices and trading strategies in this context reduce to a coupled system of matrix-valued Riccati equations. We prove the existence of a unique global solution and provide explicit asymptotic expansions that allow us to approximate the corresponding equilibrium for small transaction costs. These tractable approximation formulas make it feasible to calibrate the model to time series of prices and trading volume, and to study the cross section of liquidity premia earned by assets with higher and lower trading costs. This is illustrated by an empirical case study.

By *Xiaofei Shi, Daran Xu, and Zhanhao Zhang*

Digital Finance, Special Issue on Deep Learning in Finance | 2023 | 5(1), 113-147.

This work studies the deep learning-based numerical algorithms for optimal hedging problems in markets with general convex transaction costs. Our main focus is on how these algorithms scale with the length of the trading time horizon. Based on the comparison results of the FBSDE solver by Han, Jentzen, and E (2018) and the Deep Hedging algorithm by Buehler, Gonon, Teichmann, and Wood (2019), we propose a Stable-Transfer Hedging (ST-Hedging) algorithm, to aggregate the convenience of the leading-order approximation formulas and the accuracy of the deep learning-based algorithms. Our ST-Hedging algorithm achieves the same state-of-the-art performance in short and moderately long time horizon as FBSDE solver and Deep Hedging, and generalize well to long time horizon when previous algorithms become suboptimal. With the transfer learning technique, ST-Hedging drastically reduce the training time, and shows great scalability to high-dimensional settings. This opens up new possibilities in model-based deep learning algorithms in economics, finance, and operational research, which takes advantage of the domain expert knowledge and the accuracy of the learning-based methods.

By *S. Campbell, and T.-K. L. Wong*

SIAM Journal on Financial Mathematics | 2022 | 13 (2), 576-618

We develop a concrete and fully implementable approach to the optimization of functionally generated portfolios in stochastic portfolio theory. The main idea is to optimize over a family of rank-based portfolios parameterized by an exponentially concave function on the unit interval. This choice can be motivated by the long term stability of the capital distribution observed in large equity markets and allows us to circumvent the curse of dimensionality. The resulting optimization problem, which is convex, allows for various regularizations and constraints to be imposed on the generating function. We prove an existence and uniqueness result for our optimization problem and provide a stability estimate in terms of a Wasserstein metric of the input measure. Then we formulate a discretization which can be implemented numerically using available software packages and analyze its approximation error. Finally, we present empirical examples using CRSP data from the U.S. stock market, including the performance of the portfolios allowing for dividends, defaults, and transaction costs.

By *Marcel Nutz and Yuchong Zhang*

Mathematics of Operations Research | 2023 | 48(2): 1095-1118

We formulate a mean field game where each player stops a privately observed Brownian motion with absorption. Players are ranked according to their level of stopping and rewarded as a function of their relative rank. There is a unique mean field equilibrium, and it is shown to be the limit of associated n-player games. Conversely, the mean field strategy induces n-player ε-Nash equilibria for any continuous reward function—but not for discontinuous ones. In a second part, we study the problem of a principal who can choose how to distribute a reward budget over the ranks and aims to maximize the performance of the median player. The optimal reward design (contract) is found in closed form, complementing the merely partial results available in the n-player case. We then analyze the quality of the mean field design when used as a proxy for the optimizer in the n-player game. Surprisingly, the quality deteriorates dramatically as n grows. We explain this with an asymptotic singularity in the induced n-player equilibrium distributions.

By *Silvana Pesenti, and Sebastian Jaimungal*

SIAM J. Financial Mathematics | 2023 | accepted

We study the problem of active portfolio management where an investor aims to outperform a benchmark strategy's risk profile while not deviating too far from it. Specifically, an investor considers alternative strategies whose terminal wealth lie within a Wasserstein ball surrounding a benchmark's -- being distributionally close -- and that have a specified dependence/copula -- tying state-by-state outcomes -- to it. The investor then chooses the alternative strategy that minimises a distortion risk measure of terminal wealth. In a general (complete) market model, we prove that an optimal dynamic strategy exists and provide its characterisation through the notion of isotonic projections.

By *Carole Bernard, Silvana Pesenti, and Steven Vanduffel*

Mathematical Finance | 2023 | accepted

The 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.

By *Tobias Fissler, and Silvana Pesenti*

European Journal of Operational Research | 2023 | 307(3), 1408-1423

Since elicitable functionals typically possess rich classes of (strictly) consistent scoring functions, we demonstrate how Murphy diagrams can provide a picture of all score-based sensitivity measures. We discuss the family of score-based sensitivities for the mean functional (of which the Sobol indices are a special case) and risk functionals such as Value-at-Risk, and the pair Value-at-Risk and Expected Shortfall. The sensitivity measures are illustrated using numerous examples, including the Ishigami–Homma test function. In a simulation study, estimation of score-based sensitivities for a non-linear insurance portfolio is performed using neural nets.

# Previous Publications

**Reward Design in Risk-Taking Contests**

by *Marcel Nutz, and Yuchong Zhang*

SIAM Journal on Financial Mathematics | 2022 | 13(1), 129-146

*Short Summary: *Following the risk-taking model of Seel and Strack, $n$ players decide when to stop privately observed Brownian motions with drift and absorption at zero. They are then ranked according to their level of stopping and paid a rank-dependent reward. We study the problem of a principal who aims to induce a desirable equilibrium performance of the players by choosing how much reward is attributed to each rank. Specifically, we determine optimal reward schemes for principals interested in the average performance and the performance at a given rank. While the former can be related to reward inequality in the Lorenz sense, the latter can have a surprising shape.

**Robust Risk-Aware Reinforcement Learning**

*by Sebastian Jaimungal, Silvana M. Pesenti, Ye Sheng Wang, and Hariom Tatsat*

SIAM Journal on Financial Mathematics | 2021 | 13 (1), 213-226

*Short Summary: *We present a reinforcement learning (RL) approach for robust optimization of risk-aware performance criteria. To allow agents to express a wide variety of risk-reward profiles, we assess the value of a policy using rank dependent expected utility (RDEU). RDEU allows agents to seek gains, while simultaneously protecting themselves against downside risk. To robustify optimal policies against model uncertainty, we assess a policy not by its distribution but rather by the worst possible distribution that lies within a Wasserstein ball around it. Thus, our problem formulation may be viewed as an actor/agent choosing a policy (the outer problem) and the adversary then acting to worsen the performance of that strategy (the inner problem). We develop explicit policy gradient formulae for the inner and outer problems and show their efficacy on three prototypical financial problems: robust portfolio allocation, benchmark optimization, and statistical arbitrage.

**Teamwise Mean Field Competitions**

by *Xiang Yu, Yuchong Zhang and Zhou Zhou*

Applied Mathematics & Optimization | 2021 | 84, 903-942

*Short Summary: *This paper studies competitions with rank-based reward among a large number of teams. Within each sizable team, we consider a mean-field contribution game in which each team member contributes to the jump intensity of a common Poisson project process; across all teams, a mean field competition game is formulated on the rank of the completion time, namely the jump time of Poisson project process, and the reward to each team is paid based on its ranking. On the layer of teamwise competition game, three optimization problems are introduced when the team size is determined by: (i) the team manager; (ii) the central planner; (iii) the team members’ voting as partnership. We propose a relative performance criteria for each team member to share the team’s reward and formulate some special cases of mean field games of mean field games, which are new to the literature. In all problems with homogeneous parameters, the equilibrium control of each worker and the equilibrium or optimal team size can be computed in an explicit manner, allowing us to analytically examine the impacts of some model parameters and discuss their economic implications. Two numerical examples are also presented to illustrate the parameter dependence and comparison between different team size decision making.

**A new class of severity regression models with an application to IBNR prediction**

by *Fung, T.C., Badescu, A., and Lin, X.S.*

North American Actuarial Journal | 2021 (forthcoming)

*Short Summary:* This paper proposes a transformed Gamma logit-weighted mixture of experts (TG-LRMoE) model for severity regression.

**Cascade Sensitivity Measures**

by *Silvana Pesenti, Pietro Millossovich, and Andreas Tsanakas*

Risk Analysis | 2021 (accepted)

*Short Summary: *In risk analysis, sensitivity measures quantify the extent to which the probability distribution of a model output is affected by changes (stresses) in individual random input factors. For input factors that are statistically dependent, we argue that a stress on one input should also precipitate stresses in other input factors. We introduce a novel sensitivity measure, termed \textit{cascade sensitivity}, defined as a derivative of a risk measure applied on the output, in the direction of an input factor. The derivative is taken after suitably transforming the random vector of inputs, thus explicitly capturing the direct impact of the stressed input factor, as well as indirect effects via other inputs. Furthermore, alternative representations of the cascade sensitivity measure are derived, allowing us to address practical issues, such as incomplete specification of the model and high computational costs. The applicability of the methodology is illustrated through the analysis of a commercially used insurance risk model.

**Conditional Optimal Stopping: A Time-Inconsistent Optimization**

by *Marcel Nutz, and Yuchong Zhang*

Annals of Applied Probability | 2020 | 30(4), 1669-1692

*Short Summary:* Inspired by recent work of P.-L. Lions on conditional optimal control, we introduce a problem of optimal stopping under bounded rationality: the objective is the expected payoff at the time of stopping, conditioned on another event. For instance, an agent may care only about states where she is still alive at the time of stopping, or a company may condition on not being bankrupt. We observe that conditional optimization is time-inconsistent due to the dynamic change of the conditioning probability and develop an equilibrium approach in the spirit of R. H. Strotz’ work for sophisticated agents in discrete time. Equilibria are found to be essentially unique in the case of a finite time horizon whereas an infinite horizon gives rise to nonuniqueness and other interesting phenomena. We also introduce a theory which generalizes the classical Snell envelope approach for optimal stopping by considering a pair of processes with Snell-type properties.

**Efficient dynamic hedging for large variable annuity portfolios with multiple underlying assets**

by *Lin, X.S. and Yang, S.*

ASTIN Bulletin, The Journal of the IAA | 2020 | 50(3), 913-957

*Short Summary: *In this paper, we extend the surrogate model-assisted nest simulation approach in Lin and Yang [(2020) to efficiently calculate the total VA liability and the partial dollar Deltas for large VA portfolios with multiple underlying assets, and to perform dynamic hedging and profit and loss (P&L) analysis for the portfolio.

**Fast and efficient nested simulation for large variable annuity portfolios: A surrogate modeling approach**

by *Lin, X.S. and Yang, S.*

Insurance: Mathematics and Economics | 2020 | 91, 85-103

*Short Summary: *The nested-simulation is commonly used for calculating the predictive distribution of the total variable annuity (VA) liabilities of large VA portfolios. Due to the large numbers of policies, inner-loops and outer-loops, running the nested-simulation for a large VA portfolio is extremely time consuming and often prohibitive. In this paper, the use of surrogate models is incorporated into the nested-simulation algorithm so that the nested-simulation algorithm can be run much efficiently. The algorithm enables to accurately approximate the predictive distribution of the total VA liability at a significantly reduced running time.

**Fitting multivariate Erlang mixtures to data: A roughness penalty approach**

by *Wenyong Gui, Rongtan Huang, and X Sheldon Lin*

Journal of Computational and Applied Mathematics | 2021(accepted)

*Short Summary: *In this paper, we propose a generalized expectation conditional maximization (GECM) algorithm that maximizes a penalized likelihood with a proposed roughness penalty.

**LRMoE.jl: a software package for flexible actuarial loss modelling using mixture of experts regression model**

by *Tseung, S.C, Badescu, A., Fung, T.C., and Lin, X.S.*

Annals of Actuarial Science | 2021 (forthcoming)

*Short Summary: *This paper introduces a new julia package, LRMoE, a statistical software tailor-made for actuarial applications which allows actuarial researchers and practitioners to model and analyze insurance loss frequencies and severities using the Logit-weighted Reduced Mixture-of-Experts (LRMoE) model.

**Scenario Weights for Importance Measurement (SWIM) – an R package for sensitivity analysis**

by *Silvana Pesenti, Alberto Bettini, Pietro Millossovich, and Andreas Tsanakas*

Annals of Actuarial Science | 2021 | 1/26/2021

*Short Summary: *The SWIM package implements a flexible sensitivity analysis framework, based primarily on results and tools developed by Pesenti et al. (2019). SWIM provides a stressed version of a stochastic model, subject to model components (random variables) fulfilling given probabilistic constraints (stresses). Possible stresses can be applied on moments, probabilities of given events, and risk measures such as Value-at-Risk and Expected Shortfall. SWIM operates upon a single set of simulated scenarios from a stochastic model, returning scenario weights, which encode the required stress and allow monitoring the impact of the stress on all model components. The scenario weights are calculated to minimise the relative entropy with respect to the baseline model, subject to the stress applied. As well as calculating scenario weights, the package provides tools for the analysis of stressed models, including plotting facilities and evaluation of sensitivity measures. SWIM does not require additional evaluations of the simulation model or explicit knowledge of its underlying statistical and functional relations; hence it is suitable for the analysis of black box models. The capabilities of SWIM are demonstrated through a case study of a credit portfolio model.

**Terminal Ranking Games**

by *Erhan Bayraktar, and Yuchong Zhang*

Mathematics of Operations Research | 2021 | Ahead of Print

*Short Summary: *We analyze a mean field tournament: a mean field game in which the agents receive rewards according to the ranking of the terminal value of their projects and are subject to cost of effort. Using Schrödinger bridges we are able to explicitly calculate the equilibrium. This allows us to identify the reward functions which would yield a desired equilibrium and solve several related mechanism design problems. We are also able to identify the effect of reward inequality on the players’ welfare as well as calculate the price of anarchy.

**Time-consistent conditional expectation under probability distortion**

by *Jin Ma, Ting-Kam Leonard Wong, and Jianfeng Zhang*

Mathematics of Operations Research | 2021

*Short Summary: *When the underlying probability distorted by a weighting function, we construct a nonlinear conditional expectation such that the tower property remains valid. This construction is of interest in time-inconsistent stochastic optimization problem.

**Algorithmic Trading, Stochastic Control, and Mutually Exciting Processes**

by *Álvaro Cartea*, *Sebastian Jaimungal*, and *Jason Ricci*

SIAM Review | 2018 | Issue: 60(3), 673–703

*Short Summary:* On electronic exchanges, orders tend to induce cross-excitation in market activity, e.g., when a buy order arrives it may induce increased activity of both buy and sell orders and induce changes in the limit order book. This paper develops a detailed model of this phonomena, and takes a mathematical look at the resulting optimal control problem.

**An IBNR-RBNS insurance risk model with marked Poisson arrivals**

by *Ahn S.*, *Badescu A.*, *Cheung E.*, *Kim Y.*

Insurance: Mathematics and Economics | 2018 | Issue: 79, 26-42

*Short Summary: *A connection between Mathematical Risk Theory and Stochastic Claim Reserving

**Cover's universal portfolio, stochastic portfolio theory and the numeraire portfolio**

by *Christa Cuchiero*, *Walter Schachermayer* and *Ting-Kam Leonard Wong*

Mathematical Finance | 2018

*Short Summary:* We study Cover's universal portfolio in the context of stochastic portfolio theory, where the market portfolio is the numeraire. Under suitable conditions, we prove that the universal portfolio is asymptotically optimal.

**Exponentially concave functions and a new information geometry**

by *Soumik Pal* and *Ting-Kam Leonard Wong*

Annals of Probability | Volume 46, Number 2 (2018), 1070-1113

*Short Summary:* This paper uncovers deep connections between optimal transport and information geometry. It develops the dual geometry of L-divergence which extends the classical Bregman divergence. Our geometry can be applied to determine the optimal rebalancing frequency of portfolios.

**Trading Algorithms with Learning in Latent Alpha Models Mathematical Finance **

by *Philippe Casgrain* and *Sebastian Jaimungal*

Mathematical Finance | 2018

*Short Summary*: How does one trade when markets are driven by factors you cannot observe? This paper formulates the problem as a partial information stochastic control problem, proves various theoretical results related to the problem, solves it, uses machine learning techniques to estimate model parameters, and runs simulations to illustrate the results.