Research articles for the 2019-09-03

A Relation between Short-Term and Long-Term Arbitrage
P. Liebrich

In this work a relation between a measure of short-term arbitrage in the market and the excess growth of portfolios as a notion of long-term arbitrage is established. The former originates from "Geometric Arbitrage Theory" and the latter from "Stochastic Portfolio Theory". Both aim to describe non-equilibrium effects in financial markets. Thereby, a connection between two different theoretical frameworks of arbitrage is drawn.

A Two-Stage Market Mechanism for Electricity with Renewable Generation
Nathan Dahlin,Rahul Jain

We consider a two stage market mechanism for trading electricity including renewable generation as an alternative to the widely used multi-settlement market structure. The two stage market structure allows for recourse decisions by the market operator, which is not possible in today's markets. We allow for different generation cost curves in the forward and the real-time stage. We have considered costs of demand response programs, and black outs but have ignored network structure for the sake of simplicity. Our first result is to show existence (by construction) of a sequential competitive equilibrium (SCEq) in such a two-stage market. We then argue social welfare properties of such an SCEq. We then design a market mechanism that achieves social welfare maximization when the market participants are non-strategic.

An intelligent financial portfolio trading strategy using deep Q-learning
Hyungjun Park,Min Kyu Sim,Dong Gu Choi

Portfolio traders strive to identify dynamic portfolio allocation schemes so that their total budgets are well allocated through the investment horizon. This study proposes a novel portfolio trading strategy in which an intelligent agent is trained to identify an optimal trading action by using an algorithm called deep Q-learning. This study formulates a portfolio trading process as a Markov decision process in which the agent can learn about the financial market environment, and it identifies a deep neural network structure as an approximation of the Q-function. To ensure applicability to real-world trading, we devise three novel techniques that are both reasonable and implementable. First, the agent's action space is modeled as a combinatorial action space of trading directions with prespecified trading sizes for each asset. Second, we introduce a mapping function that can replace an initially-determined action that may be infeasible with a feasible action that is reasonably close to the original, ideal action. Last, we introduce a technique by which an agent simulates all feasible actions in each state and learns about these experiences to derive a multi-asset trading strategy that best reflects financial data. To validate our approach, we conduct backtests for two representative portfolios and demonstrate superior results over the benchmark strategies.

Are Bitcoins price predictable? Evidence from machine learning techniques using technical indicators
Samuel Asante Gyamerah

The uncertainties in future Bitcoin price make it difficult to accurately predict the price of Bitcoin. Accurately predicting the price for Bitcoin is therefore important for decision-making process of investors and market players in the cryptocurrency market. Using historical data from 01/01/2012 to 16/08/2019, machine learning techniques (Generalized linear model via penalized maximum likelihood, random forest, support vector regression with linear kernel, and stacking ensemble) were used to forecast the price of Bitcoin. The prediction models employed key and high dimensional technical indicators as the predictors. The performance of these techniques were evaluated using mean absolute percentage error (MAPE), root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R-squared). The performance metrics revealed that the stacking ensemble model with two base learner (random forest and generalized linear model via penalized maximum likelihood) and support vector regression with linear kernel as meta-learner was the optimal model for forecasting Bitcoin price. The MAPE, RMSE, MAE, and R-squared values for the stacking ensemble model were 0.0191%, 15.5331 USD, 124.5508 USD, and 0.9967 respectively. These values show a high degree of reliability in predicting the price of Bitcoin using the stacking ensemble model. Accurately predicting the future price of Bitcoin will yield significant returns for investors and market players in the cryptocurrency market.

Buy-Online-and-Pick-up-in-Store in Omnichannel Retailing
Yasuyuki Kusuda

In this paper, we extend the model of Gao and Su (2016) and consider an omnichannel strategy in which inventory can be replenished when a retailer sells only in physical stores. With "buy-online-and-pick-up-in-store" (BOPS) having been introduced, consumers can choose to buy directly online, buy from a retailer using BOPS, or go directly to a store to make purchases without using BOPS. The retailer is able to select the inventory level to maximize the probability of inventory availability at the store. Furthermore, the retailer can incur an additional cost to reduce the BOPS ordering lead time, which results in a lowered hassle cost for consumers who use BOPS. In conclusion, we found that there are two types of equilibrium: that in which all consumers go directly to the store without using BOPS and that in which all consumers use BOPS.

CO2 mitigation model for China's residential building sector
Minda Ma,Weiguang Cai

This paper aims to investigate the factors that can mitigate carbon-dioxide (CO2) intensity and further assess CMRBS in China based on a household scale via decomposition analysis. Here we show that: Three types of housing economic indicators and the final emission factor significantly contributed to the decrease in CO2 intensity in the residential building sector. In addition, the CMRBS from 2001-2016 was 1816.99 MtCO2, and the average mitigation intensity during this period was 266.12 kgCO2/household/year. Furthermore, the energy-conservation and emission-mitigation strategy caused CMRBS to effectively increase and is the key to promoting a more significant emission mitigation in the future. Overall, this paper covers the CMRBS assessment gap in China, and the proposed assessment model can be regarded as a reference for other countries and cities for measuring the retrospective CO2 mitigation effect in residential buildings.

Certifiably Pseudorandom Financial Derivatives
David Zuckerman

Arora, Barak, Brunnermeier, and Ge showed that taking computational complexity into account, a dishonest seller could strategically place lemons in financial derivatives to make them substantially less valuable to buyers. We show that if the seller is required to construct derivatives of a certain form, then this phenomenon disappears. In particular, we define and construct pseudorandom derivative families, for which lemon placement only slightly affects the values of the derivatives. Our constructions use expander graphs. We study our derivatives in a more general setting than Arora et al. In particular, we analyze arbitrary tranches of the common collateralized debt obligations (CDOs) when the underlying assets can have significant dependencies.

Dynamic Programming with State-Dependent Discounting
John Stachurski,Junnan Zhang

This paper extends the core results of discrete time infinite horizon dynamic programming theory to the case of state-dependent discounting. The traditional constant-discount condition requires that the discount factor of the controller is strictly less than one. Here we replace the constant factor with a discount factor process and require, in essence, that the process is strictly less than one on average in the long run. We prove that, under this condition, the standard optimality results can be recovered, including Bellman's principle of optimality, convergence of value function iteration and convergence of policy function iteration. We also show that the condition cannot be weakened in many standard settings. The dynamic programming framework considered in the paper is general enough to contain features such as recursive preferences. Several applications are discussed.

Fourier transform MCMC, heavy tailed distributions and geometric ergodicity
Denis Belomestny,Leonid Iosipoi

Markov Chain Monte Carlo methods become increasingly popular in applied mathematics as a tool for numerical integration with respect to complex and high-dimensional distributions. However, application of MCMC methods to heavy tailed distributions and distributions with analytically intractable densities turns out to be rather problematic. In this paper, we propose a novel approach towards the use of MCMC algorithms for distributions with analytically known Fourier transforms and, in particular, heavy tailed distributions. The main idea of the proposed approach is to use MCMC methods in Fourier domain to sample from a density proportional to the absolute value of the underlying characteristic function. A subsequent application of the Parseval's formula leads to an efficient algorithm for the computation of integrals with respect to the underlying density. We show that the resulting Markov chain in Fourier domain may be geometrically ergodic even in the case of heavy tailed original distributions. We illustrate our approach by several numerical examples including multivariate elliptically contoured stable distributions.

Hierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model
Christian Bayer,Chiheb Ben Hammouda,Raul Tempone

The rough Bergomi (rBergomi) model, introduced recently in [4], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet exhibits remarkable fit to empirical implied volatility surfaces. 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), 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.

Identifying the Discount Factor in Dynamic Discrete Choice Models
Jaap H. Abbring,Øystein Daljord

Empirical research often cites observed choice responses to variation that shifts expected discounted future utilities, but not current utilities, as an intuitive source of information on time preferences. We study the identification of dynamic discrete choice models under such economically motivated exclusion restrictions on primitive utilities. We show that each exclusion restriction leads to an easily interpretable moment condition with the discount factor as the only unknown parameter. The identified set of discount factors that solves this condition is finite, but not necessarily a singleton. Consequently, in contrast to common intuition, an exclusion restriction does not in general give point identification. Finally, we show that exclusion restrictions have nontrivial empirical content: The implied moment conditions impose restrictions on choices that are absent from the unconstrained model.

Lifetime Ruin Problem Under High-watermark Fees and Drift Uncertainty
Junbeom Lee,Xiang Yu,Chao Zhou

This paper aims to make a new contribution to the study of lifetime ruin problem by considering investment in two hedge funds with high-watermark fees and drift uncertainty. Due to multi-dimensional performance fees that are charged whenever each fund profit exceeds its historical maximum, the value function is expected to be multi-dimensional. New mathematical challenges arise as the standard dimension reduction cannot be applied, and the convexity of the value function and Isaacs condition may not hold in our ruin probability minimization problem with drift uncertainty. We propose to employ the stochastic Perron's method to characterize the value function as the unique viscosity solution to the associated Hamilton Jacobi Bellman (HJB) equation without resorting to the proof of dynamic programming principle. The required comparison principle is also established in our setting to close the loop of stochastic Perron's method.

Multiscale Asymptotic Analysis for Portfolio Optimization under Stochastic Environment
Jean-Pierre Fouque,Ruimeng Hu

Empirical studies indicate the presence of multi-scales in the volatility of underlying assets: a fast-scale on the order of days and a slow-scale on the order of months. In our previous works, we have studied the portfolio optimization problem in a Markovian setting under each single scale, the slow one in [Fouque and Hu, SIAM J. Control Optim., 55 (2017), 1990-2023], and the fast one in [Hu, Proceedings of IEEE CDC 2018, accepted]. This paper is dedicated to the analysis when the two scales coexist in a Markovian setting. We study the terminal wealth utility maximization problem when the volatility is driven by both fast- and slow-scale factors. We first propose a zeroth-order strategy, and rigorously establish the first order approximation of the associated problem value. This is done by analyzing the corresponding linear partial differential equation (PDE) via regular and singular perturbation techniques, as in the single-scale cases. Then, we show the asymptotic optimality of our proposed strategy within a specific family of admissible controls. Interestingly, we highlight that a pure PDE approach does not work in the multi-scale case and, instead, we use the so-called epsilon-martingale decomposition. This completes the analysis of portfolio optimization in both fast mean-reverting and slowly-varying Markovian stochastic environments.

Nash Bargaining Over Margin Loans to Kelly Gamblers
Alex Garivaltis

I derive practical formulas for optimal arrangements between sophisticated stock market investors (namely, continuous-time Kelly gamblers or, more generally, CRRA investors) and the brokers who lend them cash for leveraged bets on a high Sharpe asset (i.e. the market portfolio). Rather than, say, the broker posting a monopoly price for margin loans, the gambler agrees to use a greater quantity of margin debt than he otherwise would in exchange for an interest rate that is lower than the broker would otherwise post. The gambler thereby attains a higher asymptotic capital growth rate and the broker enjoys a greater rate of intermediation profit than would obtain under non-cooperation. If the threat point represents a vicious breakdown of negotiations (resulting in zero margin loans), then we get an elegant rule of thumb: $r_L^*=(3/4)r+(1/4)(\nu-\sigma^2/2)$, where $r$ is the broker's cost of funds, $\nu$ is the compound-annual growth rate of the market index, and $\sigma$ is the annual volatility. We show that, regardless of the particular threat point, the gambler will negotiate to size his bets as if he himself could borrow at the broker's call rate.

Noncooperative dynamics in election interference
David Rushing Dewhurst,Christopher M. Danforth,Peter Sheridan Dodds

Foreign power interference in domestic elections is an age-old, existential threat to societies. Manifested through myriad methods from war to words, such interference is a timely example of strategic interaction between economic and political agents. We model this interaction between rational game players as a continuous-time differential game, constructing an analytical model of this competition with a variety of payoff structures. Structures corresponding to all-or-nothing attitudes regarding the effect of the interference operations by only one player lead to an arms race in which both countries spend increasing amounts on interference and counter-interference operations. We then confront our model with data pertaining to the Russian interference in the 2016 United States presidential election contest, introducing and estimating a Bayesian structural time series model of election polls and social media posts by Russian internet trolls. We show that our analytical model, while purposefully abstract and simple, adequately captures many temporal characteristics of the election and social media activity.

On the Notions of Equilibria for Time-Inconsistent Stopping Problems in Continuous Time
Erhan Bayraktar,Jingjie Zhang,Zhou Zhou

A \emph{new} notion of equilibrium, which we call \emph{strong equilibrium}, is introduced for time-inconsistent stopping problems in continuous time. Compared to the existing notions introduced in ArXiv: 1502.03998 and ArXiv: 1709.05181, which in this paper are called \emph{mild equilibrium} and \emph{weak equilibrium} respectively, a strong equilibrium captures the idea of subgame perfect Nash equilibrium more accurately. When the state process is a continuous-time Markov chain and the discount function is log sub-additive, we show that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also prove its existence.

Optimal Information Acquisition and Consumption Under Habit Formation Preference
Yue Yang,Xiang Yu

We consider a two-stage optimal decision making involving full market information acquisition beforehand and dynamic consumption afterwards: in stage-1 from the initial time to a chosen stopping time, the investor has access to full market information to update all underlying processes subjecting to information costs; in stage-2 starting from the chosen stopping time, the investor terminates the costly full information learning but the public stock prices are still available. Therefore, during stage-2, the investor adopts the previous inputs and starts the investment and consumption based on partial observations. Moreover, the habit formation preference is employed, in which the past consumption affects the investor's current decisions. Mathematically speaking, we formulate a composite problem, in which the exterior problem is to determine the best time to initiate investment-consumption decisions and the interior problem becomes a control problem with partial information. The value function of the composite problem is characterized as the unique viscosity solution of some variational inequalities.

Optimal Stopping and Utility in a Simple Model of Unemployment Insurance
Jason S. Anquandah,Leonid V. Bogachev

Managing unemployment is one of the key issues in social policies. Unemployment insurance schemes are designed to cushion the financial and morale blow of loss of job but also to encourage the unemployed to seek new jobs more pro-actively due to the continuous reduction of benefit payments. In the present paper, a simple model of unemployment insurance is proposed with a focus on optimality of the individual's entry to the scheme. The corresponding optimal stopping problem is solved, and its similarity and differences with the perpetual American call option are discussed. Beyond a purely financial point of view, we argue that in the actuarial context the optimal decisions should take into account other possible preferences through a suitable utility function. Some examples in this direction are worked out.

Portfolio liquidation under factor uncertainty
Ulrich Horst,Xiaonyu Xia,Chao Zhou

We study an optimal liquidation problem under the ambiguity with respect to price impact parameters. Our main results show that the value function and the optimal trading strategy can be characterized by the solution to a semi-linear PDE with superlinear gradient, monotone generator and singular terminal value. We also establish an asymptotic analysis of the robust model for small amount of uncertainty and analyse the effect of robustness on optimal trading strategies and liquidation costs. In particular, in our model ambiguity aversion is observationally equivalent to increased risk aversion. This suggests that ambiguity aversion increases liquidation rates.

QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds
Igor Halperin

This paper presents a discrete-time option pricing model that is rooted in Reinforcement Learning (RL), and more specifically in the famous Q-Learning method of RL. We construct a risk-adjusted Markov Decision Process for a discrete-time version of the classical Black-Scholes-Merton (BSM) model, where the option price is an optimal Q-function, while the optimal hedge is a second argument of this optimal Q-function, so that both the price and hedge are parts of the same formula. Pricing is done by learning to dynamically optimize risk-adjusted returns for an option replicating portfolio, as in the Markowitz portfolio theory. Using Q-Learning and related methods, once created in a parametric setting, the model is able to go model-free and learn to price and hedge an option directly from data, and without an explicit model of the world. This suggests that RL may provide efficient data-driven and model-free methods for optimal pricing and hedging of options, once we depart from the academic continuous-time limit, and vice versa, option pricing methods developed in Mathematical Finance may be viewed as special cases of model-based Reinforcement Learning. Further, due to simplicity and tractability of our model which only needs basic linear algebra (plus Monte Carlo simulation, if we work with synthetic data), and its close relation to the original BSM model, we suggest that our model could be used for benchmarking of different RL algorithms for financial trading applications

Racial Disparities in Voting Wait Times: Evidence from Smartphone Data
M. Keith Chen,Kareem Haggag,Devin G. Pope,Ryne Rohla

Equal access to voting is a core feature of democratic government. Using data from millions of smartphone users, we quantify a racial disparity in voting wait times across a nationwide sample of polling places during the 2016 US presidential election. Relative to entirely-white neighborhoods, residents of entirely-black neighborhoods waited 29% longer to vote and were 74% more likely to spend more than 30 minutes at their polling place. This disparity holds when comparing predominantly white and black polling places within the same states and counties, and survives numerous robustness and placebo tests. Our results document large racial differences in voting wait times and demonstrates that geospatial data can be an effective tool to both measure and monitor these disparities.

Regulator-based risk statistics for portfolios
Xiaochuan Deng,Fei Sun

The portfolio are a critical factor not only in risk analysis, but also in insurance and financial applications. In this paper, we consider a special class of risk statistics from the perspective of regulator. This new risk statistic can be uesd for the quantification of portfolio risk. By further developing the axioms related to regulator-based risk statistics, we are able to derive dual representation for them. Finally, examples are also given to demonstrate the application of this risk statistic.

Robust no arbitrage and the solvability of vector-valued utility maximization problems
Andreas H Hamel,Birgit Rudloff,Zhou Zhou

A market model with $d$ assets in discrete time is considered where trades are subject to proportional transaction costs given via bid-ask spreads, while the existence of a num\`eraire is not assumed. It is shown that robust no arbitrage holds if, and only if, there exists a Pareto solution for some vector-valued utility maximization problem with component-wise utility functions. Moreover, it is demonstrated that a consistent price process can be constructed from the Pareto maximizer.

The Economics of Enlightenment: Time Value of Knowledge and the Net Present Value (NPV) of Knowledge Machines
Ravi Kashyap

We formulate one methodology to put a value or price on knowledge using well accepted techniques from finance. We provide justifications for these finance principles based on the limitations of the physical world we live in. To the best of our knowledge this is the first recorded attempt to put a numerical value on knowledge. The implications of this valuation exercise, which places a high premium on any piece of knowledge, are to ensure that participants in any knowledge system are better trained to notice the knowledge available from any source. Just because someone does not see a connection, does not mean that there is no connection. We need to try harder and be more open to acknowledging the smallest piece of new knowledge that might have been brought to light by anyone from anywhere about anything.

Vector Autoregressive Moving Average Model with Scalar Moving Average
Du Nguyen

We show Vector Autoregressive Moving Average models with scalar Moving Average components could be estimated by generalized least square (GLS) for each fixed moving average polynomial. The conditional variance of the GLS model is the concentrated covariant matrix of the moving average process. Under GLS the likelihood function of these models has similar format to their VAR counterparts. Maximum likelihood estimate can be done by optimizing with gradient over the moving average parameters. These models are inexpensive generalizations of Vector Autoregressive models. We discuss a relationship between this result and the Borodin-Okounkov formula in operator theory.