Multi-asset Market Research Articles

Solvability of Differential Riccati Equations and Applications to Algorithmic Trading with Signals

ABSTRACT We study a differential Riccati equation (DRE) with indefinite matrix coefficients, which arises in a wide class of practical problems. We show that the DRE solves an associated control problem, which is key to provide existence and uniqueness of a solution. As an application, we solve two algorithmic trading problems in which the agent adopts a constant absolute risk-aversion (CARA) utility function, and where the optimal strategies use signals and past observations of prices to improve their performance. First, we derive a multi-asset market making strategy in over-the-counter markets, where the market maker uses an external trading venue to hedge risk. Second, we derive an optimal trading strategy that uses prices and signals to learn the drift in the asset prices.

DeepSets and Their Derivative Networks for Solving Symmetric PDEs

Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we introduce a class of PDEs that are invariant to permutations, and called symmetric PDEs. Such problems are widespread, ranging from cosmology to quantum mechanics, and option pricing/hedging in multi-asset market with exchangeable payoff. Our main application comes actually from the particles approximation of mean-field control problems. We design deep learning algorithms based on certain types of neural networks, named PointNet and DeepSet (and their associated derivative networks), for computing simultaneously an approximation of the solution and its gradient to symmetric PDEs. We illustrate the performance and accuracy of the PointNet/DeepSet networks compared to classical feedforward ones, and provide several numerical results of our algorithm for the examples of a mean-field systemic risk, mean-variance problem and a min/max linear quadratic McKean-Vlasov control problem.

Equilibria and Information Choices in Multi-Asset Markets with Multidimensional Information

Equilibria and Information Choices in Multi-Asset Markets with Multidimensional Information

Error Estimates for Discrete Approximations of Game Options with Multivariate Diffusion Asset Prices

We obtain error estimates for strong approximations of a diffusion with a diffusion matrix $\sigma$ and a drift b by the discrete time process defined recursively X_N((n+1)/N) = X_N(n/N)+N^{1/2}\sigma(X_N(n/N))\xi(n+1)+N^{-1}b(XN(n/N)); where \xi(n); n\geq 1 are i.i.d. random vectors, and apply this in order to approximate the fair price of a game option with a diffusion asset price evolution by values of Dynkin's games with payoffs based on the above discrete time processes. This provides an effective tool for computations of fair prices of game options with path dependent payoffs in a multi asset market with diffusion evolution.

Closed-form Approximations in Multi-asset Market Making

ABSTRACT A large proportion of market making models derive from the seminal model of Avellaneda and Stoikov. The numerical approximation of the value function and the optimal quotes in these models remains a challenge when the number of assets is large. In this article, we propose closed-form approximations for the value functions of many multi-asset extensions of the Avellaneda–Stoikov model. These approximations or proxies can be used (i) as heuristic evaluation functions, (ii) as initial value functions in reinforcement learning algorithms, and/or (iii) directly to design quoting strategies through a greedy approach. Regarding the latter, our results lead to new and easily interpretable closed-form approximations for the optimal quotes, both in the finite-horizon case and in the asymptotic (ergodic) regime.

Multi-Asset Market Impact and Order Flow Commonality

The empirical finding that market movements in stock prices may be correlated with the order flow of other stocks has led to the notion of and has prompted the development of multivariate models of market impact. These models are parametrized by a matrix of impact coefficients whose off-diagonal elements are meant to capture how trades in one asset influence the price of other assets, leading to a large number of 'cross-impact' parameters which may not be identified solely based on the covariance of returns with order flow. Moreover, empirical evidence suggests that these cross-impact terms are unstable and change sign randomly over time, which poses a problem for their interpretation and use. We reexamine this empirical evidence from a causal standpoint and offer a simpler explanation for the observed correlation between the returns of an asset and the order flow imbalance of other assets, in terms of common components in order flow across stocks which may naturally arise from multi-asset trading strategies. We provide empirical evidence from order flow and price changes of NASDAQ-100 stocks to support this explanation. Our results show the main determinants of impact to be idiosyncratic order flow imbalance as well as a market order flow factor common across stocks. Additional ‘cross-impact’ terms account for less than 1% of market impact. This leads to a parsimonious approach for causal modelling of multi-asset impact, which does not require introducing any concept of cross-impact.

Size matters for OTC market makers: General results and dimensionality reduction techniques

AbstractIn most over‐the‐counter (OTC) markets, a small number of market makers provide liquidity to other market participants. More precisely, for a list of assets, they set prices at which they agree to buy and sell. Market makers face therefore an interesting optimization problem: they need to choose bid and ask prices for making money while mitigating the risk associated with holding inventory in a volatile market. Many market‐making models have been proposed in the academic literature, most of them dealing with single‐asset market making whereas market makers are usually in charge of a long list of assets. The rare models tackling multiasset market making suffer however from the curse of dimensionality when it comes to the numerical approximation of the optimal quotes. The goal of this paper is to propose a dimensionality reduction technique to address multiasset market making by using a factor model. Moreover, we generalize existing market‐making models by the addition of an important feature: the existence of different transaction sizes and the possibility for the market makers in OTC markets to answer different prices to requests with different sizes.

Impact Analysis of Financial Regulation on Multi-Asset Markets Using Artificial Market Simulations

In this study, we assessed the impact of capital adequacy ratio (CAR) regulation in the Basel regulatory framework. This regulation was established to make the banking network robust. However, a previous work argued that CAR regulation has a destabilization effect on financial markets. To assess impacts such as destabilizing effects, we conducted simulations of an artificial market, one of the computer simulations imitating real financial markets. In the simulation, we proposed and used a new model with continuous double auction markets, stylized trading agents, and two kinds of portfolio trading agents. Both portfolio trading agents had trading strategies incorporating Markowitz’s portfolio optimization. Additionally, one type of portfolio trading agent was under regulation. From the simulations, we found that portfolio optimization as each trader’s strategy stabilizes markets, and CAR regulation destabilizes markets in various aspects. These results show that CAR regulation can have negative effects on asset markets. As future work, we should confirm these effects empirically and consider how to balance between both positive and negative aspects of CAR regulation.

Statistical Arbitrage Strategy in Multi-Asset Market Using Time Series Analysis

The statistical arbitrage strategy is one of the most traditional investment strategies. There are many theoretical and empirical studies until now. However, almost all of the statistical arbitrage strategies focus on the price difference (spread) between two similar assets in the same asset class and exploit the mean reversion of spreads, i.e. pairs trading. In this study, we extend the strategy to multiple assets in the multi-asset market. Although mean-reverting portfolios were derived based on a single criterion in related researches, we derive a mean-reverting portfolio by optimizing multiple mean-reversion criteria. We expect that a mean-reverting portfolio based on multiple indicators leads to a higher return/risk. We perform an empirical analysis in multi-asset market and show the profitability of our strategy.

Historical Returns of the Market Portfolio

AbstractWe create an annual return index for the invested global multiasset market portfolio. We use a newly constructed unique data set covering the entire market of financial investors. We analyze returns and risk from 1960 to 2017, a period during which the market portfolio realized a compounded real return in U.S. dollars of 4.45%, with a standard deviation of annual returns of 11.2%. The compounded excess return was 3.39%. We publish these data on returns of the market portfolio, so they can be used for future asset pricing and corporate finance studies.Received March 4, 2019; editorial decision October 9, 2019 by Editor Jeffrey Pontiff. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.

Disposition effect and multi-asset market dynamics

PurposeAsset pricing dynamics in a multi-asset framework when investors’ trading exhibits the disposition effect is studied. The purpose of this paper is to explore asset pricing dynamics and the switching behavior among multiple assets.Design/methodology/approachThe dynamics of complex financial markets can be best explored by following agent-based modeling approach. The artificial financial market is populated with traders following two heterogeneous trading strategies: the technical and the fundamental trading rules. By simulation, the switching behavior among multiple assets is investigated.FindingsThe proposed framework can explain important stylized facts in financial time series, such as random walk price dynamics, bubbles and crashes, fat-tailed return distributions, absence of autocorrelation in raw returns, persistent long memory of volatility, excess volatility, volatility clustering and power-law tails. In addition, asset returns possess fractal structure and self-similarity features; though the switching behavior is only allowed among the asset markets.Practical implicationsThe model demonstrates stylized facts of most real financial markets. Thereafter, the proposed model can serve as a testbed for policy makers, scholars and investors.Originality/valueTo the best of knowledge, no research has been conducted to introduce the disposition effect to a multi-asset agent-based model.

Steady states, stability and bifurcations in multi-asset market models

We provide a full analytical treatment of a multi-asset market model in which speculators have the choice between two risky and one safe asset. As it turns out, the dynamics of our model is driven by a four-dimensional nonlinear map and may undergo a transcritical, flip or Neimark–Sacker bifurcation. While the first bifurcation is associated with an undervaluation of the risky assets, the latter two may trigger (complex) endogenous dynamics. To facilitate our analysis, we first study a simpler two-dimensional setup of our model in which speculators can only switch between one risky and one safe asset.

Game options with gradual exercise and cancellation under proportional transaction costs

ABSTRACTGame (Israeli) options in a multi-asset market model with proportional transaction costs are studied in the case when the buyer is allowed to exercise the option and the seller has the right to cancel the option gradually at a mixed (or randomized) stopping time, rather than instantly at an ordinary stopping time. Allowing gradual exercise and cancellation leads to increased flexibility in hedging, and hence tighter bounds on the option price as compared to the case of instantaneous exercise and cancellation. Algorithmic constructions for the bid and ask prices, and the associated superhedging strategies and optimal mixed stopping times for both exercise and cancellation are developed and illustrated. Probabilistic dual representations for bid and ask prices are also established.

Historical Returns of the Market Portfolio

Using a newly constructed unique dataset, this study is the first to document returns of the market portfolio for a long period and with a high level of detail. Our market portfolio basically contains all assets in which financial investors have invested. We analyze nominal, real, and excess return and risk characteristics of this global multi-asset market portfolio and the asset categories over the period 1960 to 2015. The global market portfolio realizes a compounded real return of 4.38% with a standard deviation of 11.6% from 1960 until 2015. In the inflationary period from 1960 to 1979, the compounded real return of the GMP is 2.27%, while this is 5.57% in the disinflationary period from 1980 to 2015. The reward for the average investor is a compounded return of 3.24%-points above the saver’s. We also compare the performance of an investor who holds the market portfolio with an investor who uses simple heuristics for the portfolio allocation. Our results suggest that the market portfolio is close to the mean-variance frontier, but our heuristic allocations achieve a significantly higher reward for risk.

Utility Maximization in a Factor Model with Constant and Proportional Costs

We study the problem of maximizing expected utility of terminal wealth under constant and proportional transactions costs in a multi-asset market in which prices are driven by a multidimensional factor process. We characterize the value function as the pointwise minimum of a set of superharmonic functions and as the unique continuous viscosity solution of the quasi-variational inequalities associated with the optimization problem. A combination of these characterizations is then used as a basis for the construction of optimal strategies.

Detection of arbitrage in a market with multi-asset derivatives and known risk-neutral marginals

In this paper we study the existence of arbitrage opportunities in a multi-asset market when risk-neutral marginal distributions of asset prices are known. We first propose an intuitive characterization of the absence of arbitrage opportunities in terms of copula functions. We then address the problem of detecting the presence of arbitrage by formalizing its resolution in two distinct ways that are both suitable for the use of optimization algorithms. The first method is valid in the general multivariate case and is based on Bernstein copulas that are dense in the set of all copula functions. The second one is easier to work with but is only valid in the bivariate case. It relies on results about improved Fréchet–Hoeffding bounds in presence of additional information. For both methods, details of implementation steps and empirical applications are provided.

AMERICAN OPTIONS WITH GRADUAL EXERCISE UNDER PROPORTIONAL TRANSACTION COSTS

American options in a multi-asset market model with proportional transaction costs are studied in the case when the holder of an option is able to exercise it gradually at a so-called mixed (randomized) stopping time. The introduction of gradual exercise leads to tighter bounds on the option price when compared to the case studied in the existing literature, where the standard assumption is that the option can only be exercised instantly at an ordinary stopping time. Algorithmic constructions for the bid and ask prices and the associated superhedging strategies and optimal mixed stopping times for an American option with gradual exercise are developed and implemented, and dual representations are established.

Cross-market volatility index with Factor-DCC

This paper proposes a new empirical methodology for computing a cross-market volatility index – coined CMIX – based on the Factor DCC-model, implemented on volatility surprises. This approach solves both problems of treating high-dimensional data and estimating time-varying conditional correlations. We provide an application to a multi-asset market data composed of equities, bonds, foreign exchange rates and commodities during 1983–2013. This new methodology may be attractive to asset managers, since it provides a simple way to hedge multi-asset portfolios with derivative contracts written on the CMIX.

Cross-market index with Factor-DCC

This paper proposes a new empirical methodology for computing a cross-market index – coined CMI – based on the Factor DCC-model. This approach solves both problems of treating high-dimensional data and estimating time-varying conditional correlations. We provide an application to a multi-asset market data composed of equities, bonds, foreign exchange rates and commodities during 1983–2013. This new methodology may be attractive to asset managers, since it provides a simple way of constructing passive portfolios customized on any asset class.

The Global Multi-Asset Market Portfolio, 1959–2012

The market portfolio contains important information for purposes of strategic asset allocation. One could consider it a natural benchmark for investors. The authors composed the invested global multi-asset market portfolio for 1990–2012 by estimating the market capitalization for equities, private equity, real estate, high-yield bonds, emerging-market debt, investment-grade credits, government bonds, and inflation-linked bonds. They also used an expanded period (1959–2012) for the main asset categories: equities, real estate, nongovernment bonds, and government bonds.