In online portfolio optimization the investor makes decisions based on new, continuously incoming information on financial assets (typically their prices). In our study we consider a learning algorithm, namely the Kiefer--Wolfowitz version of the Stochastic Gradient method, that converges to the log-optimal solution in the threshold-type, buy-and-sell strategy class.
The systematic study of this method is novel in the field of portfolio optimization; we aim to establish the theory and practice of Stochastic Gradient algorithm used on parametrized trading strategies.
We demonstrate on a wide variety of stock price dynamics (e.g. with stochastic volatility and long-memory) that there is an optimal threshold type strategy which can be learned. Subsequently, we numerically show the convergence of the algorithm. Furthermore, we deal with the typically problematic question of how to choose the hyperparameters (the parameters of the algorithm and not the dynamics of the prices) without knowing anything about the price other than a small sample.
We discuss the turnpike property for optimal investment and consumption problems. We find there exists a threshold value that determines the turnpike property for investment policy. The threshold value only depends on the Sharpe ratio, the riskless interest rate and the discount rate. We show that if utilities behave asymptotically like power utilities and satisfy some simple relations with the threshold value, then the turnpike property for investment holds. There is in general no turnpike property for consumption policy. We also provide the rate of convergence and illustrate the main results…
In this article, inspired by Shi, et al. we investigate the optimal portfolio selection with one risk-free asset and one risky asset in a multiple period setting under cumulative prospect theory (CPT). Compared with their study, our novelty is that we consider a stochastic benchmark, and portfolio constraints. We test the sensitivity of the optimal CPT-investment strategies to different model parameters by performing a numerical analysis.
The paper considers an investment timing problem appearing in real options theory. Present values from an investment project are modeled by general diffusion process. We prove necessary and sufficient conditions under which an optimal investment time is induced by threshold strategy. We study also the conditions of optimality of threshold strategy (over all threshold strategies) and discuss the connection between solutions to investment timing problem and to free-boundary problem.
In this work, we consider rule-based investment strategies for managing a defined contribution saving scheme under the Dutch pension fund testing model. We found that dynamic rule-based investment can outperform traditional static strategies, by which we mean that the pensioner can achieve the target retirement income with higher probability and limit the shortfall when target is not met. In comparison with the popular dynamic programming technique, the rule-based strategy has a more stable asset allocation throughout time and avoid excessive transactions, which may be hard to explain to the i…
We investigate the growth optimal strategy over a finite time horizon for a stock and bond portfolio in an analytically solvable multiplicative Markovian market model. We show that the optimal strategy consists in holding the amount of capital invested in stocks within an interval around an ideal optimal investment. The size of the holding interval is determined by the intensity of the transaction costs and the time horizon.
This paper treats the Merton problem how to invest in safe assets and risky assets to maximize an investor's utility, given by investment opportunities modeled by a $d$-dimensional state process. The problem is represented by a partial differential equation with optimizing term: the Hamilton-Jacobi-Bellman equation. The main purpose of this paper is to solve partial differential equations derived from the Hamilton-Jacobi-Bellman equations with a deep learning algorithm: the Deep Galerkin method, first suggested by Sirignano and Spiliopoulos (2018). We then apply the algorithm to get the soluti…
We study an optimal dividend problem for an insurer who simultaneously controls investment weights in a financial market, liability ratio in the insurance business, and dividend payout rate. The insurer seeks an optimal strategy to maximize her expected utility of dividend payments over an infinite horizon. By applying a perturbation approach, we obtain the optimal strategy and the value function in closed form for log and power utility. We conduct an economic analysis to investigate the impact of various model parameters and risk aversion on the insurer's optimal strategy.
This paper considers the problem of consumption and investment in a financial market within a continuous time stochastic economy. The investor exhibits a change in the discount rate. The investment opportunities are a stock and a riskless account. The market coefficients and discount factor switch according to a finite state Markov chain. The change in the discount rate leads to time inconsistencies of the investor's decisions. The randomness in our model is driven by a Brownian motion and a Markov chain. Following Ekeland and Pirvu we introduce and characterize the subgame perfect strategies.…
We consider an optimal investment and risk control problem for an insurer under the mean-variance (MV) criterion. By introducing a deterministic auxiliary process defined forward in time, we formulate an alternative time-consistent problem related to the original MV problem, and obtain the optimal strategy and the value function to the new problem in closed-form. We compare our formulation and optimal strategy to those under the precommitment and game-theoretic framework. Numerical studies show that, when the financial market is negatively correlated with the risk process, optimal investment m…
There is a great number of factors to take into account when building and managing an investment portfolio. It is widely believed that a proper set-up of the portfolio combined with a good, robust management strategy is the key to successful investment. In this paper, we aim at an analysis of two aspects that may have an impact on investment performance: diversity of assets and inclusion of cash in the portfolio. We also propose two new management strategies based on the MACD and RSI factors known from technical analysis. Monte Carlo simulations within the Heston model of a market are used to …
Motivated by the AIG bailout case in the financial crisis of 2007-2008, we consider an insurer who wants to maximize the expected utility of the terminal wealth by selecting optimal investment and risk control strategies. The insurer's risk process is modelled by a jump-diffusion process and is negatively correlated with the capital gains in the financial market. We obtain explicit solution to optimal strategies for various utility functions.
Demographic changes increase the necessity to base the pension system more and more on the second and the third pillar, namely the occupational and private pension plans; this paper deals with Target Date Funds (TDFs), which are a typical investment opportunity for occupational pension planners. TDFs are usually identified with a decreasing fraction of wealth invested in equity (a so-called glide path) as retirement comes closer, i.e., wealth is invested more risky the younger the saver is. We investigate whether this is actually optimal in the presence of non-tradable income risk in a stochas…
This research mainly explores the characteristics of different strategies and whether VIX INDEX positively influences the investment portfolio in any period. Our portfolio has six significant cryptocurrencies, VIX INDEX and gold. We perform parameter estimation on all raw data and bring the two types into different investment strategies, complete them effectively according to other characteristics, and compare the results. At the same time, we make two different portfolios, one contains VIX INDEX, and one does not have VIX INDEX. We use different portfolios in different portfolio strategies an…
We consider the optimal investment problem for Black-Scholes type financial market with bounded VaR measure on the whole investment interval $[0,T]$. The explicit form for the optimal strategies is found.
Optimizing Investment Strategies with Lazy Factor and Probability Weighting: A Price Portfolio Forecasting and Mean-Variance Model with Transaction Costs Approach
The finance industry has adopted machine learning (ML) as a form of quantitative research to support better investment decisions, yet there are several challenges often overlooked in practice. (1) ML code tends to be unstructured and ad hoc, which hinders cooperation with others. (2) Resource requirements and dependencies vary depending on which algorithm is used, so a flexible and scalable system is needed. (3) It is difficult for domain experts in traditional finance to apply their experience and knowledge in ML-based strategies unless they acquire expertise in recent technologies. This pape…
By exploiting a bipartite network representation of the relationships between mutual funds and portfolio holdings, we propose an indicator that we derive from the analysis of the network, labelled the Average Commonality Coefficient (ACC), which measures how frequently the assets in the fund portfolio are present in the portfolios of the other funds of the market. This indicator reflects the investment behavior of funds' managers as a function of the popularity of the assets they held. We show that $ACC$ provides useful information to discriminate between funds investing in niche markets and t…
In a collectivised pension fund, investors agree that any money remaining in the fund when they die can be shared among the survivors. We compute analytically the optimal investment-consumption strategy for a fund of $n$ identical investors with homogeneous Epstein--Zin preferences, investing in the Black--Scholes market in continuous time but consuming in discrete time. Our result holds for arbitrary mortality distributions. We also compute the optimal strategy for an infinite fund of investors, and prove the convergence of the optimal strategy as $n\to \infty$. The proof of convergence shows…
In this paper we propose a novel application of Gaussian processes (GPs) to financial asset allocation. Our approach is deeply rooted in Stochastic Portfolio Theory (SPT), a stochastic analysis framework introduced by Robert Fernholz that aims at flexibly analysing the performance of certain investment strategies in stock markets relative to benchmark indices. In particular, SPT has exhibited some investment strategies based on company sizes that, under realistic assumptions, outperform benchmark indices with probability 1 over certain time horizons. Galvanised by this result, we consider the …
We study optimal proportional reinsurance and investment strategies for an insurance company which experiences both ordinary and catastrophic claims and wishes to maximize the expected exponential utility of its terminal wealth. We propose a model where the insurance framework is affected by environmental factors, and aggregate claims and stock prices are subject to common shocks, i.e. drastic events such as earthquakes, extreme weather conditions, or even pandemics, that have an immediate impact on the financial market and simultaneously induce insurance claims. Using the classical stochastic…
We provide easily verifiable conditions for the well-posedness of the optimal investment problem for a behavioral investor in an incomplete discrete-time multiperiod financial market model, for the first time in the literature. Under two different sets of assumptions we also establish the existence of optimal strategies.
The classical optimal investment and consumption problem with infinite horizon is studied in the presence of transaction costs. Both proportional and fixed costs as well as general utility functions are considered. Weak dynamic programming is proved in the general setting and a comparison result for possibly discontinuous viscosity solutions of the dynamic programming equation is provided. Detailed numerical experiments illustrate several properties of the optimal investment strategies.
We consider a discrete-time, generically incomplete market model and a behavioural investor with power-like utility and distortion functions. The existence of optimal strategies in this setting has been shown in a previous paper under certain conditions on the parameters of these power functions. In the present paper we prove the existence of optimal strategies under a different set of conditions on the parameters, identical to the ones which were shown to be necessary and sufficient in the Black-Scholes model. Although there exists no natural dual problem for optimisation under behavioural cr…
We introduce a dynamic optimization framework to analyze optimal portfolio allocations within an information driven contagious distress model. The investor allocates his wealth across several stocks whose growth rates and distress intensities are driven by a hidden Markov chain, and also influenced by the distress state of the economy. We show that the optimal investment strategies depend on the gradient of value functions, recursively linked to each other via the distress states. We establish uniform bounds for the solutions to a sequence of approximation problems, show their convergence to t…
We study the continuous time portfolio optimization model on the market where the mean returns of individual securities or asset categories are linearly dependent on underlying economic factors. We introduce the functional $Q_γ$ featuring the expected earnings yield of portfolio minus a penalty term proportional with a coefficient $γ$ to the variance when we keep the value of the factor levels fixed. The coefficient $γ$ plays the role of a risk-aversion parameter. We find the optimal trading positions that can be obtained as the solution to a maximization problem for $Q_γ$ at any moment of tim…
We consider the portfolio choice problem for a long-run investor in a general continuous semimartingale model. We suggest to use path-wise growth optimality as the decision criterion and encode preferences through restrictions on the class of admissible wealth processes. Specifically, the investor is only interested in strategies which satisfy a given linear drawdown constraint. The paper introduces the numeraire property through the notion of expected relative return and shows that drawdown-constrained strategies with the numeraire property exist and are unique, but may depend on the financia…
We consider a spread financial market defined by the multidimensional Ornstein--Uhlenbeck (OU) process. We study the optimal consumption/investment problem for logarithmic utility functions in the base of stochastic dynamical programming method. We show a special Verification Theorem for this case. We find the solution to the Hamilton--Jacobi--Bellman (HJB) equation in explicit form and as a consequence we construct the optimal financial strategies. Moreover, we study the constructed strategy by numerical simulations.
In this paper, we study the optimal investment problem of an insurer whose surplus process follows the diffusion approximation of the classical Cramer-Lundberg model. Investment in the foreign market is allowed, and therefore, the foreign exchange rate model is considered and incorporated. It is assumed that the instantaneous mean growth rate of foreign exchange rate price follows an Ornstein-Uhlenbeck process. Dynamic programming method is employed to study the problem of maximizing the expected exponential utility of terminal wealth. By soloving the correspoding Hamilton-Jacobi-Bellman equat…
We consider the problem of optimal investment in a market with two cointegrated stocks and an agent with CRRA utility. We extend the findings of Liu and Timmermann [The Review of Financial Studies, 26(4):1048-1086, 2013] by paying special attention to when/if the associated stochastic control problem is well-posed and providing a verification result. Our new findings lead to a sharp well-posedness condition which is, surprisingly, also the necessary and sufficient condition for the optimal investment to be market-neutral (i.e. having offsetting long/short positions in the stocks). Hence, we pr…
This article examines arbitrage investment in a mispriced asset when the mispricing follows the Ornstein-Uhlenbeck process and a credit-constrained investor maximizes a generalization of the Kelly criterion. The optimal differentiable and threshold policies are derived. The optimal differentiable policy is linear with respect to mispricing and risk-free in the long run. The optimal threshold policy calls for investing immediately when the mispricing is greater than zero with the investment amount inversely proportional to the risk aversion parameter. The investment is risky even in the long ru…
In a market with stochastic investment opportunities, we study an optimal consumption investment problem for an agent with recursive utility of Epstein-Zin type. Focusing on the empirically relevant specification where both risk aversion and elasticity of intertemporal substitution are in excess of one, we characterize optimal consumption and investment strategies via backward stochastic differential equations. The supperdifferential of indirect utility is also obtained, meeting demands from applications in which Epstein-Zin utilities were used to resolve several asset pricing puzzles. The emp…
Managing investment portfolios is an old and well know problem in multiple fields including financial mathematics and financial engineering as well as econometrics and econophysics. Multiple different concepts and theories were used so far to describe methods of handling with financial assets, including differential equations, stochastic calculus and advanced statistics. In this paper, using a set of tools from the probability theory, various strategies of building financial portfolios are analysed in different market conditions. A special attention is given to several realisations of a so cal…
The aim of this work is to extend the capital growth theory developed by Kelly, Breiman, Cover and others to asset market models with transaction costs. We define a natural generalization of the notion of a numeraire portfolio proposed by Long and show how such portfolios can be used for constructing growth-optimal investment strategies. The analysis is based on the classical von Neumann-Gale model of economic dynamics, a stochastic version of which we use as a framework for the modelling of financial markets with frictions.
We designed a machine learning algorithm that identifies patterns between ESG profiles and financial performances for companies in a large investment universe. The algorithm consists of regularly updated sets of rules that map regions into the high-dimensional space of ESG features to excess return predictions. The final aggregated predictions are transformed into scores which allow us to design simple strategies that screen the investment universe for stocks with positive scores. By linking the ESG features with financial performances in a non-linear way, our strategy based upon our machine l…
We consider optimal investment problems for a diffusion market model with non-observable random drifts that evolve as an Ito's process. Admissible strategies do not use direct observations of the market parameters, but rather use historical stock prices. For a non-linear problem with a general performance criterion, the optimal portfolio strategy is expressed via the solution of a scalar minimization problem and a linear parabolic equation with coefficients generated by the Kalman filter.
In this work, we consider the optimal portfolio selection problem under hard constraints on trading volume amounts when the dynamics of the risky asset returns are governed by a discrete-time approximation of the Markov-modulated geometric Brownian motion. The states of Markov chain are interpreted as the states of an economy. The problem is stated as a dynamic tracking problem of a reference portfolio with desired return. We propose to use the model predictive control (MPC) methodology in order to obtain feedback trading strategies. Our approach is tested on a set of a real data from the radi…
Cryptocurrencies (CCs) have risen rapidly in market capitalization over the last years. Despite striking price volatility, their high average returns have drawn attention to CCs as alternative investment assets for portfolio and risk management. We investigate the utility gains for different types of investors when they consider cryptocurrencies as an addition to their portfolio of traditional assets. We consider risk-averse, return-seeking as well as diversificationpreferring investors who trade along different allocation frequencies, namely daily, weekly or monthly. Out-of-sample performance…
This paper studies the properties of the optimal portfolio-consumption strategies in a {finite horizon} robust utility maximization framework with different borrowing and lending rates. In particular, we allow for constraints on both investment and consumption strategies, and model uncertainty on both drift and volatility. With the help of explicit solutions, we quantify the impacts of uncertain market parameters, portfolio-consumption constraints and borrowing costs on the optimal strategies and their time monotone properties.
