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 time. The single-factor case is analyzed in more details. We present a simple asset allocation example featuring an interest rate which affects a stock index and also serves as a second investment opportunity. We consider two possibilities: the interest rate for the bank account is governed by Vasicek-type and Cox-Ingersoll-Ross dynamics, respectively. Then we compare our results with the theory of Bielecki and Pliska where the authors employ the methods of the risk-sensitive control theory thereby using an infinite horizon objective featuring the long run expected growth rate, the asymptotic variance, and a risk-aversion parameter similar to $γ$.
In this paper, we propose an innovative investment framework incorporating asset allocation and class diversification oriented specifically for the biotechnology industry. With growing interests and capitalization in multiple biotech markets, investors require a more dynamic method of managing their assets within individual portfolios for optimal return efficiency. By selecting a single firm representative of identified industry trends, analyzing financial metrics relevant to the suggested approaches, and assessing financial health, we developed an adaptable investment methodology. We also per…
In this paper we study the optimal investment and reinsurance problem of an insurance company whose investment preferences are described via a forward dynamic exponential utility in a regime-switching market model. Financial and actuarial frameworks are dependent since stock prices and insurance claims vary according to a common factor given by a continuous time finite state Markov chain. We construct the value function and we prove that it is a forward dynamic utility. Then, we characterize the investment strategy and the optimal proportional level of reinsurance. We also perform numerical ex…
We study an optimal consumption and investment problem in a possibly incomplete market with general, not necessarily convex, stochastic constraints. We give explicit solutions for investors with exponential, logarithmic and power utility. Our approach is based on martingale methods which rely on recent results on the existence and uniqueness of solutions to BSDEs with drivers of quadratic growth.
We consider an arbitrage-free, discrete time and frictionless market. We prove that an investor maximising the expected utility of her terminal wealth can always find an optimal investment strategy provided that her dissatisfaction of infinite losses is infinite and her utility function is non-decreasing, continuous and bounded above. The same result is shown for cumulative prospect theory preferences, under additional assumptions.
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.
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 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 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…
This work models the interconnection of company's investment managers' representations and the market attraction of its shares. The models that reflect the connection of the company's market effectiveness indices and parameters of its economic activity are created on the basis of the Mean-Variance Analysis and Regression Analysis. On another side, expert evaluation methods also clarified the same influence parameters, but it was made according to the opinion of company managers. These two evaluation rows are used when making managerial decisions.
We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic volatility and unknown stock appreciation rate. The volatility parameter is driven by an external economic factor modeled as a diffusion process of Ornstein-Uhlenbeck type with unknown drift. We use the dynamical programming approach and find an optimal financial strategy which depends on the drift parameter. To estimate the drift coefficient we observe the economic factor $Y$ in an interval $[0,T_0]$ for fixed $T_0>0$, and use sequential estimation. We show, that the consumption and …
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…
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…
This paper studies the problem of maximizing expected utility from terminal wealth in a semi-static market composed of derivative securities, which we assume can be traded only at time zero, and of stocks, which can be traded continuously in time and are modeled as locally-bounded semi-martingales. Using a general utility function defined on the positive real line, we first study existence and uniqueness of the solution, and then we consider the dependence of the outputs of the utility maximization problem on the price of the derivatives, investigating not only stability but also differentiabi…
We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynamical programming approach leads to an investigation of the Hamilton Jacobi Bellman (HJB) equation which is a highly non linear partial differential equation (PDE) of the second oder. By using the Feynman - Kac representation we prove uniqueness and smoothness of the solution. Moreover, we study the optimal convergence rate of the iter…
The most commonly accepted model for investors' preferences is expected utility theory. More recently, other theories have emerged and pose new challenges to mathematics. The present paper treats preferences of cumulative prospect theory (CPT), where an "S-shaped" utility function is considered (i.e. convex up to a certain point and concave from there on). Also, distorted probability measures are applied for calculating the utility of a given position with respect to a (possibly random) benchmark $G$. Such problems have heretofore been solved essentially for complete continuous-time market mod…
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 …
We study the problem of optimal portfolio selection in an illiquid market with discrete order flow. In this market, bids and offers are not available at any time but trading occurs more frequently near a terminal horizon. The investor can observe and trade the risky asset only at exogenous random times corresponding to the order flow given by an inhomogenous Poisson process. By using a direct dynamic programming approach, we first derive and solve the fixed point dynamic programming equation satisfied by the value function, and then perform a verification argument which provides the existence …
In this paper, we study the problem of expected utility maximization of an agent who, in addition to an initial capital, receives random endowments at maturity. Contrary to previous studies, we treat as the variables of the optimization problem not only the initial capital but also the number of units of the random endowments. We show that this approach leads to a dual problem, whose solution is always attained in the space of random variables. In particular, this technique does not require the use of finitely additive measures and the related assumption that the endowments are bounded.
We consider the problem of maximizing expected utility for a power investor who can allocate his wealth in a stock, a defaultable security, and a money market account. The dynamics of these security prices are governed by geometric Brownian motions modulated by a hidden continuous time finite state Markov chain. We reduce the partially observed stochastic control problem to a complete observation risk sensitive control problem via the filtered regime switching probabilities. We separate the latter into pre-default and post-default dynamic optimization subproblems, and obtain two coupled Hamilt…
This paper considers an optimal life insurance for a householder subject to mortality risk. The household receives a wage income continuously, which is terminated by unexpected (premature) loss of earning power or (planned and intended) retirement, whichever happens first. In order to hedge the risk of losing income stream by householder's unpredictable event, the household enters a life insurance contract by paying a premium to an insurance company. The household may also invest their wealth into a financial market. The problem is to determine an optimal insurance/investment/consumption strat…
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…
This paper is devoted to invest an optimal investment strategy for a defined-contribution (DC) pension plan under the Ornstein-Uhlenbeck (O-U) process and the loan. By considering risk-free asset, a risky asset driven by O-U process and a loan in the financial market, we firstly set up the dynamic equation and the asset market model which are instrumental in achieving the expected utility of ultimate wealth at retirement. Secondly, the corresponding Hamilton-Jacobi-Bellman(HJB) equation is derived by means of dynamic programming principle. The explicit expression for the optimal investment str…
We consider the problem of maximizing expected utility from consumption in a constrained incomplete semimartingale market with a random endowment process, and establish a general existence and uniqueness result using techniques from convex duality. The notion of asymptotic elasticity of Kramkov and Schachermayer is extended to the time-dependent case. By imposing no smoothness requirements on the utility function in the temporal argument, we can treat both pure consumption and combined consumption/terminal wealth problems, in a common framework. To make the duality approach possible, we provid…
We consider an illiquid financial market with different regimes modeled by a continuous-time finite-state Markov chain. The investor can trade a stock only at the discrete arrival times of a Cox process with intensity depending on the market regime. Moreover, the risky asset price is subject to liquidity shocks, which change its rate of return and volatility, and induce jumps on its dynamics. In this setting, we study the problem of an economic agent optimizing her expected utility from consumption under a non-bankruptcy constraint. By using the dynamic programming method, we provide the chara…
We study the two-times differentiability of the value functions of the primal and dual optimization problems that appear in the setting of expected utility maximization in incomplete markets. We also study the differentiability of the solutions to these problems with respect to their initial values. We show that the key conditions for the results to hold true are that the relative risk aversion coefficient of the utility function is uniformly bounded away from zero and infinity, and that the prices of traded securities are sigma-bounded under the numéraire given by the optimal wealth process.
We introduce an extension to Merton's famous continuous time model of optimal consumption and investment, in the spirit of previous works by Pliska and Ye, to allow for a wage earner to have a random lifetime and to use a portion of the income to purchase life insurance in order to provide for his estate, while investing his savings in a financial market comprised of one risk-free security and an arbitrary number of risky securities driven by multi-dimensional Brownian motion. We then provide a detailed analysis of the optimal consumption, investment, and insurance purchase strategies for the …
We propose a network description of large market investments, where both stocks and shareholders are represented as vertices connected by weighted links corresponding to shareholdings. In this framework, the in-degree ($k_{in}$) and the sum of incoming link weights ($v$) of an investor correspond to the number of assets held (\emph{portfolio diversification}) and to the invested wealth (\emph{portfolio volume}) respectively. An empirical analysis of three different real markets reveals that the distributions of both $k_{in}$ and $v$ display power-law tails with exponents $γ$ and $α$. Moreover,…
Stock price prediction has been an important research theme both academically and practically. Various methods to predict stock prices have been studied until now. The feature that explains the stock price by a cross-section analysis is called a "factor" in the field of finance. Many empirical studies in finance have identified which stocks having features in the cross-section relatively increase and which decrease in terms of price. Recently, stock price prediction methods using machine learning, especially deep learning, have been proposed since the relationship between these factors and sto…
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.
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…
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.
Optimal Investment and Consumption in Financial Markets with Integrated Variance Clocks
CMAP, École Polytechnique（巴黎综合理工学院）；Department of Statistics, University of Warwick（华威大学统计学系）；Institut für Stochastik und Anwendungen, University of Stuttgart（斯图加特大学随机数学与应用研究所）
In a collectivised pension fund, investors agree that any money remaining in the fund when they die can be shared among the survivors. We give a numerical algorithm to compute the optimal investment-consumption strategy for an infinite collective of identical investors with exponential Kihlstrom--Mirman preferences, investing in the Black--Scholes market in continuous time but consuming in discrete time. Our algorithm can also be applied to an individual investor. We derive an analytic formula for the optimal consumption in the special case of an individual who chooses not to invest in the fin…
We study a portfolio management problem featuring many-player and mean field competition, investment and consumption, and relative performance concerns under the forward performance processes (FPP) framework. We focus on agents using power (CRRA) type FPPs for their investment-consumption optimization problem under a common noise Merton market model. We solve both the many-player and mean field game providing closed-form expressions for the solutions where the limit of the former yields the latter. In our case, the FPP framework yields a continuum of solutions for the consumption component as …
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 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 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 study optimal investment problems under the framework of cumulative prospect theory (CPT). A CPT investor makes investment decisions in a single-period financial market with transaction costs. The objective is to seek the optimal investment strategy that maximizes the prospect value of the investor's final wealth. We obtain the optimal investment strategy explicitly in two examples. An economic analysis is conducted to investigate the impact of the transaction costs and risk aversion on the optimal investment 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.…
Automated investment managers, or robo-advisors, have emerged as an alternative to traditional financial advisors. The viability of robo-advisors crucially depends on their ability to offer personalized financial advice. We introduce a novel framework, in which a robo-advisor interacts with a client to solve an adaptive mean-variance portfolio optimization problem. The risk-return tradeoff adapts to the client's risk profile, which depends on idiosyncratic characteristics, market returns, and economic conditions. We show that the optimal investment strategy includes both myopic and intertempor…
Stochastic portfolio theory aims at finding relative arbitrages, i.e. trading strategies which outperform the market with probability one. Functionally generated portfolios, which are deterministic functions of the market weights, are an invaluable tool in doing so. Driven by a practitioner point of view, where investment decisions are based upon consideration of various financial variables, we generalize functionally generated portfolios and allow them to depend on continuous-path semimartingales, in addition to the market weights. By means of examples we demonstrate how the inclusion of addi…
We consider a financial market model driven by an R^n-valued Gaussian process with stationary increments which is different from Brownian motion. This driving noise process consists of $n$ independent components, and each component has memory described by two parameters. For this market model, we explicitly solve optimal investment problems. These include (i) Merton's portfolio optimization problem; (ii) the maximization of growth rate of expected utility of wealth over the infinite horizon; (iii) the maximization of the large deviation probability that the wealth grows at a higher rate than a…
We study investment strategy in different models of financial markets, where the investors cannot reach a perfect knowledge about available assets. The investor spends a certain effort to get information; this allows him to better choose the investment strategy, and puts a selective pressure upon assets. The best strategy is then a compromise between diversification and effort to get information.
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.
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.
We consider a financial market where the asset price follows a fractional Brownian motion. We introduce a family of investment strategies, and quantify profit possibilities for both persistent and antipersistant markets.
In this paper, we consider a numéraire-based utility maximization problem under constant proportional transaction costs and random endowment. Assuming that the agent cannot short sell assets and is endowed with a strictly positive contingent claim, a primal optimizer of this utility maximization problem exists. Moreover, we observe that the original market with transaction costs can be replaced by a frictionless shadow market that yields the same optimality. On the other hand, we present an example to show that in some case when these constraints are relaxed, the existence of shadow prices is …
In a market with one safe and one risky asset, an investor with a long horizon, constant investment opportunities, and constant relative risk aversion trades with small proportional transaction costs. We derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume. At the first order, the liquidity premium equals the spread, times share turnover, times a universal constant. Results are robust to consumption and finite horizons. We exploit the equivalence of the transaction cost market to another frictionless market, with a shadow risky …
A fractal approach to the long-short portfolio optimization is proposed. The algorithmic system based on the composition of market-neutral spreads into a single entity was considered. The core of the optimization scheme is a fractal walk model of returns, optimizing a risk aversion according to the investment horizon. The covariance matrix of spread returns has been used for the optimization and modified according to the Hurst stability analysis. Out-of-sample performance data has been represented for the space of exchange traded funds in five period time period of observation. The considered …
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.
