Optimal Portfolio Choice Problem Research Articles

An optimal portfolio choice problem with delays

An optimal portfolio choice problem with delays

Dynamic portfolio choice with uncertain rare-events risk in stock and cryptocurrency markets

In response to the unprecedented uncertain rare events of the last decade, we derive an optimal portfolio choice problem in a semi-closed form by integrating price diffusion ambiguity, volatility diffusion ambiguity, and jump ambiguity occurring in the traditional stock market and the cryptocurrency market into a single framework. We reach the following conclusions in both markets: first, price diffusion and jump ambiguity mainly determine detection-error probability; second, optimal choice is more significantly affected by price diffusion ambiguity than by jump ambiguity, and trivially affected by volatility diffusion ambiguity. In addition, investors tend to be more aggressive in a stable market than in a volatile one. Next, given a larger volatility jump size, investors tend to increase their portfolio during downward price jumps and decrease it during upward price jumps. Finally, the welfare loss caused by price diffusion ambiguity is more pronounced than that caused by jump ambiguity in an incomplete market. These findings enrich the extant literature on effects of ambiguity on the traditional stock market and the evolving cryptocurrency market. The results have implications for both investors and regulators.

Multi-period power utility optimization under stock return predictability

In this paper, we derive an analytical solution to the dynamic optimal portfolio choice problem in the case of an investor equipped with a power utility function of wealth. The results are established by solving the Bellman backward recursion under the assumption that the vector of asset returns follows a vector-autoregressive process with predictable variables. In an empirical study, the performance of the derived solution is compared with the one obtained by applying the numerical method. The comparison is performed in terms of the final wealth and its expected utility. It is documented that the application of the analytical solution to the multi-period portfolio choice problem leads to higher values of both the final wealth and the expected utility.

Robust Portfolio Choice with Sticky Wages

We present a robust version of the life-cycle optimal portfolio choice problem in the presence of labor income, as introduced in Biffis, Gozzi, and Prosdocimi [SIAM J. Control Optim., 58 (2020), pp. 1906--1938] and Dybvig and Liu [J. Econom. Theory, 145 (2010), pp. 885--907]. In particular, in Biffis, Gozzi, and Prosdocimi the influence of past wages on the future ones is modeled linearly in the evolution equation of labor income, through a given weight function. The optimization relies on the resolution of an infinite dimensional HJB equation. We improve the state of art in three ways. First, we allow the weight to be a Radon measure. This accommodates for more realistic weighting of the sticky wages, e.g., on a discrete temporal grid according to some periodic income. Second, there is a general correlation structure between labor income and stock markets. This naturally affects the optimal hedging demand, which may increase or decrease according to the correlation sign. Third, we allow the weight to change with time, possibly lacking perfect identification. The uncertainty is specified by a given set of Radon measures K, in which the weight process takes values. This renders the inevitable uncertainty on how the past affects the future, and includes the standard case of error bounds on a specific estimate for the weight. Under uncertainty averse preferences, the decision maker takes a maxmin approach to the problem. Our analysis confirms the intuition: in the infinite dimensional setting, the optimal policy remains the best investment strategy under the worst case weight.

Robust consumption and portfolio choice with derivatives trading

This paper analyzes the optimal consumption and portfolio choice problem for an ambiguity-averse investor with recursive preferences. The investor has access to both the stock and derivatives markets. The stock price process follows a stochastic volatility jump-diffusion model and the investor can have different levels of ambiguity about diffusion and jump risks, respectively. We obtain an exact analytical solution for investors with unit elasticity of intertemporal substitution of consumption and an approximate solution otherwise. We find that the optimal consumption policy is more sensitive to ambiguity aversion with respect to diffusion risks than the jump risk. The optimal exposures to diffusion and jump risks are significantly affected by the corresponding ambiguity aversions in the complete market; however, the optimal stock investment is relatively insensitive to jump misspecification in the incomplete market. We also show that taking into consideration ambiguity aversion to diffusion risks and participating in the derivatives market are essential to reduce the potential welfare loss, while the impact of jump misspecification is marginal.

Pricing of variance swap rates and investment decisions of variance swaps: Evidence from a three-factor model

This paper proposes a tractable three-factor model with self-exciting jumps for the S&P 500 index and its variance. The statistical results show that this new model empirically outperforms the two-factor models widely studied in the literature in terms of fitting variance swap rates on the S&P 500 index across maturities ranging from one month to two years over the sample period from 2008 to 2020. We also provide closed-form solution for variance swap rates and analytically solve the optimal portfolio choice problem in variance swaps for an investor with a power utility function. Unlike the optimal investment with two variance swaps in two-factor models, the investor follows a “long-short-long” trading strategy involving three swap contracts. Hence, a third swap is not redundant in our three-factor model. In particular, our portfolio optimization exercises illustrate that ignoring a third variance swap in the investment problem may incur significant economic costs in the proposed model.

Optimal portfolio choice with path dependent benchmarked labor income: A mean field model

We consider the life-cycle optimal portfolio choice problem faced by an agent receiving labor income and allocating her wealth to risky assets and a riskless bond subject to a borrowing constraint. In this paper, to reflect a realistic economic setting, we propose a model where the dynamics of the labor income has two main features. First, labor income adjusts slowly to financial market shocks, a feature already considered in Biffis et al. (2015). Second, the labor income yi of an agent i is benchmarked against the labor incomes of a population yn≔(y1,y2,…,yn) of n agents with comparable tasks and/or ranks. This last feature has not been considered yet in the literature and is faced taking the limit when n→+∞ so that the problem falls into the family of optimal control of infinite-dimensional McKean–Vlasov Dynamics, which is a completely new and challenging research field.We study the problem in a simplified case where, adding a suitable new variable, we are able to find explicitly the solution of the associated HJB equation and find the optimal feedback controls. The techniques are a careful and nontrivial extension of the ones introduced in the previous papers of Biffis et al. (2015, 0000).

Implied Ambiguity: Mean-Variance Inefficiency and Pricing Errors

We investigate the optimal portfolio choice problem for an investor who has a utility function of the smooth ambiguity model. We identify necessary and sufficient conditions for a given portfolio to be optimal for such an investor. We define the implied ambiguity of a portfolio as the smallest ambiguity aversion coefficient with which the portfolio is optimal, and the measure of ambiguity perception as the part of the variability in asset returns that can be attributed to the ambiguity. We show that there are one-to-one relations between the implied ambiguity, the Sharpe ratio, and the pricing errors when the portfolio is taken as the pricing portfolio, and that the measure of ambiguity perception is determined by the Sharpe ratio and the alpha. Based on the U.S. stock market data, we assess how ambiguity averse the representative investor is and what types of stocks the investor perceives as having more ambiguous returns than others. This paper was accepted by Manel Baucells, behavioral economics and decision analysis.

Warren Buffett versus Zvi Bodie: Should You Buy Or Sell Put Options?

Academics and investment professionals often disagree when it comes to investment advice. Legendary investor Warren Buffett is a proponent of time diversification and firmly believes that stocks are less risky in the long run. Therefore, he often sells long-term put options instead of buying them for portfolio protection. By contrast, the famous finance professor Zvi Bodie argues that time diversification is a fallacy and, therefore, his advice to fund managers is to buy long-term portfolio insurance. In this article, we consider the optimal portfolio choice problem for a loss-averse investor. First, we demonstrate that our loss-averse investor subscribes to the principle of time diversification. In particular, our investor allocates more to stocks as the investment horizon lengthens. Second, we allow our investor to trade in stocks and put options. We find that when the investment horizon is short, our investor is better off with portfolio insurance. Conversely, when the investment horizon is long, our investor sells put options. That is, our loss-averse investor prefers Buffett’s investment advice over Bodie’s. <b>TOPICS:</b>Portfolio theory, portfolio construction, risk management, derivatives, options <b>Key Findings</b> ▪ Legendary investor Warren Buffett is a proponent of time diversification, and, therefore, he often sells long-term put options instead of buying them for portfolio protection. ▪ Famous professor Zvi Bodie argues that time diversification is a fallacy and his advice to fund managers is to buy long-term portfolio insurance. ▪ In this article, we show that a loss-averse investor prefers Buffett’s investment advice over Bodie’s.

Portfolio choices: comparative statics under both expected return and volatility uncertainty

This paper studies the comparative statics of an optimal portfolio choice problem for an investor with both expected return and volatility ambiguity about the financial market. The optimal holding of the risky asset depends on risk preference, expected return and volatility ambiguity, yielding a general comparative statistics analysis for all investors with linearly growing absolute risk tolerance.

Robust Portfolio Choice With Sticky Wages

We present a robust version of the life-cycle optimal portfolio choice problem in the presence of labor income, as introduced in Biffis, Gozzi and Prosdocimi (Optimal portfolio choice with path dependent labor income: the infinite horizon case, SIAM Journal on Control and Optimization, 58(4), 1906-1938.) and Dybvig and Liu (Lifetime consumption and investment: retirement and constrained borrowing, Journal of Economic Theory, 145, pp. 885-907). In particular, in Biffis, Gozzi and Prosdocimi the influence of past wages on the future ones is modelled linearly in the evolution equation of labor income, through a given weight function. The optimization relies on the resolution of an infinite dimensional HJB equation. We improve the state of art in three ways. First, we allow the weight to be a Radon measure. This accommodates for more realistic weighting of the sticky wages, like e.g. on a discrete temporal grid according to some periodic income. Second, there is a general correlation structure between labor income and stocks market. This naturally affects the optimal hedging demand, which may increase or decrease according to the correlation sign. Third, we allow the weight to change with time, possibly lacking perfect identification. The uncertainty is specified by a given set of Radon measures $K$, in which the weight process takes values. This renders the inevitable uncertainty on how the past affects the future, and includes the standard case of error bounds on a specific estimate for the weight. Under uncertainty averse preferences, the decision maker takes a maxmin approach to the problem. Our analysis confirms the intuition: in the infinite dimensional setting, the optimal policy remains the best investment strategy under the worst case weight.

Optimal Investment Policy for Insurers under the Constant Elasticity of Variance Model with a Correlated Random Risk Process

This paper investigates the optimal portfolio choice problem for a large insurer with negative exponential utility over terminal wealth under the constant elasticity of variance (CEV) model. The surplus process is assumed to follow a diffusion approximation model with the Brownian motion in which is correlated with that driving the price of the risky asset. We first derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation and then obtain explicit solutions to the value function as well as the optimal control by applying a variable change technique and the Feynman–Kac formula. Finally, we discuss the economic implications of the optimal policy.

Bayesian mean–variance analysis: optimal portfolio selection under parameter uncertainty

The paper solves the problem of optimal portfolio choice when the parameters of the asset returns distribution, for example the mean vector and the covariance matrix, are unknown and have to be estimated by using historical data on asset returns. Our new approach employs the Bayesian posterior predictive distribution which is the distribution of future realizations of asset returns given the observable sample. The parameters of posterior predictive distributions are functions of the observed data values and, consequently, the solution of the optimization problem is expressed in terms of data only and does not depend on unknown quantities. By contrast, the optimization problem of the traditional approach is based on unknown quantities which are estimated in the second step, and lead to a suboptimal solution. We also derive a very useful stochastic representation of the posterior predictive distribution whose application not only gives the solution of the considered optimization problem, but also provides the posterior predictive distribution of the optimal portfolio return which can be used to construct a prediction interval. A Bayesian efficient frontier, the set of optimal portfolios obtained by employing the posterior predictive distribution, is constructed as well. Theoretically and using real data we show that the Bayesian efficient frontier outperforms the sample efficient frontier, a common estimator of the set of optimal portfolios which is known to be overoptimistic.

A generalized stochastic differential utility driven by G-Brownian motion

This paper introduces a class of generalized stochastic differential utility (GSDU) models in a continuous-time framework to capture ambiguity aversion on the financial market. This class of GSDU models encompasses several classical approaches to ambiguity aversion and includes new models about ambiguity aversion. For a general GSDU model, we demonstrate its continuity, monotonicity, time consistency, concavity, and homotheticity. We investigate its comparative ambiguity aversion and direction aversion under sufficient conditions. We further solve an optimal portfolio choice problem in one GSDU model as an application.

Optimal Investing after Retirement Under Time-Varying Risk Capacity Constraint

This paper explores an optimal investing problem for a retiree facing longevity risk and living standard risk. We formulate the optimal investing problem as an optimal portfolio choice problem under a time-varying risk capacity constraint. Under the specific condition on model parameters, we show that the value function is a $C^2$ solution of the HJB equation and derive the optimal investment strategy in terms of second-order ordinary differential equations. The optimal portfolio is nearly neutral to the stock market movement if the portfolio's value is at a sufficiently high level; but, if the portfolio is not worth enough to sustain the retirement spending, the retiree actively invests in the stock market for the higher expected return. In addition, we solve an optimal portfolio choice problem under a leverage constraint and show that the optimal portfolio would lose significantly in stressed markets. This paper shows that the time-varying risk capacity constraint has important implications for asset allocation in retirement.

Robust portfolio choice for a DC pension plan with inflation risk and mean-reverting risk premium under ambiguity

This paper investigates a robust optimal portfolio choice problem for a defined contribution (DC) pension plan member. The member worries about model ambiguity and aims to seek robust optimal investment strategies. Specifically, the member has a stochastic salary, considers inflation risk and invests her pension account wealth into a financial market consisting of a risk-free asset, an inflation-indexed bond and a stock whose expected return rate follows a mean-reverting process. By using the dynamic programming approach, the robust optimal investment strategy and the corresponding value function are explicitly derived, and subsequently a verification theorem is provided. Furthermore, two special cases of our portfolio model are discussed. Finally, a numerical example is presented to reveal economic implications of our theoretical results and to illustrate the effects of the model parameters on the robust optimal investment strategy. We find that ambiguity about the dynamics of the inflation-indexed bond price and the stock price and expected return rate has different influences on the robust optimal investment strategy, and that neglecting ambiguity will always lead to utility loss.

Dynamic Portfolio Choice with Annuities When the Interest Rate Is Stochastic

This paper studies the optimal life cycle consumption and portfolio choice problem taking into account annuity risk due to stochastic interest rates. When the purchase of annuities is restricted to the retirement date, the annuitant is exposed to the risk of meeting low interest rates at the purchase date. This annuity risk can be diversified by spreading annuity purchases over the whole pre-retirement period. The numerical results of our life cycle model show that such temporal diversification enhances welfare for retirees up to 9% of certainty equivalent consumption and that welfare gains increase with higher interest rate risk.

A risk‐sensitive stochastic maximum principle for fully coupled forward‐backward stochastic differential equations with applications

AbstractIn this paper, we are interested in the problem of optimal control where the system is given by a fully coupled forward‐backward stochastic differential equation with a risk‐sensitive performance functional. As a preliminary step, we use the risk neutral which is an extension of the initial control system where the admissible controls are convex, and an optimal solution exists.Then, we study the necessary as well as sufficient optimality conditions for risk sensitive performance. At the end of this work, we illustrate our main result by giving an example that deals with an optimal portfolio choice problem in financial market, specifically the model of control cash flow of a firm or project where, for instance, we can set the model of pricing and managing an insurance contract.

Optimal Portfolio Choice in a Jump-Diffusion Model with Self-Exciting

We solve the optimal portfolio choice problem for an investor who can trade a risk-free asset and a risky asset. The investor faces both Brownian and jump risks and the jump is modeled by a Hawkes process so that occurrence of a jump in the risky asset price triggers more sequent jumps. We obtain the optimal portfolio by maximizing expectation of a constant relative risk aversion (CRRA) utility function of terminal wealth. The existence and uniqueness of a classical solution to the associated partial differential equation are proved, and the corresponding verification theorem is provided as well. Based on the theoretical results, we develop a numerical monotonic iteration algorithm and present an illustrative numerical example.

Robust Portfolio Choice and Consumption with Derivative Trading Under Stochastic Volatility and Jumps

This paper analyzes the optimal consumption and portfolio choice problem for an ambiguity averse investor who has access to the stock and derivatives markets with recursive preferences. The stock process follows a stochastic volatility jump-diffusion model and the investor can have different levels of ambiguity about diffusion risks and the jump risk, respectively. We obtain an analytical solution which is exact when the investor has unit elasticity of intertemporal substitution of consumption, and approximate otherwise. We find that optimal exposures to diffusion risks and to the jump risk are significantly affected by the ambiguity aversion about the corresponding risk factors in the complete market. However, the optimal stock investment is insensitive to the ambiguity aversion about the jump risk in the incomplete market. We also find that considering ambiguity aversion with respect to diffusion risks and participating in the derivatives markets are essential to reduce the potential welfare loss, while the impact of ignoring the jump ambiguity is negligible.