Merton Portfolio Research Articles

Geometric Compound Hawkes Process and its Applications in Finance

The authors introduce a new model for a stock price, namely, geometric compound Hawkes process, and show how this model can be applied to solving many problems in finance, including European and American option pricing (perpetual American options), and Merton portfolio optimization problem. This model is a generalization of some well-known models in finance, such as Cox-Ross-Rubinstein model (1976) (geometric binomial process), Aase model (1988) (geometric compound Poisson process) and geometric Markov renewal model (2013). numerical examples are presented as well

Optimal Control Problems for Stochastic Processes with Absorbing Regime

In this paper we formulate and solve an optimal problem for Stochastic process with a regime absorbing state. The solution for this problem is obtained through a system of partial differential equations. The method is applied to obtain an explicit solution for the Merton portfolio problem when an asset has a default probability in case of a log utility.

Parameter identification for portfolio optimization with a slow stochastic factor

Abstract In this paper, we intend to identify two significant parameters – expected return and absolute risk aversion – in the Merton portfolio optimization problem under an exponential utility function where volatility is driven by a slow mean-reverting diffusion process. First, we find the approximate solution of the fully nonlinear Hamilton–Jacobi–Bellman equation for the Merton model by the stochastic asymptotic approximation method. Second, we estimate parameters – expected return and absolute risk aversion – through the approximate solution and prove the uniqueness and stability of the parameter identification problem. Finally, we provide an illustrative example to demonstrate the capacity and efficiency of our method.

Homotopy Analysis Method for Portfolio Optimization Problem Under the 3/2 Model

This paper considers the Merton portfolio optimization problem for an investor that aims at maximizing the expected power utility of the terminal wealth and intermediate consumption. Applying the homotopy analysis method, an analytical solution for value function as well as optimal strategy under the 3/2 model is derived, respectively. Compared with the existing explicit solutions for Merton problem under the 3/2 model, the formulas provide certain parameters with less requirement since the homotopy analysis method does not depend on the existence of small parameters in the equation. Finally, numerical examples are examined with the approach, and the proposed solution provides more accurate approximation as the number of terms in infinite series increases.

A note on - vs. -expected loss portfolio constraints

We consider portfolio optimization problems with expected loss constraints under the physical measure and the risk neutral measure , respectively. Using Merton's portfolio as a benchmark portfolio, the optimal terminal wealth of the -risk constraint problem can be easily replicated with the standard delta hedging strategy. Motivated by this, we consider the -strategy fulfilling the -risk constraint and compare its solution with the true optimal solution of the -risk constraint problem. We show the existence and uniqueness of the optimal solution to the -strategy fulfilling the -risk constraint, and provide a tractable evaluation method. The -strategy fulfilling the -risk constraint is not only easier to implement with standard forwards and puts on a benchmark portfolio than the -risk constraint problem, but also easier to solve than either of the - or -risk constraint problem. The numerical test shows that the difference of the values of the two strategies (the -strategy fulfilling the -risk constraint and the optimal strategy solving the -risk constraint problem) is reasonably small.

Optimal Distributional Trading Gain: State Price Density Equilibrium and Bayesian Statistics

This paper considers multiple market agents who have distinct distributional opinions about the state price density. We first determine the optimal trading positions of a utility maximizing market taker who trades Arrow-Debreu securities for prices set by the market maker. We use calculus of variations to determine the solution of this problem for a general utility function. The choice of the logarithmic utility function leads to a solution in terms of a likelihood ratio of the densities corresponding to the market taker and the market maker and the resulting optimal utility is the Kullback-Leibler divergence. In particular, we obtain a trivial solution for Merton's portfolio problem in the traditional geometric Brownian motion model and and we show its immediate extension to the multivariate case. A further extension gives a solution for the market driven by a geometric Poisson process. In a market without the market maker, the distributional opinions of market takers reach an equilibrium in the form of the linear mixture of the distributions. We show that when the the result of the outcome is observed, the profit and loss from trading updates agents' bankrolls in a Bayesian fashion, which provides one to one correspondence for the logarithmic utility maximazers' profits and Bayesian statistics. We extend these results to the continuous time setup and show that the bankrolls of agents following Merton optimal portfolio strategy evolve as a posterior distribution.

Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance

This paper is an attempt for solving operator-valued quantum stochastic optimal control problems, in Fock space. For this purpose, the dynamics of the classical system state is described by Hudson–Parthasarathy type Quantum Stochastic Differential Equation (QSDE) in Fock space and then by associating a quadratic performance criterion with the QSDE, a Quantum Stochastic Linear Quadratic Regulator (QS-LQR) optimal control problem is formulated. Also, an algorithm for solving the QS-LQR optimal control problem is designed. For solving the resulting optimal control problem, a new HJB equation is obtained. Thereby, the operator valued control process is obtained. Two theorems are proved to facilitate the algorithm. In this paper, for the first time, the optimal strategy for trading stock is designed via the presented method. For this purpose, Merton portfolio allocation problem is solved. The simulation results show that portfolio optimal performances, minimum risk and maximum return are achieved via presented method.

Optimal investment and consumption with return predictability and execution costs

We provide a closed-form solution to an optimal investment and consumption problem for a constant absolute risk aversion (CARA) agent, who faces execution costs when trading correlated risky assets with return predictability. The optimal investment strategy indicates that the agent should trade gradually toward a dynamic aim portfolio, which is an adjusted Merton portfolio with modifications to account for the persistence of the return-predicting signals and the execution costs. The optimal consumption strategy is quadratic in the return-predicting signals and linear in the agent's wealth. Our numerical studies show that the execution costs diminish the importance of asset return predictability on the agent's optimal investment strategy, thereby confirming the conjecture raised by Liu (2004). In addition, the presence of the intermediate consumption leads to a more aggressive aim portfolio than the case without consumption.

Optimal investment strategies with a minimum performance constraint

We consider the optimal investment problem of a fund manager in the presence of a minimum guarantee constraint on the fund performance. The manager receives a fee which is proportional to the liquidation value of the portfolio or of the surplus over the guarantee in case it is positive and zero otherwise, eventually augmented by a constant fee. Her remuneration is reduced through the application of a penalty if the value of the fund at maturity is below a specified-in-advance threshold (minimum guarantee). We deal with two different settings: a continuous time economy with constant instantaneous interest rate and the case where the interest rate evolves as the Vasicek model. Explicit formulas for the optimal investment strategy are presented. We compare our portfolio strategies to the Merton portfolio and to the Option Based Portfolio Insurance strategy.

Dynamic portfolio choice with return predictability and transaction costs

We derive a closed-form solution to a continuous-time optimal portfolio selection problem with return predictability and transaction costs. Specifically, we assume that asset returns are predicted by stochastic signals, and that transaction costs are of quadratic form. The agent chooses a trading strategy to maximize the expected exponential utility of his terminal wealth. Our feedback trading strategy indicates that the agent should trade gradually toward a dynamic aim portfolio, which is a weighted sum of the expected future Merton portfolios. The agent’s aim portfolio converges to the Merton portfolio as time approaches the terminal date. Our analysis offers new insights to the existing literature. First, our optimal trading strategy is affected by the volatility of return-predicting factors, while such an effect is absent in Gârleanu and Pedersen (2016). Secondly, the agent invests more into the assets with more persistent signals and with less transaction costs.

A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion

In this paper, we study general time-inconsistent stochastic control models which are driven by a stochastic differential equation with random jumps. Specifically, the time-inconsistency arises from the presence of a non-exponential discount function in the objective functional. We consider equilibrium, instead of optimal, solution within the class of open-loop controls. We prove an equivalence relationship between our time-inconsistent problem and a time-consistent problem such that the equilibrium controls for the time-consistent problem coincide with the equilibrium controls for the time-inconsistent problem. We establish two general results which characterize the open-loop equilibrium controls. As special cases, a generalized Merton's portfolio problem and a linear-quadratic problem are discussed.

Optimal Investment and Consumption with Return Predictability and Execution Costs

We provide a closed-form solution to an optimal investment and consumption problem for a constant absolute risk aversion (CARA) agent, who faces execution costs when trading correlated risky assets with return predictability. The optimal investment strategy indicates that the agent should trade gradually toward a dynamic aim portfolio, which is an adjusted Merton portfolio with modifications to account for the persistence of the return predicting signals and the execution costs. The optimal consumption strategy is quadratic in the return predicting signals and linear in the agent’s wealth. Our numerical studies show that the execution costs diminish the importance of asset return predictability on the agent’s optimal investment strategy, thereby confirming the conjecture raised by Liu (2004). In addition, the presence of the intermediate consumption leads to a more aggressive aim portfolio than the case without consumption.

Optimal control design for a class of quantum stochastic systems with financial applications

The purpose of this paper is to design an optimal quantum controller for a class of stochastic systems with application in financial problems. Dynamics of the system is prescribed via a Quantum Stochastic Differential System (QSDES) with a quantum Brownian motion on a quantum probability space. A theorem for guaranteeing the existence and uniqueness of solutions to the QSDES is proved. Additionally, a new optimal stochastic control problem is formulated and based on the necessary optimality conditions, an optimal quantum control law is designed, explicitly. Four theorems and two lemmas, for facilitating the optimal controller design algorithm, are proved. Finally, for demonstrating the applicable results, two financial problems, Merton portfolio allocation and optimal pairs trading problem are simulated by using the presented method. As the simulation results indicate, portfolio optimal performances, minimum risk and maximum return, are achieved via presented method.

High-Water Mark Fees with Stochastic Benchmark

A hedge fund manager receives high-water mark fees when the fund's relative return over a stochastic benchmark reaches a new maximum, and investments the fund in a constant investment opportunity, aiming to maximize the expected utility from fees in the long run. The manager chooses a Merton portfolio, like an investor who owns the fund if the Sharpe ratio $\nu$ of the fund investment opportunity is low. As $\nu$ increases and exceeds an endogenous threshold $\underline \nu$, the manager's effective risk aversion is shifted towards one, and the hedging component against the benchmark emerges and becomes larger, until it reaches the maximum if $\nu = \bar \nu > \underline \nu$, and keeps constant for $\nu >\bar{\nu}$. A stochastic benchmark in high-water mark fees introduces the moral hazard of risky fund investment with hedging motives, but also provides a possibility of regulating the manager's risk taking with a carefully chosen benchmark. Furthermore, the resulting optimal strategy confirms the empirical results that fund performance does not necessarily increase with the rate of performance fees.

Constructing a Long-Term Investment Portfolio (Part I)

A practical guide to our view of the current state-of-the-art in constructing and scaling a diversified, long-term investment portfolio. The goal of this note is not to break any new theoretical ground, but rather to choose the best out of many frameworks available, and delve into the key practical issues involved with solutions which are both theoretically and practically sound. In Part I, we begin by constructing a growth- and inflation-neutral beta portfolio in the spirit of Dalio et. al., but with a number of idiosyncratic implementation choices. Then we apply the Merton Portfolio Problem to chose optimal levels of consumption and leverage based on the investor's level of risk aversion. In Part II, we will discuss and work through the issues involved in overlaying alpha strategies.

Resolution of Degeneracy in Merton's Portfolio Problem

The Merton problem determines the optimal intertemporal portfolio choice by maximizing the expected utility, and is the basis of modern portfolio theory in continuous-time finance. However, its empirical performance is disappointing. The estimation errors of the expected rates of returns make the optimal policy degenerate, resulting in an extremely low (or unbounded) expected utility value for a high-dimensional portfolio. We further prove that the estimation error of the variance-covariance matrix leads to the degenerated policy of solely investing in the risk-free asset. This study proposes a constrained l1 - minimization approach to resolve the degeneracy. The proposed scheme can be implemented with simple linear programming and involves negligible additional computational time, compared to standard estimation. We prove the consistency of our framework that our estimate of the optimal control tends to be the true one. We also derive the rate of convergence. Simulation studies are provided to verify the finite-sample properties. An empirical study using S&P 500 component stock data demonstrates the superiority of the proposed approach.

Resolution of Degeneracy in Merton's Portfolio Problem

The Merton problem determines the optimal intertemporal portfolio choice by maximizing the expected utility and is the basis of modern portfolio theory in continuous-time finance. However, its empirical performance is disappointing. The estimation errors of the expected rates of returns make the optimal policy degenerate, resulting in an extremely low (or unbounded) expected utility value for a high-dimensional portfolio. We further prove that the estimation error of the variance-covariance matrix leads to the degenerated policy of solely investing in the risk-free asset. This study proposes a constrained $\ell_1$-minimization approach to resolve the degeneracy in the high-dimensional setting and stabilize the performance in the low-dimensional setting. The proposed scheme can be implemented with simple linear programming and involves negligible additional computational time, compared to standard estimation. We prove the consistency of our framework that our estimate of the optimal control tends to be the true one. We also derive the rate of convergence. Simulation studies are provided to verify the finite-sample properties. An empirical study using S&P 500 component stock data demonstrates the superiority of the proposed approach.

A Computational Method for Stochastic Optimal Control Problems in Financial Mathematics

AbstractPrinciple of optimality or dynamic programming leads to derivation of a partial differential equation (PDE) for solving optimal control problems, namely the Hamilton‐Jacobi‐Bellman (HJB) equation. In general, this equation cannot be solved analytically; thus many computing strategies have been developed for optimal control problems. Many problems in financial mathematics involve the solution of stochastic optimal control (SOC) problems. In this work, the variational iteration method (VIM) is applied for solving SOC problems. In fact, solutions for the value function and the corresponding optimal strategies are obtained numerically. We solve a stochastic linear regulator problem to investigate the applicability and simplicity of the presented method and prove its convergence. In particular, for Merton's portfolio selection model as a problem of portfolio optimization, the proposed numerical method is applied for the first time and its usefulness is demonstrated. For the nonlinear case, we investigate its convergence using Banach's fixed point theorem. The numerical results confirm the simplicity and efficiency of our method.

PORTFOLIO OPTIMIZATION AND STOCHASTIC VOLATILITY ASYMPTOTICS

We study the Merton portfolio optimization problem in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its timescales of fluctuation. This approach is tractable because it treats the incomplete markets problem as a perturbation around the complete market constant volatility problem for the value function, which is well understood. When volatility is fast mean‐reverting, this is a singular perturbation problem for a nonlinear Hamilton–Jacobi–Bellman partial differential equation, while when volatility is slowly varying, it is a regular perturbation. These analyses can be combined for multifactor multiscale stochastic volatility models. The asymptotics shares remarkable similarities with the linear option pricing problem, which follows from some new properties of the Merton risk tolerance function. We give examples in the family of mixture of power utilities and also use our asymptotic analysis to suggest a “practical” strategy that does not require tracking the fast‐moving volatility. In this paper, we present formal derivations of asymptotic approximations, and we provide a convergence proof in the case of power utility and single‐factor stochastic volatility. We assess our approximation in a particular case where there is an explicit solution.

Portfolio Optimization under Local-Stochastic Volatility: Coefficient Taylor Series Approximations and Implied Sharpe Ratio

We study the finite horizon Merton portfolio optimization problem in a general local-stochastic volatility setting. Using model coefficient expansion techniques, we derive approximations for both the value function and the optimal investment strategy. We also analyze the “implied Sharpe ratio” and derive a series approximation for this quantity. The zeroth order approximation of the value function and optimal investment strategy correspond to those obtained by [Merton, Rev. Econ. Statist., 51, pp. 247--257] when the risky asset follows a geometric Brownian motion. The first order correction of the value function can, for general utility functions, be expressed as a differential operator acting on the zeroth order term. For power utility functions, higher order terms can also be computed as a differential operator acting on the zeroth order term. While our approximations are derived formally, we give a rigorous accuracy bound for the higher order approximations in this case in pure stochastic volatility models. A number of examples are provided in order to demonstrate numerically the accuracy of our approximations.