Mean-variance Optimal Strategy Research Articles

Portfolio optimization for cointelated pairs: SDEs vs Machine learning

With the recent rise of Machine Learning (ML) as a candidate to partially replace classic Financial Mathematics (FM) methodologies, we investigate the performances of both in solving the problem of dynamic portfolio optimization in continuous-time, finite-horizon setting for a portfolio of two assets that are intertwined. In the Financial Mathematics approach we model the asset prices not via the common approaches used in pairs trading such as a high correlation or cointegration, but with the cointelation model in Mahdavi-Damghani (2013) that aims to reconcile both short-term risk and long-term equilibrium. We maximize the overall P&L with Financial Mathematics approach that dynamically switches between a mean-variance optimal strategy and a power utility maximizing strategy. We use a stochastic control formulation of the problem of power utility maximization and solve numerically the resulting HJB equation with the Deep Galerkin method introduced in Sirignano and Spiliopoulos (2018). We turn to Machine Learning for the same P&L maximization problem and use clustering analysis to devise bands, combined with in-band optimization. Although this approach is model agnostic, results obtained with data simulated from the same cointelation model gives a slight competitive advantage to the ML over the FM methodology1.

On time-consistent multi-horizon portfolio allocation

We analyse the problem of constructing multiple mean-variance portfolios over increasing investment horizons in stochastic interest rate markets. The traditional one-period mean-variance optimal portfolios of Hansen and Richard (1987) require the replication of two payoffs. When several maturities are considered, different payoffs have to be replicated each time, with an impact on transaction costs. Using martingale decomposition techniques and introducing a family of risk-adjusted measures linked to increasing maturities, we provide an intertemporal version of the traditional orthogonal decomposition of asset returns. This allows us to construct a multi-horizon mean- variance frontier that is time-consistent and requires the replication of solely two payoffs for all horizons under consideration. When transaction costs are taken into account, our time-consistent mean-variance frontier may outperform the traditional mean-variance optimal strategies in terms of Sharpe ratio. Some interesting examples of this fact come from long-term contracts as life annuities.

Portfolio Optimization for Cointelated Pairs: SDEs vs Machine Learning

With the recent rise of Machine Learning (ML) as a candidate to partially replace classic Financial Mathematics (FM) methodologies, we investigate the performances of both in solving the problem of dynamic portfolio optimization in continuous-time, finite-horizon setting for a portfolio of two assets that are intertwined. In Financial Mathematics approach we model the asset prices not via the common approaches used in pairs trading such as a high correlation or cointegration, but with the cointelation model that aims to reconcile both short-term risk and long-term equilibrium. We maximize the overall P&L with Financial Mathematics approach that dynamically switches between a mean-variance optimal strategy and a power utility maximizing strategy. We use a stochastic control formulation of the problem of power utility maximization and solve numerically the resulting HJB equation with the Deep Galerkin method. We turn to Machine Learning for the same P&L maximization problem and use clustering analysis to devise bands, combined with in-band optimization. Although this approach is model agnostic, results obtained with data simulated from the same cointelation model as FM give an edge to ML.

High-Frequency Trading with Fractional Brownian Motion

In the high-frequency limit, conditional expected increments of fractional Brownian motion converge to a white noise, shedding their dependence on the path history and the forecasting horizon, and making dynamic optimization problems tractable. We find an explicit formula for locally mean-variance optimal strategies and their performance for an asset price that follows fractional Brownian motion. Without trading costs, risk-adjusted profits are linear in the trading horizon and rise asymmetrically as the Hurst exponent departs from Brownian motion, remaining finite as the exponent reaches zero while diverging as it approaches one. Trading costs penalize numerous portfolio updates from short-lived signals, leading to a finite trading frequency, which can be chosen so that the effect of trading costs is arbitrarily small, depending on the required speed of convergence to the high-frequency limit.

The 4% strategy revisited: a pre-commitment mean-variance optimal approach to wealth management

In contrast to single-period mean-variance (MV) portfolio allocation, multi-period MV optimal portfolio allocation can be modified slightly to be effectively a down-side risk measure. With this in mind, we consider multi-period MV optimal portfolio allocation in the presence of periodic withdrawals. The investment portfolio can be allocated between a risk-free investment and a risky asset, the price of which is assumed to follow a jump diffusion process. We consider two wealth management applications: optimal de-accumulation rates for a defined contribution pension plan and sustainable withdrawal rates for an endowment. Several numerical illustrations are provided, with some interesting implications. In the pension de-accumulation context, Bengen (1994)’s [J. Financial Planning, 1994, 7, 171–180], historical analysis indicated that a retiree could safely withdraw 4% of her initial retirement savings annually (in real terms), provided that her portfolio maintained an even balance between diversified equities and U.S. Treasury bonds. Our analysis does support 4% as a sustainable withdrawal rate in the pension de-accumulation context (and a somewhat lower rate for an endowment), but only if the investor follows an MV optimal portfolio allocation, not a fixed proportion strategy. Compared with a constant proportion strategy, the MV optimal policy achieves the same expected wealth at the end of the investment horizon, while significantly reducing the standard deviation of wealth and the probability of shortfall. We also explore the effects of suppressing jumps so as to have a pure diffusion process, but assuming a correspondingly larger volatility for the latter process. Surprisingly, it turns out that the MV optimal strategy is more effective when there are large downward jumps compared to having a high volatility diffusion process. Finally, tests based on historical data demonstrate that the MV optimal policy is quite robust to uncertainty about parameter estimates.

Better than pre-commitment mean-variance portfolio allocation strategies: A semi-self-financing Hamilton–Jacobi–Bellman equation approach

We generalize the idea of semi-self-financing strategies, originally discussed in Ehrbar (1990), and later formalized in Cui et al (2012), for the pre-commitment mean-variance (MV) optimal portfolio allocation problem. The proposed semi-self-financing strategies are built upon a numerical solution framework for Hamilton–Jacobi–Bellman equations, and can be readily employed in a very general setting, namely continuous or discrete re-balancing, jump-diffusions with finite activity, and realistic portfolio constraints. We show that if the portfolio wealth exceeds a threshold, an MV optimal strategy is to withdraw cash. These semi-self-financing strategies are generally non-unique. Numerical results confirming the superiority of the efficient frontiers produced by the strategies with positive cash withdrawals are presented. Tests based on estimation of parameters from historical time series show that the semi-self-financing strategy is robust to estimation ambiguities.

Quadratic Hedging of Basis Risk

This paper examines a simple basis risk model based on correlated geometric Brownian motions. We apply quadratic criteria to minimize basis risk and hedge in an optimal manner. Initially, we derive the Föllmer–Schweizer decomposition for a European claim. This allows pricing and hedging under the minimal martingale measure, corresponding to the local risk-minimizing strategy. Furthermore, since the mean-variance tradeoff process is deterministic in our setup, the minimal martingale- and variance-optimal martingale measures coincide. Consequently, the mean-variance optimal strategy is easily constructed. Simple pricing and hedging formulae for put and call options are derived in terms of the Black–Scholes formula. Due to market incompleteness, these formulae depend on the drift parameters of the processes. By making a further equilibrium assumption, we derive an approximate hedging formula, which does not require knowledge of these parameters. The hedging strategies are tested using Monte Carlo experiments, and are compared with results achieved using a utility maximization approach.

Better than Pre-Commitment Mean-Variance Portfolio Allocation Strategies: A Semi-Self- Financing Hamilton-Jacobi-Bellman Equation Approach

We generalize the idea of semi-self-financing strategies, originally discussed in Ehrbar, Journal of Economic Theory (1990), and later formalized in em Cui et al, Mathematical Finance 22 (2012), for the pre-commitment mean-variance (MV) optimal portfolio allocation problem. The proposed semi-self-financing strategies are built upon a numerical solution framework for Hamilton-Jacobi-Bellman equations, and can be readily employed in a very general setting, namely continuous or discrete re-balancing, jump-diffusions with finite activity, and realistic portfolio constraints. We show that if the portfolio wealth exceeds a threshold, an MV optimal strategy is to withdraw cash. These semi-self-financing strategies are generally non-unique. Numerical results confirming the superiority of the efficient frontiers produced by the strategies with positive cash withdrawals are presented.Tests based on estimation of parameters from historical time series show that the semi-self-financing strategy is robust to estimation ambiguities.

Optimal Price Trading and Mean Variance Hedging

AbstractThe price of a financial claim at a fixed time can represented by arandom variable H . In an incomplete market H can be approximatedby a trading strategy known as minimum variance hedging. Minimumvariance hedging can be extended to a mean-variance optimal strat-egy where a riskier trading strategy attempts to exploit the expecteddifference between the hedging strategy and the financial claim. Thispaper shows that any mean-variance optimal hedging strategy can bedecomposed into a sum of two conceptually and mathematically dif-ferent trades. One is the minimum variance hedging strategy and theother is the mean-variance optimal price ‘direction’ trading strategythat is independent of the hedging strategy and is only dependent onthe price of the asset used for hedging. General explicit optimal price‘direction’ strategies are also formulated.Keywords: Mean-Variance Optimal Hedging, Minimum VarianceHedging, Optimal Price Trading, Trade Separation, Variance Opti-mal Martingale Measure.JEL Codes: C61, G11.

Performance Measurement of Empirical and Theoretical Pension Investment Strategies

This thesis deals with both the pension products offered by Danish pension providers and optimal saving contracts under various optimization criteria. Its approach is to analyse the investment strategies defining a given product, compare them with each other and suggest the best product given a pension saver's preferences. We develop a performance measurement methodology, which allows us to compare the products with different risks. Numerical results are obtained through a number of simulation studies. We consider pure unit-linked products, with-profit unit-linked products, mean-variance optimal strategies with pre-commitment and constant proportion portfolio insurance (CPPI) strategies with a terminal floor.

Optimal Investment and Bounded Ruin Probability: Constant Portfolio Strategies and Mean-variance Analysis

We study the continuous-time portfolio optimization problem of an insurer. The wealth of the insurer is given by a classical risk process plus gains from trading in a risky asset, modelled by a geometric Brownian motion. The insurer is not only interested in maximizing the expected utility of wealth but is also concerned about the ruin probability. We thus investigate the problem of optimizing the expected utility for a bounded ruin probability. The corresponding optimal strategy in various special classes of possible investment strategies will be calculated. For means of comparison we also calculate the related mean-variance optimal strategies.

Optimal Investment and Bounded Ruin Probability: Constant Portfolio Strategies and Mean-variance Analysis

We study the continuous-time portfolio optimization problem of an insurer. The wealth of the insurer is given by a classical risk process plus gains from trading in a risky asset, modelled by a geometric Brownian motion. The insurer is not only interested in maximizing the expected utility of wealth but is also concerned about the ruin probability. We thus investigate the problem of optimizing the expected utility for a bounded ruin probability. The corresponding optimal strategy in various special classes of possible investment strategies will be calculated. For means of comparison we also calculate the related mean-variance optimal strategies.

Along But Beyond Mean-Variance: Utility Maximization in a Semimartingale Model

It is well known that under certain assumptions the strategy of an investor maximizing his expected utility coincides with the mean-variance optimal strategy. In this paper we show that the two strategies are not equal in general and find the connection between a utility maximizing and a mean-variance optimal strategy in a continuous semimartingale model. That is done by showing that the utility maximizing strategy of a CARA investor can be expressed in terms of expectation and the expected quadratic variation of the underlying price process. It coincides with the mean-variance optimal strategy if the underlying price process is a local martingale. Keywords: mean-variance portfolios, utility maximization, dynamic portfolio selection, quadratic variation JEL classification numbers: G11, C61

Mean-variance hedging for continuous processes: New proofs and examples

Let $X$ be a special semimartingale of the form $X=X_0+M+\int d\langle M\rangle\,\widehat\lambda$ and denote by $\widehat K=\int \widehat\lambda^{\rm tr}\,d\langle M\rangle\,\widehat\lambda$ the mean-variance tradeoff process of $X$ . Let $\Theta$ be the space of predictable processes $\theta$ for which the stochastic integral $G(\theta)=\int\theta\,dX$ is a square-integrable semimartingale. For a given constant $c\in{\Bbb R}$ and a given square-integrable random variable $H$ , the mean-variance optimal hedging strategy $\xi^{(c)}$ by definition minimizes the distance in ${\cal L}^2(P)$ between $H-c$ and the space $G_T(\Theta)$ . In financial terms, $\xi^{(c)}$ provides an approximation of the contingent claim $H$ by means of a self-financing trading strategy with minimal global risk. Assuming that $\widehat K$ is bounded and continuous, we first give a simple new proof of the closedness of $G_T(\Theta)$ in ${\cal L}^2(P)$ and of the existence of the Föllmer-Schweizer decomposition. If moreover $X$ is continuous and satisfies an additional condition, we can describe the mean-variance optimal strategy in feedback form, and we provide several examples where it can be computed explicitly. The additional condition states that the minimal and the variance-optimal martingale measures for $X$ should coincide. We provide examples where this assumption is satisfied, but we also show that it will typically fail if $\widehat K_T$ is not deterministic and includes exogenous randomness which is not induced by $X$ .