We provide a new characterization of mean-variance hedging strategies in a general semimartingale market. The key point is the introduction of a new probability measure $P^{star}$ which turns the dynamic asset allocation problem into a myopic one. The minimal martingale measure relative to $P^{star}$ coincides with the variance-optimal martingale measure relative to the original probability measure $P$.
We study the design of portfolios under a minimum risk criterion. The performance of the optimized portfolio relies on the accuracy of the estimated covariance matrix of the portfolio asset returns. For large portfolios, the number of available market returns is often of similar order to the number of assets, so that the sample covariance matrix performs poorly as a covariance estimator. Additionally, financial market data often contain outliers which, if not correctly handled, may further corrupt the covariance estimation. We address these shortcomings by studying the performance of a hybrid covariance matrix estimator based on Tylers robust M-estimator and on Ledoit-Wolfs shrinkage estimator while assuming samples with heavy-tailed distribution. Employing recent results from random matrix theory, we develop a consistent estimator of (a scaled version of) the realized portfolio risk, which is minimized by optimizing online the shrinkage intensity. Our portfolio optimization method is shown via simulations to outperform existing methods both for synthetic and real market data.
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 $Q_gamma$ featuring the expected earnings yield of portfolio minus a penalty term proportional with a coefficient $gamma$ to the variance when we keep the value of the factor levels fixed. The coefficient $gamma$ 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 $Q_gamma$ 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 $gamma$.
In this paper, we investigate dynamic optimization problems featuring both stochastic control and optimal stopping in a finite time horizon. The paper aims to develop new methodologies, which are significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature, to study a managers decision. We formulate our model to a free boundary problem of a fully nonlinear equation. Furthermore, by means of a dual transformation for the above problem, we convert the above problem to a new free boundary problem of a linear equation. Finally, we apply the theoretical results to challenging, yet practically relevant and important, risk-sensitive problems in wealth management to obtain the properties of the optimal strategy and the right time to achieve a certain level over a finite time investment horizon.
A continuous-time consumption-investment model with constraint is considered for a small investor whose decisions are the consumption rate and the allocation of wealth to a risk-free and a risky asset with logarithmic Brownian motion fluctuations. The consumption rate is subject to an upper bound constraint which linearly depends on the investors wealth and bankruptcy is prohibited. The investors objective is to maximize total expected discounted utility of consumption over an infinite trading horizon. It is shown that the value function is (second order) smooth everywhere but a unique possibility of (known) exception point and the optimal consumption-investment strategy is provided in a closed feedback form of wealth, which in contrast to the existing work does not involve the value function. According to this model, an investor should take the same optimal investment strategy as in Mertons model regardless his financial situation. By contrast, the optimal consumption strategy does depend on the investors financial situation: he should use a similar consumption strategy as in Mertons model when he is in a bad situation, and consume as much as possible when he is in a good situation.
We empirically test predictability on asset price by using stock selection rules based on maximum drawdown and its consecutive recovery. In various equity markets, monthly momentum- and weekly contrarian-style portfolios constructed from these alternative selection criteria are superior not only in forecasting directions of asset prices but also in capturing cross-sectional return differentials. In monthly periods, the alternative portfolios ranked by maximum drawdown measures exhibit outperformance over other alternative momentum portfolios including traditional cumulative return-based momentum portfolios. In weekly time scales, recovery-related stock selection rules are the best ranking criteria for detecting mean-reversion. For the alternative portfolios and their ranking baskets, improved risk profiles in various reward-risk measures also imply more consistent prediction on the direction of assets in future. In the Carhart four-factor analysis, higher factor-neutral intercepts for the alternative strategies are another evidence for the robust prediction by the alternative stock selection rules.
This paper considers exponential utility indifference pricing for a multidimensional non-traded assets model, and provides two linear approximations for the utility indifference price. The key tool is a probabilistic representation for the utility indifference price by the solution of a functional differential equation, which is termed emph{pseudo linear pricing rule}. We also provide an alternative derivation of the quadratic BSDE representation for the utility indifference price.
We implement momentum strategies using reward-risk measures as ranking criteria based on classical tempered stable distribution. Performances and risk characteristics for the alternative portfolios are obtained in various asset classes and markets. The reward-risk momentum strategies with lower volatility levels outperform the traditional momentum strategy regardless of asset class and market. Additionally, the alternative portfolios are not only less riskier in risk measures such as VaR, CVaR and maximum drawdown but also characterized by thinner downside tails. Similar patterns in performance and risk profile are also found at the level of each ranking basket in the reward-risk portfolios. Higher factor-neutral returns achieved by the reward-risk momentum strategies are statistically significant and large portions of the performances are not explained by the Carhart four-factor model.
An investor trades a safe and several risky assets with linear price impact to maximize expected utility from terminal wealth. In the limit for small impact costs, we explicitly determine the optimal policy and welfare, in a general Markovian setting allowing for stochastic market, cost, and preference parameters. These results shed light on the general structure of the problem at hand, and also unveil close connections to optimal execution problems and to other market frictions such as proportional and fixed transaction costs.
In a general semimartingale financial model, we study the stability of the No Arbitrage of the First Kind (NA1) (or, equivalently, No Unbounded Profit with Bounded Risk) condition under initial and under progressive filtration enlargements. In both cases, we provide a simple and general condition which is sufficient to ensure this stability for any fixed semimartingale model. Furthermore, we give a characterisation of the NA1 stability for all semimartingale models.
