Worst case portfolio vectors and diversification effects

We consider the problem of identifying the worst case dependence structure of a portfolio X ₁,…,X _n of d-dimensional risks, which yields the largest risk of the joint portfolio. Based on a recent characterization result of law invariant convex risk measures, the worst case portfolio structure is identified as a μ-comonotone risk vector for some worst case scenario measure μ. It turns out that typically there will be a diversification effect even in worst case situations. The only exceptions arise when risks are measured by translated max correlation risk measures. We determine the worst case portfolio structure and the worst case diversification effect in several classes of examples as, e.g. in elliptical, Euclidean spherical, and Archimedean type distribution classes.     

Portfolio optimization under model uncertainty and BSDE games

We consider robust optimal portfolio problems for markets modeled by (possibly non-Markovian) Itô–Lévy processes. Mathematically, the situation can be described as a stochastic differential game, where one of the players (the agent) is trying to find the portfolio that maximizes the utility of her terminal wealth, while the other player (“the market”) is controlling some of the unknown parameters of the market (e.g., the underlying probability measure, representing a model uncertainty problem) and is trying to minimize this maximal utility of the agent. This leads to a worst case scenario control problem for the agent. In the Markovian case, such problems can be studied using the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation, but these methods do not work in the non-Markovian case. We approach the problem by transforming it into a stochastic differential game for backward stochastic differential equations (a BSDE game). Using comparison theorems for BSDEs with jumps we arrive at criteria for the solution of such games in the form of a kind of non-Markovian analogue of the HJBI equation. The results are illustrated by examples.     

Optimal consumption policies in illiquid markets

We investigate optimal consumption policies in the liquidity risk model introduced by Pham and Tankov (Math. Finance 18:613–627, 2008). Our main result is to derive smoothness C ¹ results for the value functions of the portfolio/consumption choice problem. As an important consequence, we can prove the existence of the optimal control (portfolio/consumption strategy) which we characterize both in feedback form in terms of the derivatives of the value functions and as the solution of a second-order ODE. Finally, numerical illustrations of the behavior of optimal consumption strategies between two trading dates are given.     

Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets

We consider the infinite-horizon optimal portfolio liquidation problem for a von Neumann–Morgenstern investor in the liquidity model of Almgren (Appl. Math. Finance 10:1–18, 2003). Using a stochastic control approach, we characterize the value function and the optimal strategy as classical solutions of nonlinear parabolic partial differential equations. We furthermore analyze the sensitivities of the value function and the optimal strategy with respect to the various model parameters. In particular, we find that the optimal strategy is aggressive or passive in-the-money, respectively, if and only if the utility function displays increasing or decreasing risk aversion. Surprisingly, only few further monotonicity relations exist with respect to the other parameters. We point out in particular that the speed by which the remaining asset position is sold can be decreasing in the size of the position but increasing in the liquidity price impact.      

An optimal execution problem with market impact

We study an optimal execution problem in a continuous-time market model that considers market impact. We formulate the problem as a stochastic control problem and investigate properties of the corresponding value function. We find that right-continuity at the time origin is associated with the strength of market impact for large sales; otherwise the value function is continuous. Moreover, we show the semigroup property (Bellman principle) and characterise the value function as a viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation. We present some examples where the form of the optimal strategy changes completely, depending on the amount of the trader’s security holdings, and where optimal strategies in the Black–Scholes type market with nonlinear market impact are not block liquidation but gradual liquidation, even when the trader is risk-neutral.     

Optimal and equilibrium execution strategies with generalized price impact

This paper examines the execution problems of large traders with a generalized price impact. Constructing two related models in a discrete-time setting, we solve these problems by applying the backward induction method of dynamic programming. In the first problem, we formulate the expected utility maximization problem of a single large trader as a Markov decision process and derive an optimal execution strategy. Then, in the second model, we formulate the expected utility maximization problem of two large traders as a Markov game and derive an equilibrium execution strategy at a Markov perfect equilibrium. Both of these two models enable us to investigate how the execution strategies and trade performances of a large trader are affected by the existence of other traders. Moreover, we find that these optimal and equilibrium execution strategies become deterministic when the total execution volumes of non-large traders are deterministic. We also show, by some numerical examples, the comparative statics results with respect to several problem parameters.     

Hedging structured credit products during the credit crisis: A horse race of 10 models

Pricing and hedging structured credit products poses major challenges to financial institutions. This paper puts several valuation approaches through a crucial test: How did these models perform in one of the worst periods of economic history, September 2008, when Lehman Brothers went under? Did they produce reasonable hedging strategies? We study several bottom-up and top-down credit portfolio models and compute the resulting delta hedging strategies using either index contracts or a portfolio of single-name CDS contracts as hedging instruments. We compute the profit-and-loss profiles and assess the performances of these hedging strategies. Among all 10 pricing models that we consider the Student-t copula model performs best. The dynamical generalized-Poisson loss model is the best top-down model, but this model class has in general problems to hedge equity tranches. Our major finding is however that single-name and index CDS contracts are not appropriate instruments to hedge CDO tranches.     

Focusing on the worst state for robust investing

Despite its shortcomings, the Markowitz model remains the norm for asset allocation and portfolio construction. A major issue involves sensitivity of the model's solution to its input parameters. The prevailing approach employed by practitioners to overcome this problem is to use worst-case optimization. Generally, these methods have been adopted without incorporating equity market behavior and we believe that an analysis is necessary. Therefore, in this paper, we present the importance of market information during the worst state for achieving robust performance. We focus on the equity market and find that the optimal portfolio in a market with multiple states is the portfolio with robust returns and observe that focusing on the worst market state provides robust returns. Furthermore, we propose alternative robust approaches that emphasize returns during market downside periods without solving worst-case optimization problems. Through our analyses, we demonstrate the value of focusing on the worst market state and as a result find support for the value of worst-case optimization for achieving portfolio robustness.     

Adaptive basket liquidation

We consider the infinite-horizon optimal basket portfolio liquidation problem for a von Neumann–Morgenstern investor in a multi-asset extension of the liquidity model of Almgren (Appl. Math. Finance 10:1–18, 2003) with cross-asset impact. Using a stochastic control approach, we establish a “separation theorem”: the sequence of portfolios held during an optimal liquidation depends only on the (co-)variance and (cross-asset) market impact of the assets, while the speed with which these portfolios are reached depends only on the utility function of the trader. We derive partial differential equations for both the sequence of portfolios reached during the execution and the trading speed.     

Optimal portfolio choice in the bond market

We consider the Merton problem of optimal portfolio choice when the traded instruments are the set of zero-coupon bonds. Working within a Markovian Heath–Jarrow–Morton model of the interest rate term structure driven by an infinite-dimensional Wiener process, we give sufficient conditions for the existence and uniqueness of an optimal trading strategy. When there is uniqueness, we provide a characterization of the optimal portfolio as a sum of mutual funds. Furthermore, we show that a Gauss–Markov random field model proposed by Kennedy [Math. Financ. 4, 247–258(1994)] can be treated in this framework, and explicitly calculate the optimal portfolio. We show that the optimal portfolio in this case can be identified with the discontinuities of a certain function of the market parameters.     

MDP algorithms for portfolio optimization problems in pure jump markets

We consider the problem of maximizing the expected utility of the terminal wealth of a portfolio in a continuous-time pure jump market with general utility function. This leads to an optimal control problem for piecewise deterministic Markov processes. Using an embedding procedure we solve the problem by looking at a discrete-time contracting Markov decision process. Our aim is to show that this point of view has a number of advantages, in particular as far as computational aspects are concerned. We characterize the value function as the unique fixed point of the dynamic programming operator and prove the existence of optimal portfolios. Moreover, we show that value iteration as well as Howard’s policy improvement algorithm works. Finally, we give error bounds when the utility function is approximated and when we discretize the state space. A numerical example is presented and our approach is compared to the approximating Markov chain method.      

Management of Portfolio Depletion Risk through Optimal Life Cycle Asset Allocation

Members of defined contribution (DC) pension plans must take on additional responsibilities for their investments, compared to participants in defined benefit (DB) pension plans. The transition from DB to DC plans means that more employees are faced with these responsibilities. We explore the extent to which DC plan members can follow financial strategies that have a high chance of resulting in a retirement scenario that is fairly close to that provided by DB plans. Retirees in DC plans typically must fund spending from accumulated savings. This leads to the risk of depleting these savings, that is, portfolio depletion risk. We analyze the management of this risk through life cycle optimal dynamic asset allocation, including the accumulation and decumulation phases. We pose the asset allocation strategy as an optimal stochastic control problem. Several objective functions are tested and compared. We focus on the risk of portfolio depletion at the terminal date, using such measures as conditional value at risk (CVAR) and probability of ruin. A secondary consideration is the median terminal portfolio value. The control problem is solved using a Hamilton-Jacobi-Bellman formulation, based on a parametric model of the financial market. Monte Carlo simulations that use the optimal controls are presented to evaluate the performance metrics. These simulations are based on both the parametric model and bootstrap resampling of 91 years of historical data. The resampling tests suggest that target-based approaches that seek to establish a safety margin of wealth at the end of the decumulation period appear to be superior to strategies that directly attempt to minimize risk measures such as the probability of portfolio depletion or CVAR. The target-based approaches result in a reasonably close approximation to the retirement spending available in a DB plan. There is a small risk of depleting the retiree’s funds, but there is also a good chance of accumulating a buffer that can be used to manage unplanned longevity risk or left as a bequest.     

Option market making under inventory risk

We propose a mean-variance framework to analyze the optimal quoting policy of an option market maker. The market maker’s profits come from the bid-ask spreads received over the course of a trading day, while the risk comes from uncertainty in the value of his portfolio, or inventory. Within this framework, we study the impact of liquidity and market incompleteness on the optimal bid and ask prices of the option. First, we consider a market maker in a complete market, where continuous trading in a perfectly liquid underlying stock is allowed. In this setting, the market maker may remove all risk by Delta hedging, and the optimal quotes will depend on the option’s liquidity, but not on the inventory. Second, we model a market maker who may not trade continuously in the underlying stock, but rather sets bid and ask quotes in the option and this illiquid stock. We find that the optimal stock and option quotes depend on the relative liquidity of both instruments as well as on the net Delta of the inventory. Third, we consider an incomplete market with residual risks due to stochastic volatility and large overnight moves in the stock price. In this setting, the optimal quotes depend on the liquidity of the option and on the net Vega and Gamma of the inventory.      

A Factor Allocation Approach to Optimal Bond Portfolio

This paper proposes a new method to a bond portfolio problem in a multi-period setting. In particular, we apply a factor allocation approach to constructing the optimal bond portfolio in a class of multi-factor Gaussian yield curve models. In other words, we consider a bond portfolio problem in terms of a factors’ allocation problem. Thus, we can obtain clear interpretation about the relation between the change in the shape of a yield curve and dynamic optimal strategy, which is usually hard to be obtained due to high correlations among individual bonds. We first present a closed form solution of the optimal bond portfolio in a class of the multi-factor Gaussian term structure model. Then, we investigate the effects of various changes in the term structure on the optimal portfolio strategy through series of comparative statics.     

Dynamic optimal portfolio choice in a jump-diffusion model with investment constraints

We consider the dynamic portfolio choice problem in a jump-diffusion model, where an investor may face constraints on her portfolio weights: for instance, no-short-selling constraints. It is a daunting task to use standard numerical methods to solve a constrained portfolio choice problem, especially when there is a large number of state variables. By suitably embedding the constrained problem in an appropriate family of unconstrained ones, we provide some equivalent optimality conditions for the indirect value function and optimal portfolio weights. These results simplify and help to solve the constrained optimal portfolio choice problem in jump-diffusion models. Finally, we apply our theoretical results to several examples, to examine the impact of no-short-selling and/or no-borrowing constraints on the performance of optimal portfolio strategies.     

Risk parity portfolio optimization under a Markov regime-switching framework

We formulate and solve a risk parity optimization problem under a Markov regime-switching framework to improve parameter estimation and to systematically mitigate the sensitivity of optimal portfolios to estimation error. A regime-switching factor model of returns is introduced to account for the abrupt changes in the behaviour of economic time series associated with financial cycles. This model incorporates market dynamics in an effort to improve parameter estimation. We proceed to use this model for risk parity optimization and also consider the construction of a robust version of the risk parity optimization by introducing uncertainty structures to the estimated market parameters. We test our model by constructing a regime-switching risk parity portfolio based on the Fama–French three-factor model. The out-of-sample computational results show that a regime-switching risk parity portfolio can consistently outperform its nominal counterpart, maintaining a similar ex post level of risk while delivering higher-than-nominal returns over a long-term investment horizon. Moreover, we present a dynamic portfolio rebalancing policy that further magnifies the benefits of a regime-switching portfolio.     

Iterated VaR or CTE measures: A false good idea?

The purpose of this paper is twofold. Firstly, we consider different risk measures in order to determine the solvency capital requirement of a pension fund. Secondly, we illustrate the impact of the time horizon of long-term guarantee products on these capital. We consider a financial market modelled by a common Black–Scholes–Merton model. We neglect the mortality and underwriting risks by assuming that the pension fund is fully hedged against these risks, which allows us to keep understandable and tractable formulæ (the longevity risk will be a part of future researches). A portfolio is built in this market according to different strategies and the pension fund offers a fixed guaranteed rate on a certain time horizon. We begin with well-known static risk measures (value at risk and conditional tail expectation measures) and then we consider their natural dynamic generalization. In order to be time consistent, we consider their iterated versions by a backward iterations scheme. Within the dynamic setting, we show that solvency capital can be expensive and that attention must be paid to the safety level considered.