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arXiv2013

Theory of Performance Participation Strategies

The purpose of this article is to introduce, analyze and compare two performance participation methods based on a portfolio consisting of two risky assets: Option-Based Performance Participation (OBPP) and Constant Proportion Performance Participation (CPPP). By generalizing the provided guarantee to a participation in the performance of a second risky underlying, the new strategies allow to cope with well-known problems associated with standard portfolio insurance methods, like e.g. the CPPI cash lock-in. This is especially an issue in times of market crisis. However, the minimum guaranteed portfolio value at the end of the investment horizon is not deterministic anymore, but subject to systematic risk instead. With respect to the comparison of the two strategies, various criteria are applied such as comparison of terminal payoffs and payoff distributions. General analytical expressions for all moments of both performance participation strategies as well as standard OBPI and CPPI are derived. Furthermore, dynamic hedging properties are examined, in particular classical delta hedging.

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arXiv2013

Continuous-time Mean-Variance Portfolio Selection with Stochastic Parameters

This paper studies a continuous-time market {under stochastic environment} where an agent, having specified an investment horizon and a target terminal mean return, seeks to minimize the variance of the return with multiple stocks and a bond. In the considered model firstly proposed by [3], the mean returns of individual assets are explicitly affected by underlying Gaussian economic factors. Using past and present information of the asset prices, a partial-information stochastic optimal control problem with random coefficients is formulated. Here, the partial information is due to the fact that the economic factors can not be directly observed. Via dynamic programming theory, the optimal portfolio strategy can be constructed by solving a deterministic forward Riccati-type ordinary differential equation and two linear deterministic backward ordinary differential equations.

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arXiv2013

Optimal investment and price dependence in a semi-static market

This paper studies the problem of maximizing expected utility from terminal wealth in a semi-static market composed of derivative securities, which we assume can be traded only at time zero, and of stocks, which can be traded continuously in time and are modeled as locally-bounded semi-martingales. Using a general utility function defined on the positive real line, we first study existence and uniqueness of the solution, and then we consider the dependence of the outputs of the utility maximization problem on the price of the derivatives, investigating not only stability but also differentiability, monotonicity, convexity and limiting properties.

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arXiv2013

Unified Framework of Mean-Field Formulations for Optimal Multi-period Mean-Variance Portfolio Selection

The classical dynamic programming-based optimal stochastic control methods fail to cope with nonseparable dynamic optimization problems as the principle of optimality no longer applies in such situations. Among these notorious nonseparable problems, the dynamic mean-variance portfolio selection formulation had posted a great challenge to our research community until recently. A few solution methods, including the embedding scheme, have been developed in the last decade to solve the dynamic mean-variance portfolio selection formulation successfully. We propose in this paper a novel mean-field framework that offers a more efficient modeling tool and a more accurate solution scheme in tackling directly the issue of nonseparability and deriving the optimal policies analytically for the multi-period mean-variance-type portfolio selection problems.

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arXiv2013

Investment and Consumption with Regime-Switching Discount Rates

This paper considers the problem of consumption and investment in a financial market within a continuous time stochastic economy. The investor exhibits a change in the discount rate. The investment opportunities are a stock and a riskless account. The market coefficients and discount factor switch according to a finite state Markov chain. The change in the discount rate leads to time inconsistencies of the investor's decisions. The randomness in our model is driven by a Brownian motion and a Markov chain. Following Ekeland and Pirvu we introduce and characterize the subgame perfect strategies. Numerical experiments show the effect of time preference on subgame perfect strategies and the pre-commitment strategies.

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arXiv2014

Portfolio Optimization under Partial Information with Expert Opinions: a Dynamic Programming Approach

This paper investigates optimal portfolio strategies in a market where the drift is driven by an unobserved Markov chain. Information on the state of this chain is obtained from stock prices and expert opinions in the form of signals at random discrete time points. As in Frey et al. (2012), Int. J. Theor. Appl. Finance, 15, No. 1, we use stochastic filtering to transform the original problem into an optimization problem under full information where the state variable is the filter for the Markov chain. The dynamic programming equation for this problem is studied with viscosity-solution techniques and with regularization arguments.

q-fin.PM↗

arXiv2014

Dynamic Credit Investment in Partially Observed Markets

We consider the problem of maximizing expected utility for a power investor who can allocate his wealth in a stock, a defaultable security, and a money market account. The dynamics of these security prices are governed by geometric Brownian motions modulated by a hidden continuous time finite state Markov chain. We reduce the partially observed stochastic control problem to a complete observation risk sensitive control problem via the filtered regime switching probabilities. We separate the latter into pre-default and post-default dynamic optimization subproblems, and obtain two coupled Hamilton-Jacobi-Bellman (HJB) partial differential equations. We prove existence and uniqueness of a globally bounded classical solution to each HJB equation, and give the corresponding verification theorem. We provide a numerical analysis showing that the investor increases his holdings in stock as the filter probability of being in high growth regimes increases, and decreases his credit risk exposure when the filter probability of being in high default risk regimes gets larger.

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arXiv2015

The General Structure of Optimal Investment and Consumption with Small Transaction Costs

We investigate the general structure of optimal investment and consumption with small proportional transaction costs. For a safe asset and a risky asset with general continuous dynamics, traded with random and time-varying but small transaction costs, we derive simple formal asymptotics for the optimal policy and welfare. These reveal the roles of the investors' preferences as well as the market and cost dynamics, and also lead to a fully dynamic model for the implied trading volume. In frictionless models that can be solved in closed form, explicit formulas for the leading-order corrections due to small transaction costs are obtained.

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arXiv2013

A liability tracking approach to long term management of pension funds

We propose a long term portfolio management method which takes into account a liability. Our approach is based on the LQG (Linear, Quadratic cost, Gaussian) control problem framework and then the optimal portfolio strategy hedges the liability by directly tracking a benchmark process which represents the liability. Two numerical results using empirical data published by Japanese organizations are served: simulations tracking an artificial liability and an estimated liability of Japanese organization. The latter one demonstrates that our optimal portfolio strategy can hedge his or her liability.

q-fin.PM↗

arXiv2014

Reinvestigating the Uncovered Interest Rate Parity Puzzle via Analysis of Multivariate Tail Dependence in Currency Carry Trades

The currency carry trade is the investment strategy that involves selling low interest rate currencies in order to purchase higher interest rate currencies, thus profiting from the interest rate differentials. This is a well known financial puzzle to explain, since assuming foreign exchange risk is uninhibited and the markets have rational risk-neutral investors, then one would not expect profits from such strategies. That is uncovered interest rate parity (UIP), the parity condition in which exposure to foreign exchange risk, with unanticipated changes in exchange rates, should result in an outcome that changes in the exchange rate should offset the potential to profit from such interest rate differentials. The two primary assumptions required for interest rate parity are related to capital mobility and perfect substitutability of domestic and foreign assets. Given foreign exchange market equilibrium, the interest rate parity condition implies that the expected return on domestic assets will equal the exchange rate-adjusted expected return on foreign currency assets. However, it has been shown empirically, that investors can actually earn arbitrage profits by borrowing in a country with a lower interest rate, exchanging for foreign currency, and investing in a foreign country with a higher interest rate, whilst allowing for any losses (or gains) from exchanging back to their domestic currency at maturity. Therefore trading strategies that aim to exploit the interest rate differentials can be profitable on average. The intention of this paper is therefore to reinterpret the currency carry trade puzzle in light of heavy tailed marginal models coupled with multivariate tail dependence features in the analysis of the risk-reward for the currency portfolios with high interest rate differentials and low interest rate differentials.

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arXiv2015

Fluctuation Analysis for the Loss From Default

We analyze the fluctuation of the loss from default around its large portfolio limit in a class of reduced-form models of correlated firm-by-firm default timing. We prove a weak convergence result for the fluctuation process and use it for developing a conditionally Gaussian approximation to the loss distribution. Numerical results illustrate the accuracy and computational efficiency of the approximation.

math.PR↗

arXiv2013

Optimal initiation of a GLWB in a variable annuity: no arbitrage approach

This paper offers a financial economic perspective on the optimal time (and age) at which the owner of a Variable Annuity (VA) policy with a Guaranteed Living Withdrawal Benefit (GLWB) rider should initiate guaranteed lifetime income payments. We abstract from utility, bequest and consumption preference issues by treating the VA as liquid and tradable. This allows us to use an American option pricing framework to derive a so-called optimal initiation region. Our main practical finding is that given current design parameters in which volatility (asset allocation) is restricted to less than 20%, while guaranteed payout rates (GPR) as well as bonus (roll-up) rates are less than 5%, GLWBs that are in-the-money should be turned on by the late 50s and certainly the early 60s. The exception to the rule is when a non-constant GPR is about to increase (soon) to a higher age band, in which case the optimal policy is to wait until the new GPR is hit and then initiate immediately. Also, to offer a different perspective, we invert the model and solve for the bonus (roll-up) rate that is required to justify delaying initiation at any age. We find that the required bonus is quite high and more than what is currently promised by existing products. Our methodology and results should be of interest to researchers as well as to the individuals that collectively have over \$1 USD trillion in aggregate invested in these products. We conclude by suggesting that much of the non-initiation at older age is irrational (which obviously benefits the insurance industry.)

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arXiv2013

Mirror and Synchronous Couplings of Geometric Brownian Motions

The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and show that, unlike in the case of Brownian motion, the optimality fails in general even if the geometric Brownian motions are martingales. On the other hand, we prove that in the cases of the ergodic average and the infinite time horizon criteria, the mirror coupling and the synchronous coupling are always optimal for general (possibly non-martingale) geometric Brownian motions. We show that the two couplings are efficient if and only if they are optimal over a finite time horizon and give a conjectural answer for the efficient couplings when they are suboptimal.

math.PR↗

arXiv2013

Double Whammy - How ICT Projects are Fooled by Randomness and Screwed by Political Intent

The cost-benefit analysis formulates the holy trinity of objectives of project management - cost, schedule, and benefits. As our previous research has shown, ICT projects deviate from their initial cost estimate by more than 10% in 8 out of 10 cases. Academic research has argued that Optimism Bias and Black Swan Blindness cause forecasts to fall short of actual costs. Firstly, optimism bias has been linked to effects of deception and delusion, which is caused by taking the inside-view and ignoring distributional information when making decisions. Secondly, we argued before that Black Swan Blindness makes decision-makers ignore outlying events even if decisions and judgements are based on the outside view. Using a sample of 1,471 ICT projects with a total value of USD 241 billion - we answer the question: Can we show the different effects of Normal Performance, Delusion, and Deception? We calculated the cumulative distribution function (CDF) of (actual-forecast)/forecast. Our results show that the CDF changes at two tipping points - the first one transforms an exponential function into a Gaussian bell curve. The second tipping point transforms the bell curve into a power law distribution with the power of 2. We argue that these results show that project performance up to the first tipping point is politically motivated and project performance above the second tipping point indicates that project managers and decision-makers are fooled by random outliers, because they are blind to thick tails. We then show that Black Swan ICT projects are a significant source of uncertainty to an organisation and that management needs to be aware of.

q-fin.GN↗

arXiv2015

Dynamic robust duality in utility maximization

A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: (i) The optimal terminal wealth $X^*(T) : = X_{φ^*}(T)$ of the problem to maximize the expected $U$-utility of the terminal wealth $X_φ(T)$ generated by admissible portfolios $φ(t), 0 \leq t \leq T$ in a market with the risky asset price process modeled as a semimartingale; (ii) The optimal scenario $\frac{dQ^*}{dP}$ of the dual problem to minimize the expected $V$-value of $\frac{dQ}{dP}$ over a family of equivalent local martingale measures $Q$, where $V$ is the convex conjugate function of the concave function $U$. In this paper we consider markets modeled by Itô-Lévy processes. In the first part we use the maximum principle in stochastic control theory to extend the above relation to a \emph{dynamic} relation, valid for all $t \in [0,T]$. We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process, $0 \leq t \leq T$. In the terminal time case $t=T$ we recover the classical duality connection above. We get moreover an explicit relation between the optimal portfolio $φ^*$ and the optimal measure $Q^*$. We also obtain that the existence of an optimal scenario is equivalent to the replicability of a related $T$-claim. In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a similar dynamic relation between them. In particular, we show how to get from the solution of one of the problems to the other. We illustrate the results with explicit examples.

q-fin.PM↗

arXiv2013

Balancing small fixed and proportional transaction cost in trading strategies

Transaction costs appear in financial markets in more than one form. There are several results in the literature on small proportional transaction cost and not that many on fixed transaction cost. In the present work, we heuristically study the effect of both types of transaction cost by focusing on a portfolio optimization. Here we assume the presence of fixed transaction cost and that there is a balance between fixed and proportional transaction cost, such that none of them dominates the other, asymptotically. We find out that the deviation of value function, when the fixed transaction cost is $\varepsilon$, from the Merton value function, without transaction cost, is of order $\varepsilon^1/2$ which is different from the pure proportional cost of $\varepsilon^2/3$. Based on this, we propose an expansion for the value function in terms of powers of $\varepsilon^1/2$.

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arXiv2013

On the Dividend Strategies with Non-Exponential Discounting

In this paper, we study the dividend strategies for a shareholder with non-constant discount rate in a diffusion risk model. We assume that the dividends can only be paid at a bounded rate and restrict ourselves to the Markov strategies. This is a time inconsistent control problem. The extended HJB equation is given and the verification theorem is proved for a general discount function. Considering the pseudo-exponential discount functions (Type I and Type II), we get the equilibrium dividend strategies and the equilibrium value functions by solving the extended HJB equations.

q-fin.PM↗

arXiv2013

Relative Robust Portfolio Optimization

Considering mean-variance portfolio problems with uncertain model parameters, we contrast the classical absolute robust optimization approach with the relative robust approach based on a maximum regret function. Although the latter problems are NP-hard in general, we show that tractable inner and outer approximations exist in several cases that are of central interest in asset management.

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