HEDGING AND PORTFOLIO OPTIMIZATION UNDER TRANSACTION COSTS: A MARTINGALE APPROACH12

We derive a formula for the minimal initial wealth needed to hedge an arbitrary contingent claim in a continuous-time model with proportional transaction costs; the expression obtained can be interpreted as the supremum of expected discounted values of the claim, over all (pairs of) probability measures under which the “wealth process” is a supermartingale. Next, we prove the existence of an optimal solution to the portfolio optimization problem of maximizing utility from terminal wealth in the same model, we also characterize this solution via a transformation to a hedging problem: the optimal portfolio is the one that hedges the inverse of marginal utility evaluated at the shadow state-price density solving the corresponding dual problem, if such exists. We can then use the optimal shadow state-price density for pricing contingent claims in this market. the mathematical tools are those of continuous-time martingales, convex analysis, functional analysis, and duality theory.     

UNIVERSAL INVESTMENT IN MARKETS WITH TRANSACTION COSTS

In this paper we investigate growth optimal investment in two-asset discrete-time markets with proportional transaction costs and no distributional assumptions on the market return sequences. We construct a policy with growth rate at least as large as any interval policy. Since interval policies are ε-optimal for independent and identically distributed (i.i.d.) markets ( Iyengar 2002 ), it follows that our policy when employed in an i.i.d. market is able to "learn" the optimal interval policy and achieve growth optimality; in other words, it is a universal growth optimal policy for i.i.d. markets.     

PORTFOLIO OPTIMIZATION WITH JUMPS AND UNOBSERVABLE INTENSITY PROCESS

We consider a financial market with one bond and one stock. The dynamics of the stock price process allow jumps which occur according to a Markov-modulated Poisson process. We assume that there is an investor who is only able to observe the stock price process and not the driving Markov chain. The investor's aim is to maximize the expected utility of terminal wealth. Using a classical result from filter theory it is possible to reduce this problem with partial observation to one with complete observation. With the help of a generalized Hamilton–Jacobi–Bellman equation where we replace the derivative by Clarke's generalized gradient, we identify an optimal portfolio strategy. Finally, we discuss some special cases of this model and prove several properties of the optimal portfolio strategy. In particular, we derive bounds and discuss the influence of uncertainty on the optimal portfolio strategy.     

NO ARBITRAGE UNDER TRANSACTION COSTS, WITH FRACTIONAL BROWNIAN MOTION AND BEYOND

We establish a simple no-arbitrage criterion that reduces the absence of arbitrage opportunities under proportional transaction costs to the condition that the asset price process may move arbitrarily little over arbitrarily large time intervals.
We show that this criterion is satisfied when the return process is either a strong Markov process with regular points, or a continuous process with full support on the space of continuous functions. In particular, we prove that proportional transaction costs of any positive size eliminate arbitrage opportunities from geometric fractional Brownian motion for H ∈ (0, 1) and with an arbitrary continuous deterministic drift.     

CONSTRAINED OPTIMIZATION WITH RESPECT TO STOCHASTIC DOMINANCE: APPLICATION TO PORTFOLIO INSURANCE

We are concerned with a classic portfolio optimization problem where the admissible strategies must dominate a floor process on every intermediate date (American guarantee). We transform the problem into a martingale, whose aim is to dominate an obstacle, or equivalently its Snell envelope. The optimization is performed with respect to the concave stochastic ordering on the terminal value, so that we do not impose any explicit specification of the agent's utility function. A key tool is the representation of the supermartingale obstacle in terms of a running supremum process. This is illustrated within the paper by an explicit example based on the geometric Brownian motion.     

MEAN–VARIANCE PORTFOLIO OPTIMIZATION WITH STATE‐DEPENDENT RISK AVERSION

The objective of this paper is to study the mean–variance portfolio optimization in continuous time. Since this problem is time inconsistent we attack it by placing the problem within a game theoretic framework and look for subgame perfect Nash equilibrium strategies. This particular problem has already been studied in Basak and Chabakauri where the authors assumed a constant risk aversion parameter. This assumption leads to an equilibrium control where the dollar amount invested in the risky asset is independent of current wealth, and we argue that this result is unrealistic from an economic point of view. In order to have a more realistic model we instead study the case when the risk aversion depends dynamically on current wealth. This is a substantially more complicated problem than the one with constant risk aversion but, using the general theory of time‐inconsistent control developed in Björk and Murgoci, we provide a fairly detailed analysis on the general case. In particular, when the risk aversion is inversely proportional to wealth, we provide an analytical solution where the equilibrium dollar amount invested in the risky asset is proportional to current wealth. The equilibrium for this model thus appears more reasonable than the one for the model with constant risk aversion.     

A CONVEX STOCHASTIC OPTIMIZATION PROBLEM ARISING FROM PORTFOLIO SELECTION

A continuous-time financial portfolio selection model with expected utility maximization typically boils down to solving a (static) convex stochastic optimization problem in terms of the terminal wealth, with a budget constraint. In literature the latter is solved by assuming a priori that the problem is well-posed (i.e., the supremum value is finite) and a Lagrange multiplier exists (and as a consequence the optimal solution is attainable). In this paper it is first shown that, via various counter-examples, neither of these two assumptions needs to hold, and an optimal solution does not necessarily exist. These anomalies in turn have important interpretations in and impacts on the portfolio selection modeling and solutions. Relations among the non-existence of the Lagrange multiplier, the ill-posedness of the problem, and the non-attainability of an optimal solution are then investigated. Finally, explicit and easily verifiable conditions are derived which lead to finding the unique optimal solution.     

CONTINUOUS-TIME MEAN-VARIANCE PORTFOLIO SELECTION WITH BANKRUPTCY PROHIBITION

A continuous-time mean-variance portfolio selection problem is studied where all the market coefficients are random and the wealth process under any admissible trading strategy is not allowed to be below zero at any time. The trading strategy under consideration is defined in terms of the dollar amounts, rather than the proportions of wealth, allocated in individual stocks. The problem is completely solved using a decomposition approach. Specifically, a (constrained) variance minimizing problem is formulated and its feasibility is characterized. Then, after a system of equations for two Lagrange multipliers is solved, variance minimizing portfolios are derived as the replicating portfolios of some contingent claims, and the variance minimizing frontier is obtained. Finally, the efficient frontier is identified as an appropriate portion of the variance minimizing frontier after the monotonicity of the minimum variance on the expected terminal wealth over this portion is proved and all the efficient portfolios are found. In the special case where the market coefficients are deterministic, efficient portfolios are explicitly expressed as feedback of the current wealth, and the efficient frontier is represented by parameterized equations. Our results indicate that the efficient policy for a mean-variance investor is simply to purchase a European put option that is chosen, according to his or her risk preferences, from a particular class of options.     

NO MARGINAL ARBITRAGE OF THE SECOND KIND FOR HIGH PRODUCTION REGIMES IN DISCRETE TIME PRODUCTION–INVESTMENT MODELS WITH PROPORTIONAL TRANSACTION COSTS

We consider a class of production–investment models in discrete time with proportional transaction costs. For linear production functions, we study a natural extension of the no‐arbitrage of the second kind condition introduced by Rásonyi. We show that this condition implies the closedness of the set of attainable claims and is equivalent to the existence of a strictly consistent price system under which the evaluation of future production profits is strictly negative. This allows us to discuss the closedness of the set of terminal wealth in models with nonlinear production, functions which may admit arbitrages of the second kind for low production regimes but not marginally for high production regimes.     

OPTIMAL CONSUMPTION AND PORTFOLIO DECISIONS WITH PARTIALLY OBSERVED REAL PRICES

We consider optimal consumption and portfolio investment problems of an investor who is interested in maximizing his utilities from consumption and terminal wealth subject to a random inflation in the consumption basket price over time. We consider two cases: (i) when the investor observes the basket price and (ii) when he receives only noisy observations on the basket price. We derive the optimal policies and show that a modified Mutual Fund Theorem consisting of three funds holds in both cases. The compositions of the funds in the two cases are the same, but in general the investor's allocations of his wealth into these funds will differ. However, in the particular case when the investor has constant relative risk-aversion (CRRA) utility, his optimal investment allocations into these funds are also the same in both cases.     

NEWS-GENERATED DEPENDENCE AND OPTIMAL PORTFOLIOS FOR n STOCKS IN A MARKET OF BARNDORFF-NIELSEN AND SHEPHARD TYPE

We consider Merton's portfolio optimization problem in a Black and Scholes market with non-Gaussian stochastic volatility of Ornstein–Uhlenbeck type. The investor can trade in n stocks and a risk-free bond. We assume that the dependence between stocks lies in that they partly share the Ornstein–Uhlenbeck processes of the volatility. We refer to these as news processes, and interpret this as that dependence between stocks lies solely in their reactions to the same news. The model is primarily intended for assets that are dependent, but not too dependent, such as stocks from different branches of industry. We show that this dependence generates covariance, and give statistical methods for both the fitting and verification of the model to data. Using dynamic programming, we derive and verify explicit trading strategies and Feynman–Kac representations of the value function for power utility.     

RECOVERING PORTFOLIO DEFAULT INTENSITIES IMPLIED BY CDO QUOTES

We propose a stable nonparametric algorithm for the calibration of “top‐down” pricing models for portfolio credit derivatives: given a set of observations of market spreads for collateralized debt obligation (CDO) tranches, we construct a risk‐neutral default intensity process for the portfolio underlying the CDO which matches these observations, by looking for the risk‐neutral loss process “closest” to a prior loss process, verifying the calibration constraints. We formalize the problem in terms of minimization of relative entropy with respect to the prior under calibration constraints and use convex duality methods to solve the problem: the dual problem is shown to be an intensity control problem, characterized in terms of a Hamilton–Jacobi system of differential equations, for which we present an analytical solution. Given a set of observed CDO tranche spreads, our method allows to construct a default intensity process which leads to tranche spreads consistent with the observations. We illustrate our method on ITRAXX index data: our results reveal strong evidence for the dependence of loss transitions rates on the previous number of defaults, and offer quantitative evidence for contagion effects in the (risk‐neutral) loss process.