# Abstracts

**Optimal Fund Management With Gradual Contributions**

*Igor Melichercik (Comenius University, Slovakia)*

Tuesday June 3, 14:00-14:30 | session 2.2 | Portfolio Optimization | room CD

We consider optimal investment for an individual pension savings plan in receipt of gradual contributions against which one cannot borrow, using expected power utility as the optimality criterion. It is well known that in the presence of credit constraints the Samuelson paradigm of investment in constant proportions out of total wealth (including current savings and future contributions) no longer applies. Instead, the optimal investment gives rise to so-called stochastic lifestyling, whereby for low levels of accumulated capital it is optimal to invest fully in stocks and then gradually switch to safer assets as the level of savings increases. We highlight in this paper that not only does the leverage between risky and safe assets change but also the actual mix of the risky assets varies over time.

Risky assets are modeled using a multidimensional Itô process. In addition there is a risk-free asset with deterministic value process. We formulate a problem of dynamic stochastic programming with credit constraints and present the corresponding Hamilton-Jacobi-Bellman (HJB) equation.

The fully nonlinear HJB equation is transformed (using Riccati transformation) into equivalent quasi-linear parabolic equation for which we prove (under natural conditions) the existence and uniqueness of the solution.

Since the computation of the fully optimal strategy is prohibitively technical for practitioners, we propose a quasi-optimal strategy involving only a static constrained quadratic programme, easily implementable in a spreadsheet. We show numerically that the CQP strategy is practically indistinguishable from the optimal investment in terms of its welfare implications. We provide an explicit formula which helps visualize the lifestyling effect and further lowers the technical barrier towards its implementation.

Finally we examine the welfare effect of the credit constraint and the role of leverage in potentially alleviating the effects of borrowing restrictions. The welfare effect of credit constraint is moderate for high levels of risk aversion, but it is highly significant for moderate and low risk aversion. In the latter case we conclude that plausible levels of leverage are only a partial substitute to investing the present value of all future contributions up-front.

**A Class of Incomplete Markets with Optimal Portfolio in Closed Form**

*Francesco Menoncin (Brescia University, Italy)*

Tuesday June 3, 14:30-15:00 | session 2.2 | Portfolio Optimization | room CD

We study an investor maximizing the expected power utility of his terminal wealth. Some closed form solutions for such a problem have been found in the literature in a framework characterized a follows: (i) there is one state variable (typically the riskless interest rate or the risk premium) following a mean-reverting process either with constant or with linear volatility, (ii) there is one risky asset, (iii) a bond may exist.

The existence (and uniqueness) of the optimal portfolio is usually approached by establishing the existence (and uniqueness) of a viscosity solution to the Hamilton-Jacobi-Bellman equation deriving from the stochastic optimal control problem. Here, instead, we find sufficient conditions for easily checking whether there exists a closed form solution to an optimal portfolio problem when there exist both a set of (stochastic) state variables and a set of risky assets. Furthermore, we provide the algebraic form of this exact solution.

These sufficient conditions must hold on some combinations of drift and diffusion coefficients of state variables and risky assets. More precisely, we are able to demonstrate that all the exact solutions available in the literature for incomplete markets can be traced back to a framework where both state variables and risky assets follow mean-reverting processes with either constant or linear volatility. In such a framework, the value function solving the optimization problem has either a log-quadratic or a log-linear form whose exponent solves a Riccati differential equation. When the coefficients of this differential equation are constant (which is the most common case in the literature), then we are able to find a closed for solution for the optimal portfolio.

Furthermore, we derive the properties of the “linear” solution and we show that the absolute values of portfolio composition are monotonic functions of time. The direction of this monotonicity can be easily checked looking at the sign of a parameter. Finally, we present the structure of an incomplete financial market with one risky asset and two state variables which allows for a closed form solution according to our previous results. This is something new in the literature, since all the existing papers which present optimal portfolios in closed form, take into account only one state variable (at least to our knowledge).

**A Weak Discrete American-Type Stochastic Target Problem and its Application**

*Geraldine Bouveret (Imperial College London, UK)*

Tuesday June 3, 15:00-15:30 | session 2.2 | Portfolio Optimization | room CD

We study a stochastic target problem with a controlled probability of success on a set of deterministic dates. Proceeding as in [1] we can suitably increase the state space and the controls to reduce the problem to a more standard stochastic target one. More precisely we can reduce the problem to a problem of super-replication of a Bermudean option. However the increased controls are then valued in an unbounded set. Nevertheless we can deduce the related dynamic programming equation. We then apply our results to the so-called quantile hedging example of [2]. We can then extend their result to the case where the constraint holds on a set of deterministic dates and find a pseudo-explicit solution.

[1] Bouchard B., Elie R. and Touzi N., (2009). Stochastic target problems with controlled loss. SIAM Journal on Control and Optimization, 48 (5), pp. 3123-3150.

[2] Follmer H. and Leukert P., (1999). Quantile Hedging. Finance and Stochastics, 3, pp. 251-273.

**Consumption and portfolio optimization with stochastic differential utility**

*Thomas Seiferling (University of Kaiserslautern, Germany)*

Tuesday June 3, 15:30-16:00 | session 2.2 | Portfolio Optimization | room CD

In this talk we present a general solution of the optimal consumption and portfolio selection problem for an investor with recursive preferences of Epstein-Zin type in an incomplete market. Analytic solutions for special parameterizations have previously been obtained by Chacko and Viceira (2005) and Kraft, Seifried and Steffensen (2013). We approach the optimization problem via the associated Hamilton-Jacobi-Bellman (HJB) partial differential equation. First it is shown that solutions of the HJB equation that satisfy a boundedness condition provide the solution to the corresponding consumption-portfolio optimization problem. For this novel verification theorem for SDU in incomplete markets, utility gradient inequalities similar to those of Schroder and Skiadas (1999) are used in combination with HJB methods. Finally we employ a fixed point argument to construct a classical solution of the HJB meeting the required boundedness conditions. More precisely, generalizing the approach of Berdjane and Pergamenshchikov (2013), we study the Feynman-Kac representation mapping $\Phi$ that is associated to a power transform of the HJB equation. A fixed point argument yields a fixed point of $\Phi$ in the space of continuous functions as a limit of iterations of $\Phi$. Using the probabilistic representation of this solution we are able to deduce convergence in $C^{0,1}$. This not only yields a theoretical optimality result, but also leads directly to an efficient method for the numerical computation of optimal strategies by iteratively solving linear parabolic PDEs. Our proposed method exhibits superlinear convergence. We illustrate our results for various popular models, including the Heston stochastic volatility model.