Abstracts

Black and Litterman (1992) developed a framework that produces more accurate and stable assets' returns estimates as inputs to Markowitz's (1952) mean-variance portfolio optimization (MV). Within the Black-Litterman (BL) model, investors combine a prior vector, based on market equilibrium returns obtained through MV reverse optimization, and their views on the future returns of the assets to arrive at a posterior vector. Previous studies have demonstrated that BL helps to overcome the shortcomings of traditional MV.
Das et al (2010) proposed a mental accounting (MA) framework in which they integrate MV and behavioral portfolio theory (BPT) from Shefrin and Statman (2000). In MA investors divide their wealth in subportfolios, each having a different goal stated as a threshold return and maximum probability of not reaching it. Hence, instead of specifying a single risk-aversion coefficient, in MA investors can intuitively set multiple VaR-like constraints for their mental accounts.
We integrate BL and MA to achieve a full integration between standard and behavioral portfolio theory starting from parameter specification to optimization and to monitoring. The objective is to build portfolios that deliver a strong risk-adjusted performance as a consequence of sensible and intuitive processes. We compute the prior vector by reverse optimization using the MA equations, with market capitalization weights and a market threshold return and probability as inputs. The level of confidence on the views is treated as in Idzorek (2004). Once the posterior vector is computed through the BL equation, the weights of each subportfolio are calculated constrained by the investor's goals. Finally, we obtain the aggregated portfolio with the selected distribution of wealth.
A preliminary out-of-sample analysis with the 10 S&P 500 Sector Indices from Jan/2001 until Sep/2013 indicated that the performance of the proposed strategy is superior compared to the market portfolio. The investor's views were set as the historical means of each asset (with a 50\% confidence) and we used three mental accounts. The BL-MA strategy produced an annualized Sharpe ratio and 5\% monthly VaR and CVaR of 0.33, -6.44\%, and -9.22\%, while the market portfolio returned 0.24, -7.81\%, and -10.09\%. The break-even transaction cost of the BL-MA strategy was 0.3\%. The BL-MA strategy is then capable of overperforming its prior even in the presence of naïve views and when a high confidence level is assigned to them.

We present an extension of the Barndorff-Nielsen-Shephard stochastic volatility model class, where the jumps in the asset price and the jumps in the asset volatility are partially disentangled. For certain special cases, where jump dependence is established by linear or time change constructions, we deduce the characteristic function in a semi-closed form. In case of jumps following a compound Poisson processes with exponential jumps, we derive the weak-link Gamma-OU-BNS model, which has a true closed-form characteristic function and yields the Gamma-OU-BNS model in a limit case.

Recently, there were series of papers, where the solution of optimal stopping problems for Lévy processes and random walks when the reward is a monotone function were found in terms of the maximum/minimum of the process. In the present paper we use the integral transform method and develop it further to the case of non-monotone functions. We present a constructive method on how to solve optimal stopping problems (i.e. show how to find the optimal stopping boundary) for a fairly general reward function $g$. The key ingredient here is to introduce the integral transform $\mathcal{A}^{\eta(x)}$ based on the random variable $\eta(x) = \argmax g_{0 \leq t \leq e_q}(x+ X_s) - x$.
We find the stopping region as those arguments at which the function $\mathcal{A}^{\eta(x)}\{g\}$ (where $\mathcal{A}^{\eta(x)}$ is a form of an Esscher-Laplace transform) is non-negative, and the continuation region as those arguments at which the function $\mathcal{A}^\eta\{g\}$ is negative. Our algorithm to find the solution to the optimal stopping problem is the following: firstly, we introduce an auxiliary random variable $\eta(x)$ pathwise tracking the value of $X_t$ that achieved the running maximum of $g(x+X)$. Secondly, we use ${\eta(x)}$ to define the integral transform $\mathcal{A}^{\eta(x)}$. The integral transform maps the reward function $g=g(\cdot)$ into function $\mathcal{A}^{\eta(x)}\{g\} (\cdot)$ for each $x$. Finally, we find the stopping region $S$ as those arguments $(x,y)$ at which $\mathcal{A}^{\eta(x)}\{g\}$(y) is non-negative. The optimal strategy is to stop whenever $(x,x) \in S$, and to continue observations while $(x,x) \notin S$. We illustrate this approach with some examples. The proposed method is computationally attractive as it does not require solving differential or integro-differential equations which appear if the traditional methods are used.

The introduction of Knightian uncertainty in mathematical models for finance has recently renew the attention on foundational issues such as the existence of option pricing rules or super-hedging theorems. In this context an agent allows for the possibility of describing a certain market with a probabilistic model but he can’t be sure about a single reference probability, hence, a whole class of reference probabilities is taken into account. Although the literature on this topic is rapidly growing it is not clear yet what is a good notion of arbitrage opportunity and the consequent properties on martingale measures. In the present work we slightly change the point of view avoiding the necessity of fixing a subset of probability measures, which might be problematic in some settings. Given a certain market model, described by a discrete time stochastic process, we study the intrinsic properties of the model, independently on any reference probability. In particular two questions arises naturally: 1) which markets are feasible, in the sense that the properties of the market are nice for most probabilistic models? 2) which are the markets that exhibits no arbitrage for most probabilistic models?
A good notion for “most” probabilistic models is clearly needed and it is undertaken in this work from a topological point of view. Different results are obtained depending on the coarseness of the topology under consideration.
A key property is the existence of martingale measures. An important difference from the classical case is that we are not providing equivalent conditions in terms of absence of arbitrage opportunities, we are studying, instead, the structural properties of the market needed for the existence of such measures.
Once this minimal property of the market is guaranteed, we introduce definitions of arbitrage as trading strategies that work for most probabilistic models, in the spirit of the rest of the work. Additional properties on the martingale measures are retrieved under the No arbitrage hypothesis.