Outperforming a Benchmark with $α$-Bregman Wasserstein divergence

Abstract

We consider the problem of active portfolio management, where an investor seeks the portfolio with maximal expected utility of the difference between the terminal wealth of their strategy and a proportion of the benchmark's, subject to a budget and a deviation constraint from the benchmark portfolio. As the investor aims at outperforming the benchmark, they choose a divergence that asymmetrically penalises gains and losses as well as penalises underperforming the benchmark more than outperforming it. This is achieved by the recently introduced $α$-Bregman-Wasserstein divergence, subsuming the Bregman-Wasserstein and the popular Wasserstein divergence. We prove existence and uniqueness, characterise the optimal portfolio strategy, and give explicit conditions when the divergence constraints and the budget constraints are binding. We conclude with a numerical illustration of the optimal quantile function in a geometric Brownian motion market model.

Figures (5)

• Figure 1: Integrand of the $\alpha$-BW divergence for the Power family $\phi_p$ from the quantile function ${\Breve{G}}$ given in \ref{['eq:modified-bench']} to the benchmark quantile ${\Breve{F}}_{X_T^{{\boldsymbol{{\pi}}}}}$. Top left panel: $p \in \{1.6, 1.8, 2, 2.2, 2.4\}$ and $\alpha = 0.5$. Top right panel: $\alpha \in \{0.01, 0.25, 0.5\}$ and $p = 1.6$. Bottom left panel: $\alpha \in \{0.01, 0.25, 0.5\}$ and $p = 2$. Bottom right panel: $\alpha \in \{0.01, 0.25, 0.5\}$ and $p = 2.4$.
• Figure 2: Optimal density functions for the Power Bregman generator $\phi_p$. Left: $p = 2$ (one-half the squared Wasserstein distance). Right: $p = 1.6$,. Common parameters $\alpha = 0.25$, $x_0 = 1$, $c = 0.9$, $\gamma = \frac{1}{2}$, and ${\varepsilon} = \frac{1}{2}$.
• Figure 3: Optimal density function $g{^*}$ of \ref{['opt:main-prime']} when both constraints are binding for different choices of $p \in \{1.6, 1.8, 2, 2.2, 2.4\}$ of the Power Bregman generator. Left $\alpha = 0.1$, right $\alpha = 0.5$. Common parameters are $x_0 = 1$, $c = 0.9$, $\gamma = \frac{1}{2}$, and ${\varepsilon} = 0.5$.
• Figure 4: Optimal density function $g{^*}$ of \ref{['opt:main-prime']} when both constraints are binding for different choices of $\alpha \in \{0.1, 0.5, 0.9\}$. Left $p = 1.6$, middle $p = 2$, right $p = 2.4$. To the left of the red dashed vertical line, the strategies underperform the benchmark, i.e., ${{\Breve{F}}_{Y}}(\cdot)> {\Breve{G}}_{{\boldsymbol{{\eta}}}*, \alpha}(\cdot)$. To the right of the red dashed vertical line, the strategies outperform the benchmark. The red lines are numerically same for ${\Breve{G}}_{{\boldsymbol{{\eta}}}*, \alpha}$, $\alpha\in \{0.1, 0.5, 0.9\}$ and fixed $p$. Common parameters are $c = 0.9$, $x_0 = 1$, $\gamma = \frac{1}{2}$, and ${\varepsilon} = 0.5$.
• Figure 5: Optimal density function $g{^*}$ of \ref{['opt:main-prime']} when both constraints are binding for different choices of $c$. Left $c \in \{0.5, 0.7, 0.9\}$ and $x_0 = 1$. Right $(c, x_0) \in \{(0.8, 0.9), (0.85, 0.95), (0.9, 1)\}$. Common parameters are $\alpha = 0.1$, $p=1.6$, $\gamma = \frac{1}{2}$, and ${\varepsilon} = 0.5$.

Theorems & Definitions (58)

• Definition 2.1
• Definition 2.2: $\alpha$-Bregman-Wasserstein divergence - Pesenti2024ORL
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Example 2.1: Geometric Brownian motion market model
• Definition 2.3
• Proposition 2.1: Budget constraint
• proof