Article Contents
Article Contents

A stochastic gradient descent algorithm to maximize power utility of large credit portfolios under Marshall–Olkin dependence

• *Corresponding author: Matthias Scherer
• A vector of bankruptcy times with Marshall–Olkin multivariate exponential distribution implies a simple, yet reasonable, continuous-time model for dependent credit-risky assets with an appealing trade-off between tractability and realism. Within this framework, the maximization of expected power utility of terminal wealth requires the optimization of a concave function on a polygon, a numerical problem whose complexity grows exponentially in the number of considered assets. We demonstrate how this seemingly impractical numerical problem can be solved reliably and efficiently in order to prepare the model for practical use cases. To this end, we resort to a specifically designed factor construction for the Marshall–Olkin distribution that separates dependence parameters from idiosyncratic parameters, and we develop a tailor-made stochastic gradient descent algorithm with random constraint projections for the model's numerical implementation. Finally, we explain a new method to include transaction costs and apply the model in a real-world, high-dimensional example.

Mathematics Subject Classification: Primary: 91B16, 91B70; Secondary: 49M05, 60J76.

 Citation:

• Figure 1.  The function $K \mapsto -U_p(K)$ for different values of $K$

Figure 2.  Visualization of the SGD algorithm with step width $\Delta_n = 1/\sqrt{n+1}$ for parameters $d = 2$, $p = 0$, $\mathit{\boldsymbol{\eta}} = (0.038, 0.065)$, $\mathit{\boldsymbol{\kappa}} = (0.3, 0.15)$, $\lambda_{\{1\}} = 0.021$, $\lambda_{\{2\}} = 0.041$, $\lambda_{\{1, 2\}} = 0.009$. The domain is specified by $\mathit{\boldsymbol{\ell}} = (0, 0)$, $\mathit{\boldsymbol{u}} = (0.35, 0.35)$, and $\epsilon = 0.8$. Here, the arithmetic average of the last $m_A = 100$ iterates is returned. Further, the stochastic gradient in each iteration is computed as the arithmetic average over $m_S$ iid simulations of $\mathit{\boldsymbol{F}}(\mathit{\boldsymbol{x}}^{(n)}, S)$ in step $n$. Top left: $\delta = \epsilon = 0.8$, $m_S = 1$. Top right: $\delta = 0.2$, $m_S = 1$. Bottom left: $\delta = 0.8$, $m_S = 20$. Bottom right: $\delta = 0.2$, $m_S = 20$

Figure 3.  Optimal portfolios computed via the SGD algorithm in an example with $d = 2$, $p = -1$, and $\epsilon = 0.1$. The model parameters are $\mathit{\boldsymbol{\eta}} = (0.018, 0.025)$, $\mathit{\boldsymbol{\kappa}} = (0.3, 0)$, $\lambda_{\{1\}}\approx 0.0199$, $\lambda_{\{2\}} \approx 0.0099$, $\lambda_{\{1, 2\}} \approx 0.0201$. In the cases with transaction costs it is assumed that $\eta_{i, \pm} = \eta_i\, (1 \pm c/2)$ for $c \in \{0, 0.025, 0.05, 0.075, 0.125, 0.25, 0.5\}$, as depicted in the plot. The contour lines correspond to the objective function without transaction costs

Figure 4.  Left: Histogram of the values $\mu_{i, \pm} = \eta_{i, \pm}-\Lambda_{\{i\}}\, (1-\kappa_i)$, $i \in [d]$; but only such $\mu_{i, +} \geq -2\%$ and only such $\mu_{i, -} \leq 2\%$. Right: Portfolio weights of the top $15$ holdings in the optimal portfolio $\mathit{\boldsymbol{x}}_{\ast}$ for different risk aversion parameters $p \in \{0, -2, -5\}$

Figure 5.  Left: The top 15 holdings of for risk aversion $p = -2$ in the right plot of Figure 4, together with their weights in four further runs of the same SGD algorithm. Right: Runtime in minutes for the SGD algorithm with $N = 100, 000$, $\epsilon = \delta = 0.6$, $m_A = m_S = 0$, with increasing number of assets $d \in [10, 410]$

Table 1.  Enhanced parameters when transaction costs are taken into account

 old asset $i$ new asset $(i, +)$ new asset $(i, -)$ default intensity $\Lambda_{\{i\}}$ $\Lambda_{\{(i, +)\}}=\Lambda_{\{i\}}$ $\Lambda_{\{(i, -)\}}=\Lambda_{\{i\}}$ recovery rate $\kappa_i$ $\kappa_{i, +}=\kappa_i$ $\kappa_{i, -}=\kappa_i$ factor weight $w_j^{(i)}$ $w_j^{(i, +)}=w_j^{(i)}$ $w_j^{(i, -)}=w_j^{(i)}$ yield $\eta_i$ $\eta_{i, +}<\eta_i$ $\eta_{i, -}>\eta_i$ lower restriction $\ell_i$ $\ell_{i, +}=\max\{\ell_i, 0\}\geq 0$ $\ell_{i, -}=\min\{\ell_i, 0\} \leq 0$ upper restriction $u_i$ $u_{i, +}=\max\{u_i, 0\} \geq 0$ $u_{i, -}=\min\{u_i, 0\} \leq 0$

Table 2.  One-parametric families of Laplace transforms $\varphi$. The column "$J \sim$" indicates how the jump distribution is simulated efficiently. We denote by $U \sim \mathcal{U}(0, 1)$ a random variable that is uniform on $(0, 1)$, and by $E$ an independent standard exponential random variable

 Name $\varphi(x)$ $\theta \in$ $\varphi^{-1}(y)$ $J \sim$ constant $e^{-\theta\, x}$ $(0, \infty)$ $-\log(y)/\theta$ $\theta$ exponential $\frac{\theta}{x+\theta}$ $(0, \infty)$ $\theta\, \frac{1-y}{y}$ $E/\theta$ stable $e^{-x^{\theta}}$ $(0, 1)$ $(-\log(y))^{\frac{1}{\theta}}$ $\sin( \theta\, \pi\, U)\, \Big( \frac{\cos\big(\pi\, U\, (1-\theta)-\pi/2\big)}{E\, \cos\big(\pi(U-1/2)\big)^{1/(1-\theta)}}\Big)^{\frac{1-\theta}{\theta}}$ gamma $(1+x)^{-\theta}$ $(0, \infty)$ $y^{-\frac{1}{\theta}}-1$ $\Gamma(\theta, 1)$, see [21, p. 240 ff] inv. Gauss 1 $e^{-\theta\, (\sqrt{2\, x+1}-1)}$ $(0, \infty)$ $\frac{1}{2}\, \Big\{\Big( 1-\frac{\log(y)}{\theta}\Big)^2-1 \Big\}$ IG$(\theta, 1)$, see [21, p. 245] inv. Gauss 2 $e^{-(\sqrt{2\, x+\theta^2}-\theta)}$ $(0, \infty)$ $\frac{1}{2}\, \big\{\big( \theta-\log(y)\big)^2-\theta^2 \big\}$ IG$(1, \theta)$, see [21, p. 245]
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