# A Risk-Based Approach for Asset Allocation with A Defaultable Share

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## Abstract

**:**

## 1. Introduction

## 2. The Model Dynamics

- (
**A1**) - There exists a $\mathbb{G}$-adapted (intensity) process $\left\{\lambda \right(t\left)\right|t\in \mathcal{T}\}$ such that$$\begin{array}{c}\hfill M\left(t\right):=N\left(t\right)-{\int}_{0}^{t}{\mathbf{1}}_{\{\tau >s\}}\lambda \left(s\right)ds\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}t\in \mathcal{T}\phantom{\rule{4pt}{0ex}},\end{array}$$
- (
**A2**) - Every $\mathbb{G}$-local martingale is an $\mathbb{F}$-local martingale.

**A2**) is usually called the H-hypothesis, and under this condition ${W}_{1}$ and ${W}_{2}$ are also two $(\mathbb{F},\mathcal{P})$-Brownian motions.

- (1)
- ${\mathcal{K}}_{n}$ is the set of $\mathbb{F}$-progressively measurable, ${\Re}^{n}$-valued processes on the product space $\mathcal{T}\times \Omega $;
- (2)
- ${\mathcal{L}}_{n}^{\infty}\left(\mathcal{F}\left(t\right)\right)$ is the set of $\mathcal{F}\left(t\right)$-measurable, ${\Re}^{n}$-valued, essentially bounded random variables;
- (3)
- ${\mathcal{L}}_{n}^{p}\left(\mathcal{F}\left(t\right)\right)$ is the set of $\mathcal{F}\left(t\right)$-measurable, ${\Re}^{n}$-valued random variables $\mathbf{\zeta}$ such that$$\begin{array}{c}\hfill \mathrm{E}\left[\right|\left|\mathbf{\zeta}\right|{|}^{p}]<\infty \phantom{\rule{4pt}{0ex}};\end{array}$$
- (4)
- ${\mathcal{S}}_{n}^{\infty}(0,T)$ is the set of $\mathbb{F}$-adapted, ${\Re}^{n}$-valued, essentially bounded, càdlàg processes;
- (5)
- ${\mathcal{S}}_{n}^{p}(0,T)$ is the set of $\mathbb{F}$-adapted, ${\Re}^{n}$-valued, càdlàg processes $\mathbf{f}(t,\omega )$ such that$$\begin{array}{c}\hfill \mathrm{E}\left[\underset{t\in \mathcal{T}}{\mathrm{sup}}{\left|\right|\mathbf{f}\left(t\right)\left|\right|}^{p}\right]<\infty \phantom{\rule{4pt}{0ex}};\end{array}$$
- (6)
- ${\mathcal{H}}_{n}^{p}(0,T)$ is the set of $\mathbb{F}$-predictable, ${\Re}^{n}$-valued processes $\mathbf{h}(t,\omega )$ such that$$\begin{array}{c}\hfill \mathrm{E}\left[{\left({\int}_{0}^{T}{\left|\right|\mathbf{h}\left(t\right)\left|\right|}^{2}dt\right)}^{\frac{p}{2}}\right]<\infty \phantom{\rule{4pt}{0ex}};\end{array}$$
- (7)
- ${\mathcal{N}}_{n}^{p}(0,T)$ is the set of $\mathbb{F}$-predictable, ${\Re}^{n}$-valued processes $\mathbf{k}(t,\omega )$ such that$$\begin{array}{c}\hfill \mathrm{E}\left[{\left({\int}_{0}^{\tau \wedge T}{\left|\right|\mathbf{k}\left(t\right)\left|\right|}^{2}dt\right)}^{\frac{p}{2}}\right]<\infty \phantom{\rule{4pt}{0ex}}.\end{array}$$

- (1)
- $\mathit{\pi}$ is $\mathbb{F}$-predictable;
- (2)
- $\mathit{\pi}\left(t\right)\in {\mathbf{U}}_{1}$, for a.a. $(t,\omega )\in \mathcal{T}\times \Omega $, where ${\mathbf{U}}_{1}$ is a compact subset of ${\Re}^{2}$;
- (c)
- ${\pi}_{2}\left(t\right)\gamma \left(t\right)<1$, for a.a. $(t,\omega )\in \mathcal{T}\times \Omega $.

## 3. Risk-Based Asset Allocation Problem

**Definition**

**1.**

- (1)
- Cash additivity (or cash invariance) : if $X\in \mathcal{X}$ and $K\in \Re $, then $\rho (X+K)=\rho \left(X\right)-K$.
- (2)
- Monotonicity : for any ${X}_{1},{X}_{2}\in \mathcal{X}$, if ${X}_{1}\left(\omega \right)\le {X}_{2}\left(\omega \right)$, for all $\omega \in \Omega $, then $\rho \left({X}_{1}\right)\ge \rho \left({X}_{2}\right)$.
- (3)
- Convexity : for any ${X}_{1},{X}_{2}\in \mathcal{X}$ and $a\in (0,1)$, then $\rho (a{X}_{1}+(1-a){X}_{2})\le a\rho \left({X}_{1}\right)+(1-a)\rho \left({X}_{2}\right)$.

**Definition**

**2.**

- (1)
- Cash sub-additivity : if $X\in \mathcal{X}$ and $K\in {\Re}^{+}$, then $\mathcal{R}(X+K)\ge \mathcal{R}\left(X\right)-K$.
- (2)
- Monotonicity : for any ${X}_{1},{X}_{2}\in \mathcal{X}$, if ${X}_{1}\left(\omega \right)\le {X}_{2}\left(\omega \right)$, for all $\omega \in \Omega $, then $\mathcal{R}\left({X}_{1}\right)\ge \mathcal{R}\left({X}_{2}\right)$.
- (3)
- Convexity : for any ${X}_{1},{X}_{2}\in \mathcal{X}$ and $a\in (0,1)$, then $\mathcal{R}(a{X}_{1}+(1-a){X}_{2})\le a\mathcal{R}\left({X}_{1}\right)+(1-a)\mathcal{R}\left({X}_{2}\right)$.

**Theorem**

**1.**

- (1)
- $\mathit{\psi}$ is $\mathbb{F}$-predictable;
- (2)
- $\mathit{\psi}\left(t\right)\in {\mathbf{U}}_{2}$, for a.a. $(t,\omega )\in \mathcal{T}\times \Omega $, where ${\mathbf{U}}_{2}$ is a compact subset of ${\Re}^{4}$;
- (3)
- $0\le \varphi \left(t\right)\le C$ for some constant C, for a.a. $(t,\omega )\in \mathcal{T}\times \Omega $;
- (4)
- ${\theta}_{3}\left(t\right)>-1$, for a.a. $(t,\omega )\in \mathcal{T}\times \Omega $;

## 4. The BSDE Approach to the Game Problem

**Definition**

**3.**

**Theorem**

**2.**

- (i)
- $\left\{g(t,\omega ,0,\mathbf{0},0)\right|t\in \mathcal{T}\}\in {\mathcal{H}}_{1}^{2}(0,T)$;
- (ii)
- the Lipschitz condition: for each $(t,\omega ,{y}_{1},{\mathbf{z}}_{1},{l}_{1}),(t,\omega ,{y}_{2},{\mathbf{z}}_{2},{l}_{2})\in \mathcal{T}\times \Omega \times \Re \times {\Re}^{2}\times \Re $, there exists a constant $K\ge 0$ such that$$\begin{array}{c}\hfill |g(t,{y}_{1},{\mathbf{z}}_{1},{l}_{1})-g(t,{y}_{2},{\mathbf{z}}_{2},{l}_{2})|\le K(|{y}_{1}-{y}_{2}|+||{\mathbf{z}}_{1}-{\mathbf{z}}_{2}\left|\right|+|{l}_{1}-{l}_{2}|{\mathbf{1}}_{\{\tau >t\}}\sqrt{\lambda \left(t\right)})\phantom{\rule{4pt}{0ex}}.\end{array}$$

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

- (a)
- ${\xi}_{1}\ge {\xi}_{2}$, a.e., a.s.,
- (b)
- ${g}_{1}(t,{Y}_{2}\left(t\right),{\mathbf{Z}}_{2}\left(t\right),{L}_{2}\left(t\right))\ge {g}_{2}(t,{Y}_{2}\left(t\right),{\mathbf{Z}}_{2}\left(t\right),{L}_{2}\left(t\right))$, a.e., a.s., and
- (c)
- for each $(t,y,\mathbf{z})\in \mathcal{T}\times \Re \times {\Re}^{2}$ and ${l}_{1},{l}_{2}\in \Re $, where $({l}_{1}-{l}_{2}){\mathbf{1}}_{\{\tau >t\}}\lambda \left(t\right)\ne 0$, the following inequality holds:$$\begin{array}{c}\hfill \frac{{g}_{1}(t,y,\mathbf{z},{l}_{1})-{g}_{2}(t,y,\mathbf{z},{l}_{2})}{({l}_{1}-{l}_{2}){\mathbf{1}}_{\{\tau >t\}}\lambda \left(t\right)}>-1\phantom{\rule{4pt}{0ex}},\end{array}$$

**Lemma**

**1.**

**Proof.**

**Theorem**

**5.**

## 5. Particular Cases

#### 5.1. Quadratic Penalty Function

- Pre-default case: $\tau >t$$$\begin{array}{ccc}\hfill {\varphi}^{*}\left(t\right)& =& r\left(t\right)+\lambda \left(t\right)\phantom{\rule{4pt}{0ex}},\hfill \\ \hfill {\theta}_{1}^{*}\left(t\right)& =& -\frac{{\sigma}_{1}\left(t\right)\left[(1-{\rho}^{2}\left(t\right)){\sigma}_{2}^{2}\left(t\right)+{\gamma}^{2}\left(t\right)\lambda \left(t\right)\right]({\mu}_{1}\left(t\right)-r\left(t\right))-\rho \left(t\right)(1-{\rho}^{2}\left(t\right)){\sigma}_{1}^{2}\left(t\right){\sigma}_{2}\left(t\right)({\mu}_{2}\left(t\right)-r\left(t\right))}{{\sigma}_{1}^{2}\left(t\right)\left[{(1-{\rho}^{2}\left(t\right))}^{2}{\sigma}_{2}^{2}\left(t\right)+(1+{\rho}^{2}\left(t\right)){\gamma}^{2}\left(t\right)\lambda \left(t\right)\right]}\phantom{\rule{4pt}{0ex}},\hfill \\ \hfill {\theta}_{2}^{*}\left(t\right)& =& \frac{\rho \left(t\right){\sigma}_{1}\left(t\right)\left[(1-{\rho}^{2}\left(t\right)){\sigma}_{2}^{2}\left(t\right)-{\gamma}^{2}\left(t\right)\lambda \left(t\right)\right]({\mu}_{1}\left(t\right)-r\left(t\right))-(1-{\rho}^{2}\left(t\right)){\sigma}_{1}^{2}\left(t\right){\sigma}_{2}\left(t\right)({\mu}_{2}\left(t\right)-r\left(t\right))}{{\sigma}_{1}^{2}\left(t\right)\left[{(1-{\rho}^{2}\left(t\right))}^{2}{\sigma}_{2}^{2}\left(t\right)+(1+{\rho}^{2}\left(t\right)){\gamma}^{2}\left(t\right)\lambda \left(t\right)\right]}\phantom{\rule{4pt}{0ex}},\hfill \\ \hfill {\theta}_{3}^{*}\left(t\right)& =& \frac{\gamma \left(t\right)\left[(1+{\rho}^{2}\left(t\right)){\sigma}_{1}^{2}\left(t\right)({\mu}_{2}\left(t\right)-r\left(t\right))-2\rho \left(t\right){\sigma}_{1}\left(t\right){\sigma}_{2}\left(t\right)({\mu}_{1}\left(t\right)-r\left(t\right))\right]}{{\sigma}_{1}^{2}\left(t\right)\left[{(1-{\rho}^{2}\left(t\right))}^{2}{\sigma}_{2}^{2}\left(t\right)+(1+{\rho}^{2}\left(t\right)){\gamma}^{2}\left(t\right)\lambda \left(t\right)\right]}\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$$
- Post-default case: $\tau \le t$$$\begin{array}{ccc}\hfill {\varphi}^{*}\left(t\right)& =& r\left(t\right)\phantom{\rule{4pt}{0ex}},\hfill \\ \hfill {\theta}_{1}^{*}\left(t\right)& =& -\frac{{\mu}_{1}\left(t\right)-r\left(t\right)}{{\sigma}_{1}\left(t\right)(1+{\rho}^{2}\left(t\right))}\phantom{\rule{4pt}{0ex}},\hfill \\ \hfill {\theta}_{2}^{*}\left(t\right)& =& -\frac{\rho \left(t\right)({\mu}_{1}\left(t\right)-r\left(t\right))}{{\sigma}_{1}\left(t\right)(1+{\rho}^{2}\left(t\right))}\phantom{\rule{4pt}{0ex}},\hfill \\ \hfill {\theta}_{3}^{*}\left(t\right)& =& \frac{{\mu}_{2}\left(t\right)-r\left(t\right)}{\gamma \left(t\right)\lambda \left(t\right)}-\frac{2\rho \left(t\right){\sigma}_{2}\left(t\right)({\mu}_{1}\left(t\right)-r\left(t\right))}{\gamma \left(t\right)\lambda \left(t\right){\sigma}_{1}\left(t\right)(1+{\rho}^{2}\left(t\right))}\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$$

- Pre-default case: $\tau >t$$$\begin{array}{ccc}\hfill {\pi}_{1}^{*}\left(t\right)& =& \frac{\left[(1+{\rho}^{2}\left(t\right)){\sigma}_{2}^{2}\left(t\right)+{\gamma}^{2}\left(t\right)\lambda \left(t\right)\right]({\mu}_{1}\left(t\right)-r\left(t\right))-2\rho \left(t\right){\sigma}_{1}\left(t\right){\sigma}_{2}\left(t\right)({\mu}_{2}\left(t\right)-r\left(t\right))}{{\sigma}_{1}^{2}\left(t\right)\left[{(1-{\rho}^{2}\left(t\right))}^{2}{\sigma}_{2}^{2}\left(t\right)+(1+{\rho}^{2}\left(t\right)){\gamma}^{2}\left(t\right)\lambda \left(t\right)\right]}\phantom{\rule{4pt}{0ex}},\hfill \\ \hfill {\pi}_{2}^{*}\left(t\right)& =& \frac{(1+{\rho}^{2}\left(t\right)){\sigma}_{1}^{2}\left(t\right)({\mu}_{2}\left(t\right)-r\left(t\right))-2\rho \left(t\right){\sigma}_{1}\left(t\right){\sigma}_{2}\left(t\right)({\mu}_{1}\left(t\right)-r\left(t\right))}{{\sigma}_{1}^{2}\left(t\right)\left[{(1-{\rho}^{2}\left(t\right))}^{2}{\sigma}_{2}^{2}\left(t\right)+(1+{\rho}^{2}\left(t\right)){\gamma}^{2}\left(t\right)\lambda \left(t\right)\right]}\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$$
- Post-default case: $\tau \le t$$$\begin{array}{ccc}\hfill {\pi}_{1}^{*}\left(t\right)& =& \frac{{\mu}_{1}\left(t\right)-r\left(t\right)}{{\sigma}_{1}^{2}\left(t\right)(1+{\rho}^{2}\left(t\right))}\phantom{\rule{4pt}{0ex}},\hfill \\ \hfill {\pi}_{2}^{*}\left(t\right)& =& 0\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$$

- The investor should decrease (resp. increase) his investment in the ordinary share after default, i.e., $\Delta {\pi}_{1}^{*}\left(t\right)>0$ (resp. $\Delta {\pi}_{1}^{*}\left(t\right)<0$), if the Sharpe ratios of the ordinary share and the defaultable security satisfy the following condition:$$\begin{array}{cc}& \frac{{\mu}_{1}\left(t\right)-r\left(t\right)}{{\sigma}_{1}\left(t\right)}>\left[\rho \left(t\right)+\frac{1}{\rho \left(t\right)}\right]\frac{{\mu}_{2}\left(t\right)-r\left(t\right)}{{\sigma}_{2}\left(t\right)}\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}\forall t\in \mathcal{T}\phantom{\rule{4pt}{0ex}},\hfill \\ \hfill (\mathrm{resp}.& \frac{{\mu}_{1}\left(t\right)-r\left(t\right)}{{\sigma}_{1}\left(t\right)}<\left[\rho \left(t\right)+\frac{1}{\rho \left(t\right)}\right]\frac{{\mu}_{2}\left(t\right)-r\left(t\right)}{{\sigma}_{2}\left(t\right)}\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}\forall t\in \mathcal{T})\phantom{\rule{4pt}{0ex}};\hfill \end{array}$$
- The investor should maintain his investment in the ordinary share after default, i.e., $\Delta {\pi}_{1}^{*}\left(t\right)=0$, if the Sharpe ratios of the ordinary share and the defaultable security satisfy the following condition:$$\begin{array}{c}\hfill \frac{{\mu}_{1}\left(t\right)-r\left(t\right)}{{\sigma}_{1}\left(t\right)}=\left[\rho \left(t\right)+\frac{1}{\rho \left(t\right)}\right]\frac{{\mu}_{2}\left(t\right)-r\left(t\right)}{{\sigma}_{2}\left(t\right)}\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}\forall t\in \mathcal{T}\phantom{\rule{4pt}{0ex}}.\end{array}$$

#### 5.2. Sub-Additive Coherent Risk Measure

- (1)
- The default risk is assumed to be zero (the martingale part M has been diversified away);
- (2)
- The diffusion risk is offset by holding opposite positions in ${S}_{1}$ and ${S}_{2}$;
- (3)
- The discount risk is hedged by investing the outstanding proportion of wealth into the money market account.

## 6. Numerical Examples on the Self-Exciting Threshold Diffusion

#### 6.1. Self-Exciting Threshold Diffusion Model

**Case I:**The price process of the bond B is governed by:

**Case II:**The price process of the bond B is governed by:

**Case I:**The process of instantaneous correlation coefficients $\left\{\rho \right(t\left)\right|t\in \mathcal{T}\}$ is governed by:

**Case II:**The process of instantaneous correlation coefficients $\left\{\rho \right(t\left)\right|t\in \mathcal{T}\}$ is governed by:

#### 6.2. Simulation Procedures and Numerical Results

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix: Proof of Theorem 3

**Proof.**

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2 | One reviewer points out that in practice, defaultable bonds could have non-zero recoverable value and that along the line of thought of the Duffie-Lando model (see Duffie and Lando 2001), there could be different opinions on whether an equity may be valued zero if default is not certain. While we reckon that these could be practically relevant issues, we posit that the abstraction that the price of the defaultable security is zero after default may not be unreasonable from the theoretical perspective. |

**Figure 1.**Simulated log returns ${Y}_{1}$ and ${Y}_{2}$ for the risky share and the defaultable security. Panel (

**a**) for the ordinary share and Panel (

**b**) for the defaultable security.

**Figure 2.**Simulated prices for the risky share ${S}_{1}$ and the defaultable security ${S}_{2}$. Panel (

**a**) for the ordinary share and Panel (

**b**) for the defaultable security.

**Figure 3.**Simulated optimal proportions invested in ${S}_{1}$ for the cases without and with delay. Panels (

**a**) and (

**b**) for the optimal proportions invested in the ordinary share with and without time delay.

**Figure 4.**Simulated optimal proportions invested in ${S}_{2}$ for the cases without and with delay. Panel (

**a**) and (

**b**) for the optimal proportions invested in the defaultable security with and without time delay.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shen, Y.; Siu, T.K.
A Risk-Based Approach for Asset Allocation with A Defaultable Share. *Risks* **2018**, *6*, 14.
https://doi.org/10.3390/risks6010014

**AMA Style**

Shen Y, Siu TK.
A Risk-Based Approach for Asset Allocation with A Defaultable Share. *Risks*. 2018; 6(1):14.
https://doi.org/10.3390/risks6010014

**Chicago/Turabian Style**

Shen, Yang, and Tak Kuen Siu.
2018. "A Risk-Based Approach for Asset Allocation with A Defaultable Share" *Risks* 6, no. 1: 14.
https://doi.org/10.3390/risks6010014