# Default Ambiguity

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Intensity-Based Models

**Remark**

**1.**

#### Ambiguity in Intensity-Based Models

**Lemma**

**1.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

## 3. Absence of Arbitrage under Ambiguity

**Definition**

**1.**

- (i)
- for every $\mathbb{P}\in \mathcal{P}$ we have that ${(\mathsf{\Phi}\xb7X)}_{{T}^{*}}\ge 0$, $\mathbb{P}$-almost surely, and
- (ii)
- for at least one $\mathbb{P}\in \mathcal{P}$ it holds that $\mathbb{P}({(\mathsf{\Phi}\xb7X)}_{{T}^{*}}>0)>0$.

**Theorem**

**1.**

**Proof.**

## 4. Ambiguity on the Default Intensity

#### 4.1. Dynamic Defaultable Term Structures

**Assumption**

**1.**

- (i)
- the initial forward curve is measurable, and integrable on $[0,{T}^{*}]$:$${\int}_{0}^{{T}^{*}}\left|f(0,u)\right|du<\infty ,$$
- (ii)
- the drift parameter $a(\omega ,s,t)$ is $\mathbb{R}$-valued $\mathcal{O}\otimes \mathcal{B}$-measurable and integrable on $[0,{T}^{*}]$:$${\int}_{0}^{{T}^{*}}{\int}_{0}^{{T}^{*}}\left|a(s,t)\right|dsdt<\infty ,$$
- (iii)
- the volatility parameter $b(\omega ,s,t)$ is ${\mathbb{R}}^{d}$-valued, $\mathcal{O}\otimes \mathcal{B}$-measurable, and$$\underset{s,t\le {T}^{*}}{sup}\parallel b(s,t)\parallel <\infty .$$
- (iv)
- Let ${r}_{t}$ be the short rate process at time t, for $0\le t\le {T}^{*}$. With probability one, it holds that$$0<f(t,t)-{r}_{t},\phantom{\rule{2.em}{0ex}}0\le t\le {T}^{*}.$$

**Lemma**

**2.**

#### 4.2. Absence of Arbitrage without Ambiguity on the Default Intensity

**Proposition**

**1.**

- (i)
- $f(t,t)={r}_{t}+{\lambda}_{t},$
- (ii)
- the drift condition$$\begin{array}{c}\hfill \overline{a}(t,T)=\frac{1}{2}{\u2225\overline{b}(t,T)\u2225}^{2},\end{array}$$holds $dt\otimes dQ$-almost surely for $0\le t\le T\le {T}^{*}$ on $\{\tau >t\}$.

**Proof.**

#### 4.3. Absence of Arbitrage with Ambiguity on the Default Intensity

**Theorem**

**2.**

- (i)
- ${E}^{{\mathbb{P}}^{\lambda}}\left[{Z}_{{T}^{*}}^{*}{z}_{{T}^{*}}^{{\theta}^{*}}\right]=1$,
- (ii)
- the drift condition$$\overline{a}(t,T)=\frac{1}{2}{\u2225\overline{b}(t,T)\u2225}^{2}-\phantom{\rule{4pt}{0ex}}\overline{b}(t,T){\theta}_{t}^{*},\phantom{\rule{1.em}{0ex}}0\le t\le T\le {T}^{*}$$

**Proof.**

## 5. Examples

**Example**

**1.**

## 6. Ambiguity on the Recovery

#### 6.1. Fractional Recovery without Ambiguity

**Remark**

**4.**

**Proposition**

**2.**

- (i)
- $f(t,t)={r}_{t}+{g}_{t},$
- (ii)
- the drift condition$$\begin{array}{c}\hfill \overline{a}(t,T)=\frac{1}{2}{\u2225\overline{b}(t,T)\u2225}^{2},\phantom{\rule{1.em}{0ex}}0\le t\le T\le {T}^{*},\end{array}$$holds $dt\otimes d\mathbb{Q}$-almost surely.

**Proof.**

**Example**

**2.**

#### 6.2. Fractional Recovery with Ambiguity

**Theorem**

**3.**

- (i)
- there exists an $\mathbb{F}$-progressive ${\theta}^{*}$ such that ${E}^{\prime}\left[{z}_{{T}^{*}}^{{\theta}^{*}}\right]=1$,
- (ii)
- there exist ${\mu}^{*}$ and ${h}^{*}(t,x)$, such that $E\left[{L}^{{\mu}^{*},{h}^{*}}\right]=1$ and$${g}_{t}^{*}=\int (x-1){\mu}_{t}^{*}{h}^{*}(t,x){K}_{t}^{\mu ,h}\left(dx\right),\phantom{\rule{1.em}{0ex}}0\le t\le {T}^{*},$$$dt\otimes d{P}^{\prime}$-almost surely, and
- (iii)
- the drift condition$$\overline{a}(t,T)=\frac{1}{2}{\u2225\overline{b}(t,T)\u2225}^{2}-\overline{b}(t,T){\theta}_{t}^{*},\phantom{\rule{1.em}{0ex}}0\le t\le T\le {T}^{*},$$holds $dt\otimes d{P}^{\prime}$-almost surely.

**Proof.**

**Remark**

**5.**

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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1 | The risk that an agent fails to fulfil contractual obligations. Example of an instrument bearing credit risk is a corporate bond. |

2 | The amount that the owner of a defaulted claim receives upon default. |

3 | |

4 | Augmentation can be done in a standard fashion with respect to ${\mathbb{P}}^{\prime}$. |

**Figure 1.**This figure shows the solution of the nonlinear PDE in Equation (14) with boundary condition $\psi \left(y\right)={(y-{K}_{1}+x)}^{+}-2{(y-{K}_{2}+x)}^{+}+{(y-{K}_{3}+x)}^{+}$, ${K}_{1}=-0.2$, ${K}_{2}=0.3$, ${K}_{3}=0.8$, and $x\in [-0.5,0.7]$ is depicted on the x-axis of the plot. The dashed lines show the solution for the lower bound (upper bound, respectively), i.e., for the constants $\underline{\lambda}=0.1$ and $\overline{\lambda}=0.5$. The upper and lower solid lines show the upper and lower price bounds.

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**MDPI and ACS Style**

Fadina, T.; Schmidt, T.
Default Ambiguity. *Risks* **2019**, *7*, 64.
https://doi.org/10.3390/risks7020064

**AMA Style**

Fadina T, Schmidt T.
Default Ambiguity. *Risks*. 2019; 7(2):64.
https://doi.org/10.3390/risks7020064

**Chicago/Turabian Style**

Fadina, Tolulope, and Thorsten Schmidt.
2019. "Default Ambiguity" *Risks* 7, no. 2: 64.
https://doi.org/10.3390/risks7020064