*Article* **Risk Factor Evolution for Counterparty Credit Risk under a Hidden Markov Model**

**Ioannis Anagnostou 1,2,\* and Drona Kandhai 1,2**


Received: 31 March 2019; Accepted: 5 June 2019; Published: 12 June 2019

**Abstract:** One of the key components of counterparty credit risk (CCR) measurement is generating scenarios for the evolution of the underlying risk factors, such as interest and exchange rates, equity and commodity prices, and credit spreads. Geometric Brownian Motion (GBM) is a widely used method for modeling the evolution of exchange rates. An important limitation of GBM is that, due to the assumption of constant drift and volatility, stylized facts of financial time-series, such as volatility clustering and heavy-tailedness in the returns distribution, cannot be captured. We propose a model where volatility and drift are able to switch between regimes; more specifically, they are governed by an unobservable Markov chain. Hence, we model exchange rates with a hidden Markov model (HMM) and generate scenarios for counterparty exposure using this approach. A numerical study is carried out and backtesting results for a number of exchange rates are presented. The impact of using a regime-switching model on counterparty exposure is found to be profound for derivatives with non-linear payoffs.

**Keywords:** Counterparty Credit Risk; Hidden Markov Model; Risk Factor Evolution; Backtesting; FX rate; Geometric Brownian Motion

#### **1. Introduction**

One of the main factors that amplified the financial crisis of 2007–2008 was the failure to capture major risks associated with over-the-counter (OTC) derivative-related exposures (Basel Committee on Banking Supervision 2010a). Counterparty exposure, at any future time, is the amount that would be lost in the event that a counterparty to a derivative transaction would default, assuming zero recovery at that time. Banks are required to hold regulatory capital against their current and future exposures to all counterparties in OTC derivative transactions.

A key component of the counterparty exposure framework is modeling the evolution of underlying risk factors, such as interest and exchange rates, equity and commodity prices, and credit spreads. Risk Factor Evolution (RFE) models are, arguably, the most important part of counterparty exposure modeling, since small changes in the underlying risk factors may have a profound impact on the exposure and, as a result, on the regulatory and economic capital buffers. It is, therefore, crucial for financial institutions to put significant effort in the design and calibration of RFE models and, in addition, have a sound framework in place in order to assess the forecasting capability of the model.

Although the Basel Committee on Banking Supervision has stressed the importance of the ongoing validation of internal models method (IMM) for counterparty exposure (Basel Committee on Banking Supervision 2010b), there are no strict guidelines on the specifics of this validation process. As a result, there is some degree of ambiguity regarding the regulatory requirements that financial institutions are expected to meet. In an attempt to reduce this ambiguity, Anfuso et al. (2014) introduced a complete framework for counterparty credit risk (CCR) model backtesting which is compliant with Basel III and

the new Capital Requirements Directives (CRD IV). A detailed backtesting framework for CCR models was also introduced by Ruiz (2014), who expanded the corresponding framework for Value-at-Risk (VaR) models by the Basel Committee (Basel Committee on Banking Supervision 1996).

The most ubiquitous model for the evolution of exchange rates is Geometric Brownian Motion (GBM). Under GBM, the exchange rate dynamics are assumed to follow a continuous-time stochastic process, in which the returns are log-normally distributed. Although simplicity and tractability render GBM a particularly popular modeling choice, it is generally accepted that it cannot adequately describe the empirical facts exhibited by real exchange rate returns (Boothe and Glassman 1987). More specifically, exchange rate returns can be leptokurtic, exhibiting tails that exceed those of the normal distribution. As a result, a scenario-generation framework based on GBM may assign unrealistically low probabilities to extreme scenarios, leading to the under-estimation of counterparty exposure and, consequently, regulatory and economic capital buffers.

The main reason for the inability of GBM to produce return distributions with realistically heavy tails is the assumption of constant drift and volatility parameters. In this paper, we present a way to address this limitation without entirely departing from the convenient GBM framework. We propose a model where the GBM parameters are allowed to switch between different states, governed by an unobservable Markov process. Thus, we model exchange rates with a hidden Markov model (HMM) and generate scenarios for counterparty exposure using this approach.

A HMM is a mathematical model in which the system being modeled is assumed to follow a Markov chain whose states are hidden from the observer. HMMs have a broad range of applications, in speech recognition (Juang and Rabiner 1991), computational biology (Krogh et al. 1994), gesture recognition (Wilson and Bobick 1999), and in other areas of artificial intelligence and pattern recognition (Ghahramani 2001). HMMs have gained significant popularity in the mathematical and computational finance fields. The application of HMMs in financial and economic time-series was pioneered by Hamilton in Hamilton (1988; 1989). Since then, a significant amount of literature has been published, focusing on the ability of HMMs to reproduce stylized facts of asset returns (Bulla et al. 2011; Nystrup et al. 2015; Rydén et al. 1998), asset allocation (Ang and Bekaert 2004; Guidolin and Timmermann 2007; Nystrup et al. 2015), and option pricing (Bollen 1998; Guo 2001; Naik 1993).

Our paper expands the counterparty exposure literature by introducing a hidden Markov model for the evolution of exchange rates. We provide a detailed description of HMMs and their estimation process. In our numerical experiments, we use GBM and HMM to generate scenarios for the Euro against two major and two emerging currencies. We perform a thorough backtesting exercise, based on the framework proposed by Ruiz (2014), and find similar performances for GBM and a two-state HMM. Finally, we use the generated scenarios to calculate credit exposure for foreign exhange (FX) options, and find significant differences between the two models, which are even more pronounced for deep out-of-the-money instruments.

The remainder of the paper is organized as follows. Section 2 provides the fundamentals of HMMs, along with the algorithms for determining their parameters from data. Section 3 gives background information on modeling the evolution of exchange rates. Section 4 outlines the framework for performance evaluation of RFE models. A numerical study is presented in Section 5. Finally, in Section 6, we draw conclusions and discuss future research directions.

#### **2. An Introduction to Hidden Markov Models**

The hidden Markov model (HMM) is a statistical model in which a sequence of observations is generated by a sequence of unobserved states. The hidden state transitions are assumed to follow a first-order Markov chain. The theory of hidden Markov models (HMMs) originates from the work of Baum et al. in the late 1960s (Baum and Petrie (1966), Baum and Eagon (1967)). In the rest of this section, we introduce the theory of hidden Markov models (HMMs), following Rabiner (1990).

#### *2.1. Formal Definition of a HMM*

In order to formally define a hidden Markov model (HMM), the following elements are required:


$$a\_{i\rangle} = P\left[q\_{t+i} = X\_{\rangle} | q\_t = X\_{\bar{i}}\right], \quad 1 \le i, j \le N. \tag{1}$$

4. The observation symbol probability distribution in state *j*, *B* = {*bj*(*k*)}, where

$$b\_j(k) = P\left[v\_k \text{ at } t | q\_t = X\_j\right], \quad 1 \le j \le N, 1 \le k \le M. \tag{2}$$

5. The initial distribution of the hidden states, *π* = {*πi*}, where

$$\pi\_i = P\left[q\_1 = X\_i\right], \quad 1 \le i \le N. \tag{3}$$

The parameter set of the model is denoted by *λ* = (*A*, *B*, *π*). A graphical representation of a hidden Markov model with two states and three discrete observations is given by Figure 1.

**Figure 1.** A hidden Markov model (HMM) with two states and three discrete observations, where *aij* is the probability of transition from state *Xi* to state *Xj* and *bj*(*k*) is the emission probability for symbol *vk* in state *Xj*.

In the case where there are an infinite amount of symbols for each hidden state, *vk* is omitted and the observation probability *bj*(*k*), conditional on the hidden state *Xj*, can be replaced by

$$b\_j(O\_t) = P(O\_t | q\_t = X\_j).$$

If the observation symbol probability distributions are Gaussian, then *bj*(*Ot*) = *φ*(*Ot*|*uj*, *σj*), where *φ*(·) is the Gaussian probability density function, and *uj* and *σ<sup>j</sup>* are the mean and standard deviation of the corresponding state *Xj*, respectively. In that case, the parameter set of the model is *λ* = (*A*, *u*, *σ*, *π*), where *u* and *σ* are vectors of means and standard deviations, respectively.

#### *2.2. The Three Basic Problems for HMMs*

The idea that HMMs should be characterized by three fundamental problems originates from the seminal paper of Rabiner (1990). These three problems are the following:

**Problem 1** (Likelihood)**.** *Given the observation sequence O* = *O*1*O*<sup>2</sup> ... *OT and a model λ* = (*A*, *B*, *π*)*, how do we compute the conditional probability P*(*O*|*λ*) *in an efficient manner?*

**Problem 2** (Decoding)**.** *Given the observation sequence O* = *O*1*O*<sup>2</sup> ... *OT and a model λ, how do we determine the state sequence Q* = *q*1*q*<sup>2</sup> ... *qT which optimally explains the observations?*

**Problem 3** (Learning)**.** *How do we select model parameters λ* = (*A*, *B*, *π*) *that maximize P*(*O*|*λ*)*?*

*2.3. Solutions to the Three Basic Problems*

#### 2.3.1. Likelihood

Our objective is to calculate the likelihood of a particular observation sequence, *O* = *O*1*O*<sup>2</sup> ··· *OT*, given the model *λ*. The most intuitive way of doing this is by summing the joint probability of *O* and *Q* for all possible state sequences *Q* of length *T*:

$$P(O|\lambda) = \sum\_{\text{all }Q} P(O|Q\_\prime \lambda) \cdot P(Q|\lambda). \tag{4}$$

The probability of a particular observation sequence *O*, given a state sequence *Q* = *q*1*q*<sup>2</sup> ··· *qT*, is

$$\begin{aligned} P(O|Q,\lambda) &= \prod\_{t=1}^{T} P(O\_t|q\_{t\prime}\lambda) \\ &= \quad b\_{\mathbb{H}\_1}(O\_1) \cdot b\_{\mathbb{H}\_2}(O\_2) \cdots \cdot b\_{\mathbb{H}\_T}(O\_T) \end{aligned} \tag{5}$$

as we have assumed that the observations are independent. The probability of a state sequence *Q* can be written as

$$P(Q|\lambda) = \pi\_{\mathfrak{q}\_1} a\_{\mathfrak{q}\_1 \mathfrak{q}\_2} a\_{\mathfrak{q}\_2 \mathfrak{q}\_3} \cdot \cdot a\_{\mathfrak{q}\_{T-1} \mathfrak{q}\_T}.\tag{6}$$

The joint probability of *O* and *Q* is the product of the above two terms; that is,

$$P(O, Q | \lambda) = P(O | Q, \lambda) \cdot P(Q | \lambda). \tag{7}$$

Although the calculation of *P*(*O*|*λ*) using the above definition is rather straightforward, the associated computational cost is huge.

Thankfully, a dynamic programming approach, called the Forward Algorithm, can be used instead.

Consider the forward variable *αi*(*t*), defined as

$$a\_t(i) = P(O\_1O\_2 \cdots \cdot O\_t, q\_t = X\_i|\lambda). \tag{8}$$

We can solve for *αt*(*i*) inductively using Algorithm 1.

#### **Algorithm 1** The Forward Algorithm.

1. Initialization:

$$
\pi\_1(i) = \pi\_i b\_i(O\_1), \quad 1 \le i \le N. \tag{9}
$$

2. Induction:

$$a\_{t+1}(j) = \left[\sum\_{i=1}^{N} a\_t(i)a\_{ij}\right] b\_j(O\_{t+1}), \quad 1 \le \quad t \le T-1$$

$$1 \le \quad j \le N. \tag{10}$$

3. Termination:

$$P(O|\lambda) = \sum\_{i=1}^{N} \alpha\_T(i). \tag{11}$$

Correspondingly, we can define a backward variable *βt*(*i*) as

$$\beta\_l(i) = P(O\_{t+1}O\_{t+2}\cdots O\_T | q\_l = X\_i, \lambda). \tag{12}$$

Again, we can solve for *βt*(*i*) inductively using Algorithm 2.

#### **Algorithm 2** The Backward Algorithm.

1. Initialization:

$$\beta\_T(i) = 1, \quad 1 \le i \le N. \tag{13}$$

2. Induction:

$$\beta\_t(i) = \sum\_{j=1}^{N} a\_{ij} \, b\_j(O\_{t+1}) \beta\_{t+1}(j), \quad \quad t = \quad T - 1, T - 2, \dots, 1$$

$$1 \quad \le \quad i \le N. \tag{14}$$

#### 2.3.2. Decoding

In order to identify the best sequence *Q* = {*q*1*q*<sup>2</sup> ··· *qT*} for the given observation sequence *O* = {*O*1*O*<sup>2</sup> ··· *OT*}, we need to define the quantity

$$\delta\_{\mathbf{l}}(i) = \max\_{q\_1, q\_2, \dots, q\_{l-1}} P(q\_1 q\_2 \cdots q\_l = i, O\_1 O\_2 \cdots \cdot O\_l | \lambda). \tag{15}$$

By induction, we have

$$\delta\_{t+1}(j) = \left[ \max\_{i} \delta\_{t}(i) a\_{i\hat{j}} \right] \cdot b\_{j}(O\_{t+1}). \tag{16}$$

To actually retrieve the state sequence, it is necessary to keep track of the argument which maximizes Equation (16), for each *t* and *j*. We do so via the array *ψt*(*j*). The complete procedure for finding the best state sequence is presented in Algorithm 3.

#### **Algorithm 3** Viterbi algorithm.

1. Initialization:

$$\begin{aligned} \delta\_1(i) &= \pi\_i b\_i(O\_1)\_\prime \quad 1 \le i \le N \quad \tag{17} \\ \psi\_1(i) &= \, ^\prime 0. \end{aligned} \tag{18}$$

2. Recursion:

$$\delta\_t(j) = \max\_{1 \le i \le N} \left[ \delta\_{t-1}(i) a\_{ij} \right] b\_j(O\_t), \quad 2 \le t \le T$$

$$1 \le j \le N \tag{19}$$

$$\psi\_t(j) = \underset{1 \le i \le N}{\text{arg}\max} \left[ \delta\_{t-1}(i) a\_{ij} \right] \qquad \qquad 2 \le t \le T$$

$$1 \le j \le N.\tag{20}$$

3. Termination:

$$\begin{array}{rcl}P^\* &=& \max\_{1 \le i \le N} \left[\delta\_T(i)\right] \\ q\_T^\* &=& \arg\min\_{1 \le i \le N} \left[\delta\_T(i)\right]. \end{array} \tag{21}$$

4. Sequence back-tracking:

$$q\_t^\* = \psi\_{t+1}(q\_{t+1}^\*), \quad t = T - 1, T - 2, \cdots, 1. \tag{22}$$

#### 2.3.3. Learning

The model which maximizes the probability of an observation sequence *O*, given a model *λ* = (*A*, *B*, *π*), cannot be determined analytically. However, a local maximum can be found using an iterative algorithm, such as the Baum-Welch method or the expectation-maximization (EM) method (Dempster et al. 1977). In order to describe the iterative procedure of obtaining the HMM parameters, we need to define *ξt*(*i*, *j*), the probability of being at the state *Xi* at time *t*, and the state *Xj* at time *t* + 1, given the model and observation sequence; that is,

$$\mathbf{g}\_t(\mathbf{i}, \mathbf{j}) = P(q\_t = X\_{\mathbf{i}\prime} q\_{t+1} = X\_{\mathbf{j}} | O\_{\prime}\prime \lambda). \tag{23}$$

Using the earlier defined forward and backward variables, *ξt*(*i*, *j*) can be rewritten as

$$\xi\_t(i,j) = \frac{a\_l(i)a\_{l\bar{j}}b\_{\bar{j}}(O\_{l+1})\beta\_{l+1}(j)}{P(O|\lambda)}.\tag{24}$$

We define

$$\gamma\_t(i) = \sum\_{j=i}^{N} \xi\_t(i, j) \tag{25}$$

as the probability of being in state *Xi* at time *t*. It is clear that

$$\sum\_{t=i}^{T-1} \gamma\_t(i) \quad = \text{ expected number of transitions from } X\_i \text{, and} \tag{26}$$

$$\sum\_{t=i}^{T-1} \zeta\_t(i,j) \quad = \text{ expected number of transitions from } X\_i \text{ to } X\_j. \tag{27}$$

Using these formulas, the parameters of a HMM can be estimated, in an iterative manner, as follows:

$$
\begin{aligned}
\hat{\pi}\_{i}&=&\gamma\_{1}(i)=\text{expected number of times in state }X\_{i}\text{ at time }t=1;\tag{28}
\\ \hat{\pi}\_{\hat{\eta}}&=&\underbrace{\sum\_{t=i}^{T-1}\hat{\xi}\_{t}(i,j)}\_{\begin{subarray}{c}t=i\\t\end{subarray}}\\ &=&\underbrace{\begin{subarray}{c}\text{expected number of transitions from }X\_{i}\text{ to }X\_{j}\\\text{expected number of transitions from }X\_{i}\end{subarray}}\_{\begin{subarray}{c}t=i\\t\end{subarray}};\tag{29}
\end{split}
\tag{29}
$$

$$
\begin{aligned}
\hat{\mu}\_{j}(k)&=&\underbrace{\sum\_{t=1}^{T}\mathbf{1}\_{\{O\_{t}=\upsilon\_{k}\}}\gamma\_{t}(j)}\_{\begin{subarray}{c}t=1\\t\end{subarray}}\\ &=&\underbrace{\begin{subarray}{c}\text{expected number of times in state }j\text{ and observing symbol }\upsilon\_{k}\\\text{expected number of times in state }j\end{subarray}}\_{\begin{subarray}{c}\text{expected number of times in state }j\end{subarray}};\tag{30}
\end{aligned}\tag{30}
$$

If *λ* = (*A*, *B*, *π*) is the current model and *λ*ˆ = (*A*ˆ, *B*ˆ, *π*ˆ) is the re-estimated one, then it has been shown, by Baum and Eagon (1967); Baum and Petrie (1966), that *<sup>P</sup>*(*O*|*λ*ˆ) <sup>≥</sup> *<sup>P</sup>*(*O*|*λ*).

In case the observation probabilities are Gaussian, the following formulas are used to update the model parameters *u* and *σ*:

$$\hat{\mu}\_{\hat{l}} = \underbrace{\sum\_{t=1}^{T} \gamma\_{t}(\boldsymbol{j}) \boldsymbol{O}\_{\boldsymbol{t}}}\_{\text{t} = 1} \tag{31}$$

$$\vartheta\_{j} = \sqrt{\frac{\sum\_{t=1}^{T} \gamma\_{t}(j)(O\_{t} - u\_{j})^{2}}{\sum\_{t=1}^{T} \gamma\_{t}(j)}}.\tag{32}$$

#### **3. Modelling the Evolution of Exchange Rates**

As discussed in the introduction, the first step in calculating the future distribution of counterparty exposure is the generation of scenarios using the models that represent the evolution of the underlying market factors. These factors typically include interest and exchange rates, equity and commodity prices, and credit spreads. This article is concerned with the modeling of exchange rates.

#### *3.1. Geometric Brownian Motion*

In mathematical finance, the Geometric Brownian Motion (GBM) model is the stochastic process which is usually assumed for the evolution of stock prices (Hull 2009). Due to its simplicity and tractability, GBM is also a widely used model for the evolution of exchange rates.

A stochastic process, *St*, is said to follow a GBM if it satisfies the following stochastic differential equation:

$$dS\_t = \mu S\_t dt + \sigma S\_t dW\_{t\prime} \tag{33}$$

where *Wt* is a Wiener process, and *μ* and *σ* are constants representing the drift and volatility, respectively.

The analytical solution of Equation (33) is given by:

$$S\_t = S\_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W\_t\right). \tag{34}$$

With this expression in hand, and knowing that *Wt* ∼ *N*(0, *t*), one can generate scenarios simply by generating standard normal random numbers.

#### *3.2. A Hidden Markov Model for Drift and Volatility*

One of the main shortcomings ofthe GBM model is that, due to the assumption of constant drift and volatility, some important characteristics of financial time-series, such as volatility clustering and heavy-tailedness in the return distribution, cannot be captured. To address these limitations, we consider a model with an additional stochastic process. The observations of the exchange rates are assumed to be generated by a discretised GBM, in which both the drift and volatility parameters are able to switch, according to the state of an unobservable process which satisfies the Markov property. In other words, the conditional probability distribution of future states depends solely upon the current state, not on the sequence of states that preceded it. The observations also satisfy a Markov property with respect to the states (i.e., given the current state, they are independent of the history).

Thus, we consider a hidden Markov model with Gaussian emissions *λ* = (*A*, *u*, *σ*, *π*), as was presented in Section 2.1. We denote the hidden states by *X* = {*X*1, *X*2, ... , *XN*}, and the state at time *t* as *qt*. The unobservable Markov process governs the distribution of the log-return process *<sup>Y</sup>* <sup>=</sup> {*Y*2,...,*YT*}, where *Yt* <sup>=</sup> log *St St*−<sup>1</sup> , *t* = 2, . . . , *T*. The dynamics of *Y* are then as follows:

$$Y\_t = u(q\_t) + \sigma(q\_t) Z\_{t\prime} \tag{35}$$

where *<sup>u</sup>*(*qt*) = *<sup>μ</sup>*(*qt*) <sup>−</sup> *<sup>σ</sup>*2(*qt*) 2 ! and *Zt* ∼ *N*(0, 1) are independent standard normal random numbers.

The transition probabilities of the hidden process, as well as the drift and volatility of the GBM, can be estimated from a series of observations, using the algorithms presented in Section 2. The number of hidden states has to be specified in advance. In many practical applications, the number of hidden states can be determined based on intuition. For example, stock markets are often characterized as "bull" or "bear", based on whether they are appreciating or depreciating in value. A bull market occurs when returns are positive and volatility is low. On the other hand, a bear market occurs when returns are negative and volatility is high. It would, therefore, be in line with intuition to assume that stock market observations are driven by a two-state process. The number of states can also be determined empirically; for example, using the Akaike information criterion (AIC) or the Bayesian information criterion (BIC). Once the model parameters have been estimated, scenarios can be generated by generating the hidden Markov chain and sampling the log-returns from the corresponding distributions.

#### **4. RFE Model Performance Evaluation**

#### *4.1. Backtesting*

In this sub-section, we give a brief overview of a framework for the backtesting of RFE models. For a more detailed description, the reader is referred to Ruiz (2014). Backtesting is the process of comparing the distributions given by the RFE models with the realized history of the corresponding risk factors. In accordance with regulatory requirements, RFE models have to be backtested at

multiple forecasting horizons, making use of various distributional tests (Basel Committee on Banking Supervision 2010b).

To test whether a set of realizations can reasonably be modeled as arising from a specific distribution, we use the Probability Integral Tranform (PIT) (see (Diebold et al. 1997)), defined as

$$F(\mathbf{x}\_n) = \int\_{-\infty}^{\mathbf{x}\_n} f(u) du,\tag{36}$$

where *xn* is the realization of a random variable and *f*(·) is its predicted density. Note that, if one applies PIT using the true density of *xn* to construct a set of values, it follows that the distribution of the constructed set will simply be U(0, 1). As a result, one is able to evaluate the quality of the model *f*(·) for *xn*, simply by measuring the distance between the distribution of the constructed set and U(0, 1).

For a given set of realizations *xti* of the risk factor to be tested, we set a starting point *tstart* and an ending point *tend*. The size of the backtesting window is then *Tb* = *tend* − *tstart*. If the time horizon over which we want to test our model is Δ, we proceed as follows:


This exercise yields a set {*Fi*}*<sup>K</sup> <sup>i</sup>*=1, where *K* is the number of steps taken. As mentioned previously, if the empirical distribution of the realizations is the same as the predicted distribution, then the constructed set {*Fi*}*<sup>K</sup> <sup>i</sup>*=<sup>1</sup> will be uniformly distributed.

In order to measure the distance *d* between the distribution of the constructed set and U(0, 1) we can use a number of metrics, such as:

The Anderson–Darling metric:

$$d\_{AD} = \int\_{-\infty}^{\infty} \left( F\_{\mathbf{f}}(\mathbf{x}) - F(\mathbf{x}) \right)^2 \omega(F(\mathbf{x})) dF(\mathbf{x})$$

$$\omega(\mathbf{x}) = \frac{1}{\overline{\mathbf{x}(1-\mathbf{x})}'} \tag{37}$$

the Cramer–von Mises metric:

$$d\_{\rm CVM} = \int\_{-\infty}^{\infty} \left( F\_{\rm f}(\mathbf{x}) - F(\mathbf{x}) \right)^{2} \omega(F(\mathbf{x})) dF(\mathbf{x})$$
  $\omega(\mathbf{x}) = 1$ , or 
$$\omega(\mathbf{x}) = \frac{1}{2}, \text{ or }$$

the Kolmogorov–Smirnov metric:

$$d\_{KS} = \sup\_{\mathbf{x}} |F\_{\mathbf{c}}(\mathbf{x}) - F(\mathbf{x})|\_{\prime} \tag{39}$$

where *Fe* is the empirical and *F* is the theoretical cumulative distribution function. Note that each of these metrics provides a single distance value ˜*<sup>d</sup>* between the distribution of the realized set and <sup>U</sup>(0, 1). To obtain an understanding of whether this distance is acceptable, we simulate time-series using the model being tested. Although the simulated time-series will follow the model by definition, there will still be some distance, *d*, due to numerical noise. By repeating this experiment a sufficiently large number of times (say, *<sup>M</sup>*), we can obtain a set {*di*}*<sup>M</sup> <sup>i</sup>*=<sup>1</sup> and approximate, numerically, its cumulative distribution function *ψ*(*d*). With *ψ*(*d*) in hand, we can assess the distance ˜*d*, as follows: If ˜*d* falls in a range with high probability with respect to *ψ*(*d*), then the probability of the model being accurate is

high. By defining *dy* and *dr* as the 95th and the 99.99th percentiles, respectively, we can obtain three color bands for model performance:


An example of the three-color scoring scheme is shown in Figure 2. The backtesting process can be carried out for a set of time horizons, and for every horizon a single result can be produced, in terms of a probability *ψ*( ˜*d*) and a color band.

#### *4.2. Long-Term Percentiles of Distribution Cones*

Backtesting provides a statistical judgement of the performance of the model for relatively short-term forecast horizons. Assessing the distribution cones of the risk factor evolution provides insight into the behavior of the model for long forecast horizons. The high and low percentiles of the distribution cones need to be compared to the long-term percentiles of the observed risk factor data. Please note that assessing the long-term percentiles needs expert judgement to some extent, as it is difficult to statistically unambiguously state what the long-term percentile of a distribution cone should be.

**Figure 2.** Examples of the three-color scoring scheme. The model in the example receives yellow scores for all three metrics, as *dy* < ˜*d* < *dr*.

#### **5. Numerical Experiments**

#### *5.1. Overview of Data Selections*

In order to evaluate the performance of the HMM approach, we used the foreign exchange rates of the Euro against two G10 and two emerging-market currencies. We used daily observations, between 1 January 2004 and 31 December 2016, for the following FX rates:


#### *5.2. Selection of the Number of Hidden States*

Choosing the appropriate number of hidden states for a HMM is not a trivial task. Two commonly used criteria for model comparison are the Akaike information criterion (AIC):

$$\text{AICC} = -2\log L + 2p,\tag{40}$$

and the Bayesian information criterion (BIC):

$$\text{BIC} = -2\log L + p\log T,\tag{41}$$

where *L* is the likelihood of the fitted model, *p* is the number of free parameters in the model, and *T* denotes the number of observations (Zucchini et al. 2016). The number of free parameters in a HMM with a Gaussian distribution for each hidden state is:

$$p = N^2 + 2N - 1,\tag{42}$$

where *N* is the number of hidden states. Thus, in both criteria, the second term is a penalty term which increases with increasing *N*. Compared to the AIC, the penalty term of the BIC has more weight when *T* > *e*<sup>2</sup> and, therefore, the BIC often favors models with fewer parameters than the AIC does.

A bank that uses internal models to measure exposure for capital purposes must use at least three years of historical data for calibration, where the parameters have to be updated quarterly or more frequently, if market conditions warrant. During the course of backtesting, re-calibration of the RFE model parameters needs to be done at the same frequency as for production to make the re-calibration effects visible (Basel Committee on Banking Supervision 2010b). Consequently, in the backtesting exercise that follows (in Section 5.3), we use calibration blocks of three years and move the block forward by one quarter every time.

To choose the appropriate number of hidden states, we calibrate HMMs with 2–5 states for each of the three-year blocks and calculate the AIC and BIC. The results are shown in Figures 3–6. Based on the AIC results, the performance of HMMs with 2, 3, 4, or 5 states is almost the same for the emerging-market currencies. For USD/EUR, models with higher number of hidden states seem to perform better while, for GBP/EUR, the two-state model is preferable. However, based on the BIC, the HMM with two states is the best candidate for all four currency pairs. Therefore, we focus on the HMM with two states for the rest of our numerical experiments.

**Figure 3.** Akaike information criterion (AIC) and Bayesian information criterion (BIC) for HMMs calibrated using USD/EUR time-series.

**Figure 4.** AIC and BIC for HMMs calibrated using GBP/EUR time-series.

**Figure 5.** AIC and BIC for HMMs calibrated using RUB/EUR time-series.

**Figure 6.** AIC and BIC for HMMs calibrated using MXN/EUR time-series.

#### *5.3. Model Backtesting*

We applied the backtesting algorithm (presented in Section 4) using observations between 1 January 2004 and 31 December 2016 for the selected FX rates. We used a calibration window *Tc* of three years with quarterly re-calibration (*δ<sup>c</sup>* = 3 months). The length of the backtesting window was *Tb* = 10 years and we tested model performance for time horizons Δ of length 1 week, 2 weeks, 1 month, and 3 months.

In order to generate scenarios of length Δ, the following steps were taken. At every time point *t* with 1 ≤ *t* ≤ Δ and, given the current hidden state *qt* = *Xi*, the next hidden state *qt*+<sup>1</sup> = *Xj* was chosen using the transition probability matrix *A*. The observation *Ot* was then generated, according to the corresponding emission probability distribution *bj*. The initial hidden state *q*<sup>0</sup> was assumed to be the most probable state at the end of the learning procedure.

It is important to note that the backtesting procedure provides a statistical assessment of the model performance for relatively short-term forecast horizons. For instance, a backtesting window *Tb* of 10 years and a time horizon Δ = 1 year would translate to only 10 independent points. As a result, the statistical relevance of the backtesting exercise would be limited. In order to gain an insight into model behavior for longer forecast horizons, we consider the distribution cones of the risk factor evolution. The high and the low percentiles of the distribution cones are compared to observed risk factor data.

In the following, we discuss the results obtained for each of the FX rates.

#### 5.3.1. USD/EUR

Table 1 summarizes the results of the backtesting exercise for USD/EUR, in terms of probabilities as well as color bands. When the forecasting horizon was 1 week, both the GBM and the two-state HMM scored yellow under the Anderson–Darling and the Cramerthree–von Mises metrics, and green under the Kolmogorov–Smirnov metric. For the two-week forecasting horizon, both models obtained a yellow score under all three metrics. Finally, for the longer horizons (1 and 3 months), both models performed significantly better, with green scores under all metrics. The backtesting results do not indicate any notable difference in performance between the HMM and the GBM.

**Table 1.** Backtesting results for USD/EUR with calibration window *Tc* = 3 years, frequency of re-calibration *δ<sup>c</sup>* = 3 months, and backtesting window *Tb* = 10 years. GBM, Geometric Brownian Motion.


In order to gain an insight into the performance of the models for longer time horizons, we present, in Figure 7, the 5th and 95th percentiles of the forecast distributions for a horizon of 7 years, between 2011 and end of 2016. It can be seen that the HMM gave slightly more conservative forecasts, compared to the GBM, but the realized time-series fell within the 90% probability region under both models, at the end of the 7 year period.

**Figure 7.** Percentiles of long-term distribution cones for USD/EUR under GBM and HMM with two states.

#### 5.3.2. GBP/EUR

The backtesting results for GBP/EUR are summarized in Table 2. The two models achieved similar performance when the time horizon was 2 weeks or longer. When the forecasting horizon was 2 weeks, both models scored yellow. In the 1-month horizon, both models had green scores under the Anderson–Darling and Cramer–von Mises metrics, and a yellow score under the Kolmogorov–Smirnov metric. The scores were green for both models under all metrics when the time horizon was 3 months. The greatest difference between the two models was observed for the 1-week forecasting horizon, where the two-state model performed significantly better, scoring green under all three metrics, while the corresponding scores for GBM were yellow.

**Table 2.** Backtesting results for GBP/EUR with calibration window *Tc* = 3 years, frequency of re-calibration *δ<sup>c</sup>* = 3 months, and backtesting window *Tb* = 10 years.


Figure 8 shows the 5th and 95th percentiles of the forecast distributions between 2011 and end of 2016. Similarly to the results for USD/EUR, HMM gave slightly more conservative forecasts and the realized time-series fell within the 90% probability region under both models, at the end of the 7 year period. However, in 2016, the realized time-series fell outside the 95th percentile of the GBM distribution, while it was still within this bound for the HMM.

**Figure 8.** Percentiles of long-term distribution cones for GBP/EUR under GBM and HMM with two states.

#### 5.3.3. RUB/EUR

Table 3 presents the results of the backtesting exercise for RUB/EUR. It can be seen that both GBM and HMM did not perform very well when the forecasting horizon was 1 week, with HMM having yellow scores under every metric. The results were similar for the 2 week forecasting horizon. In the longer time horizons, however, both models performed better. HMM outperformed the one-state model GBM, achieving green scores in the 1-month horizon. The scores were green for both models when the forecasting horizon was 3 months.

**Table 3.** Backtesting results for RUB/EUR with calibration window *Tc* = 3 years, frequency of re-calibration *δ<sup>c</sup>* = 3 months, and backtesting window *Tb* = 10 years.


Figure 9 shows the percentiles of the long-term distribution cones for RUB/EUR. It is clear that the difference between GBM and HMM was more pronounced, with the HMM yielding significantly more conservative forecasts. The realized time-series was close to the 95th percentile of the GBM distribution until mid-2014, exceeding it on a number of occasions in 2011 and in 2013. Despite a sharp decline in 2015, the realized time-series remained above the 5th percentile for both models throughout the 7 year period.

**Figure 9.** Percentiles of long-term distribution cones for RUB/EUR under GBM and HMM with two states.

#### 5.3.4. MXN/EUR

Table 4 summarizes the results of the backtesting exercise for MXN/EUR, in terms of scores as well as color bands. Both HMM and GBM had yellow scores for the shorter time horizons (1 and 2 weeks), under all metrics. The models performed better for the longer time horizons (1 and 3 months), achieving green scores. Figure 10 shows the long-term distribution cones. Similar to the the GBP/EUR case, we do not observe a clear difference in performance between GBM and HMM with two states.

**Figure 10.** Percentiles of long-term distribution cones for MXN/EUR under GBM and HMM with two states.


**Table 4.** Backtesting results for MXN/EUR, with a 3-year calibration window, quarterly re-calibration, and a 10-year backtesting window.

#### *5.4. Impact on Credit Exposure: A Case Study for FX Options*

#### 5.4.1. Exposure at Default (EAD)

Prior to presenting the case study on FX options, we provide a brief introduction to credit exposure calculation. For a more detailed description, the reader is referred to Zhu and Pykhtin (2007) and Gregory (2012).

When a financial institution is permitted to use the IMM to calculate credit exposure, the following steps need to be taken:


The outcome of this process is a set of realizations of credit exposure at each exposure date in the future. One can then estimate the expected exposure EE*<sup>k</sup>* as the average exposure at future date *tk*, where the average is taken across all simulated scenarios of the relevant risk factors.

The Expected Positive Exposure (EPE) is defined as the weighted average of the EE over the first year

$$\text{EPE} = \sum\_{k=1}^{\min(\mathbf{1})} \text{EE}\_k \times \Delta t\_{k\prime} \tag{43}$$

where the weights <sup>Δ</sup>*tk* = *tk* − *tk*−<sup>1</sup> are the proportion that an individual expected exposure represents over the entire one-year time horizon.

In order to account for potential non-conservative aging effects, a modification is necessary. First, an Effective EE profile is obtained from the EE profile by adding the non-decreasing constraint for maturities below one year. Effective EE can be calculated, recursively, as follows:

$$\text{Effective EE}\_k = \max\left\{ \text{Effective EE}\_{k-1} - \text{EE}\_k \right\},\tag{44}$$

where the current date is denoted as *t*<sup>0</sup> and EE0 equals the current exposure.

Effective EPE can, then, be calculated from the Effective EE profile, in the same way that EPE is calculated from the EE profile:

$$\text{Effective EPE} = \sum\_{k=1}^{\min(1\text{ year, maturity})} \text{Effective EE}\_k \times \Lambda t\_k. \tag{45}$$

Finally, the Exposure at Default (EAD) is the product of a multiplier *α* and the Effective EPE

$$\text{EAD} = \mathfrak{a} \times \text{Efficiency EPE.} \tag{46}$$

The multiplier *α*, introduced by Picoult (2002), is a correction coefficient that accounts for wrong-way risk. Under the IMM, *α* is fixed at a rather conservative level of 1.4. However, banks using the IMM have an option to use their own estimate of *α*, with the prior approval of the supervisor and a floor of 1.2.

#### 5.4.2. Results

In order to study the impact of using a two-state HMM, instead of a GBM, on regulatory and economic capital, we consider the case of FX call options on the RUB/EUR rate. The rationale behind this choice was that the Russian currency suffered a crisis in 2014, which will be included in our calibration data set.

Our starting date was 2 Januray 2016. We estimated the parameters of a GBM and a two-state HMM, using three years of data (between January 2013 and December 2015). Following the methodology presented in Section 5.4.1, we generated market scenarios for the following set of future exposure dates:

$$\{t\_k\}\_{k=1}^9 = \{1 \text{ week, } 2 \text{ weeks, } 3 \text{ weeks, } 4 \text{ weeks, } 2 \text{ months, } 3 \text{ months, } 6 \text{ months, } 9 \text{ months, } 1 \text{ year}\}.\tag{47}$$

For each generated scenario and each exposure date, option valuation was performed using the Garman–Kohlhagen model (Garman and Kohlhagen (1983)).

The value of a call option at time *t* is given by the analytical formula

$$\mathbf{C}\_{l} = \mathbf{S}\_{l}\mathbf{e}^{-r\_{f}(T-t)}\mathbf{N}(\mathbf{x} + \sigma\sqrt{T-t}) - \mathbf{K}\mathbf{e}^{-r\_{d}(T-t)}\mathbf{N}(\mathbf{x})\_{\prime} \tag{48}$$

where

$$\alpha \equiv \frac{\ln \left( S\_t / K \right) + \left( r\_d - r\_f - \left( \sigma^2 / 2 \right) \right) \left( T - t \right)}{\sigma \sqrt{T - t}},$$

*St* it the spot price of the deliverable currency at time *t* (domestic units per foreign unit),

*K* is the strike price of the option (domestic units per foreign unit),

*T* − *t* is the time to maturity,

*rd* is the domestic risk-free interest rate,

*rf* is the foreign risk-free interest rate,

*σ* is the volatility of the spot currency price, and

*N*(·) is the cumulative normal distribution function.

Note that, in the formula, both spot and strike price are quoted in units of domestic currency per unit of foreign currency. As a result, the option price will be in the same units, as well. In order to obtain the market value of a position in such an option, it is necessary to multiply by a notional amount Λ in the foreign currency.

In our example, the foreign and domestic currencies are RUB and EUR, respectively. In order to achieve a candid comparison of the two RFE models for the exchange rate, we do not consider interest rate and volatility as risk factors for FX options. Instead, we make the simplistic assumptions of *rd* = *rf* = 0 and constant volatility *σ* = 0.15 (equal to the supervisory volatility for foreign exchange options in the standardised approach, see Basel Committee on Banking Supervision (2014)). The notional amount Λ is set to RUB 100,000,000. The spot RUB/EUR exchange rate on 2 January 2016 was *S*<sup>0</sup> = 0.01263.

The credit exposure values for out-of-the-money (OTM) call options on the RUB/EUR exchange rate, for a range of strike prices, are illustrated in Figure 11a. The impact of using a two-state HMM, instead of a GBM, is shown in Figure 11b. These results are summarized in Table 5. It is clear that exposure values under HMM exceeded the exposure values under GBM markedly for deep-out-the-money options. This difference would have a direct impact on how these positions would be capitalized against counterparty default, with a difference that could exceed 400% for the strike price *K* = 0.023. It is also important to note that, given the exchange rate movements over recent years, it is not unrealistic for the moneyness of such options to change dramatically, leading to large unexpected losses. For in-the-money call options, the two models produced identical exposure values. Thus, these results are omitted from this paper.

**Figure 11.** Credit exposure values for out-of-the-money (OTM) call options on the RUB/EUR exchange rate (**a**) and the impact of using a two-state HMM, instead of a GBM (**b**).


**Table 5.** Credit exposure values for out-of-the-money (OTM) options on the RUB/EUR exchange rate.

#### **6. Conclusions**

In this paper, we presented a hidden Markov model for the evolution of exchange rates with regards to counterparty exposure. In the proposed model, the observations of the exchange rates were assumed to be generated by a discretized GBM, in which both the drift and volatility parameters are able to switch, according to the state of a hidden Markov process. The main motivation of using such a model is the fact that GBM can assign unrealistically low probabilities to extreme scenarios, leading to the under-estimation of counterparty exposure and the corresponding capital buffers. The proposed model is able to produce distributions with heavier tails and capture extreme movements in exchange rates without entirely departing from the convenient GBM framework.

We generated exchange rate scenarios for four currency pairs: USD/EUR, GBP/EUR, RUB/EUR, and MXN/EUR. A risk factor evolution model backtesting exercise was performed, in line with Basel III requirements, and the the percentiles of the long-term distribution cones were obtained. The performances of the one-state and two-state models (GBM and the two-state HMM, respectively) were found to be very similar, with the two-state model HMM being slightly more conservative. However, when the generated scenarios were used to calculate exposure profiles for options on the RUB/EUR exchange rate, we found significant differences between the results of the two models. These differences were even more pronounced for deep out-of-the-money options.

Our study highlights some of the limitations of backtesting as a tool for comparing the performance of RFE models. Backtesting can be a useful way to objectively assess model performance. However, it can only be performed over short time horizons; with our available data, we could perform a statistically sound test of modeling assumptions for a time horizon of maximum length three months. It is, therefore, important to put effort into the interpretation of backtesting results, before they are translated into conclusions about model performance. Our results show how two models with similar performances in a backtesting exercise can result in very different exposure values and, consequently, in very different regulatory and economic capital buffers. This can lead to regulatory arbitrage and potentially weaken financial stability and, further, turn into a systemic risk.

The research presented in this paper can be extended in a number of ways, such as considering the evolution of risk factors other than exchange rates. Another topic worthy of investigation is the enhancement of the backtesting framework presented by Ruiz (2014), by considering statistical tests similar to the ones presented by Berkowitz (2001) and Amisano and Giacomini (2007). Finally, an interesting research direction is the development of an agent-based simulation model with heterogeneous modeling approaches, with regards to the RFE models. This model could potentially give valuable insights into the impact of heterogeneous models in financial stability.

**Author Contributions:** Both authors conceived and planned the research. Ioannis Anagnostou performed the numerical experiments. Both authors discussed the results and contributed to the final version of the manuscript.

**Funding:** This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement no. 675044 (http://bigdatafinance.eu/), Training for Big Data in Financial Research and Risk Management.

**Acknowledgments:** The authors are grateful to Jori Hoencamp, Steven van Haren, Marcel Boersma, and the reviewers for the useful comments.

**Conflicts of Interest:** The opinions expressed in this article are solely those of the authors and do not represent in any way those of their current and past employers. The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### **References**


#### *Risks* **2019**, *7*, 66


c 2019 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/).

## *Article* **Default Ambiguity**

#### **Tolulope Fadina <sup>1</sup> and Thorsten Schmidt 1,2,3,***<sup>∗</sup>*


Received: 29 March 2019; Accepted: 17 May 2019; Published: 10 June 2019

**Abstract:** This paper discusses ambiguity in the context of single-name credit risk. We focus on uncertainty in the default intensity but also discuss uncertainty in the recovery in a fractional recovery of the market value. This approach is a first step towards integrating uncertainty in credit-risky term structure models and can profit from its simplicity. We derive drift conditions in a Heath–Jarrow–Morton forward rate setting in the case of ambiguous default intensity in combination with zero recovery, and in the case of ambiguous fractional recovery of the market value.

**Keywords:** model ambiguity; default time; credit risk; no-arbitrage; reduced-form HJM models; recovery process

#### **1. Introduction**

Recently, an increasing amount of literature focuses on uncertainty as it relates to financial markets. The problem is that, the probability distribution of randomness in these markets is unknown. Typically, the unknown distribution is either estimated by statistical methods or calibrated to given market data by means of a model for the financial market. For example, in credit risk, the default probability is not observed, hence, have to be estimated from observable data. These methods introduce a large *model risk*.

Already, Knight (1921) pointed towards a formulation of risk which is able to treat such challenges in a systematic way. He was followed by Ellsberg (1961), who called random variables with known probability distribution *certain*, and those where the probability distribution is not known *uncertain*.

In this paper, we address these problems by constructing a model such that the parameters are characterised by uncertainty. Then, a single probability measure in a classical model is replaced by a family of probability measures, that is, a full class of models.

Following the modern literature in the area, we will call the feature that the probability distribution is not entirely fixed or, cannot be modelled by a single probability measure, *ambiguity*. This area has recently renewed the attention of researchers in mathematical finance to fundamental subjects such as arbitrage conditions, pricing mechanisms, and super-hedging. In equity markets, volatility uncertainty plays a crucial role, and has been extensively investigated, see for example, Avellaneda et al. (1995); Denis and Martini (2006); Lyons (1995); Vorbrink (2014). A major difficulty in this setting is that volatility uncertainty is characterized (at least in continuous time) by probabilities measures being mutually singular. Thus, the classical fundamental theorem of asset pricing fails to justify the no-arbitrage conditions, and new techniques are demanded, see Bayraktar and Zhang (2013); Bouchard and Nutz (2015); Burzoni et al. (2017), and Biagini et al. (2017).

In this paper, we introduce the concept of ambiguity to *defaultable term structure models*. The starting point for term structure models are typically bond prices of the form

$$P(t,T) = e^{-\int\_{t}^{T} f(t,\mu)d\mu}, \qquad 0 \le t \le T\_\prime \tag{1}$$

where (*f*(*t*, *T*))0≤*t*≤*<sup>T</sup>* is the instantaneous forward rate and *T* is the maturity time. This follows the seminal approach proposed in Heath et al. (1992). The presence of credit risk<sup>1</sup> in the model introduces an additional factor known as the default time. In this setting, bond prices are assumed to be absolutely continuous with respect to the maturity of the bond. This assumption is typically justified by the argument that, in practice, only a finite number of bonds are liquidly traded and the full term structure is obtained by interpolation, thus is smooth. There are two classical approaches to model market default risk: the *structural approach* Merton (1974) and the *reduced-form approach* (see for example, Artzner and Delbaen (1995); Duffie et al. (1996); Lando (1994) for some of the first works in this direction).

Structural models of credit risk describe the modelling of credit events specific to a particular corporate firm. Here, the underlying state is the value of a firm's assets which is observable. Default time is define as the first time the firm's asset value falls below a certain barrier level (for example, its liabilities). Hence, default is not a surprise. The approach links the default events to the firm's economic fundamentals, consequently, default time is endogenously within the model. However, the assumption that the firm's asset value is observable is often too strong in real applications, see Duffie and Lando (2001) and the survey article Frey and Schmidt (2011). Structural models in credit risk have been studied under many different viewpoints, see Black and Cox (1976); Frey and Schmidt (2009 2012); Gehmlich and Schmidt (2018); Geske (1977); Kim et al. (1993); Leland and Toft (1996); Merton (1974).

In comparison to the structural approach, reduced form approaches take a less stringent viewpoint regarding the mechanisms leading to default and model default events via an additional random structure. This additional degree of freedom together with their high tractability led to a tremendous success of this model class in credit risk modelling. For more details on the reduced form approach, we refer to Bielecki and Rutkowski (2002), and the references therein. Structural models can be embedded into (generalized) reduced form models as pointed out in Bélanger et al. (2004) and Fontana and Schmidt (2018).

Reduced-form models typically postulate that default time is totally inaccessible and, consequently, bond prices are absolutely continuous with respect to the maturity. Under the assumption of zero *recovery*2, this implies that credit risky bond prices *P*(*t*, *T*) are given by

$$P(t,T) = \mathbb{1}\_{\{\tau > t\}} e^{-\int\_t^T f(t,u)du} \tag{2}$$

with *τ* denoting the *random default time*. Since *τ* is totally inaccessible, it has an intensity *λ*. For example, if the intensity is constant, the default time is the first jump of a Poisson process with constant arrival rate *λ*. More generally, *λ<sup>t</sup>* may be viewed as the conditional rate of occurrence of default at time *t*, given information up to that time. In a situation where the owner of a defaultable claim recovers part of its initial investment upon default, the associated survival process <sup>1</sup>{*τ*>*t*} in (2), is replaced by a semimartingale. The quantity of the investment recovered is the so-called *recovery*.

Under ambiguity, we suggest that there is some prior information at hand which gives a upper and lower bounds on the *intensity*. The implicit assumption that the probability distribution of default is known is quite restrictive. Thus, we analyse our problem in a multiple priors model which describe

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

<sup>2</sup> The amount that the owner of a defaulted claim receives upon default.

uncertainty about the "true probability distribution". By means of the Girsanov theorem, we construct the set of priors from the reference measure. The assumption is that all priors are equivalent.

In view of our framework, it is only important to acknowledge that a rating class provides an estimate of the one-year default probability in terms of a confidence interval. Also estimates for 3-, and 5-year default probabilities can be obtained from the rating migration matrix. Thus, leading to a certain amount of model risk.

The aim of this paper is to incorporate ambiguity into the context of single-name credit risk. We focus on ambiguity on the default intensity, and also discuss ambiguity on the recovery.

The main results are as follows: we obtain a necessary and sufficient condition for a reference probability measure to be a local martingale measure for credit risky bond markets under default ambiguity, thereby ensuring the absence of arbitrage in a sense to be precisely specified below. Furthermore, we consider the case where we have partial information on the amount that the owner of a defaulted claim receives upon default.

The next section of this paper introduces homogeneous ambiguity, and its example. Section 3 introduces the fundamental theorem of asset pricing (FTAP) under homogeneous ambiguity. In Sections 4 and 6, we derive the no-arbitrage conditions for defaultable term structure models with zero-recovery, and fractional recovery of the market value, in our framework. We conclude in section 7.

#### **2. Intensity-Based Models**

Intensity-based models are the most used model class for modeling credit risk (see (Bielecki and Rutkowski 2002, Chapter 8) for an overview of relevant literature). The default intensity, however, is difficult to estimate and therefore naturally carries a lot of uncertainty. This has led to the emergence of rating agencies which, since the early 20th century, estimate bond's credit worthiness3.

Modeling of credit risk has up to now incorporated uncertainty in the default intensity in a systematic way. On the other side, a number of Bayesian approaches exist, utilizing filtering technologies (see, for example, Duffie and Lando (2001); Frey and Schmidt (2009), among many others).

Here, we introduce an alternative treatment of the lack of precise knowledge of the default intensity based on the concept of *ambiguity* following the seminal ideas from Frank Knight in Knight (1921).

Uncertainty in our setting will be captured through a family of probability measures P replacing the single probability measure P in classical approaches. Intuitively, each P represents a model and the family P collects models which we consider equally likely.

In this spirit, working with a single P, or with a set <sup>P</sup> = {P} which contains only one element, is in a one-to-one correspondence to assuming that the parameters of the underlying processes are exactly known. In financial markets, this is certainly not the case and ambiguity helps to incorporate this uncertainty into the used models.

We consider throughout a fixed finite time horizon *T*∗ > 0. In light of our discussion above, let (Ω, F ) be a measurable space and P be a set of probability measures on the measurable space (Ω, F ). In particular, there is no fixed and known measure P (except in the special case where P contains only one element which we treat en passant).

Intensity-based default models correspond to the case where the ambiguity is *homogeneous*, i.e., there is a measure P such that P ∼ P for all P ∈ <sup>P</sup>. Here, P ∼ P means that P and P are *equivalent*, that is, they have the same nullsets. The reference measure P only has the role of fixing events of measure zero for all probability measures under consideration. Intuitively, this means there is no ambiguity on these events of measure zero. In the following, *E* is the expectation with respect to the reference measure P .

<sup>3</sup> For a historical account, see Sylla (2002): John Moody founded the first rating agency in 1909, in the United States.

**Remark 1.** *As a consequence of the equivalence of all probability measures in* P*, all equalities and inequalities will hold almost-surely with respect to any probability measure* P ∈ <sup>P</sup>*, or, respectively, to* P *.*

#### *Ambiguity in Intensity-Based Models*

In this section, we introduce ambiguity in intensity-based models. Our goal is not the most general approach in this setting: we rather focus on simpler, but still practically highly relevant cases. For a more general treatment, we refer the reader to Biagini and Zhang (2017). The main mathematical tool we use here is enlargement of filtrations and we refer the reader to Aksamit and Jeanblanc (2017) for further details and a guide to the literature.

Assume that under P we have a *d*-dimensional Brownian motion *W* with canonical and augmented4 filtration <sup>F</sup> = (F*t*)0≤*t*≤*T*<sup>∗</sup> and a standard exponential random variable *<sup>τ</sup>*, independent of F*T*<sup>∗</sup> , that is, P (*t* < *τ*|F*t*) = exp(−*t*), 0 ≤ *t* ≤ *T*∗. The Brownian motion *W* has the role of modelling market movements and general information, excluding default information. We therefore call F the *market filtration* in the following. The filtration <sup>G</sup> = (G*t*)0≤*t*≤*T*<sup>∗</sup> includes default information and is obtained by a progressive enlargement of F with *τ*, i.e.,

$$\mathcal{H}\_t = \bigcap\_{\mathfrak{e} > 0} \sigma(\mathbb{1}\_{\{t \ge \tau\}}, W\_{\mathfrak{e}} : 0 \le \mathfrak{s} \le t + \mathfrak{e}), \qquad 0 \le t \le T^\*.$$

To finalize our setup, we assume that F = G*T*<sup>∗</sup> .

Note that up to now, everything has been specified under the reference measure P and nothing was said about the concrete models we are interested in (except about the nullsets). These models will now be introduced using the Girsanov theorem, i.e., by changing from P to the measures we are interested in.

Consequently, the next step is to construct measures P*<sup>λ</sup>* with appropriate processes *λ* – under P*λ*, the default time *τ* will have the intensity *λ*. More precisely, assume that *λ* is some positive process which is predictable with respect to the market filtration, F. Define the density process *Z<sup>λ</sup>* by

$$Z\_t^\lambda := \begin{cases} \exp\left(\int\_0^t (1-\lambda\_s)ds\right), & t < \tau\\ \lambda\_\tau \exp\left(\int\_0^\tau (1-\lambda\_s)ds\right) & t \ge \tau. \end{cases} \tag{3}$$

Note that *Z<sup>λ</sup>* is a G-local martingale and corresponds to a Girsanov-type change of measure (see Theorem VI.2.2 in Brémaud (1981)). If *E* [*Z<sup>λ</sup> <sup>T</sup>*<sup>∗</sup> ] = 1 we obtain an equivalent measure <sup>P</sup>*<sup>λ</sup>* <sup>∼</sup> <sup>P</sup> via

$$\mathbb{P}^{\mathbb{A}}(A) := E'[\mathbb{1}\_A Z\_{T^\*}^{\mathbb{A}}] \quad \forall A \in \mathcal{P}. \tag{4}$$

Under the measure P*λ*, *τ* has intensity *λ*: more precisely, this means that the process

$$M\_t^\lambda := \mathbb{1}\_{\{t \le \tau\}} - \int\_0^{t \wedge \tau} \lambda\_s ds, \qquad 0 \le t \le T^\*,\tag{5}$$

is a P*λ*-martingale.

Now we introduce a precise definition of ambiguity on the default intensity which is very much in spirit of the *G*-Brownian motion in Peng (2010): we consider an interval [*λ*, *λ*] ⊂ (0, ∞) where *λ* and *λ* denote lower (upper) bounds in the default intensity. Intuitively, we include all possible

<sup>4</sup> Augmentation can be done in a standard fashion with respect to P .

intensities lying in these bounds in our family of models P. More precisely, we define the set of density generators *H* by

$$H := \{ \lambda : \lambda \text{ is } \mathbb{F}\text{-preichlet and } \mathbb{P}'(\underline{\lambda} \le \lambda\_t \le \overline{\lambda}, \quad t \in [0, T^\*]) = 1 \}.$$

Ambiguity on the default intensity is now covered by considering the concrete family of probability measures

$$\mathcal{A}^{\mathcal{P}} := \{ \mathbb{P}^{\lambda} : \lambda \in H \}. \tag{6}$$

In the following, we will always consider this P. First, we observe that this set is convex.

**Lemma 1.** P *is a convex set.*

**Proof.** Consider P*λ* , P*λ* ∈ P and *α* ∈ (0, 1). Then,

$$
\alpha \mathbb{P}^{\lambda'}(A) + (1 - \alpha) \mathbb{P}^{\lambda''}(A) = E' \left[ \mathbb{1}\_A(\alpha Z\_{T^\*}^{\lambda'} + (1 - \alpha) Z\_{T^\*}^{\lambda''}) \right].
$$

Now consider the (well-defined) intensity *λ*, given by

$$\int\_0^t \lambda\_s ds := t - \log \left[ \alpha e^{\int\_0^t (1 - \lambda\_s') ds} + (1 - \alpha) e^{\int\_0^t (1 - \lambda\_s'') ds} \right],$$

0 ≤ *t* ≤ *T*∗. Then,

$$
\alpha Z\_{T^\*}^{\lambda'} + (1 - \mathfrak{a}) Z\_{T^\*}^{\lambda''} = Z\_{T^\*}^{\lambda}
$$

such that by (4), <sup>P</sup>*<sup>λ</sup>* <sup>∼</sup> <sup>P</sup> refers to an equivalent change of measure. Finally, we have to check that *λ* ∈ *H*, which means that *λ* ∈ [*λ*, *λ*], 0 ≤ *t* ≤ *T*∗: note that

$$\begin{aligned} t - \log\left[\alpha e^{\int\_0^t (1-\lambda\_s')ds} + (1-\alpha)e^{\int\_0^t (1-\lambda\_s')ds} \right] \\ \leq t - \log\left[\alpha e^{\int\_0^t (1-\overline{\lambda})ds} + (1-\alpha)e^{\int\_0^t (1-\overline{\lambda})ds} \right] \\ \leq t - t(1-\overline{\lambda}) = \overline{\lambda}t, \end{aligned}$$

and *λ<sup>t</sup>* ≤ *λ* follows. Similarly, *λ* ≤ *λ<sup>t</sup>* and the claim follows since *t* was arbitrary.

**Remark 2.** *Intuitively, the requirement λ* > 0 *states that there is always a positive risk of experiencing a default, which is economically reasonable. Technically it has the appealing consequence that all considered measures in* P *are equivalent.*

It turns out that the set of possible densities will play an important role in connection with measure changes. In this regard, we define *admissible measure changes* with respect to P by

> <sup>A</sup> := {*λ*<sup>∗</sup> : *<sup>λ</sup>*<sup>∗</sup> is positive, F-predictable and *<sup>E</sup>* [*Zλ*<sup>∗</sup> *<sup>T</sup>*<sup>∗</sup> ] = 1}.

The associated Radon-Nikodym derivatives *Zλ*<sup>∗</sup> *<sup>T</sup>*<sup>∗</sup> for *λ*<sup>∗</sup> ∈ A are the possible Radon-Nikodym derivatives for equivalent measure changes.

**Remark 3.** *It is of course possible to consider an ambiguity setting more general than the specific one in* (6)*. One possibility is to consider only a subset of* P*. Another possibility is to allow the bounds λ and λ to depend on time, or even on the state of the process – this latter case is important for considering affine processes under uncertainty and we refer to Fadina et al. (2019) for further details. In Section 5, we consider indeed such a more general setting.*

#### **3. Absence of arbitrage under ambiguity**

Absence of arbitrage and the respective generalizations, *no free lunch* (NFL), and *no free lunch with vanishing risk* (NFLVR), are well established concepts when the underlying probability measure P is known and fixed. Here, we give a small set of sufficient conditions for absence of arbitrage extended to the setting with ambiguity. In this regard, consider, a fixed set P of probability measures on the measurable space (Ω, <sup>F</sup> ). In addition, let <sup>G</sup> = (G*t*)0≤*t*≤*T*<sup>∗</sup> be a right-continuous filtration.

Discounted price processes of the traded assets are given by a finite dimensional G-semimartingale *<sup>X</sup>* = (*Xt*)0≤*t*≤*T*<sup>∗</sup> . The semimartingale property holds equivalently in any of the filtration <sup>G</sup><sup>+</sup> or the augmentation of G+, see (Neufeld and Nutz 2014, Proposition 2.2). It is well known that then *X* is a semimartingale for all P ∈ <sup>P</sup>.

A self-financing trading strategy is a predictable and *X*-integrable process Φ and the associated discounted gains process is given by the stochastic integral of Φ with respect to *X*,

$$(\Phi \cdot X)\_t = \int\_0^t \Phi\_{\mathfrak{U}} dX\_{\mathfrak{U}\_t} \qquad 0 \le t \le T^\*.$$

Intuitively, an *arbitrage* is an admissible self-financing trading strategy which starts from zero initial wealth, has non-negative pay-off under all possible future scenarios, hence for all *P* ∈ P, there is at least one *P* ∈ P, such that the pay-off is positive. This is formalized in the following definition, compare for example Vorbrink (2014). As usual a trading strategy is called *a*-admissible, if (Φ · *X*)*<sup>t</sup>* ≥ −*a* for all 0 ≤ *t* ≤ *T*∗.

**Definition 1.** *A self-financing trading strategy* Φ *is called* P-arbitrage *if it is a-admissible for some a* > 0 *and*


Since all probability measures P ∈ <sup>P</sup> are considered as possible, a <sup>P</sup>-arbitrage is a riskless trading strategy for all possible models (i.e., for all P ∈ <sup>P</sup>) while it is a profitable strategy for at least one scenario (i.e., for at least one P ∈ <sup>P</sup>).

The main tool for classifying arbitrage free markets will be local martingale measures, even in the setting with ambiguity. In this regard, we call a probability measure Q a *local martingale measure* if *X* is a Q-local martingale.

It is well-known that *no arbitrage* or, more precisely, *no free lunch with vanishing risk* (NFLVR) in a market where discounted price processes are locally bounded semimartingales is equivalent to the existence of an equivalent local martingale measure (ELMM), see Delbaen and Schachermayer (1994, 1998). The technically difficult part of this result is to show that a precise criterion of absence of arbitrage implies the existence of an ELMM. In the following, we will not aim at such a deep result under ambiguity, but utilize the easy direction, namely that existence of an ELMM implies the absence of arbitrage as formulated below.

From the classical *fundamental theorem of asset pricing* (FTAP), the following result follows easily.

**Theorem 1.** *If, for every* P ∈ <sup>P</sup> *there exists an equivalent local martingale measure* Q(P)*, then there is no arbitrage in the sense of Definition 1.*

**Proof.** Indeed, assume on the contrary that there is an arbitrage Φ with respect to some measure P ∈ <sup>P</sup> which we fix for the remainder of the proof. If there exists an ELMM Q(P) then <sup>Φ</sup> would be an arbitrage strategy together with an ELMM, a contradiction to the classical FTAP.

This (sufficient) condition directly corresponds to the existing results in the literature (see, for example, Biagini et al. (2017)) where arbitrages of the first kind are studied under the additional assumption of continuity for the traded assets.

#### **4. Ambiguity on the default intensity**

Our aim is to discuss dynamic term structure models under default risk with ambiguity on the default intensity. The relevance of this issue has, for example, already been reported in Riedel (2015). Here, we take this as motivation to propose a precise framework taking ambiguity on the default intensity into account.

#### *4.1. Dynamic defaultable term structures*

We specialize the considerations of absence of arbitrage in section 3 to defaultable bond markets. Recall that we have a filtration G at hand and that *τ* is the G-stopping time at which the company defaults. We define the *default indicator process H* by

$$H\_t = \mathbf{1}\_{\{t \ge \tau\} \prime} \qquad 0 \le t \le T^\*.$$

The associated *survival process* is 1 − *H*. A credit risky bond with a maturity time *T* ≤ *T*<sup>∗</sup> is a contingent claim promising to pay one unit of currency at *T*. We denote the price of such a bond at time *t* ≤ *T* by *P*(*t*, *T*). If no default occurs prior to *T*, *P*(*T*, *T*) = 1. We will first consider *zero recovery*, i.e., assume that the bond loses its total value at default. Then *P*(*t*, *T*) = 0 on {*t* ≥ *τ*}.

Besides zero recovery, we only make the weak assumption that bond-prices prior to default are positive and absolutely continuous with respect to maturity *T*. This follows the well-established approach by Heath et al. (1992). More formally, we assume that

$$P(t,T) = \mathbb{1}\_{\{\tau > t\}} \exp\left(-\int\_{t}^{T} f(t,u) du\right) \qquad 0 \le t \le T. \tag{7}$$

The initial forward curve *T* → *f*(0, *T*) is then assumed to be sufficiently integrable and the *forward rate processes f*(·, *T*) are assumed to follow Itô processes satisfying

$$f(t, T) = f(0, T) + \int\_0^t a(s, T)ds + \int\_0^t b(s, T)dW\_{s\prime} \tag{8}$$

for 0 ≤ *t* ≤ *T*. Recall that *W* was chosen to be a Brownian motion. We denote by O the optional *σ*-algebra and by B the Borel *σ*-algebra.

**Assumption 1.** *We require the following technical assumptions:*

*(i) the initial forward curve is measurable, and integrable on* [0, *T*∗]*:*

$$\int\_{0}^{T^{\*}} |f(0, u)| du < \infty$$

*(ii) the drift parameter a*(*ω*,*s*, *<sup>t</sup>*) *is* R*-valued* <sup>O</sup> ⊗ <sup>B</sup>*-measurable and integrable on* [0, *<sup>T</sup>*∗]*:*

$$\int\_{0}^{T^\*} \int\_{0}^{T^\*} |a(s, t)| ds dt < \infty,$$

*(iii) the volatility parameter b*(*ω*,*s*, *<sup>t</sup>*) *is* <sup>R</sup>*d-valued,* <sup>O</sup> <sup>⊗</sup> <sup>B</sup>*-measurable, and*

$$\sup\_{s,t \le T^\*} \|b(s,t)\| < \infty.$$

*(iv) Let rt be the short rate process at time t, for* 0 ≤ *t* ≤ *T*∗*. With probability one, it holds that*

$$0 < f(t, t) - r\_{t \prime} \qquad 0 \le t \le T^\*.$$

Set for 0 ≤ *t* ≤ *T* ≤ *T*∗,

$$\begin{aligned} \overline{a}(t,T) &= \int\_{\mathfrak{f}}^{T} a(t,u) du, \\ \overline{b}(t,T) &= \int\_{\mathfrak{f}}^{T} b(t,u) du. \end{aligned}$$

**Lemma 2.** *Under Assumption 1 it holds that,*

$$\int\_{t}^{T} f(t, u) du = \int\_{0}^{T} f(0, u) du + \int\_{0}^{t} \overline{a}(\cdot, u) du + \int\_{0}^{t} \overline{b}(\cdot, u) dW\_{u} - \int\_{0}^{t} f(u, u) du$$

*for* 0 ≤ *t* ≤ *T* ≤ *T*∗*, almost surely.*

This follows as in Heath et al. (1992): for the case *W* is a Brownian motion, this is Lemma 6.1 in Filipovi´c (2009). This result could also be generalized where *W* is replaced by a semimartingale with absolutely continuous characteristics, see Proposition 5.2 in Björk et al. (1997). Note that the strong condition (iii) of uniform boundedness of *b* in Assumption 1 is needed for the application of the stochastic Fubini theorem.

#### *4.2. Absence of arbitrage without ambiguity on the default intensity*

The first step towards the study of term structure models with default ambiguity is the study of absence of arbitrage in (classical) intensity-based dynamic term structure models. Consider *λ* = (*λt*)0≤*t*≤*T*<sup>∗</sup> <sup>∈</sup> <sup>A</sup> and the probability measure <sup>P</sup>*λ*. Then, the dual predictable projection *<sup>H</sup><sup>p</sup>* of *<sup>H</sup>* is given by *H<sup>p</sup> <sup>t</sup>* <sup>=</sup> *<sup>t</sup>*∧*<sup>τ</sup>* <sup>0</sup> *<sup>λ</sup>sds* (under <sup>P</sup>*λ*). Moreover, the Doob-Meyer decomposition yields that

$$\mathcal{M}^{\lambda} := H - \int\_0^{\cdot \wedge \tau} \lambda\_s ds$$

is P*λ*-martingale, compare equation (5).

For discounting, we use the bank account. Its value is given by a stochastic process starting with 1 which is then upcounted by the short rate *r*, i.e., the value process of the bank account is *γ*(*t*) = exp( *t* <sup>0</sup> *rsds*) with an G-predictable process *<sup>r</sup>*.

In the bond market context considered here, a measure Q is called local martingale measure if, for any maturity *T* ∈ (0, *T*∗], the discounted bond price process for the bond with maturity *T* is a Q-local martingale. Then, we obtain the following result.

**Proposition 1.** *Assume that Assumption 1 holds. Consider a measure* Q *on* (Ω, F )*, such that M<sup>λ</sup> is a* Q*-martingale, that W is a* Q*-Brownian motion and that* Q( *T*<sup>∗</sup> <sup>0</sup> |*rs*|*ds* < <sup>∞</sup>) = <sup>1</sup>*. Then* Q *is a local martingale measure if and only if*


$$\vec{a}(t,T) = \frac{1}{2} \left\| \mathcal{F}(t,T) \right\|^2 \rho$$

*holds dt* ⊗ *dQ-almost surely for* 0 ≤ *t* ≤ *T* ≤ *T*<sup>∗</sup> *on* {*τ* > *t*}*.*

**Proof.** We set *<sup>E</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>H</sup>* and *<sup>F</sup>*(*t*, *<sup>T</sup>*) = exp <sup>−</sup> *<sup>T</sup> <sup>t</sup> <sup>f</sup>*(*t*, *<sup>u</sup>*)*du* . Then (7) can be written as *P*(*t*, *T*) = *E*(*t*)*F*(*t*, *T*). Integrating by part yields

$$dP(t,T) = F(t-,T)dE(t) + E(t-)dF(t,T) + d[E,F(\cdot,T)]\_{t=0}$$

For {*t* < *τ*},

$$\begin{split}dP(t,T) &= P(t-\prime,T)\left(-\lambda\_{l}dt + \left(f(t,t) + \frac{1}{2}\left\|\overline{b}(t,T)\right\|^{2} - \overline{a}(t,T)\right)dt\right) \\ &- P(t-\prime,T)\left(dM^{\lambda} + \overline{b}(t,T)dW\_{t}\right). \end{split} \tag{9}$$

The discounted bond price process is a local martingale if and only if the predictable part in the semimartingale decomposition vanishes, i.e.,

$$\left\| f(t, t) - r\_t - \lambda\_t - \bar{a}(t, T) + \frac{1}{2} \left\| \tilde{b}(t, T) \right\| \right\|^2 = 0. \tag{10}$$

Letting *T* = *t* we obtain (i) and (ii) and the result follows.

#### *4.3. Absence of arbitrage with ambiguity on the default intensity*

Next, we derive the no-arbitrage conditions for the forward rate in term of the intensity and the short rate, and also the conditions for the drift and volatility parameters, under ambiguity on the default intensity. In this regard, we require a bit more structure: we assume that the setting detailed in section 2 holds, in particular, we consider the family of probability measures P constructed in equation (6). Recall that for all <sup>P</sup>*<sup>λ</sup>* <sup>∈</sup> <sup>P</sup>, *<sup>W</sup>* is a Brownian motion and that <sup>P</sup>*<sup>λ</sup>* <sup>∼</sup> <sup>P</sup> . We may, for the moment, safely assume that the market filtration F satisfies the usual conditions under P .

For a generic real-valued, <sup>F</sup>-progressive process *<sup>θ</sup>* = (*θt*)*t*≥0, let the process *<sup>z</sup><sup>θ</sup>* = (*z<sup>θ</sup> <sup>t</sup>*)0≤*t*≤*T*<sup>∗</sup> be given as the unique strong solution of

$$dz\_t^\theta = \theta\_t z\_t^\theta d\mathsf{W}\_{t\prime} \quad z\_0^\theta = 1. \tag{11}$$

Then, *z<sup>θ</sup>* is a continuous local martingale. If *E* [*zθ <sup>T</sup>*<sup>∗</sup> ] = 1, we can define a probability measure <sup>P</sup>˜ *<sup>θ</sup>* by letting

$$\tilde{\mathbb{P}}^{\theta}(A) := E'[\mathbb{1}\_A z\_{T^\*}^{\theta}], \quad \forall A \in \mathcal{F}, \tag{12}$$

just as in equation (4). Under <sup>P</sup>˜ *<sup>θ</sup>* the process *<sup>W</sup>*˜ <sup>=</sup> *<sup>W</sup>* <sup>−</sup> · <sup>0</sup> *θsds* is a Brownian motion, i.e., *W* itself became a Brownian motion with drift *θ*, see Theorem 5.1 in Chapter 3 of Karatzas and Shreve (1998).

Moreover, set *<sup>λ</sup>*˜ *<sup>t</sup>* := (*f*(*t*, *<sup>t</sup>*) <sup>−</sup> *rt*) · *<sup>λ</sup>*−<sup>1</sup> *<sup>t</sup>* , *<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*∗]. Note that under Assumption 1, *<sup>λ</sup>*˜ is positive (which is necessary for an equivalent change of measure). The associated density is abbreviated by

$$Z^\*\_{\Gamma^\*} = Z^{\bar{1}}\_{\Gamma^\*}.$$

**Theorem 2.** *Consider* <sup>P</sup>*<sup>λ</sup>* <sup>∈</sup> <sup>P</sup>*. Under Assumption 1, there exists an ELMM to* <sup>P</sup>*λ, if there exists an* F*-progressive process θ*<sup>∗</sup> *such that*


$$d(t, T) = \frac{1}{2} \parallel \bar{b}(t, T) \parallel^2 - \bar{b}(t, T)\theta\_{t \prime}^\* \quad 0 \le t \le T \le T^\*$$

*holds dt* <sup>⊗</sup> *<sup>d</sup>*P*λ-almost surely on* {*<sup>t</sup>* <sup>&</sup>lt; *<sup>τ</sup>*}*.*

Intuitively, the theorem states that for the probability measure P*λ*, we find an ELMM if we are able to perform an equivalent change of measure (condition (i)) in such a way that under the new measure the drift condition holds for the Brownian motion with drift *θ*∗ (condition (ii)).

**Proof.** We start from some <sup>P</sup>*<sup>λ</sup>* <sup>∈</sup> <sup>P</sup> and fix this measure in the following. This means that, under P*λ*, *W* is a Brownian motion and *τ* has intensity *λ*. In the search for an ELMM we are looking for an equivalent measure P<sup>∗</sup> which satisfies the conditions of Proposition 1.

In this regard, note the following: by its definition, (11), together with condition (i), *zθ*<sup>∗</sup> is a density process for a change of measure via the Girsanov theorem for Itô processes, see Theorem 5.1 in Chapter 3 of Karatzas and Shreve (1998). Moreover, by (3) together with (i), *Z*<sup>∗</sup> = *Zλ*˜ is the density for the change in intensity from *λ* under P*<sup>λ</sup>* to the intensity given by *λ*<sup>∗</sup> *<sup>t</sup>* := (*f*(*t*, *t*) − *rt*)0≤*t*≤*T*<sup>∗</sup> , see (Brémaud 1981, Theorem VI.2.T2). We set

$$d\mathbb{P}^\* := Z\_{\mathbb{T}^\*}^\* z\_{\mathbb{T}^\*}^{\theta^\*} d\mathbb{P}^\lambda.$$

According to Theorem 3.40 in Chapter III of Jacod and Shiryaev (2003), this refers to a Girsanov-type (and equivalent) change of measure. Moreover, *W*∗ *<sup>t</sup>* <sup>=</sup> *Wt* <sup>−</sup> *<sup>t</sup>* <sup>0</sup> *θ*<sup>∗</sup> *<sup>s</sup> ds*, 0 ≤ *<sup>t</sup>* ≤ *<sup>T</sup>*∗, is a P∗-Brownian motion and *Mλ*<sup>∗</sup> is a P∗-martingale.

We now show that P<sup>∗</sup> is also a local martingale measure. Recall from (9) that, under P*λ*,

$$\frac{dP(t,T)}{P(t-,T)} = \left(f(t,t) + \frac{1}{2}\left\|\overline{b}(t,T)\right\|^2 - \overline{a}(t,T)\right)dt - dH\_{\mathbb{H}} - \overline{b}(t,T)dW\_{\mathbb{H}}$$

for {*<sup>t</sup>* <sup>&</sup>lt; *<sup>τ</sup>*}; here *Ht* <sup>=</sup> <sup>1</sup>{*t*≥*τ*} is the default indicator. We introduce the martingales *<sup>W</sup>*<sup>∗</sup> and *<sup>M</sup>λ*<sup>∗</sup> into this representation: note that

$$\begin{aligned} \frac{dP(t,T)}{P(t-,T)} &= \left(-\lambda^\*(t) + f(t,t) + \frac{1}{2} \left\| \overline{b}(t,T) \right\|^2 - \overline{a}(t,T) - \overline{b}(t,T)\theta\_t^\* \right) dt \\ &- d\mathcal{M}\_t^{\lambda^\*} - \overline{b}(t,T)dW\_t^\* \end{aligned}$$

again on {*t* < *τ*}. Since, by assumption, −*λ*<sup>∗</sup> *<sup>t</sup>* + *f*(*t*, *t*) = *rt*, together with the drift condition (ii), we obtain for the discounted bond price process *P*˜(*t*, *T*) = *P*(*t*, *T*)/*γt*,

$$d\tilde{P}(t,T) = \tilde{P}(t-\prime,T) \cdot \left(-dM\_t^{\Lambda^\*} - \tilde{b}(t,T)dW\_t^\*\right),$$

which is a P∗-local martingale and the proof is finished.

#### **5. Examples**

The setting proposed above can, in dimension one, be directly linked to a special case of the non-linear affine processes introduced in Fadina et al. (2019). Indeed, note that for a progressive process *λ*, the integral

$$X\_t := \mathfrak{x} + \int\_0^t \lambda\_s ds\_\prime \qquad 0 \le t \le T^\*.$$

is a special semimartingale. Moreover, there are affine bounds on drift and volatility (the bound of the volatility is simply zero) since

$$
\underline{\lambda} \le \lambda\_t \le \overline{\lambda}\_r
$$

such that *X* is a non-linear affine process.

The major advantage of this setting is that numerical methods via non-linear PDE come into reach. More precisely, Theorem 4.1 in Fadina et al. (2019) shows that whenever *ψ* is Lipschitz, the non-linear expectation

$$\mathcal{E}\left[\psi(X\_{T^\*})\right] := \sup\_{\mathbb{P}\in \beta^\bullet} E^\mathbb{P}[\psi(X\_{T^\*})] \tag{13}$$

can be expressed as viscosity solution of the fully non-linear PDE

$$\begin{cases} -\partial\_t v(t, \mathbf{x}) - G\left(\mathbf{x}, \partial\_x v(t, \mathbf{x})\right) = 0 & \text{on } [0, T^\*) \times [\underline{\lambda}, \overline{\lambda}], \\ v(T, \mathbf{x}) = \psi(\mathbf{x}) & \mathbf{x} \in [\underline{\lambda}, \overline{\lambda}], \end{cases} \tag{14}$$

where *G* is defined by

$$G(x, p) := \sup\_{\lambda \in [\underline{\lambda}, \overline{\lambda}]} \{\lambda p\} \tag{15}$$

and *v*(0, *x*) = E[*ψ*(*XT*<sup>∗</sup> )] (the dependency on *x* arises through *X*<sup>0</sup> = *x*).

Clearly, when *p* is either strictly positive (hence *∂xv*(*t*, *x*)) or negative, then the supremum in (15) is immediate and the PDE (14) can be solved using standard methods. This means that the solution to the non-linear expectation is obtained simply by the upper bound *λ*¯ (or the lower bound *λ*, respectively). Such a condition holds if *ψ* is monotone. The more general case has to be solved using numerical methods and we provide a simple example now.

**Example 1.** *Consider a butterfly on XT*<sup>∗</sup> *, i.e., the derivative with the payoff*

$$\psi(\mathbf{x}) = (\mathbf{x} - \mathbf{K}\_1)^+ - 2(\mathbf{x} - \mathbf{K}\_2)^+ + (\mathbf{x} - \mathbf{K}\_3)^+,$$

*where we choose K*<sup>1</sup> = −0.2*, K*<sup>2</sup> = 0.3*, and K*<sup>3</sup> = 0.8*. Moreover, let λ* = 0.1 *and λ* = 0.5*. Then the upper and lower price bounds for the butterfly are shown in Figure 1 (the upper prices are given by the nonlinear expectation in equation* (13)*, while the lower prices are obtained by replacing the supremum in* (13) *by an infimum).*

**Figure 1.** This figure shows the solution of the nonlinear PDE in Equation (14) with boundary condition *<sup>ψ</sup>*(*y*)=(*<sup>y</sup>* <sup>−</sup> *<sup>K</sup>*<sup>1</sup> <sup>+</sup> *<sup>x</sup>*)<sup>+</sup> <sup>−</sup> <sup>2</sup>(*<sup>y</sup>* <sup>−</sup> *<sup>K</sup>*<sup>2</sup> <sup>+</sup> *<sup>x</sup>*)<sup>+</sup> + (*<sup>y</sup>* <sup>−</sup> *<sup>K</sup>*<sup>3</sup> <sup>+</sup> *<sup>x</sup>*)+, *<sup>K</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>0.2, *<sup>K</sup>*<sup>2</sup> <sup>=</sup> 0.3, *<sup>K</sup>*<sup>3</sup> <sup>=</sup> 0.8, and *x* ∈ [−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 *λ* = 0.1 and *λ* = 0.5. The upper and lower solid lines show the upper and lower price bounds.

#### **6. Ambiguity on the recovery**

A detailed study of bond markets beyond zero recovery is often neglected, the high degree of uncertainty about the recovery mechanism being a prime reason for this. This motivates us to take some time for developing a deeper understanding of a suitable recovery model under ambiguity.

We start from the observation that intensity-based models always need certain recovery assumptions, as for example, zero recovery, fractional recovery of treasury, and fractional recovery of par value, see (Bielecki and Rutkowski 2002, Chapter 8). We have so far considered the case where the credit risky bond becomes worthless at default (zero recovery). In the following, we will consider *fractional recovery of market value* where the credit risky bond looses a fraction of its market value upon default. Other recovery models can be treated in a similar fashion.

#### *6.1. Fractional recovery without ambiguity*

Fractional recovery of market value (RMV) is specified through a marked point process (*Tn*, *Rn*)*n*≥<sup>1</sup> where the stopping times (*Tn*) denote the default times and *Rn* ∈ (0, 1] denotes the associated fractional recovery. Let

$$R\_t = \prod\_{T\_n \le t} R\_{n\_t} \qquad 0 \le t \le T^\*. \tag{16}$$

denote the *recovery process*. Then, *R* is non-increasing, positive with *R*<sup>0</sup> = 1. The recovery process replaces the default indicator in (7). More precisely, we assume that the family of defaultable bond prices under RMV satisfy

$$P\_R(t, T) = R\_t \exp\left(-\int\_t^T f(t, u) du\right), \qquad 0 \le t \le T \le T^\*. \tag{17}$$

**Remark 4.** *If a default occurs at t* = *Tn, the bond looses a random fraction qt* = 1 − *RTn* ∈ [0, 1) *of its pre-default value. Thus, the value* (1 − *qt*)*P*(*t*−, *T*) *is immediately available to the bond owner at default. It is still subject to default risk because of the possible future defaults occurring at Tn*+1, *Tn*+2,... *.*

First, we state a generalization of Proposition 1 to this setting. To this end, we require more structure and continue in the setting of the section 2. Assume that the marked point process (*Tn*, *Rn*)*n*≥<sup>1</sup> is independent from *W* and standard in the following sense: the random times (*Tn*) are the jumping times from a Poisson process with intensity one, and the recovery values (*Rn*) are independent from (*Tn*) and *W*, and uniformly distributed in [*r*,*r*¯] ⊂ (0, 1].

The filtration <sup>G</sup> = (G*t*)0≤*t*≤*T*<sup>∗</sup> is obtained by a progressive enlargement of <sup>F</sup> with default and recovery information (given by *R*), i.e.,

$$\mathcal{H}\_t = \bigcap\_{\mathfrak{e} > 0} \sigma(R\_{\mathfrak{s}\prime} W\_{\mathfrak{s}} : 0 \le \mathfrak{s} \le t + \mathfrak{e}), \qquad 0 \le t \le T^\*.$$

We assume again F = G*T*<sup>∗</sup> . As next step, we introduce measure changes for the marked point process. Let

$$\Phi\_t = \sum\_{T\_n \le t} R\_{n\prime} \qquad 0 \le t \le T^\* \cdot t$$

This implies that the defaultable bond prices under RMV, *PR*(0, *t*), at maturity time *t* of this bond, pays out Φ*t*, the accumulated fractional default losses. Then, Φ is a special semimartingale w.r.t. G. Let

$$\mu^{\Phi}(dt, dx) = \sum\_{n \ge 1} \delta\_{(T\_n, R\_n)}(dt, dx)$$

denote the associated jump measure and let *ν*Φ(*dt*, *dx*) denote its compensator, see Chapter II.1 in Jacod and Shiryaev (2003) or Chapter VIII.1 in Brémaud (1981). Note that *<sup>ν</sup>*Φ(*dt*, *dx*) = <sup>1</sup>{*x*∈[*r*,*r*]}(*<sup>r</sup>* <sup>−</sup> *<sup>r</sup>*)−1*dxdt*.

We introduce the densities

$$L^{\mu\_{\rm h}} = \left(\prod\_{T\_n \le T^\*} \mu\_{T\_n} h(T\_{n\_\star} R\_\hbar)\right) \cdot \epsilon^{\int\_0^{T^\*} \int\_\mathbb{L}^{\overline{\pi}} (1 - \mu\_t h(t, \mathbf{x})) (\overline{\pi} - \underline{\pi})^{-1} d\mathbf{x} \, dt}\_{\prime} \tag{18}$$

where the predictable process *<sup>μ</sup>* is positive and, for any *<sup>x</sup>* ∈ [*r*,*r*], the G-predictable process (*h*(*t*, *<sup>x</sup>*))0≤*t*≤*T*<sup>∗</sup> is also positive. If *<sup>E</sup>*[*LT*<sup>∗</sup> ] = 1, we can define the equivalent measure <sup>P</sup>*μ*,*<sup>h</sup>* by

$$d\mathbb{P}^{\mu,h} = L\_{T^\*} d\mathbb{P}'.\tag{19}$$

By A ∗ we denote all pairs (*μ*, *h*) which satisfy the above properties. Then, the compensator of the jump measure *μ*<sup>Φ</sup> under *Pμ*,*<sup>h</sup>* is

$$
\mu\_l h(t, \mathbf{x}) \nu^{\Phi}(dt, d\mathbf{x}) = \mu\_l h(t, \mathbf{x}) \mathbb{1}\_{\{\mathbf{x} \in [\underline{x} \overline{\mathcal{I}}]\}} (\overline{\tau} - \underline{\mathbf{r}})^{-1} d\mathbf{x} dt =: K\_t^{\mu, h}(d\mathbf{x}) dt,\tag{20}
$$

see T10 in Section VIII.3 of Brémaud (1981). Next, we compute the compensator of *R*. We obtain from (16) that

$$R\_t - \int\_0^t \int R\_{s-} (\mathbf{x} - \mathbf{1}) K\_s^{\mu, \mathbf{h}} (d\mathbf{x}) ds, \qquad 0 \le t \le T^\* \tag{21}$$

is a P*μ*,*h*-martingale. For a G-progressive process *g*, we denote

$$M^{\mathfrak{g}} = \mathcal{R} + \int\_0^\cdot \mathcal{R}\_{s-} \mathcal{g}\_s ds.$$

**Proposition 2.** *Assume that Assumption 1 holds and let g be a positive and* G*-predictable process. Consider a measure* Q *on* (Ω, F )*, such that M<sup>g</sup> is a* Q*-martingale, W is a* Q*-Brownian motion, and* Q( *T*<sup>∗</sup> <sup>0</sup> |*rs*|*ds* < ∞) = 1*. Then* Q *is a local martingale measure if and only if*


$$\vec{a}(t,T) = \frac{1}{2} \left\| \vec{b}(t,T) \right\|^2, \quad 0 \le t \le T \le T^\*.$$

*holds dt* ⊗ *<sup>d</sup>*Q*-almost surely.*

**Proof.** We generalize the proof of Proposition 1 to the case of RMV. To this end, let *F*(*t*, *T*) = exp <sup>−</sup> *<sup>T</sup> <sup>t</sup> <sup>f</sup>*(*t*, *<sup>u</sup>*)*du* . Then (17) reads *PR*(*t*, *T*) = *R*(*t*)*F*(*t*, *T*). Integrating by part yields

$$dP\_{\mathbb{R}}(t, T) = F(t - \prime, T)dR(t) + \mathcal{R}(t - \prime)dF(t, T) + d[\mathcal{R}, F(\cdot, T)]\_{t - \mathcal{R}}$$

Note that, by assumption,

$$M\_t^\mathcal{S} = R\_t + \int\_0^t R\_{s-} \mathcal{g}\_s ds\_s \qquad 0 \le t \le T^\*,$$

is a *Q*-martingale and that [*R*, *F*(., *T*)] = 0 since *R* is of finite variation and *F*(., *T*) is continuous. Hence, by Lemma 2,

$$\begin{split}dP\_{\mathcal{R}}(t,\mathcal{T}) &= P\_{\mathcal{R}}(t-\mathcal{T},\mathcal{T}) \left( -\mathcal{g}\_{t} + f(t,\mathfrak{t}) + \frac{1}{2} \left\| \overline{\mathfrak{b}}(t,\mathcal{T}) \right\|^{2} - \overline{\mathfrak{a}}(t,\mathcal{T}) \right) dt \\ &+ P\_{\mathcal{R}}(t-\mathcal{T}) \left( r\_{t} dt + dM^{\mathbb{S}} - \overline{\mathfrak{b}}(t,\mathcal{T}) dW\_{t} \right), \end{split} \tag{22}$$

and we obtain the result as in the proof of Proposition 1.

**Example 2.** *A classical example is when the defaults* (*Tn*) *arrive at rate λ* > 0*, and the recovery values* (*Rn*) *are i.i.d. Then,* (*<sup>x</sup>* <sup>−</sup> <sup>1</sup>)*ν*Φ(*dt*, *dx*) = *<sup>λ</sup>E*[*R*<sup>1</sup> <sup>−</sup> <sup>1</sup>]*dt. We obtain that the instantaneous forward rate of the defaultable bond f*(*t*, *t*) *equals rt* + *λE*[1 − *R*1]*. In the case of zero recovery, we recover f*(*t*, *t*) = *rt* + *λ, and, in the case of full recovery (the case without default risk), f*(*t*, *t*) = *rt.*

#### *6.2. Fractional recovery with ambiguity*

We introduce ambiguity in this setting by changing from the standardized measure P to various appropriate measures via the Girsanov theorem. We also generalize the setting for ambiguity from the quite specific P to a general set of probability measures P∗ here, see Remark 3. The reason for this is also economic: while bounding the intensity from above and below seems to be quite plausible, an upper/lower bound on the recovery (i.e., on (*Rn*)) sounds too strong for some applications.

Recall that A ∗ was the set of all candidates (*μ*, *h*) which induce the measure changes via (19). Ambiguity is introduced by the set P∗ of probability measures satisfying

$$
\mathcal{Q} \neq \mathcal{P}^\* \subset \{ \mathbb{P}^{\mu, h} : (\mu, h) \in \mathcal{A}^\* \}. \tag{23}
$$

If P∗ contains only one probability measure, we are in the classical setting, otherwise there is ambiguity in the market. Measure changes from P*μ*,*<sup>h</sup>* to a new measure are done via the density *Lμ*∗,*h*<sup>∗</sup> (see (18)) where, as above, *μ*∗, *h*∗(., *x*), *x* ∈ [*r*,*r*] are positive and progressive. Recall the definition of the density *z<sup>θ</sup>* in (11).

**Theorem 3.** *Let g*∗ *<sup>t</sup>* := *f*(*t*, *t*) − *rt*, *t* ∈ [0, *T*∗]*, and assume that Assumption 1 holds. Then there exists an ELMM for* <sup>P</sup>*μ*,*<sup>h</sup>* <sup>∈</sup> <sup>P</sup><sup>∗</sup> *if*


$$\mathbf{g}\_t^\* = \int (\mathbf{x} - \mathbf{1}) \mu\_t^\* h^\*(t, \mathbf{x}) K\_t^{\mu, \mathbf{h}}(d\mathbf{x}), \quad 0 \le t \le T^\* \mu$$

*dt* ⊗ *dP -almost surely, and*

*(iii) the drift condition*

$$d(t, T) = \frac{1}{2} \parallel \bar{b}(t, T) \parallel^2 - \bar{b}(t, T)\theta\_t^\*, \quad 0 \le t \le T \le T^\*.$$

*holds dt* ⊗ *dP -almost surely.*

Absence of arbitrage in this general ambiguity setting can now be classified, thanks to Theorem 1 as follows: if an ELMM exists for each <sup>P</sup>*μ*,*<sup>h</sup>* <sup>∈</sup> <sup>P</sup>∗, then the market is free of arbitrage in the sense of Definition 1.

**Proof.** Fix <sup>P</sup>*μ*,*<sup>h</sup>* <sup>∈</sup> <sup>P</sup>∗. We can define an equivalent measure <sup>P</sup><sup>∗</sup> <sup>∼</sup> <sup>P</sup>*μ*,*<sup>h</sup>* by

$$d\mathbb{P}^\* := L^{\mu^\* \mathcal{J} \mathfrak{r}^\*} z\_{T^\*}^{\theta^\*} d\mathbb{P}^{\mu, \mathfrak{h}}.$$

with *μ*∗ and *h*∗ as in (ii). According to Theorem 3.40 in Chapter III of Jacod and Shiryaev (2003), this refers to a Girsanov-type (and equivalent) change of measure. Moreover, *<sup>W</sup>*<sup>∗</sup> <sup>=</sup> *<sup>W</sup>* <sup>−</sup> · <sup>0</sup> *θ*<sup>∗</sup> *<sup>s</sup> ds* is a P∗-Brownian motion. Next, note that the compensator of the jump measure *μ*<sup>Φ</sup> under P<sup>∗</sup> computes, according to T10 in Section VIII.3 in Brémaud (1981), to

$$\nu^\*(dt, dx) := \mu\_t^\* h^\*(t, \mathfrak{x}) \mathcal{K}\_t^{\mu\_J}(dx) dt$$

with *Kμ*,*<sup>h</sup> <sup>t</sup>* from Equation (20). This implies that

$$M\_t^{\mathbb{g^\*}} = R\_{\mathbb{s}-}(x-1)\nu^\*(ds, dx) = R\_l - \int\_0^t R\_{\mathbb{s}-} \mathbb{g}\_s^\* ds \tag{24}$$

is a P∗-martingale.

Now, we show that P<sup>∗</sup> is indeed a martingale measure: from (22) we obtain that

$$\frac{dP\_R(t,T)}{P\_R(t-,T)} = \left(f(t,t) + \frac{1}{2}\left\|\overline{b}(t,T)\right\|^2 - \overline{a}(t,T)\right)dt + dR\_t - \overline{b}(t,T)dW\_t.$$

It follows that

$$\begin{split} \frac{dP\_{\mathbb{R}}(t,T)}{dP\_{\mathbb{R}}(t-\mathsf{,}\prime T)} &= \left( f(t,t) + \frac{1}{2} \left\| \overline{b}(t,\prime T) \right\|^{2} - \overline{a}(t,\prime T) - \overline{b}(t,\prime T) \theta\_{\mathrm{t}}^{\*} \right) dt + dM\_{\mathrm{t}}^{\mathbb{R}} - \overline{b}(t,\prime T) dW\_{\mathrm{t}}^{\*} \\ &= r\_{\mathrm{t}} dt + dM\_{\mathrm{t}}^{\mathbb{R}} - \overline{b}(t,\prime T) d\mathcal{W}\_{\mathrm{t}}^{\*}, \end{split}$$

by the definition of *g*<sup>∗</sup> and the drift condition (iii). Hence, discounted bond prices are P∗-local martingales and the proof is finished.

**Remark 5.** *We can view zero recovery in the above setting by assuming that* P (*R*<sup>1</sup> = 0) = 1 *and letting τ* = *T*1*. Note that this case is excluded in RMV setting, since, under this assumption, at the first default all prices drop to zero and further defaults can not occur.*

#### **7. Conclusions**

This paper provides a first step towards including ambiguity in intensity based models for credit risk. Many research questions are still open: first, the extension of constant boundaries *λ*, *λ* to time-dependent, or, as in Fadina et al. (2019), state-dependent boundaries. Second, the extension to two or more defaultable assets, where default dependence comes into play. Third, estimation and determination of the bounds by statistical methods or calibration to market data. We hope that our paper provides the foundation for future works in these directions.

**Author Contributions:** All authors contributed equally to the paper. The author appearance is in alphabetical order.

**Funding:** Financial support by Carl-Zeiss-Stiftung, and German Research Foundation (DFG) via CRC 1283 is gratefully acknowledged. We also thank the Freiburg Institute of Advanced Studies (FRIAS) for its hospitality and financial support.

**Conflicts of Interest:** The authors declare no conflict of interest.

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