A new family of modified Gaussian copulas for market consistent valuation of government guarantees

In this section, we deal with the evaluation of the theoretical up-front value V(t) of a defaultable guarantee contract as in (1), using the standard Gaussian copula model first introduced by Li (2001) to financial problems. Without loss of generality, we set \(t=0\) hereafter and denote \(V(0)=V\).

To assess V, we need the joint distribution of the default times of the parties (\(\tau _1,\tau _2\)) involved in the contract that, according to Sklar’s theorem can be obtained by combining suitable marginal distributions with an appropriate copula function (see, e.g., Meucci 2011, 2011b).

In this analysis we assume constant default intensities for the parties (\(\lambda _1,\lambda _2\)). Therefore, the default time \(\tau _i\) for the \(i-th\) party follows a negative exponential marginal distribution. Specifically, the marginal cumulative distribution of the default time \(\tau _i\) for the \(i-th\) party at time \({\tilde{t}} >0\), namely \(F_{i}({\tilde{t}})\), is given by (see, e.g., Lando 1998)On the other hand, to describe the dependence structure between default times of the parties (\(\tau _1,\tau _2\)), we use the Gaussian copula. Note that the Gaussian copula has become a market reference for credit risk modelling due to its simplicity and tractability. Indeed, the use of the Gaussian copula is particularly appreciated among practitioners since this copula only needs the linear correlation parameter \(\rho\) as input. Then, according to Sklar’s theorem, the joint distribution of the default times \((\tau _{1}\),\(\tau _{2})\in {\mathbb {R}}_{+}^{2}\) is obtained bywhere \((u_{1},u_{2})\in \left[ 0,1\right] ^{2}\) are two random vectors generated from a Gaussian copula, and \(F_1^{-1}\), \(F_2^{-1}\) are the inverse functions of the marginal default times distributions of the parties as in (2). Thus, we can writeIn the following sections, we report numerical results on the theoretical up-front value V as in (1), using the Gaussian model. As we shall see, on the one hand, we can evaluate the up-front value V of a defaultable guarantee contract as a function of the correlation parameter \(\rho\), which can be easily viewed as the market correlation between default intensities. On the other hand, for \(\lambda _1 < \lambda _2\), which represents a typical situation, the Gaussian model determines up-front values V that are not monotone with respect to the correlation parameter \(\rho\), and, therefore, it does not satisfy the so-called conditions of consistency described in the Introduction.

We provide here numerical results on the theoretical up-front value V, as in equation (1), of a defaultable guarantee with maturity \(T=3\) years, guaranteed liability \(N=100\) euros, and constant recovery rate \(R=0.4\). We assume that the dependence structure is described by a Gaussian copula with correlation parameter \(\rho\), while the marginal distributions of the default times of the parties are represented by exponential functions with intensities \(\lambda _1\) (guarantor), \(\lambda _2\) (borrower). We evaluate the up-front value V for different levels of the parties’ default intensities (\(\lambda _{1}\), \(\lambda _{2}\)) and for various levels of the copula correlation parameter \(\rho\) by using \(5 \times 10^{5}\) Monte Carlo simulations. Furthermore, we assume a constant spot rate \(r=0.01\). The reference levels of the borrower and of the guarantor intensities have been fixed considering the sovereign and bank CDS market prices as follows: \(\lambda _{1}=\) 0.005, 0.01, 0.02, 0.04, 0.06, 0.2; \(\lambda _{2}=\) 0.005, 0.01, 0.02, 0.04, 0.06, 0.2. As far as financial market evidence is concerned, these levels correspond to single-name CDS sovereign and bank implied default probabilities observed in the market since 2005 (see Bassan and Mottura 2015). Indeed, empirical evidence shows that credit default swaps spreads can be used as an explanatory variable to assess a sovereign’s creditworthiness (Janus et al. 2013). Even in the banking sector, CDS spreads are significantly related to the credit risk of bank corporations (Kanagaretnam et al. 2016).

In Table 1 and Fig. 1, we report and show the computational results obtained from this empirical setup. As already pointed out, we expect that for a fixed borrower’s creditworthiness \(\lambda _2\) and correlation level \(\rho\), the value V of the guarantee decreases with the weakness of the guarantor \(\lambda _1\). Conversely, for a fixed guarantor’s creditworthiness \(\lambda _1\) and correlation level \(\rho\), V increases with the weakness of the borrower \(\lambda _2\).

Looking at the computational results, we note that when the correlation parameter \(\rho\) is equal to 0, the up-front value V depends, by definition, only on the marginal risk features of the parties. Accordingly, the values V decrease with the default risk of the guarantor \(\lambda _1\) when fixing the risk of the borrower \(\lambda _2\), and clearly increase in the opposite case. For intermediate levels of correlation, when the default risk of the guarantor is higher than that of the borrower (i.e., \(\lambda _1>\lambda _2\)), the value V always decreases when the correlation parameter \(\rho\) increases. On the contrary, if the default risk of the guarantor is lower than that of the borrower (i.e., \(\lambda _1<\lambda _2\)), the value V is not monotonically decreasing w.r.t. the copula’s correlation parameter \(\rho\). Indeed, V is first decreasing and then, around some correlation levels, starts increasing. This contradictory behaviour can be seen both in Fig. 1 (on the left) and in Table 1. More precisely, in Table 1 we mark in italics the ”non-acceptable” cases where the up-front value V of the guarantee increases when the parameter \(\rho\) increases. We call “maximum acceptable correlation” \({\bar{\rho }}_{G}\), the maximum level of the copula correlation parameter for which the Gaussian copula (G) model gives realistic results. In other words, the Gaussian model works appropriately for \(\rho \le {\bar{\rho }}_{G}\). Furthermore, when the correlation parameter \(\rho\) is equal to 1, if the default risk of the guarantor is higher than that of the borrower, then V approaches 0. Conversely, if the borrower has a default intensity higher than that of the guarantor (that is the typical situation), then the guarantee’s value V reaches its maximum value. This result is exactly the opposite of what one should expect.

Considering the combination between the Gaussian copula and the marginal default time distributions (\(F_1,F_2\)) with intensities (\(\lambda _1,\lambda _2\)) as in (2) gives rise to random default times (\(\tau _1,\tau _2\)) that can be chronologically distant when the copula’s correlation parameter \(\rho\) is equal to 1 (Morini 2011). This phenomenon appears in contradiction with the financial meaning of perfect positive correlation. Indeed, for \(\rho =1\), the parties should have the same behaviour in terms of default.

This aspect can be shown by using equation (4). We observe that when the copula’s correlation parameter \(\rho\) is high (\(\rho \rightarrow {1}\)) the realisations of (\(u_1,u_2\)) tend to be equal, i.e., \(u_1=u_2=u\). Thus, for \(\rho =1\), we havehence, the following relation between the default times (\(\tau _1,\tau _2\)) holdsAs a consequence of (6), the party that defaults first is always the one with the higher default intensity. This implies that if the default intensity of the guarantor is higher than that of the borrower (i.e., \(\lambda _1>\lambda _2\)), for \(\rho =1\) the guarantee’s value V approaches to 0, and V is always decreasing with respect to the correlation parameter \(\rho\). Conversely, if the borrower is the party with the higher default intensity (i.e., \(\lambda _1<\lambda _2\)), when \(\rho =1\), then V reaches its maximum value, hence failing the conditions of consistency.

In Fig. 2 we show the surface of the “maximum acceptable correlation” \({\bar{\rho }}_{G}\) for the Gaussian model, as a function of different values of the default intensities of the guarantor \(\lambda _1\) and of the borrower \(\lambda _2\). We observe that the yellow region, where \(\lambda _1>\lambda _2\), represents values of \({\bar{\rho }}_{G}\) equal to 1. This means that when the guarantor’s default intensity is higher than that of the borrower, the Gaussian model works appropriately for all \(\rho \in [0,1]\). In this case, the up-front value of the guarantee V is indeed monotonically decreasing with respect to the correlation parameter \(\rho\) and approaches 0 when \(\rho =1\) (see Fig. 1 on the right). This result is in line with the conditions of consistency. Differently, the darker regions of Fig. 2 identify values of \({\bar{\rho }}_{G}\) less than 1. In particular, the dark blue area represents values of \({\bar{\rho }}_{G}\) that are close to 0. Therefore, in the typical situation where the guarantor is less risky than the borrower (i.e., \(\lambda _1<\lambda _2\)), the Gaussian model is not able to give realistic results in terms of V. In this case, the up-front value of the guarantee V is not monotonically decreasing on the copula’s correlation parameter \(\rho\), as we should expect, but for \(\rho \ge {\bar{\rho }}_{G}\) increases (see Fig. 1 on the left).

We suppose thatwhere A and B are appropriate constants so that \(g(0,a,c)=0\) and \(g(1,a,c)=1\). Employing some algebraic manipulations, it is straightforward to show thatThe constant c represents the value of \(\eta _{\lambda }\) of the sigmoid’s midpoint, while a determines the steepness of the curve. To support intuition in Fig. 3, we show the graph of the adjusted default intensity \({\hat{\lambda }}_{1}(\eta _{\lambda },a,c)\) as a function of the “adjustment parameter” \(\eta _{\lambda }\) for several fixed values of the constants a and c.

It is important to notice that the selection of the sigmoid curve in (11) is not the only possible one. However, this choice fits well with our problem in that the sigmoid function is one of the most versatile and general representations of the adjustment term. Indeed, such a function ranges in a bounded interval, hence including also the situation of no adjustment as a corner case; moreover, it depends on three parameters, whose values allow a large variety of shapes for such a curve.

In the next sections, we will provide some numerical experiments on the Modified Gaussian model. To this aim, we will need to suitably calibrate the parameters of g in (11). We now explain the rationale behind the calibration exercise.

We notice that the shape of the curve is remarkably dependent on the selection of parameters a and c. Reasonably, the selected parameters go in the direction of having a strong similarity between the outcomes of the Gaussian model and those of the Modified Gaussian one when the Gaussian model satisfies the conditions of consistency. Under this guide, we take the steepness parameter \(a=1\) and the sigmoid’s mid-point parameter c equal to the “maximum acceptable correlation” \({\bar{\rho }}_{G}\).

For what concerns the calibration of \(\eta _\lambda\), we first detect the feasible variation range of such a parameter which allows the fulfilment of the conditions of consistency. Then, two possible solutions could be adopted. The first one is based on expert judgement, considering that the value of \(\eta _{\lambda }\) is a strategic (exogenous) variable in the hand of the “valuation” agent. In the European insurance sector, this is a way admitted and widely used in the Solvency II framework, in particular, when adopting the internal model approach (see European Parliament 2009). In the second one, as proposed below, it is \(\eta _{\lambda }=\rho\). In this case, the value of the parameter \(\rho\) can be calibrated using market data. More precisely, once the model is given, the values of \(\rho\) can be implicitly computed by cross-sectional analysis of quoted “multiname” CDS prices at the evaluation date (see Duffie 2004).

Finally, we notice that the three conditions of consistency define constraints on the domain of admissibility of the function’s parameters, significantly limiting the “discretion” of the expert’s judgment; even more so in the case \(\eta _{\lambda }=\rho\) in which, as already observed, the degrees of freedom of judgment are further reduced.

We provide here numerical results of the theoretical up-front value V of a defaultable guarantee as a function of \(\rho\) and \(\eta _{\lambda }\) assuming the default times \((\tau _{1}\),\(\tau _{2})\) as in (10). More precisely, we set the maturity \(T=3\) years, the guaranteed liability \(N=100\) euros, and the recovery rate \(R=0.4\). The values V are computed for different levels of the market default intensities of the parties \(\lambda _{1}\) and \(\lambda _{2}\) by using \(5 \times 10^{5}\) Monte Carlo simulations. Furthermore, we assume (i) a constant spot rate \(r = 0.01\), and (ii) for the sigmoid function g the steepness parameter a equal to one and the sigmoid’s mid-point parameter c equal to the “maximum acceptable correlation” \({\bar{\rho }}_{G}\), in order to obtain values of the MG model as close as possible to those of the Gaussian model (i.e., for calibration purpose).

In Figs. 4 and 5 we show the defaultable guarantee contract value surfaces V obtained from the MG model, given the values of the default intensities, by varying the dependence parameter \(\rho\) and the adjustment parameter \(\eta _{\lambda }\). The values of \(\lambda _1\) and \(\lambda _2\) are fixed as follows: \(\lambda _1=0.06\) and \(\lambda _2=0.2\) (Fig. 4); \(\lambda _1=0.15\) and \(\lambda _2=0.2\) (Fig. 5).

 Note that when the adjustment parameter \(\eta _{\lambda }\) is equal to 0, the up-front values V of the guarantee coincide with those obtained from the standard Gaussian model as a function of \(\rho\). The up-front values V are decreasing as the parameter \(\rho\) increases only for a range of feasible values of the adjustment parameter \(\eta _{\lambda }\). Furthermore, the range of feasible values of \(\eta _{\lambda }\) decreases when \(\rho\) increases. For instance, when \(\lambda _1=0.06\) and \(\lambda _2=0.2\): if \(\rho =0.90\) then the range of feasible values of \(\eta _{\lambda }\) is (0.70, 1); if \(\rho =0.95\) then the range of feasible values of \(\eta _{\lambda }\) is (0.84, 1). We stress here that: (i) in general, the value of \(\eta _{\lambda }\) to be used for the specific valuation is a strategic (exogenous) variable in the hand of the calculation agent; (ii) to be consistent, the value of \(\eta _{\lambda }\) has to belong to the \(\eta _{\lambda }\) feasible range. In other words, by means of \(\eta _{\lambda }\), the conditions of consistency reduce the range of possible values of the guarantee.

In this section, given the parameter \(\rho\) and within the \(\eta _{\lambda }\) feasible range values, we focus on the particular case when \(\eta _{\lambda } = \rho\) that in the contour plots at the bottom of Figs. 4 and 5 corresponds to the bisectors. Note that it is a case where \(\eta _{\lambda }\) can be interpreted as an “observable” variable. Using the same experimental setup of Sects. 3.2 and 4.2, we report here numerical results for the following reference levels of the borrower and the guarantor default intensities: (i) for \(\lambda _{2}=0.2\), \(\lambda _{1}=0.005\), 0.01, 0.03, 0.06, 0.1, 0.15 (Fig. 6); (ii) for \(\lambda _{1}=0.01\), \(\lambda _{2}=0.03\), 0.05, 0.10,0.13, 0.15, 0.2 (Fig. 7).

Figs. 6 and 7 show the computational results obtained for the MG model (on the right) and the standard Gaussian model (on the left).

We notice that (in line with the conditions of consistency (1) and (3) described in the Introduction) when the correlation parameter \(\rho\) is equal to 0 (i.e., \(\eta _{\lambda }=\rho =0\)) the up-front value V of the guarantee depends only on the marginal risk features of the parties (\(\lambda _1,\lambda _2\)). Thus, in this case, the values V obtained for the MG model are equal to those obtained for the Gaussian model. When the correlation parameter \(\rho\) is equal to 1 (i.e., \(\eta _{\lambda }=\rho =1\)) in the MG model the value V always approaches to 0. As requested, for intermediate levels of correlation, using the MG model, the value V of the guarantee always decreases when the correlation parameter \(\rho\) increases, differently from the Gaussian model when \(\lambda _1 < \lambda _2\).

Similar numerical results are reported in Table 2. In line with condition 2 of the conditions of consistency, we observe that, given \(\rho\) and \(\lambda _2\), the values of the guarantee are decreasing with the intensity of default of the guarantor \(\lambda _1\), while they are increasing with \(\lambda _2\).

For the regulator, it might be useful to identify the “threshold value” of \(\rho\), say \(\rho ^*\), such that the value of the guarantee issued by the guarantor becomes ineffective for \(\rho >\rho ^*\), i.e., the value of the guarantee collapses to 0 (see Figs. 6 and 7). The proposed model and the parameter selection might lead to consider the threshold level \(\rho ^*\) as a contractual quantity in the agreement between guarantor and debtor.