In this section we first lay the foundations in brief for pricing of risky debt in a general reduced form setting. Then we add the local currency debt into the picture and discuss the risky spreads. We conclude by derivation and analysis of the no-arbitrage conditions.
2.1 Risky Bonds Under Marked Point Process
The first task is to model default in a suitable way. We start with the most general formulation and then modify it appropriately. We consider a filtered probability space \((\varOmega ,\left( G_{t}\right) _{t\ge 0},P)\) which supports an n-dimensional Brownian motion \( W^{P}=(W_{1},W_{2},\ldots ,W_{n})\) under the objective probability measure P and a marked point process \(\mu :\left( \varOmega ,B(R^{+}),\varepsilon \right) \rightarrow R^{+}\) with markers \(\left( \tau _{i}\text {,}\;X_{i}\right) \) representing the jump times and their sizes in a measurable space \(\left( E,\varepsilon \right) \), where \(E=\left[ 0,1\right] \) and by \(\varepsilon \) we denote the Borel subsets of E. We assume that \(\mu (\omega ;dt,dx)\) has a separable compensator of the form:
$$\begin{aligned} \upsilon :\left( \varOmega ,B(R^{+}),\varepsilon \right) \rightarrow R^{+}\quad \text { and }\quad \upsilon \left( \omega ;dt,dx\right) =h(\omega ;t)F_{t}(\omega ;dx)dt \text {,} \end{aligned}$$
where \(h(\omega ;t)=\int _{R^{+}}\upsilon \left( \omega ;t,dx\right) \) is a \( G_{t\text { }}\)measurable intensity and the marks have a conditional distribution of the jumps of \(F_{t}(\omega ;dx).\) Thus, we have the identity \(\int _{E}F_{t}\left( \omega ;dx\right) =1\). Furthermore, we can define the total loss function \(L(\omega ;t)=\int _{0}^{t}\int _{E}l(\omega ;s,x)\mu (\omega ;ds,dx)\) and the recovery \(R(\omega ;t)=1-\int _{0}^{t}\int _{E}l(\omega ;s,x)\mu (\omega ;ds,dx)\). The function \( l(\omega ;t,x)\) scales the marks in a suitable way, and having control over it, we can define it such that our model is tractable enough. We define also the sum of the jumps by \(S(\omega ;t)=\int _{0}^{t}\int _{E}x\mu (\omega ;ds,dx)\) and their number by \(N(\omega ;t)=\int _{0}^{t}\int _{E}\mu (\omega ;ds,dx).\)
Effectively, the marked point process as a sequence of random jumps \(\left( \tau _{i}\text {,}X_{i}\right) \) is characterized by the probability measure \( \mu (\omega ;dt,dx),\) which gives the number of jumps with size dx in a small time interval of dt. The compensator \(\upsilon \left( \omega ;t,dx\right) \) provides a full probability characterization of the process. It incorporates in itself two effects. On one hand, we have the intensity \({}\), which gives the conditional probability of jump of the process in a small time interval of dt incorporating the whole market information up to t. On the other hand, we have the conditional distribution \(F_{t}(\omega ;dx)\) of the markers X in case of a jump realization.
We can look at the jumps of the marked point process as sequential defaults of an obligor at random times \(\tau _{i}\) that lead to losses \(X_{i}\) at each of them. They can also be considered a set of restructuring events leading to losses for the creditors. Under this general setting, the prices of the riskless and risky bonds are given by:
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Riskless bond:
$$\begin{aligned} P(t,T)=E^{Q}\left( \exp \left( -\int _{t}^{T}r(s)ds\right) |G_{t}\right) =\exp \left( -\int _{t}^{T}f(t,s)ds\right) \end{aligned}$$
(1)
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Risky bond:
$$\begin{aligned} P^{*}(t,T)= & {} E^{Q}\left( \exp \left( -\int _{t}^{T}r(s)ds\right) R(\omega ;T)|G_{t}\right) \nonumber \\= & {} R(t)\exp \left( -\int _{t}^{T}f^{*}(t,s)ds\right) , \end{aligned}$$
(2)
where r(t), f(t, T), and \(f^{*}(t,T)\) are the riskless spot, riskless forward, and risky forward rates respectively.
Depending on how we specify the convention of recovery, we can get further simplification of the formulas. However, this should be well motivated and come either from the legal definitions of the debt contracts or their economic grounding.
Under a recovery of market value (RMV) setting, default is a percentage mark down, q, from the previous recovery. So we have \(R(\omega ;\tau _{i})=\left( 1-q(\omega ;\tau _{i},X_{i})\right) R(\omega ;\tau _{i}-)\) and \( l(\omega ;\tau _{i})\) has the form \(l(\omega ;\tau _{i})=\) \(-q(\omega ;\tau _{i},X_{i})\times R(\omega ;\tau _{i}-).\) This definition allows us to write:
$$\begin{aligned} \mu (\omega ,dt,dx)= & {} \mathop {\displaystyle \sum }\limits _{s>0}1_{\left\{ \varDelta N(\omega ,s)\ne 0\right\} }\delta _{\left( s,\varDelta N(\omega ,s)\right) }(dt,dx) \\ dR(\omega ;t)= & {} -R(\omega ;t)\int _{E}q\left( \omega ;t,x\right) \mu (\omega ;dt,dx);R(\omega ;0)=1 \end{aligned}$$
and if we assume no jumps of the intensity and the risk-free rate at default times (contagion effects), we have no change for the risk-free bond pricing formula and for the risky one and as in [13] we get:
$$\begin{aligned} P^{*RMV}(t,T)= & {} E^{Q}\left( \exp \left( -\int _{t}^{T}r(s)ds\right) R(\omega ;T)|G_{t}\right) \nonumber \\= & {} R(t)E^{Q}\left( \exp \left( -\int _{t}^{T}(r(s)+h(s)\int _{E}q\left( \omega ;s,x\right) F_{s}(dx))ds\right) |G_{t}\right) \nonumber \\= & {} R(t)\exp \left( -\int _{t}^{T}f^{*RMV}(t,s)ds\right) \end{aligned}$$
(3)
Note that within this setting there is no “last default”. The intensity is defined for the whole marked point process and not just for a concrete single default time, so it does not go to zero after default realizations. This combined with the fact that intensity is continuous makes the market filtration \(G_{t}\) behave like a background filtration in the pricing formulas. So we can avoid using the generalized Duffie, Schroder, and Skiadas [7] formula. Furthermore, we can denote \(q_{e}(t)=\int _{E}q\left( \omega ;t,x\right) F_{t}(dx)\) to be the expected loss. So we have that the pricing formula is dependent on the generalized intensity \(h(t)q_{e}(t).\) Due to the multiplicative nature of the last expression, only from market information, as discussed in Schönbucher (2003), we cannot distinguish between the pure intensity effect h(t) and the recovery induced one \( q_{e}(t)\).
Under a recovery of par (RP) setting, in case of default, the recovery is a separate fixed or random quantity independent of the default indicator and the risk-free rate. So we have \(E=\{0,1-R\left( \omega \right) \}\) and \( \upsilon \left( \omega ;dt,dx\right) =h(\omega ;t)(1-R_{e})dt\) with \( R_{e}=E^{Q}(R\left( \omega \right) \mid G_{t}).\) Since we have just one jump, we can write:
$$\mu (\omega ,dt,dx)=1_{\left\{ \varDelta N(\omega ,t)\ne 0\right\} }\delta _{\left( t,\varDelta N(\omega ,t)\right) }(dt,dx)$$
The bond price is:
$$\begin{aligned} \ P^{*RP}(t,T)= & {} E^{Q}\left( \exp \left( -\int _{t}^{T}r(s)ds\right) \left( R\left( \omega \right) 1_{\{\tau \le T\}}+1_{\{\tau>T\}}\right) |G_{t}\right) \nonumber \\= & {} 1_{\{\tau >t\}}\exp \left( -\int _{t}^{T}f^{*RP}(t,s)ds\right) \end{aligned}$$
(4)
In contrast to RMV, here, as discussed in Schönbucher (2003), he can distinguish between the pure intensity and recovery induced effects.
2.2 Model Formulation
In this section we develop our HJM model for pricing of local and foreign currency bonds of a risky country. However, before this being done formally, it is essential to elaborate on the nature of the problem. Although we do not put here explicitly macrofinancial structure, but just proxy it by jumps and correlations, it, by all means, stays in the background and must be conceptually considered.
2.2.1 General Notes
A risky emerging market country can have bonds denominated both in local and foreign currency that give rise to two risky yield curves and risky spreads—credit and currency. Generally, the latter arise due to the possibility of the respective credit events to occur and their severity. To investigate them, formal assumptions are needed both on their characteristics and interdependence.
We will consider that the two types of debt have different priorities. The country is first engaged to meeting the foreign debt obligation from its limited international reserves. The impossibility of this being done leads to default or restructuring. In both cases, we have a credit event according to the ISDA classification. The foreign debt has a senior status. The spread that arises reflects the credit risk of the country. It is a function of: (1) the probability of the credit event to occur; (2) the expected loss given default; (3) the risk aversion of the market participants to the credit event.
The domestic debt economically stands differently. It reflects the priority of the payments in hard currency and it incurs instantly the losses in case of default of the country. So this debt is the first to be affected by a default and is subordinated. Technically, the credit event can be avoided under a flexible exchange rate regime because the country can always make a debt monetization and pay the amounts due in local currency taking advantage of the fact that there is no resource constraint on it. However, the price for this is inflation pick-up and exchange rate depreciation. This leads to real devaluation of the domestic debt. It is exactly the seigniorage and the dilution effect that cause the loss in the value.Footnote 2 This resembles the case of a firm issuing more equity to avoid default. The spread of the domestic debt over the foreign one forms the currency spread. Its nature is very broad, and it is not only due to the currency mismatch. Namely, it is a function of: (1) the probability of the credit event to occur and the need for monetization; (2) the negative side effect of the credit event on the exchange rate by a sudden depreciation of the latter; (3) the volatility of the exchange rate; (4) the expected depreciation of the exchange rate without taking into consideration the monetization; (5) the risk aversion of market participants to the credit event and the need for monetization, the sudden exchange rate depreciation and its size; (6) the risk aversion of the market participants to the volatility of the exchange rate. All these effects are captured by our model.
2.2.2 Multi-currency Risky Bonds Model
We use the setting of Sect. 2.1 modified to a multi-currency debt. Firstly, we consider the case of no monetization and then analyze the case with monetization. Secondly, to avoid using an additional marked point process, and thus a second intensity, the default on the foreign debt is modeled indirectly. Namely, we assume that default on domestic debt leads to default on foreign debt, but due to the different priority of the two, we have just different losses incurred, respectively recoveries. This means that by controlling recoveries we control default and the inherent subordination without imposing too much structure. If the default on the domestic debt is so strong that it leads to a default on the foreign debt as well, we incur zero recovery on the domestic debt and some positive one on the foreign debt. If the insolvency is mild, we have a loss only on the domestic debt, so we incur some positive recovery on the domestic debt and a full recovery on the foreign debt. Thirdly, for notational purposes, we take as a benchmark Germany and EUR as the base hard currency. Lastly, we employ the recovery of market value assumption. The reason for this is twofold. On one hand, in that way, we are consistent with the HJM methodology of Schönbucher [12] for a single risky curve under RMV and produce parsimonious no-arbitrage conditions for the extension to a multi-curve environment. On the other hand, as pointed out in Bonnaud et al. [5], for bonds denominated in a different currency than the numerator employed in discounting, the RMV assumption should be the working engine. Their argument is exactly as ours above, in case of default, the sovereign would rather dilute by depreciating the exchange rate and thus the remaining cash flows of the bond produce in essence the RMV structure. Moreover, rather than using EUR denominated bonds, we could take advantage of the CDS quotes and produce synthetic bonds having an RMV recovery structure. Using them is actually preferable for empirical work since major academic studies argue that it is the CDS market that first captures the market information about the credit risk stance of the risky sovereign. Furthermore, with a few exceptions, if the EM sovereigns have in most cases both well developed local currency treasury markets and are subject to CDS quotation, they do have only few Eurobonds outstanding. Figure 1 represents the typical situation the risky sovereign faces.
Mathematical formulation We continue with the model setup. Firstly, we give the suitable notation and assumptions. Then we move to the derivation of the no-arbitrage conditions and the pricing.
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Notation
\(f_{EUR}(t,T)\)—nominal forward rate, EUR, Ger.
\(f_{EUR}^{*}(t,T)\)—nominal forward rate, EUR, EM
\(f_{LC}^{*}(t,T)\)—nominal forward rate in LC, EM
\(r_{EUR}(t)\)—nominal short rate, EUR, Ger.
\(r_{EUR}^{*}(t)\)—nominal short rate, EUR, EM
\(r_{LC}^{*}(t)\)—nominal short rate in LC, EM
\(h_{EUR}^{*}(t,T)=f_{EUR}^{*}(t,T)-f_{EUR}(t,T)\)—credit spr., EM
\(h_{LC,EUR}^{*}(t,T)=f_{LC}^{*}(t,T)-f_{EUR}^{*}(t,T)\)—currency spr., EM
\(h_{LC}^{*}(t,T)=f_{LC}^{*}(t,T)-f_{EUR}(t,T)\)—general currency spr., EM
\(P_{EUR}(t,T)=\exp (-\int _{t}^{T}f_{EUR}(t,s)ds)\)—bond, EUR, Ger.
\(P_{f,EUR}^{*}(t,T)=R_{f,EUR}(t)\exp (-\int _{t}^{T}f_{EUR}^{*}(t,s)ds)\)—for. bond price., EUR, EM
\(P_{d,LC}^{*}(t,T)=R_{d,LC}(t)\exp (-\int _{t}^{T}f_{LC}^{*}(t,s)ds)\)—dom. bond price., LC, EM
\(B_{EUR}(t)=\exp (\int _{0}^{t}r_{EUR}(s)ds)\)—bank account, EUR, Ger.
\(B_{f,EUR}^{*}(t)=R_{f,EUR}(t)\exp (\int _{0}^{t}r_{EUR}^{*}(s)ds)\)—for. bank account, EUR, EM
\(B_{d,LC}^{*}(t)=R_{d,LC}(t)\exp (\int _{0}^{t}r_{LC}^{*}(s)ds)\)—dom. bank account, LC, EM
X(t)—exchange rate, EUR for 1 LC, \(\widetilde{X}(t)\)—exchange rate, LC for 1 EUR
\(R_{f,EUR}(t)\)—bond recovery, EUR, EM, \(R_{d,LC}(t)\)—bond recovery, LC, EM
We use the asterisk to denote risk, the first letter (d or f) to denote domestic or foreign debt, and finally the currency of denomination is shown as EUR or LC.Footnote 3
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Currency denominations
\(P_{d,EUR}^{*}(t,T)=X(t)P_{d,LC}^{*}(t,T)\)—dom. bond, EUR
\(P_{f,LC}^{*}(t,T)=\widetilde{X}(t)P_{f,EUR}^{*}(t,T)\)—for. bond, LC
\(B_{d,EUR}^{*}(t)=X(t)B_{d,LC}^{*}(t)\)—dom. bank account, EUR
\(B_{f,LC}^{*}(t)=\widetilde{X}(t)B_{f,EUR}^{*}(t)\)—for. bank account, LC
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Intensities
$$ \begin{array}{ll} \text {Foreign debt, EUR:} &{} \\ { \ \text {Intensity:}} &{} {h}_{EUR}{(t)=h(t)} \\ { \ \text {Compensator:}} &{} {h}_{EUR}{(t)q} _{e,EUR}{(t)=h(t)\int _{E}q}_{f,EUR}\left( \omega ;t,x\right) {F }_{t}{(dx)} \\ \text {Domestic debt, LC:} &{} \\ { \ \text {Intensity:} } &{} {h}_{LC}{(t)=h(t)} \\ \ { \text {Compensator:}} &{} {h}_{LC}{(t)q}_{e,LC}{\small (t)=h(t)\int _{E}q}_{d,LC}\left( \omega ;t,x\right) {F}_{t}{(dx) } \end{array} $$
The compensator (generalized intensity) characterizes default. Controlling in a suitable way the recovery, we can control the compensator and thus the default event. We turn attention now to the dynamics of the instruments under consideration.
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Forward rates
\(df_{EUR}(t,T)=\alpha _{EUR}(t,T)dt+\mathop {\textstyle \sum }\nolimits _{i=1}^{n}\sigma _{EUR,i}(t,T)dW_{i}^{P}(t)\)
\(df_{EUR}^{*}(t,T)=\alpha _{EUR}^{*}(t,T)dt+\mathop {\textstyle \sum }\nolimits _{i=1}^{n}\sigma _{EUR,i}^{*}(t,T)dW_{i}^{P}(t)\)
\(+{\int _{E}}\delta _{EUR}^{*}(x,t,T)\mu (dx,dt)\)
\(df_{LC}^{*}(t,T)=\alpha _{LC}^{*}(t,T)dt+\mathop {\textstyle \sum }\nolimits _{i=1}^{n}\sigma _{LC,i}^{*}(t,T)dW_{i}^{P}(t)\)
\(+{\int _{E}}\delta _{LC}^{*}(x,t,T)\mu (dx,dt) \)
We assume that in case of default there is a market turmoil leading to a jump in both curves. At maturity T, the EUR curve jumps by a size of \( {\int _{E}}\delta _{EUR}^{*}(x,t,T)\mu (dx,dt),\) and that of the local currency by \({\int _{E}}\delta _{LC}^{*}(x,t,T)\mu (dx,dt).\) The terms \(\delta _{EUR}^{*}(x,t,T)\) and \(\delta _{LC}^{*}(x,t,T)\) show the jump sizes of the respective curves for every maturity. As indicated at the beginning of the section, everywhere we will work under the market filtration \(G_{t}\) so both the Brownian motions and the point process are adapted to it.
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Recoveries
\(\frac{dR_{f,EUR}(t)}{R_{f,EUR}(t)}=-\int _{E}q_{f,EUR}(x,t)\mu (dx,dt)\)
\(\frac{dR_{d,LC}(t)}{R_{d,LC}(t)}=-\int _{E}q_{d,LC}(x,t)\mu (dx,dt) \)
After each default we have a devaluation of the respective bond by an expected value of \({\int _{E}}q_{f/d}(x,t)\mu (dx,dt)\). The stochasticity of the loss is captured by the random jump size q(., .) as elaborated in Sect. 2.1.
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Bank accounts
\(\frac{dB_{EUR}(t)}{B_{EUR}(t)}=r_{EUR}(t)dt\)
\(\frac{dB_{f,EUR}^{*}(t)}{B_{f,EUR}^{*}(t)}=r_{EUR}^{*}(t)dt-\int _{E}q_{f,EUR}(x,t)\mu (dx,dt)\)
\(\frac{dB_{d,LC}^{*}(t)}{B_{d,LC}^{*}(t)}=r_{LC}^{*}(t)dt -\int _{E}q_{d,LC}(x,t)\mu (dx,dt) \)
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Exchange rate
\(\frac{dX(t)}{X(t)}=\)
\(\alpha _{X}(t)dt+\mathop {\textstyle \sum }\nolimits _{i=1}^{n}\sigma _{X,i}(t)dW_{i}^{P}(t)-{\small \int _{E}}\delta _{X}(x,t)\mu (dx,dt) \)
We assume that in case of default the market turmoil causes an exchange rate devaluation by \({\int _{E}}\delta _{X}(x,t)\mu (dx,dt).\)
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Bond prices
\(P_{EUR}(t,T)=\exp (-\int _{t}^{T}f_{EUR}(t,s)ds)=E^{Q^{f}}(\exp (-\int _{t}^{T}r_{EUR}(s)ds)|G_{t})\)
\(P_{f,EUR}^{*}(t,T)=R_{f,EUR}(t)\exp (-\int _{t}^{T}f_{EUR}^{*}(t,s)ds)\)
\(=E^{Q^{f}}(\exp (-\int _{t}^{T}r_{EUR}(s)ds)R_{f,EUR}(T)|G_{t})\)
\(P_{d,EUR}^{*}(t,T)=P_{d,LC}^{*}(t,T)X(t)=R_{d,LC}(t)X(t)\exp (-\int _{t}^{T}f_{LC}^{*}(t,s)ds)\)
\(=E^{Q^{f}}(\exp (-\int _{t}^{T}r_{EUR}(s)ds)R_{d,LC}(T)X(T)|G_{t}) \)
It must be emphasized that the effects of exchange rate, recovery, and the expected devaluation sizes are incorporated in the respective forward rates of the bonds. Furthermore, the expectations are taken under \(Q^{f}\), the foreign risk-neutral measure.
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Arbitrage
Under standard regularity conditions, for the system to be free of arbitrage, all traded assets denominated in euro must have a rate of return \( r_{EUR}\) under \(Q^{f}\). This means that the processes:
$$\begin{aligned} \frac{P_{EUR}(t,T)}{B_{EUR}(t)},\frac{B_{f,EUR}^{*}(t)}{B_{EUR}(t)}, \frac{P_{f,EUR}^{*}(t,T)}{B_{EUR}(t)},\frac{B_{d,LC}^{*}(t)X(t)}{ B_{EUR}(t)},\frac{P_{d,LC}^{*}(t,T)X(t)}{B_{EUR}(t)} \end{aligned}$$
must be local martingales under \(Q^{f}\). For our purposes being martingales would be enough.
Taking the stochastic differentials of the upper expressions, omitting the technicalities to the appendix, we can get the respective no-arbitrage conditions.
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Spreads:
$$\begin{aligned} r_{EUR}^{*}(t)-r_{EUR}(t)=h(t)\varphi _{q_{f,EUR}}(t) \end{aligned}$$
(5.1)
$$\begin{aligned} \begin{array}{l} r_{LC}^{*}(t)-r_{EUR}^{*}(t)=-\alpha _{X}(t)-\phi (t)\sigma _{X}(t) \nonumber \\ +h(t)(\varphi _{\delta _{X}}(t)-\varphi _{q_{d,LC},\delta _{X}}(t)+\varphi _{q_{d,LC}}(t)-\varphi _{q_{f,EUR}}(t)) \end{array} \end{aligned}$$
(5.2)
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Drifts:
\(\alpha _{EUR}(t,T)=\sigma _{EUR}(t,T)\int _{t}^{T}\sigma _{EUR}(t,v)dv-\sigma _{EUR}(t,T)\phi (t)\)
\(\alpha _{EUR}^{*}(t,T)=\sigma _{EUR}^{*}(t,T)\int _{t}^{T}\sigma _{EUR}^{*}(t,v)dv-\sigma _{EUR}^{*}(t,T)\phi (t)\)
\(+h_{EUR}(t)\varphi _{\theta _{EUR}^{*}}^{q_{f,EUR},\delta _{X}}(t)\)
\(\alpha _{LC}^{*}(t,T)=\sigma _{LC}^{*}(t,T)\int _{t}^{T}\sigma _{LC}^{*}(t,v)dv-\sigma _{LC}^{*}(t,T)\phi (t)-\sigma _{LC}^{*}(t,T)\sigma _{X}(t,T)\)
\(+h_{LC}(t)\varphi _{\theta _{LC}^{*}}^{q_{d,LC},\delta _{X}}(t)\),
where we have used the notation:
\(\theta _{EUR}^{*}=\exp (-\int _{t}^{T}\delta _{EUR}^{*}(x,t,s)ds)\) , \(\theta _{LC}^{*}=\exp (-\int _{t}^{T}\delta _{LC}^{*}(x,t,s)ds)\)
\(\varphi _{a.b,\ldots }^{x,y,\ldots }(t)=\int _{E}(ab\ldots )((1-x)(1-y)\ldots )\varPhi (t,x)F_{t}(dx) \)
and used vector notation and scalar products where necessary for simplicity.
By \(\varPhi (t,x)\) and \(\phi (t)\) we denoted the Girsanov’s kernels of the counting process and the Brownian motion respectively when changing the probability measure from P to \(Q^{f}.\) The term \(\varphi (t)\) represents the scaled expected jump sizes of the counting process. We can give the interpretation that \(\phi (t)\) is the market price of diffusion risk and \( \varphi (t)\) is the market price of jump risk. Parametrizing the volatilities and the market prices of risk, as well as imposing suitable dynamics for h(t), we give a full characterization of our system. Furthermore, the intensity could be a function of the underlying processes of the rates, so we could get correlation between the intensity, the interest rates, and the exchange rate.
Spreads diagnostics from a reduced form point of view It is important to give a deeper interpretation of the no-arbitrage conditions and see which factors drive the credit and currency spreads. Despite the heavy notation, the analysis actually goes fluently. The drift equations give the modified HJM drift restrictions. The slight change from the classical riskless case is due to the jumps that arise. Equation (5.1) shows that the credit risk is proportional to the intensity of default and the scaled expected LGD by the coefficient controlling the risk aversion. The higher they are, the higher the spread is. Equation (5.2) gives the currency spread. It arises due to two main reasons. Firstly, the intensity of default and the difference between the two LGDs in local currency and euro, scaled by the coefficient for the risk aversion, act as in the previous case. They also make explicit the subordination. Secondly, the expected local currency depreciation, its volatility, and the risk aversion to diffusion risk act similarly to the standard uncovered interest parity (UIP) relationship. The higher they are, the higher the spread is. It is both important and interesting to note that inflation does not appear directly and it influences the spreads, as the next section shows, only through a secondary channel.
Monetization The analysis so far considered a loss of \(1-R_{d,LC}(T)\) on default of the domestic debt. However, if a full monetization is applied, then we would have \(R_{d,LC}(T)=1\) and thus \(\varphi _{q_{d,LC}}(t)=0\) and \(\varphi _{q_{d,LC},\delta _{X}}(t)=0\). If such a monetary injection is neutral to nominal values, it is certainly not to real ones. Devaluation arises due to the negative market sentiment following the default and the higher amount of money in circulation. Its effect can be measured differently based on what we take as a base—the price index or the exchange rate. Most naturally, we can expect both of them to depreciate due to the structural macrolinks that exist between these variables. For quantifying the amount we would need a macromodel which is beyond the scope of the reduced form model presented. The latter only shows what characteristics the market prices in general without imposing concrete macrolinks among them. Depending on what the base is, we would have a direct estimation of certain type of indicators and an indirect one of the rest up to their structural influence on the former. If the inflation is taken as a base, then we would have the comparison of inflation indexed bonds to the non-indexed ones. The spread between them would give an estimate for the expected inflation. Unfortunately, such an analysis is unrealistic due to the fact that such bonds are issued very rarely by emerging market countries. If the exchange rate is taken as a base, then we would have the comparison of domestic debt bonds to foreign debt bonds. The spread between them would give an estimate for the currency risk and the devaluation effect. The estimate for the inflation would be indirect and based on hypothetical structural links.
Whether the country would monetize or declare a formal default is based on strategic considerations. It is a matter of structural analysis which option it would take. By all means, its decision is priced. In case of default, the pricing formula is Eq. (5.2). In case of monetization, we would have a jump in the exchange rate. Let us denote its size by \(\widehat{\delta }_{X}\). It will be different from the no-monetization one, \(\delta _{X}\), due to the different regimes that are followed, and we would thus get:
$$\begin{aligned} r_{LC}^{*}(t)-r_{EUR}^{*}(t)=h(t)(\varphi _{\widehat{\delta } _{X}}(t)-\varphi _{q_{f,EUR}}(t))-\alpha _{X}(t)-\phi (t)\sigma _{X}(t) \end{aligned}$$
(6)
There is no a priori no-arbitrage argument that \(\varphi _{\widehat{\delta }_{X}}(t)=\varphi _{\delta _{X}}(t)-\varphi _{q_{d,LC},\delta _{X}}(t)+\varphi _{q_{d,LC}}(t)\) must hold so that the two scenarios are equivalent.Footnote 4 The only information we get from the market is an estimate for the generalized intensity being \(h(t)\varphi _{ \widehat{\delta }_{X}}(t)\) or \(h(t)(\varphi _{\delta _{X}}(t)-\varphi _{q_{d,LC},\delta _{X}}(t)+\varphi _{q_{d,LC}}(t))\) but not knowing which possible scenario will be realized.