Modelling sovereign debts

Hello all! I hope you're all doing well.

Along the recent history several countries defaulted their debt obligations. The impact of each of these defaults is enormous not only for the debtor itself but also for the entire financial net associated with this country. So, given this impact how one could model the corresponding decision process (default or not) in the fairest possible way? This post intends to say something on this question. 

The reader interested in deeper details and more reading materials, I recommend this paper and references therein published in 2018 in the Journal of Derivatives. Beyond the elucidating introduction, this paper evaluates the consequence of different actions taken by the creditor.

A simple put option, as a first approach

Consider a country with a debt obligation $D$ that should be payed at $T$. This debt should be taken from the financial output of the country denoted by $S_t$, therefore if the debt is payed the country wealth will be $S_T - D$. On the other hand, in case the country defaults a complex cost takes place, including trustfulness downgrade, political costs, investment evasion, among several others. The effect of all the complex default cost is a reduction in the wealth function of the country. For sake of simplicity, assume the impact is punctual in time, reducing the country wealth function, therefore $S_T(1-c)$, where $c$ encompass this complex default cost. 

The decision between pay the debt or default is then represented by the function: $W_T=max\{S_T - D, S_T(1-c)\}$, which can be rewritten $W_T=S_T - D+max\{0, D-c S_T\}$. Assuming a (continuously compounded) risk free $r$, the wealth function at a time earlier than $T$ can be given by:

$W_t=e^{-r(T-t)}\{E[S_t | F(t) ] - D + E[ D - c S_T | F(t)] \}$

where $E[...| F(t)]$ denotes the expectation value operator within a risk free measure, considering a filtration $F(t)$. The first term with the expectation value operator resembles the pricing model of a future contract. The second term is simply a put option. Some authors (like this and this) model $S_t$ as a geometric Brownian motion (GBM), which leads to a Black-like formula.

Improvements

Once we have written the simple model, we can try to relax some strong assumptions. From my point of view, GBM is not adequate to represent the dynamics of $S_t$. A more general approach could be the use of a shift-log normal that can include both log normal and normal behaviors, which is essential to include the possibility of negative values of $S_t$. Other interesting possibility is to model $S_t$ as a mean reversion process, whose periodicity could be related to commodity production, for instance. The second interesting improvement that could be done is to consider a extended effect of the default cost for a determined period and not punctual in time. For instance, one could model $c=\gamma e^{-\delta t}$, where $\gamma$ and $\delta$ are adjusting parameters.

Personally, I think this theme is quite interesting and has possibility for some further exploration and perhaps a small publication (why not?). If you, reader, is keen to work on it, get in contact! 

Peace profound for all of you.

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Diogo de Moura Pedroso

LinkedIn: www.linkedin.com/in/diogomourapedroso