In this paper we use Merton’s (1976) jump diffusion model and Heston- Nandi stochastic volatility model (2000) for pricing options when the underlying asset is driven by a mixed diffusion-jump process or GARCH volatility process to compute the monthly default probabilities of a bond issuer whose income is uncertain with high volatility in tax collection. In particular, we analize the case of a sovereign bond issued by the Mexican government in United States Dollars (to ensure the existence of default risk). The proposed methodology is based on concepts such as: previous leverage, income generation, non-recurring expenses, term and loan size (traditionally used in the calculation of probabilities of default), which provides an alternative methodology for computing

En este trabajo se utiliza el modelo de Merton (1976) de valuación de opciones y el modelo de volatilidad estocástica Heston-Nandi (2000) cuando el activo subyacente sigue un proceso de difusión con saltos para calcular las probabilidades mensuales de incumplimiento de un bono cuyo emisor tiene ingresos inciertos con alta volatilidad en la recaudación de impuestos. En particular se ilustra el caso de un bono soberano emitido por el gobierno mexicano en dólares americanos (para asegurar la existencia de riesgo de incumplimiento). La metodología propuesta incorpora los conceptos de: apalancamiento previo, capacidad de generación de ingresos, gastos no recurrentes, plazo y tamaño del préstamo (tradicionalmente usados en el cálculo de probabilidades de incumplimiento), lo que provee una metodología alternativa para el cálculo o

Endless governmental financial needs make authorities seek for funds through debt and taxation; this being their major current concern. In order to obtain the required funding, most federal governments issue debt instruments. These instruments may be backed with some specific assets such as natural resources or government owned companies, but in some cases, they have no collateral except for the fact of being issued as sovereign debt.

As any other debtor, governments are subject to default, even if bonds are issued in its own currency, despite governmental monopoly on primary money emission due to central bank constraints^{1}

Regardless of whether we are dealing with a company or a government, debt payment is conditioned on debtor’s capacity to generate enough resources to meet his financial commitments. Fulfilling them guarantees countries and companies an easy and low cost access to debt markets, reducing thus debtor’s incentives to default, so nonpayment becomes an unusual occurrence.

Although rare, defaults play an important role in the sovereign bond market due to their size. Defaults may spread across the financial system with negative political effects.

Financial markets have created instruments to hedge credit risk. An example is Credit Default Swaps (CDS), which intend to deal with credit risk in a similar manner to insurance. Even though CDS were created to cover corporate debt, these instruments may easily be used to cover sovereign debt if the financial system is deep enough (due to debt size) or if they are used in a deposit insurance framework.^{2}

CDS works identically to its corporate counterpart. For simplicity, suppose that a country issues a sovereign bond rated BBB by some rating agency and it is bought by a single bank, called NonRiskyBank, which may be concerned about the country's capability to pay its debt. For hedging this exposure, a NonRiskyBank may enter in a deal with a counterpart called RiskyDeals who receives a regular payment,

There are several CDS valuation methods for the above described mechanism. Most of them can be grouped in two competing approaches: reduced form and non arbitrage methods. Reduced form models may be considered as bond survival models; these models concede an expected value to risky bonds given a default probability,^{3}

The reduced form CDS valuation method relies heavily on default rate probabilistic models, these include from pure structural models to pure reduced default rate probabilistic models.

Normality and martingale assumptions are crucial in most probabilistic CDS valuation models, in order to get a stochastic process for the default probability, in all cases they give an expected value for defaultable bonds on each coupon payment date and subtract it from the risk free bond price, in order to get an average value for the default event.

A non arbitrage approach is explained by

Complete market approach implies that only market based factors influence default on credit risky bonds, since those factors cannot be diversified, therefore they are fully discounted by market expectations. This statement is relaxed when the existence of different credit risk levels is introduced. These levels are clearly explained as a market answer to the adverse selection problem,

Our model is based on the Hull and White CDS valuation model as a starting point (non arbitrage statement), and then a set of path dependent default probabilities is estimated based on Merton’s jump pricing options model (non Gaussian structural model), using as input a GARCH forecast of debtor’s future income and its volatility. Anything lying outside the 2 confidence interval is considered a jump and it is included in the Merton’s equation. The use of these models, considering debt payment as a strike price, transforms our model into a structural one, since debtor’s income generating capability (future cash flow) is replicated by the model. This part of our methodology may be easily adapted to incorporate debtor's intrinsic variables, like

In order to show our method's consistency we also calculate the default probabilities using the ^{4}^{5}

where

This probability model may be used directly, since it incorporates the impact of the underlying asset GARCH process in the “moneyness" measurement. As it will be explained later, the resulting probability is slightly smaller than the one obtained by our method, because we incorporate the GARCH effect on each of Merton's probability estimation (using current data as in Heston-Nandi model), this methodology will be widely explained below.

The paper has been divided into four sections. The next section explains the possibility of modeling the default rate by means of the Merton's jump- diffusion model using debtor's income cash flow instead of considering debt as a derivative on debtor's assets. This is a key statement since it stresses that the expected income is the main uncertainty source, avoiding the use of the complex two equation nonlinear system as in the traditional Merton debt valuation model (1974) or any related estimation based on Merton's jump diffusion model (1976). The second section is devoted to explain briefly Hull and White's CDS valuation model and its interaction with previously explained probability models. Also, the role of the recovery rate calculation,

As previously mentioned, our model provides a set of default rates based on debtor’s income flow predictions, instead of those given by a non linear two equation system as in the traditional ^{6}

The predictive power associated to derivative’s distribution is directly inherited from other models like ^{7}

Following the previous statement, we must emphasize that bankruptcy will be considered when debtor’s income is not enough to fulfill its commitments on a certain (expiration) date, and not when debtor’s assets are smaller than his/her liabilities. At this point, our model resembles that of Credit Suisse Financial Products (1997) because it only considers a default as a credit event. According to

Therefore, the combination of Bernoulli draws and Poisson jumps frameworks that gave rise to Credit+ may be considered as a jump-diffusion Stochastic Differential Equation (SDE) for debtor’s cash flow, when regarded as a continuous function. This will be held as our main assumption, it also relates our work to some previous models that perform well, therefore it allows us to include their most important features into our framework. Furthermore, the same fact permits us to stress that we do not need to include a non linear equation system, because we are using income, I, rather than market prices for calculations. Therefore, we can use a jump-diffusion SDE for debtor’s income, as

Where μ is the instantaneous mean for income σ is its volatilly and v is the expected jum size. Here, we have two sources of uncertainity, the Brownian moton, dW_{t} and the Poisson jum, dN_{t}. The Stochastic Differential Formula in Equation (1.a) must be defined in an augmented probability space with an augmented filtration (

If the Heston-Nandi approach is used, a similar process in followed, since it implicity states that the underlying asset (debtor´s income) emulates an Instant Equation, given by

While its volatility is a mean reversion process given by

where _{
t
} is a Brownian motion associated to volatility. This Brownian motion may or may not be correlated to the underlying asset volatility; _{t}, dW_{t}), obviously it is set on its own probability space given by

We must state clearly that all probabilities resulting from this model were calculated using current data, so results are estimated from today to a certain point in the future, all of this included in the characteristic function,

And those resulting from the Heston-Nandi process, given by

The main difference results in a volatility change through time and the effect of this volatility in the option’s vega first derivative,

Differences between both SDPE may be eliminated with the correct jump tunning. In fact income jumps may be regarded as the average of volatility clusters given by

In other words, the log normal jump diffusion probability stated in

As postulated above, we use jump-diffusion SDE as stated in (1. a) to model debtor’s income,

Merton shows that this probability scenario matches the probability of an "in-the-money” option, so default probability is given by the probability complement. This is an important issue because we are only interested in the option probability associated, not in its financial interpretation because we are not modeling jump-option premium valuation, _{
M
}

where ^{8}^{2} is the debtor's income volatility, δ^{
2
} is the variance of the log normal distribution defined in Equation (1), and _{
BSn
} is the

with the appropriate substitutions, it follows

And

As stated previously, default probabilities estimations link our model to the Credit Metrics and Credit^{+}, but also set some theoretical constraints that must be clearly stated. The most important constraint is the assumption that debtor's income follows a log normal multivariate distribution with independent variables. This may not necessarily be true for a troubled debtor. Moreover, this assumption isolates debtor’s income from other agents, complicating substantially the modeling of massive defaults effects that often impact the credit markets during recession periods. Due to capital markets global integration, massive default component is important, both to corporate debt valuation and to sovereign debtors. Recent events such as the European sovereign debt crisis confirm this assertion.

This deficiency in our model is partially overcome by the jumps and stochastic volatility included in the income modeling.^{9}

Diffusion Probability Model, there is no difference whether default probabilities are estimated in a high or low volatility environment, but as explained above, according to the proposed methodology, it incorporates volatility cluster effects on default probabilities ("moneyness” probability complement) as an alternative and easier method than the Heston-Nandi approach.

We overcome the fixed volatility problem by including a GARCH model for debtor’s income forecasts, as stated in

Following _{
t
}

Where the ARMA(p,g) model for the mean considers p autoregressive parameters, Φ_{i}., and _{i}, while the ARMA for the variance has _{i}, and _{
i
}
_{
t
}
_{
t ≥1} , which is consistent with the Martingale framework where the Merton’s model arose.^{10}

Although the forecast specification improves the correlation weakness discussed before, some GARCH family methods may deliver better results for some specific scenarios. For example, consider a medium or long run crisis environments, in this case an E-GARCH volatility specification captures the volatility behavior better. Other special case is the TARCH model used in overshooting expectations scenarios with "relatively” common shocks. These GARCH approaches may be easily incorporated in the original proposed model.^{11}

An important remark, since default probabilities are the core in CDS estimation, is that they must mirror macroeconomic environment and the debtor’s capacity to fulfill his financial commitments when economic conditions change. This feature is the main improvement of our model compared with pure reduced or structural models. Our model captures intrinsic characteristics of debtor’s income forecasts, and relates them with macroeconomic environment when income’s volatility and average size jumps are included.

The use of a CDS for a sovereign bond immunization is essentially an answer to the monitoring problem on this kind of bond, since there isn’t a mechanism by which a sovereign nation may be forced to maintain certain fiscal policies in order to ensure debt payment. Actually, creditor’s monitoring turns into an information mechanism that may stop credit flows into a nation, but such mechanism could be as effective as CDS spreads, due to government’s commitment to be considered able to honor its debts in order to obtain new funding.

The use of the CDS spreads makes monitoring cheaper for small credit risk products investors, also allows them to diversify their portfolios with instruments that are not usually available to them. Other CDS advantage regarding sovereign debt is the possibility of risk transference among a large set of investors, through standardized contracts in big enough markets.

A market, as described in this paper, may avoid traditional problems on corporate CDS such as incentives to tear down the value associated to CDS,^{12}

Once explained the default probabilities estimations and its implications on the overall model, the CDS calculation method may be explained. Due to its simplicity and widely extended use in financial transaction, the

The proposed methodology was chosen over the Swap Market Model developed by

As the reader may notice below, our model partially resembles the one proposed by

Hull and White CDS valuation model estimates the fair payment that generates equilibrium between the present value of a risky bond expected loss (the amount received by bond holder in case of default) and a set of payments made to the insurer (the short position on CDS). This mean that CDS value is given by CDS expected payment,

Expanding Equation (5), an explicit form for Hull-White CDS valuation equation is found, hence

Where _{
i
} represents default probability on last payment, given by the sum of probabilities of each period; _{
t
} denotes the discount factor for a risky bond from today, t=0, to default time or _{
t
} represents discount factor for a risky bond from past coupon payment, _{
i-t
} to, _{
i
}
_{
t
} is the discount factor for a riskless bond from today, t=0, to any given coupon payment; _{
t
} denotes default probability on _{
t
} represents accrued interest on the risky bond at t.

It is important to notice that on initial time, CDS value must be zero since we are on an arbitrage free environment, but when credit or interest rate conditions change CDS value may also change. Actually, one of the paper objectives is to estimate the regular payment amount, w*, given^{13}

This approach assumes that there is always a portfolio that may resemble risk and expected returns of any financial instrument. This implies that complete market assumption is attained, thus there is a single risk neutral probability set that allows achieving the equilibrium condition.

The only minor change that we are proposing for the Hull and White CDS valuation method is that the recovery rate estimation, _{
ij
}
_{
i
} hence

The recovery rate allows us to partially capture the change in the recovery rate when the loan credit quality deteriorates, or when the whole transition probabilities curve changes due to major economic movements.

With the previous analysis for default probabilities estimation and CDS valuation, we are able to show the proposed method performance on a hypothetical USD denominated Mexican bond issued by the Mexican government. We chose this particular case since Mexico has access to global debt markets, it has no recent default history, it faces volatility problems on tax collection.

Public information on tax collection and previously issued USD denominated debt is available. The first step to carry out our calculations is to gather tax collecting data from INEGI's^{14}^{15}

After performing a KPSS stationary test, we conclude that the first seasonal differences for tax income were stationary, the resulting correlogram was examined for that stationary time series and it was concluded that the process could be modeled as driven by an ARMA (1,2,10; 1,2) process. Results are shown in

Z
Const
4078,16
1210,9
3,3679
0,00076
***
phi_l
0,268489
0,0524338
5,1205
<0,00001
***
phi_2
0,548326
0,0840448
6,5242
<0,00001
***
phi_l0
0,114372
0,0523673
2,1840
0,02896
**
phi_1
0,260583
0,118093
2,2066
0,02734
**
theta_2
-0,376471
0,0981241
-3,8367
0,00012
***
theta_l
-0,635104
0,0954546
-6,6535
<0,00001
***
Dep. Variable mean.
4255,638
Deppendent D.T.
6007,021
Innovation mean
51,44147
Innovations D.T.
4926,207
Log-likelihood
-2947,998
Akaike Criterion
5911,997
Schwarz Criteron
5941,547
Hannan-Quinn Criterion
5923,827
Residuals normality test
Null Hypothesis: Normally distributed error
Test Statistic: Squared Chi (2) = 170,545
P value = 9,26234e-038
ARCH test, order 12 -
Null Hypothesis: there is no GARCH effect
Test Statistic: LM = 78,3302
P Value = P(Squared Chi (12) > 78,3302) = 8,58422e-012

This estimation may not be parsimonious but reflects some GARCH components that clearly appear in the ARCH effects test. When those conditional volatility components are incorporated into the model, they correct the second order dependence effect and provide well fitted results.

The ARMA-GARCH model used to get Merton’s default probabilities is similar to that established in Equation (4). Parameter estimation is showed in ^{16}

Const
332.545
109.509
3.0367
0.00239
***
sdlngresosl
0.368048
0.0653258
5.6340
<0.00001
***
sd_Ingresos_2
0.333049
0.0781017
4.2643
0.00002
***
sdlngresolO
0.141471
0.0541922
2.6105
0.00904
***
alpha(l)
0.160258
0.0616832
2.5981
0.00937
***
alpha(2)
0.170527
0.0809477
2.1066
0.03515
**
Dep. variable Mean 4379.363
Dep. variable D.T.
6073.599
Log-likelihood
-2711.740
Akaike Criterion
5443.480
Schwarz Criterion
5480.075
Hannan-Quinn Criterion
5458.147

Before estimating default probability, we must emphasize that this ARMA- GARCH model has a non explosive variance, (its coefficients are non negative and their sum is smaller than the unit), they also preserve the fitting properties showed by the previous ARMA model as shown in

With this model estimation at hand, debtor’s (Mexican government) income in MXpesos is converted to USDollars, in order to avoid the use of seigniorage^{17}

All forthcoming calculations were made by using R Statistical Software.^{18}^{19}^{th}, 2010 (valuation date). Additionally, a linear interpolation for the implied interest rate on time ^{20}

A hypothetical monthly compounded fixed rate coupon bond issued for infrastructure development was used to show the adjustment capability of the proposed method in a realistic environment. Due to the nature of the bond, historic infrastructure expenses during the last four years were considered,^{21}^{22}

Therefore, a 140,666.6 million monthly payment in Mexican pesos is considered as a 24 period annuity of 10,480, 696,395.09 USD. With this information and an interest rate linear interpolation for USD risk free interest rates, we can calculate the default rate for each monthly payment, _{
2
}
_{
t+j
}
^{23}

As we showed in Equation (3) a B-S option estimation is required, so we used a slightly modified version of Option R package^{24}

This default probability estimation method incorporates the income/debt coverage ratio, as a traditional structural model variable. Also, the proposed method incorporates income forecasted volatility and interpolated credit risk free interest rate, offering a partial view on income environment. These adjustments create the set of probability defaults showed on

Corresponding probabilities were estimated using the Heston-Nandi framework in order to show their resemblance to the curve of jump diffusion probabilities using as input the ARIMA-GARCH forecasts, calculated with the proposed method. The Heston-Nandi estimations were made using a modified Heston Nandi option original function included in the Options package in R.^{25}

Once the jump diffusion - ARIMA-GARCH probabilities were estimated, the next step is to use Hulls & White CDS valuation formula converting the annuity present value into an equivalent coupon bond. The bond nominal value was estimated considering the current coupon rate of a previously issued Mexican bond as our coupon rate^{26}^{27}

A remarkon the annuity associated interest rate must be made, since it is usually calculated using a single rate for all the annuity lifetime. In our model, we used an average rate derived from the interpolated rates until bond's time to maturity. Obviously, this calculation can be refined using a cubic spline interpolation for interest rates and with them estimate the Net Present Value (NPV) for each payment. This procedure is hardly used since results are about the same but the estimation strategy is far more complex.

Also, an expected value for the recovery rate as in Equation (7) is required. It was estimated based on

With all these elements at hand, eventually CDS valuation on this hypothetical bond can be performed, resulting in a fair payment amount of 982,061,812 USD^{28}

The payment is slightly different from the normal distribution suggested payment due to the jumps existence, the payment was also modified by volatility induced by the GARCH approximation and the interest rate interpolation. These features have not been considered by traditional CDS valuation methods.

In this paper we have used the jump-diffusion risk neutral default probabilities models in a CDS valuation, we did this by using the complement of an "in-the-money” Merton’s option pricing probability model where the debtor’s

income is taken as the underlying asset and the expected payment is considered as the strike price. We also incorporate debtor’s income volatility into the default probabilities by using ARMA-GARCH income forecasted volatility as the underlying asset standard deviation in a jump diffusion option valuation framework. All modifications to traditional default probabilities models, usually fixed by rating agencies, as in the Hull and White CDS valuation method, result in a hybrid default probability and CDS valuation algorithm that incorporates some structural variables, as expected payment coverage, following a jump-diffusion underlying process.

Along the paper we showed that our simple jump diffusion probability series provided with ARIMA-GARCH forecasted volatilities are similar (but larger for small coverage ratios) to those obtained with the Heston-Nandi model since they arose from similar stochastic differential partial equations and the volatility process is the same, nevertheless it was calculated in different ways; outside for the jump diffusion probabilities and in the model for the Heston-Nandi ones.

The proposed method is consistent with traditional credit default probability estimation methods like Credit Metrics or Credit", and with the risk neutral valuation used in H&W method. Despite all of these advantages, the log normal jump modeling of outliers may be improved, and perhaps it can be modeled by extreme value jump-diffusion processes. This is left as a future research as well as the use of these default probabilities in a reduced CDS valuation model or in a copula based probability model.

This is true in countries where the central bank is independent of the government.

Similar to those used to hedge public deposits on private banks in most countries; an example is the Mexican Banks Savings Protection Institute (IPAB).

This probability may be re-evaluated on each valuation period as in Credit Risk+.

This means that the option price is above zero, therefore it will be exercided.

A complete explanation is given in the Heston and Nandi paper. We considered it an ad hoc model, since the characteristic distribution function,

This was proven using a naive predictor that inherits the normal distribution but does not come from any non linear system. This naive predictor behaves slightly worse than its non linear counterparts.

In the case of Merton'sjump diffusion model we use the Nd_{2}), while in the Heston- Nandi model we use the P_{2} probability described at the beginning of the paper.

We must remember that the bond is issued in a foreign currency, so all calculations are in that currency.

As in all time series modeling, jump diffusion SDE modeling assumes that variable's stochastic process will maintain its interaction with the rest of the economy, which may not be true.

We should remember that Brownian motion is a stationary process.

We consider that this methodological approach is not appropriate for Mexico's case because of the relative macroeconomic stability in the last decade. TARCH approach may be suitable for a stopping crisis scenario.

A person can lend some money to a company, buy a CDS for its debt and short their shares while they forget monitoring the company. If it defaults, the loan is hedged and they may have a profit from short selling.

Interested readers may analyze all calculations and assumptions behind this equation on Hulls and White's paper.

Instituto Nacional de Estadística Geografía e Informática, US Census Bureau Mexican's counterpart.

Banco de Información Económica, http://www.inegi.org.mx/sistemas/bie/

The residuals tests for this model are not shown but are available on request.

This is the difference between the cost of producing Money and its value on economy.

Freely available in

Published by Mexico's Central Bank on its web page.

This information was downloaded from their respective central banks.

Information retrieved from federal government web site

Assuming two years as the remaining time for current Federal Government Administration.

This is for getting an annual default rate.

Diethelm Wuertz and many others, the package can be downloaded from

This rate was taken from a USD

The algorithm used for calculations are available on request.