Appendix 2: Calibration Procedure: Grid Search Method

We calibrate the model to bond and CDS data for the following ratings: A, BBB and BB. Our calibration procedure is similar in spirit to Buhler and Trapp (2010), however since our model involves many parameters it is worth discussing in details the calibration and the dierent robustness checks carried throughout the analysis. As mentioned before, we assume a CIR process (equation (5.6)) for default intensity and a Gaussian process (equation (5.7)) for liquidity intensities, this gives us a total of 8 parameters (µ,β, σ,ηb, ηask, ηbid,ηgb,ηsys).

• We initialize a 8-dimensional grid for the drift and the diusion parameters listed above. j represents the grid number and ithe grid point in the grid j.

(a) The initialized vector (µij, βij, σij, ηijb, ηaskij , ηijbid, η gb ij, η

sys

ij ) denes the pa-

rameters in each grid point i in gridj.

(b) In each grid point i, we initialize the factor matrix Hijk where k counts

the number of iterations. k will be dierent across the ratings.

(c) The calibration is done at cross-sectional level. Assuming that the ini- tialized parameters in step (a) are true, for each time t (where t= 1,2, ...., T

) we numerically specify the parameters (λij, γijb, γijask, γijbid, γ gb ij, γ

sys

ij )(t) that

minimize the sum of squared dierences between the theoretical values and the observed prices (i.e. CDS and bond yield spreads both in basis points). Step (c) is performed by assuming Hijk to be an identity matrix, this leads

to a one-to-one relationship between the correlated and the independent in- tensities.

(5.38)

where P_nmod represents the model price, P_nobs the observed values and n the

number of cross-sectional units.

(d) In order to determine the true factor matrix Hijk, we need to minimize

the element-wise-sum of the squared dierences between the assumed latent factors in step (a) and the empirical variance, covariance and autocovariance. We denote this sum as Mijk(Hijk).

(e) To minimizeMijk(Hijk) we need to numerically change the parameters of Hijk. We stop this iteration when the max-norm of Hˆijkopt and Hˆ_ijkopt−1 is

smaller than 0.01. Due to the high number of parameters in the factor matrix, we try many minimizations with dierent guessed values in order to compare the outputs and ensure that the global minimum is picked.

ˆ

Hijkopt =argminM_ijk(H_ijk)

(5.39)

(f) We use the true factor matrix _Hˆ

ijkopt and the new intensities

(ˆλijkopt,γˆb

ijkopt,γˆ_ijkaskopt,γˆ_ijkbidopt,γˆ

gb ijkopt,γˆ

sys

ijkopt)(t) as new inputs, compute the dif-

ference between the theoretical and observed values and nally sum the dif- ferences across the time series.

Sij = ΣTt=1Σ N n=1(P mod n −P obs n ) 2 (5.40)

We do the same operation across each grid pointiin gridj and then take the

minimum value of grid j39.

S_jopt=min{S1,j, S2,j..., SI,j}

(5.41)

(g) We pick the minimum value of step (f), use it as a ner local grid and move to the second grid. We do the same analysis until we reach grid 8. We stop this iteration across the grids when there is no further improvement in the tting.

• As a robustness check instead of doing a grid search and changing individu- ally each parameter in the vector (µij, βij, σij, ηijb, ηijask, ηijbid, η

gb ij, η

sys

ij ) as we have

done above (from step(a) to step(g)), we take the optimal vector of parameters ( ˆµij,βˆij,σˆij,ηˆijb,ηaskijˆ ,ηˆijbid,

ˆ

η_ijgb,η_ijˆsys) retrieved from step(g), put it as initial guess in a new optimization and run the algorithm in one go. In this procedure, each iter- ation will require the algorithm to come back to step(a) and redo the calibration by changing (µij, βij, σij, ηbij, ηijask, ηbidij , η

gb ij, η

sys

ij ) at the same time. This method

ensures that our results are robust. The optimal parameters that we get are no dierent from the one obtained in step(g).

We t the model to the data with a very small error. We obtain a mean error of 0.305,

39_{Although grid search method is a robust technique because it allows to optimize non-linear problems, it suers from}

the fact that one has to know the acceptable parameter space before starting the optimization, otherwise convergence is not guaranteed. Therefore, before initializing the vector(µij, βij, σij, ηbij, ηaskij , ηijbid, η

gb ij, η

sys

ij )at step(a), we rst use

0.35 and 0.369 bps for both bond and CDS spreads for rating A, BBB and BB respec- tively. In Figure 5.2, we plot the model-implied bond and CDS spreads with the actual data. Table 5.1 shows the values of the optimal parameters( ˆµij,βˆij,σˆij,ηˆijb,ηaskijˆ ,ηˆijbid,

ˆ

η_ijgb,η_ijˆsys). Rating BBB and BB have parameters that are in the same range. However rating A have higher volatility (i.e. ηl). Although in theory, rating A should have lower volatility

we found that it does not converge at values that are in the same range as those of rating BBB and BB. Therefore, in our calibration exercise, we expand the grid by increment- ing progressively the grid bounds of the volatility parametersηl_{. We stop increasing the}

grid when the inner iteration at step(e) retrieve the optimal factor sensitivity matrix ˆ

5.7.3 Appendix 3: Dynamic Interactions Between CDS and Bond Markets: