1. Comparison of Estimation Methods of Structural Models of Credit Risk
MS&E 347 Term Project
Stanford University
June 2009
Jeff Blokker, Shafigh Mehraeen, Won Chase Kim, Bobak Javid, and John Weng

2. Structural Models
Structural models refer to models that look at the evolution of the capital structure of the firm to evaluate their credit risk.
Merton’s model (1974) was the first modern credit risk model that was considered a structural model.
It assumes the capital structure of the firm is composed of equity St and a zero coupon bond of value Dt with face value F.
Then the asset value of the firm is the sum of the equity and debt.
Assumptions
No transaction costs, no bankruptcy costs, no taxes,
infinite divisibility of assets, unrestricted borrowing and lending,
constant interest rate
GBM of firm’s asset value.

3. Merton’s Model
If the value of the firm at the maturity date T is less than K then the firm will be unable to repay the debt.
The payoff structure at T is:

4. Merton’s Model
The firm’s equity St represents a European call option on the firm’s assets with maturity T.
The Bond represents a risk free loan F with maturity T plus selling a European put option with strike F and maturity T
Merton’s model assumes that the firm can only default at time T.
The value of the firm is assumed to follow the SDE
With the volatility of the firm’s asset value, a constant interest rate r, and risk neutral Brownian motion

5. Merton’s Model
Applying the Black Scholes equation to the equity value of the firm yields
To implement Merton’s model we need an estimate of :
Volatility of the asset value -
Drift of the asset value -

6. First Passage Model
The first passage model is an extension of the Merton model
Default at any time T1 < T if the asset value Vt crosses the barrier K.

7. First Passage Model At T the value of the equity is
This is a Down and Out call option with formula
when F>=K
when F<K

8. Model Calibration
To implement the first passage model we need an estimate of
Asset volatility -
Default barrier - K
Drift -
We compare three methods for calibration:
Inversion Method
MLE
Iterative Method - KMV

9. Inversion Method for Merton’s model for First Passage model
From Ito’s formula we get
Comparing coefficients of the two SDE equations we conclude that
where f is a simple call option (Merton)
or down-and-out call option (First Passage model)

10. Maximum Likelihood Estimate (MLE)
Proposed by Duan (1994)
Given a time sequence of equity values , we can estimate a time sequence of asset values , volatility , drift , and the barrier K.
We denote the probability density function for the equity value at ti given the equity value at ti-1 and the parameter vector .
Then the log-likelihood is given by
Using the previously defined function and assuming it is differentiable and invertible we can write
where is the P-density of Vt given Vt-1.

11. Maximum Likelihood Estimate (MLE)
MLE for the Merton’s Model
Letting be the time between observations
where

12. Maximum Likelihood Estimate (MLE)
MLE for the First Passage Model

13. Iteration Method - KVM Estimation of and
Asset values Vt are implied from equity value
Returns and
Volatility
Drift
Repeat until convergence.
Equivalent to EM algorithm and asymptotically converges to ML
For the Merton’s model, much faster than ML
For the First Passage model, no analytical formula.

14. Monte Carlo Simulation Environment
Asset value paths are generated by GBM with constant parameters
V0=1.5
F = 1.0
K/F = 0.8 or 1.2
T = 2
volatility = 0.3
Drift = 0.1
R = 5%
2500 samples generated and down-sampled to 250 per year
To reduce bias (In reality, we only observe daily equity values)
Only keep the value process which does not default
Converted to equity value paths by BS formula (call or DOC)
Use equity paths in each model to recover parameters

15. Results – Merton Model Merton Model Method Mean ML 0.2984 0.1275 STD
0.0174
0.2615
Inversion
0.3245
0.1366
0.0347
0.2632
Iterative
0.2992
0.1278
0.0175
0.2617
Volatility
Drift

16. Results –First Passage Model F>=K
Cox Model (F>K)
Method
Mean ML
0.2957
0.1362
0.7426
STD
0.0224
0.2576
0.2257
Inversion
0.3468
0.1607
0.4330
0.0848
0.2579
0.1524
Volatility
Drift
Default Barrier

17. Results –First Passage Model F<K
Cox Model (F<K)
Method
Mean ML
0.3033
0.2729
1.1894
STD
0.0516
0.2313
0.1364
Inversion
0.6641
0.6632
0.2211
0.2243
0.2925
0.1982
Volatility
Drift
Default Barrier

18. DJIA Merton Model ML
Volatility
Drift
Equity to Debt ratio (S/F) 3M
0.1527
0.2818
5.6689
ALCOA
0.1698
0.3005
1.0928
PHILIP MORRIS
0.1689
0.2322
1.2633
AMERICAN EXPRESS
0.0672
0.0896
0.3730
AIG
0.0671
0.0358
0.2926
BOEING
0.1037
0.1091
0.5879
CATERPILLAR
0.1143
0.2688
0.7583
CITI
0.0398
0.0660
0.2073
DU PONT
0.1375
0.0704
1.5864
EXXON
0.1316
0.1435
3.0847 GE
0.0820
0.0902
0.5388 GM
0.0152
0.0364
0.0599 HP
0.2499
0.1958
1.7116
HONEYWELL INTERNATIONAL
0.1522
0.1959
1.1639
IBM
0.1510
0.1132
1.9118
INTEL
0.3384
0.6756
CHASE
0.0224
0.0436
0.0865
JOHNSON & JOHNSON
0.1882
8.6067
MCDONALDS
0.2055
0.2989
1.8293
MERCK
0.1983
4.0999
MICROSOFT
0.2722
0.0637
PFIZER
0.2007
0.1386
7.3363
SBC
0.1848
1.2807
COCA COLA
0.1810
0.1469
8.4725
HOME DEPOT
0.2829
0.3646
8.2828
PROCTER & GAMBLE
0.1076
0.1285
4.2341
UNITED TECHNOLOGIES
0.1426
0.2803
1.6363
Verizon
0.1240
0.7291
WAL MART
0.1825
0.0471
4.8501
DISNEY
0.1856
0.2096
1.4929

19. DJIA (2003) - Cox Model ML
Volatility
Drift
Barrier Level
Barrier to Debt ratio (K/F) 3M
0.1527
0.2818
0.5625
0.5878
ALCOA
0.1698
0.3005
1.4516
0.7102
PHILIP MORRIS
0.1689
0.2322
4.8873
0.7004
AMERICAN EXPRESS
0.0672
0.0896
6.4314
0.4375
AIG
0.0671
0.0358
0.8367
BOEING
0.1037
0.1091
2.3736
0.5186
CATERPILLAR
0.1143
0.2688
1.5075
0.5371
CITI
0.0398
0.0660
0.9163
DU PONT
0.1375
0.0704
1.4721
0.5567
EXXON
0.1316
0.1435
6.7947
0.8492 GE
0.0820
0.0901
0.5073 GM
0.0152
0.0364
0.6336 HP
0.2499
0.1956
2.6907
0.7619
HONEYWELL INTERNATIONAL
0.1522
0.1959
1.2275
0.6426
IBM
0.1510
0.1132
4.6293
0.6127
INTEL
0.3384
0.6756
1.1676
1.3008
CHASE
0.0224
0.0436
0.1467
JOHNSON & JOHNSON
0.1882
1.6901
0.9232
MCDONALDS
0.2054
0.2990
1.1378
0.8107
MERCK
0.1983
2.7054
0.8988 MS
0.2722
0.0637
2.3658
1.4921
PFIZER
0.2007
0.1386
3.8797
1.4332
SBC
0.1848
4.7038
0.7418
COCA COLA
0.1810
0.1469
0.7987
0.6134
HOME DEPOT
0.2092
0.2534
4.7739
5.6025
PROCTER & GAMBLE
0.1076
0.1285
0.9861
0.3515
UNITED TECHNOLOGIES
0.1426
0.2803
1.2504
0.5882
Verizon
0.1240
9.0159
0.6522
WAL MART
0.1825
0.0471
5.2550
1.0602
DISNEY
0.1855
0.2096
2.0560
0.7540

20. Empirical analysis: an example
From the model we can calculate corporate default probability

21. Conclusion
Three estimation methods are compared for two structural credit models
For Merton’s model, ML and KMV are equivalent and superior to inversion
For the first passage model, ML is the only option but estimation of barrier is not an easy problem.
Drift estimation is also difficult but it is out of our interest
When K/F is small, two models does not make much difference
Further research must be done for benefits of the first passage model
Results from this projects can be extended for various applications
Default probability estimation
Term structure of credit spread