Comparison of Estimation Methods of Structural Models of Credit Risk

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

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 firmwill 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 Volatility Drift

16. Results –First Passage Model F>=K Drift Volatility Default Barrier

17. Results –First Passage Model F<K Volatility Default Barrier Drift

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