Modelling Default Correlations in a Two-Firm Model by Dynamic Leverage Ratios Following Jump Diffusion Processes

Transcription

1 Modelling Default Correlations in a Two-Firm Model by Dynamic Leverage Ratios Following Jump Diffusion Processes Presented by: Ming Xi (Nicole) Huang Co-author: Carl Chiarella University of Technology, Sydney CEF 2009, Sydney, Australia July 15-17, 2008

2 1 Credit Risk Analysis for Default Correlations Gaussian copula model Reduced-form approach Structural approach Nicole M.X. Huang The Two-Firm Model w/ Jumps 1

3 Table 1. Taxonomy of classical structural models & the two-firm model. Models Firm value V Default threshold D Interest rate Merton (1974) dv/v = µdt + σdz bond face value D=F deterministic LS (1995) dv/v = µdt + σdz a constant D=K Vasicek Zhou (2001a) dv i /V i = µ i dt + σ i dz i,(i =1,2) a time-dependent case D=e µ i t K i deterministic Zhou (2001b) dv/v = (µ λ q k q )dt + σdz D=e µt K Vasicek +(Y 1)dq Briys & de dv/v = rdt + σ( p 1 ρ 2 dz V follows dynamic of interest rate Hull & White Varenne (1997) +ρdz r ) D=ξF B(r, t) Collin-Dufresne & dv/v = (r δ)dt + σdz k=ln D, k is mean-reverting Vasicek Goldstein (2001) dk = λ(ln V ν k) dt Hui et al. (2006) dv/v = µ V (t)dt + σ V (t)dz V k is mean-reverting & stochastic Hull & White dk = [µ(t) + λ(t)(ln V k) σ 2 (t)/2]dt + σ(t)dz Hui et al. (2004) L D/V ; dl/l = µ L (t)dt + σ L (t)dz L, default barrier L b Hull & White One-firm w/ jumps dl/l = (µ λ q k q )dt + σdz + (Y 1)dq Vasicek Two-firm w/ jumps dl i /L i = (µ i λ qi k qi )dt + σ i dz i + (Y i 1)dq, (i = 1, 2) Nicole M.X. Huang The Two-Firm Model w/ Jumps 2

4 2 Motivation To capture sudden external shocks, we extend the Hui et al. (2004) dynamic leverage ratio model by assuming that the dynamics of the leverage ratio follow a jump-diffusion process. Extend the one-firm model to the two-firm situation, thereby capturing the surprise risk of default in a group of firms. Study and compare the impact of jumps on single defaults and default correlations. Nicole M.X. Huang The Two-Firm Model w/ Jumps 3

5 3 Presentation Overview Review the one-firm model with dynamic leverage ratios Dynamic leverage ratios model with jumps Choice of parameters Impact of jumps on individual default probabilities Fitting the S&P historical individual rates Framework of the two-firm model with jumps Impact of jumps on default correlations Conclusions Nicole M.X. Huang The Two-Firm Model w/ Jumps 4

6 4 Hui et al. (2004): A Dynamic Leverage Ratio Model Hui et al. (2004) propose an alternative model for corporate bond pricing based on the dynamic leverage ratio L (defined as the total debt to the market-value capitalization of the firm) as dl = α(t)ldt + σ L (t)ldz L. (1) In the model, they use the White & Hull (1990) interest rate model dr = κ(t) [θ(t) r] dt + σ r (t)dz r. (2) Nicole M.X. Huang The Two-Firm Model w/ Jumps 5

7 5 Dynmaic Leverage Ratio Model with Jumps Extend Hui et al. (2004) model to incorporte jump risks. Assume that L follows the jump-diffusion process dl L = (µ L λ q k q )dt + σ L dz L + (Y 1)dq, (3) µ L and σ L are the instantaneous expected drift rate and variance conditional on no jumps. Z is the standard Wiener process and q is a Poisson counting process with intensity λ q. (Y 1) is the random variable percentage change in the leverage ratio level if the Poisson event occurs with the expected mean value k q. The distribution function of ln Y is chosen as normal distribution with the jump size mean µ q and the jump size volatility σ q. Nicole M.X. Huang The Two-Firm Model w/ Jumps 6

8 5.1 Choice of Parameters The choice of parameters for the pure diffusion component is based on Hui et al. (2004). The choice of jump intensity and jump size volatility is based on Zhang & Melink (2007), who extend the Zhou (2001) to the multi-firm case. The number of time steps used is 36,500 per year, and the number of paths used is M = 500, 000. The confidence limits e.g. for a CCC-rated firm, when M = 500, 000 with 95% confidence PD(t) lies between PD exact and PD exact Nicole M.X. Huang The Two-Firm Model w/ Jumps 7

9 70 PD Jump size mean PD(%) CCC-rated firm with µ q = 0.3 CCC-rated firm with µ q = 0.5 A-rated firm with µ q = 0.5 A-rated firm with µ q = Time Figure 1: The impact of the jump size mean on the default probability for CCC and A-rated firms. When the average jump size increases, the PD for the CCC-rated firm declines, while the PD for the A-rated firm rises. Nicole M.X. Huang The Two-Firm Model w/ Jumps 8

10 This may due to the high initial leverage level of low quality firms, so the possibility of jumping down to a lower leverage level is higher than jumping up to a higher leverage level, therefore the default probability declines over time. A contrasting effect is at work for the good quality firms, because of the low initial leverage level, the probability of jumping up to a higher leverage level is more than that of jumping down to a lower leverage level, so that the default probability rises over time. Similar results for the effects of jump size volatility and jump intensity. How do the jump-diffusion model and pure diffusion model differ when calibrated to S&P historical default rates? Nicole M.X. Huang The Two-Firm Model w/ Jumps 9

11 65 PD fit CCC S&P data GBM with opitmal dirft= JP with optimal jump size mean= PD(%) Time Figure 2: Comparing the best fit to the observed S&P default probabilities with the jump-diffusion model (JP) and with the pure diffusion model (GBM) for a CCC-rated firm. Nicole M.X. Huang The Two-Firm Model w/ Jumps 10

12 14 12 PD fit BBB S&P data GBM with opitmal dirft=0.002 JP with optimal jump size mean= PD(%) Time Figure 3: Comparing the best fit to the observed S&P default probabilities with the jump-diffusion model (JP) and with the pure diffusion model (GBM) for a BBB-rated firm. Nicole M.X. Huang The Two-Firm Model w/ Jumps 11

13 PD fit AA S&P data GBM with opitmal dirft=0.08 JP with optimal jump size mean= PD(%) Time Figure 4: Comparing the best fit to the observed S&P default probabilities with the jump-diffusion model (JP) and with the pure diffusion model (GBM) for a AA-rated firm. Nicole M.X. Huang The Two-Firm Model w/ Jumps 12

14 A summary of the comparison of two processes on individual default probabilities fit to S&P historical data Fit to S&P data GBM optimal JP optimal CCC-rated firms Fits well Fits well BBB-rated firms No model fits well AA-rated firms Underestimate Fits well The results seems to reflect the fact that for non-investment grade firms, default of firms is driven by either gradual diffusion or jumps, while for investment grade firms, the default is mainly driven by jumps. It will be of interest to see how this effect works out when we come to consider the effect of jump-diffusion dynamics of the leverage ratios on default correlation among two firms. Nicole M.X. Huang The Two-Firm Model w/ Jumps 13

15 6 Two-Firm Model with Jump-Diffusion Processes Extend to the two-firm situation. The two-firm model is based on the consideration of a financial instrument: a credit-linked note that is exposed to the default risk of the note issuer (Issuer B) and the reference obligor (Bond C) A typical structure of a credit-linked note is in the figure below Investor A Proceeds (for a CLN) Interest on note CE: CEP No CE: par Issuer B (Note Issuer) Proceeds invest in Bond C Return Bond C (Reference Obligor) Nicole M.X. Huang The Two-Firm Model w/ Jumps 14

16 Let L 1 and L 2 denote the leverage ratios of the note issuer and the reference obligor, and their dynamics are described by dl i L i = (µ i λ qi k qi )dt + σ i dz i + (Y i 1)dq, (i = 1, 2) (4) µ i and σ i are the instantaneous expected drift rates and variances conditional on no jumps. Z 1 and Z 2 are Wiener processes and their increaments are assumed to be correlated with E[dZ 1 dz 2 ] = ρ 12 dt. q is the Poisson counting process defined in previous section with the intensity λ q, here we assume that the jump event affects both firms so the same intensity of the jump arrivals λ q applies to both. The jump sizes Y 1, Y 2 and their distributions are assumed to be independent, their log-distributions are chosen as normal distributions with jump size mean µ qi and jump size volatilities σ qi. Nicole M.X. Huang The Two-Firm Model w/ Jumps 15

17 0.1 DC Jump size mean DC µ q = 0.3,ρ 12 = 0.5 µ q = 0.3,ρ 12 = 0.5 µ q = 0.3,ρ 12 = 0.5 µ q = 0.3,ρ 12 = Time Figure 5: The impact of jump size mean on the DC for CCC-A paired firms. The DC as the average jump size for +ve correlated firms. This is due to the fact that, when the jump event occurs, the L i jump to higher values on average with a larger µ qi, and so move closer to the b L i. So, if one firm has defaulted, this is a signal that the L i of the other firm moves in the same direction (because of ρ 12 > 0) to the b L i, since the L i of the other firm is already close to the default barrier with the larger µ qi, thus the default of one firm will be a signal that the Nicole other M.X. Huang firm is likely to default. The Two-Firm Model w/ Jumps 16

18 One the other hand, when firms are negatively correlated (ρ 12 < 0), the default correlation (in absolute value) increases as the average jump size decreases. the argument is similar to the case of ρ 12 > 0 when the jump event occurs, the leverage ratios jump to lower values on average with a small jump size mean, and away from the default barrier therefore, if one firm has defaulted, then leverage ratio of the other firm moves in the opposite direction (because of ρ 12 < 0) away from the default barrier, since the leverage ratio of this firm is already far from the default barrier due to the small jump size mean thus combining these two effects, the default of one firm will make it less likely on average that the other firm will default. Nicole M.X. Huang The Two-Firm Model w/ Jumps 17

19 0.08 DC Jump size volatility 0.06 DC σq 2 = 0.25,ρ 12 = 0.5 σq 2 = 0.5,ρ 12 = 0.5 σq 2 = 0.25,ρ 12 = 0.5 σq 2 = 0.5,ρ 12 = Time Figure 6: The impact of the jump size volatility on the DC for CCC-A paired firms. The DC as the jump size variance. Nicole M.X. Huang The Two-Firm Model w/ Jumps 18

20 0.12 DC Jump intensity DC λ q = 0.5,ρ 12 = 0.5 λ q = 0.1,ρ 12 = 0.5 λ q = 0.5,ρ 12 = 0.5 λ q = 0.1,ρ 12 = Time Figure 7: The impact of the jump intensity on the DC for CCC-A paired firms. The DC as the jump intensity for +ve correlated firm, while when firms are ve correlated, a higher jump intensity shifts the default correlation to positive values. Nicole M.X. Huang The Two-Firm Model w/ Jumps 19

21 This may be due to the fact that, when the frequency of arrival of the jump event increases, the frequency of the L i jumping to higher values is increased, and this drives the leverage ratios closer to the default barrier. The leverage ratio of this firm is already very close to the default barrier (because of the high frequency of jump arrivals), thus the default of one firm will be more likely to signal the possibility of the default of the other firm. Nicole M.X. Huang The Two-Firm Model w/ Jumps 20

22 6.1 Overview Jump-diffusion processes Impact on PD Impact on JSP Impact on DC CCC A +ρ 12 ρ 12 jump size volatilities jump intensity jump size means Table 1: A summary of the impact on individual defaults, joint survival probabilities and default correlations for jump-diffusion processes. Nicole M.X. Huang The Two-Firm Model w/ Jumps 21

23 7 Conclusions This research program has set up the generalized theoretical model for default correlation based on structural approach incorporating stochastic interest rate with dynamic leverage ratios following the two types of processes to capture the different features of firms: following simple continuous diffusion processes (GBM) facing both default risk of gradual diffusion and unforeseen external shocks (jump-diffusion processes) Study and compare the impact of jumps on individual default probabilities and default correlations. Nicole M.X. Huang The Two-Firm Model w/ Jumps 22

24 8 Future Research However, these findings are based on a study of the impact of parameters The values of the parameters need to be obtained by some robust econometric estimation methodology from market data, this will certainly be a key topic for future research. On the other hand, the results of this research could be used as a benchmark for assessing different copula functions used in valuing default correlations. Nicole M.X. Huang The Two-Firm Model w/ Jumps 23