Theoretical Problems in Credit Portfolio Modeling 2

Theoretical Problems in Credit Portfolio Modeling 2 David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Jiaotong University(SJTU) November 3, 2017 Presented at the University of South California 2 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 3

Overview of Credit Portfolio Problems Credit portfolio risk management for general financial institutions Managing credit risk at portfolio level David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 4

Overview of Credit Portfolio Problems Credit portfolio risk management for general financial institutions Managing credit risk at portfolio level C-VAR Rating agency credit portfolio modeling Binomial expansion SIV model 4 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 4

Overview of Credit Portfolio Problems Credit portfolio risk management for general financial institutions Managing credit risk at portfolio level C-VAR Rating agency credit portfolio modeling Binomial expansion SIV model Credit derivative Pricing and hedging Corporate ABS including subprime mortgage 4 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 5

General Issues and Modeling Approach Need a method to deal with number of names from 5 to 100 or 200. This excludes the method of modeling aggregate loss directly. General Modeling Approach: Bottom-up, top-down and some in between. David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 6

General Issues and Modeling Approach Need a method to deal with number of names from 5 to 100 or 200. This excludes the method of modeling aggregate loss directly. Need to incorporate all single name information completely for both pricing and hedging purpose General Modeling Approach: Bottom-up, top-down and some in between. 6 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 6

General Issues and Modeling Approach Need a method to deal with number of names from 5 to 100 or 200. This excludes the method of modeling aggregate loss directly. Need to incorporate all single name information completely for both pricing and hedging purpose Need to introduce correlation General Modeling Approach: Bottom-up, top-down and some in between. 6 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 6

General Issues and Modeling Approach Need a method to deal with number of names from 5 to 100 or 200. This excludes the method of modeling aggregate loss directly. Need to incorporate all single name information completely for both pricing and hedging purpose Need to introduce correlation Need to have efficient computation: imagine managing a portfolio of thousands portfolio-type trades with thousands of underlying credits or collateral assets. General Modeling Approach: Bottom-up, top-down and some in between. 6 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 7

Copula Credit Portfolio Model In the current copula function approach to credit, we start with creating individual credit curves independently from each other and the rest of the market. The credit curve can be described by its survival time distribution with distribution function, Fi (t), and the density function, fi (t). Then we simply use a copula function C(u 1, u 2,..., u n ) to combine the marginal distribution functions, and claim that F (t 1, t 2,..., t n ) = C(F 1 (t 1 ), F 2 (t 2 ),..., F n (t n )) as the joint risk neutral distribution of τ 1, τ 2,..., τ n. David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 8

Current Market Practice We use one correlation Gaussian copula function along with base correlation method. One correlation Gaussian copula function F (t 1,..., t n ) = Φ n ( Φ 1 (F 1 (t 1 )), Φ 1 (F 2 (t 2 )),..., Φ 1 (F n (t n )); ρ ) David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 9

Current Market Practice We use one correlation Gaussian copula function along with base correlation method. One correlation Gaussian copula function F (t 1,..., t n ) = Φ n ( Φ 1 (F 1 (t 1 )), Φ 1 (F 2 (t 2 )),..., Φ 1 (F n (t n )); ρ ) Base Correlation: To price each tranche as a difference of two equity tranches, and attach a different equity tranche correlation for different detachment point. max(l(t) K T L, 0) max(l(t) K T U, 0) 9 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 10

How well or how bad does the model do in practice? How a Formula Ignited Market That Burned Some Big Investors - Wall Street Journal, September 12, 2005. David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 11

How well or how bad does the model do in practice? How a Formula Ignited Market That Burned Some Big Investors - Wall Street Journal, September 12, 2005. Hedge Fund Losses, Credit Derivatives and Dr. Li s Copula?Financial Engineering News, Nov/Dec 2005. 11 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 11

How well or how bad does the model do in practice? How a Formula Ignited Market That Burned Some Big Investors - Wall Street Journal, September 12, 2005. Hedge Fund Losses, Credit Derivatives and Dr. Li s Copula?Financial Engineering News, Nov/Dec 2005. CDO Valuation: A Look at Fact and Fiction, By Robert Jarrow, Li Li, Mark Mesler, Don van Deventer, December 19, 2007. Recipe for Disaster: The Formula That Killed Wall Street, Felix Salmon, Wired Magazine, February 28, 2009. 11 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 11

How well or how bad does the model do in practice? How a Formula Ignited Market That Burned Some Big Investors - Wall Street Journal, September 12, 2005. Hedge Fund Losses, Credit Derivatives and Dr. Li s Copula?Financial Engineering News, Nov/Dec 2005. CDO Valuation: A Look at Fact and Fiction, By Robert Jarrow, Li Li, Mark Mesler, Don van Deventer, December 19, 2007. Recipe for Disaster: The Formula That Killed Wall Street, Felix Salmon, Wired Magazine, February 28, 2009. The formula that felled Wall Street, Financial Times, Jones, Sam (April 24, 2009). 11 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 11

How well or how bad does the model do in practice? How a Formula Ignited Market That Burned Some Big Investors - Wall Street Journal, September 12, 2005. Hedge Fund Losses, Credit Derivatives and Dr. Li s Copula?Financial Engineering News, Nov/Dec 2005. CDO Valuation: A Look at Fact and Fiction, By Robert Jarrow, Li Li, Mark Mesler, Don van Deventer, December 19, 2007. Recipe for Disaster: The Formula That Killed Wall Street, Felix Salmon, Wired Magazine, February 28, 2009. The formula that felled Wall Street, Financial Times, Jones, Sam (April 24, 2009). The formula that felled Wall Street? The Gaussian Copula and the Material Cultures of Modeling, Donald Mackenzie and Taylor Spears, June 2012 11 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 12

Correlation Parameter Compared to Moodys BET Use a sample transaction as an example David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 13

Correlation Parameter Compared to Moodys BET Use a sample transaction as an example It has a total name of 120 with a diversity score of 64, average rating of Baa2 and 1.35% default probability over 5-year period 13 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 13

Correlation Parameter Compared to Moodys BET Use a sample transaction as an example It has a total name of 120 with a diversity score of 64, average rating of Baa2 and 1.35% default probability over 5-year period Use a normal copula function with constant correlation number to match loss distribution up to the first two moments; It is found that the correlation in normal copula function is 6.9% 13 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 14

Probability Comparison Between BET and Copula Function Comparision Between BET and C 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.00 5.00 10.00 15.00 20.00 Loss Amount David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 15

Problems with Gaussian Copula Not completely theoretically justified. David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 16

Problems with Gaussian Copula Not completely theoretically justified. Many copula functions could be used and have been suggested: t-coplua, inverse Gaussian etc.. 16 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 16

Problems with Gaussian Copula Not completely theoretically justified. Many copula functions could be used and have been suggested: t-coplua, inverse Gaussian etc.. How could we simulate one period return and use the result to decide default in 5 or even 10 years? We have trouble to calibrate to the market from time to time. It has too strong correlation, this can be seen from conditional perspective. 16 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 16

Problems with Gaussian Copula Not completely theoretically justified. Many copula functions could be used and have been suggested: t-coplua, inverse Gaussian etc.. How could we simulate one period return and use the result to decide default in 5 or even 10 years? We have trouble to calibrate to the market from time to time. It has too strong correlation, this can be seen from conditional perspective. Hedge performance for certain tranches even though not fully investigated, but is pretty poor based on our experience. 16 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 16

Problems with Gaussian Copula Not completely theoretically justified. Many copula functions could be used and have been suggested: t-coplua, inverse Gaussian etc.. How could we simulate one period return and use the result to decide default in 5 or even 10 years? We have trouble to calibrate to the market from time to time. It has too strong correlation, this can be seen from conditional perspective. Hedge performance for certain tranches even though not fully investigated, but is pretty poor based on our experience. Valuation theory from financial economics: replication approach and equilibrium aapproach. 16 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 17

Replication of nth-to-default CDS Contracts (I) We introduce the following notations: m: the number of names in the basket. n: the order of the defaults specified in the swap. τ i : the time of the i-th basket entity with S i (t), F i (t), f i (t) denoting its survival distribution, cumulative distribution, and probability density function respectively. τ i,m : the ith order statistics for m survival times, τ 1, τ 2,, τ m, τ 1,m τ 2,m..., τ m,m. We denote the distribution function, density function or survival function for the order statistics as F n,m (t), f n,m (t), and S n,m (t). : the ith default for m 1 survival times, τ 1, τ 2,, τ s 1, τ s+1,, τ m, excluding τ s. τ (s) i,m 1 David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 18

Replication of nth-to-default CDS Contracts (II) Two Names: F 2,2 (t) = F 1 (t) + F 2 (t) F 1,2 (t) In general, market is not complete as we do not have the David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 19

Replication of nth-to-default CDS Contracts (II) Two Names: F 2,2 (t) = F 1 (t) + F 2 (t) F 1,2 (t) General Recursive Formula for order statistics (Sathe and Dixit(1990)) m r F r+1,m (t)+(m r) F r,m (t) = Fr,m 1(t), i for 1 r m 1, i=1 In general, market is not complete as we do not have the David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 19

Replication of nth-to-default CDS Contracts (II) Two Names: F 2,2 (t) = F 1 (t) + F 2 (t) F 1,2 (t) General Recursive Formula for order statistics (Sathe and Dixit(1990)) m r F r+1,m (t)+(m r) F r,m (t) = Fr,m 1(t), i for 1 r m 1, Every order can be expressed as the combinations of the first orders of equal or less names ( Balakrishnan et al.(1992)) m ( ) j 1 F n:m = ( 1) j+m n+1 H 1:j (x) m n j=m n+1 H 1:j = i=1 1 i 1 <i 2...<i m j m F (i 1,i 2,...i m j ) 1:j (x) In general, market is not complete as we do not have the David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 20

The Valuation of the nth-to-default CDS Contracts PV (loss side) = = = = For 2 n m, m i=1 m i=1 m i=1 m i=1 T N(1 R i ) D(t)P(N (i) t = n 1, τ i = t) 0 T N(1 R i ) D(t)P(N t = n, τ i = t) 0 T N(1 R i ) D(t)P(τ n,m t, τ i = t) 0 T N(1 R i ) D(t)P(τ n,m t τ i = t)f i (t) dt. 0 (n 1)P(τ n,m x τ i = t)+(m n + 1)P(τ n 1,m x τ i = t) = m P(τ (s) n 1,m 1 x τ i = t). s=1 David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 21

Merton Model (1974) The firm s asset value V t is specified to be GBM under the P measure. That is dv (t) V (t) = µdt + σdw t (1) The distance of default is defined as follows: X (t) = log [V (t)] log(d) σ X (t) is a Brownian motion with a unit variance parameter and a constant drift of m = (µ σ2 2 )/σ. Under the Merton model the default occurs at the maturity date T if (2) q(t ) = P [X (T ) 0] = Φ [B(T )] (3) where B(T ) = X (0) + mt T (4) David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 22

Merton Model (1974) Under risk neutral distribution we have µ = r. The risk neutral default probability using the Merton model is the same as equation (3) with the change of m to m = (r σ2 2 )/σ. That is, q (T ) = P [X (T ) 0] = Φ [B (T )] (5) where B (T ) = X (0) + m T T (6) David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 23

Default Probability: Physical Measure and Risk Neutral Measure From (3) and (5), the relationship between the risk neutral and nature or physical default probabilities is give as follows: ( q (T ) = Φ Φ 1 [q(t )] + µ r ) T. σ So the general relationship between the CDF functions under two measures is as follow: ( F (t) = Φ Φ 1 [F (t)] + µ r ) t. (7) σ David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 24

Wang Transform Intuitions of Wang Transform: F (x) = Φ ( Φ 1 [F (x)] + λ ). (8) For any non-normal financial risk, we don t know how to make measure change. David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 25

Wang Transform Intuitions of Wang Transform: F (x) = Φ ( Φ 1 [F (x)] + λ ). (8) For any non-normal financial risk, we don t know how to make measure change. To transform risk into a uniform risk using its distribution function F (X ) 25 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 25

Wang Transform Intuitions of Wang Transform: F (x) = Φ ( Φ 1 [F (x)] + λ ). (8) For any non-normal financial risk, we don t know how to make measure change. To transform risk into a uniform risk using its distribution function F (X ) To transform it into a normal risk using normal inverse function, Φ 1 (F (X )) 25 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 25

Wang Transform Intuitions of Wang Transform: F (x) = Φ ( Φ 1 [F (x)] + λ ). (8) For any non-normal financial risk, we don t know how to make measure change. To transform risk into a uniform risk using its distribution function F (X ) To transform it into a normal risk using normal inverse function, Φ 1 (F (X )) To make a measure change by changing its mean Φ 1 (F (X )) + λ 25 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 25

Wang Transform Intuitions of Wang Transform: F (x) = Φ ( Φ 1 [F (x)] + λ ). (8) For any non-normal financial risk, we don t know how to make measure change. To transform risk into a uniform risk using its distribution function F (X ) To transform it into a normal risk using normal inverse function, Φ 1 (F (X )) To make a measure change by changing its mean Φ 1 (F (X )) + λ To use normal or other CDF to make it into a CDF function G ( Φ 1 (F (X )) + λ ) 25 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 26

Default Probability Risk Neutral Distribution: Single Name Nature and Risk Neutral Default Probability 100% 80% 60% Default Prob (P) Default Prob (Q) 40% 20% 0% 0 10 20 30 40 50 60 70 80 Time Figure: Nature and Risk Neutral Default Probabilities David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 27

Bühlmann Theorem Bühlmann (1980) studied a risk exchange problem among a set of agents. Each agent has an exponential utility function u(w) = e λ i w for i = 1, 2,..., m and faces a risk of potential loss X j. In the equilibrium model, Bühlmann obtained the equilibrium pricing formula: π(x ) = E [ηx ], η = e λx E [e λx ], (9) where Z = m i=1 X i is the aggregate risk and λ is given by λ 1 = m i=1 λ 1 i, λ j > 0. (10) The parameter λ can be interpreted as the risk aversion index for a representative agent in the economy. David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 28

Measure Change in Multiasset Valuation Multivariate equity options: how to price a multivariate equity option such as the marginal risk neutral distribution is preserved? CDO squared pricing: How to price a CDO squared transaction such as the skew from each baby CDO tranche is matched? A. To use single name risk neutral distributions and a copula function to form a joint risk neutral distribution B. To have joint risk neutral transformation and then see how individual risk neutral distribution is. David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 29

Risk Neutralized with Joint Normal Distribution Under joint Geometrical Brownian motion, the marginal distribution of the risk neutral distribution is necessarily the univariate risk neutral distribution. dv i (t) [ ] V i (t) = µ idt + σ i dw i (t), E W (t)w (t) T = Σt ( dq t dp = exp λ(s), dw (s) 1 t ) λ(s), λ(s) ds t 0 2 t 0 λ(t) = [λ 1, λ 2,..., λ n ], λ i = µ i r σ i David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 30

Marginal Risk Neutral Distribution [ ] ( dq 1 dq = E dp 1 dp W 1(t) = exp λ 1 (W 1 (t) W 1 (t 0 ) 1 ) 2 λ2 1(t t 0 ) David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 31

Measure Changes for Risks specified by Normal Copula Under physical measure each firm survival time distribution is linked to asset return distribution as follows ( ) Xi (t) µ i (t) F (t) = Φ σ i t X i (t) and X j (t) are joint Wiener process We do the following: We first make a measure change to X i by changing only its mean Then we translate the measure change from X i to the joint measure change of survival times David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 32

Definition of Gaussian Copula for Stochastic Processes Definition (Gaussian Copula for Stochastic Processes) For n dimensional stochastic processes X (t), X (0) = 0 follows a Gaussian copula with correlation coefficient Σ ρ if there exists a standard Brownian motion vector W (t) = (W 1 (t), W 2 (t),..., W n (t)) whose correlation coefficient is Σ ρ, W (0) = 0 which makes X (t) joint distribution function to satisfy the following condition for any t > t 0 > 0 P(X 1 (t) < x 1,..., X n (t) < x n X 1 (t 0 ) = x 01,..., X n (t 0 ) = x 0n ) =P(W 1 (t) < w 1,..., W n (t) < w n W 1 (t 0 ) = w 01,..., W n (t 0 ) = w 0n ) where w i = Φ 1 (Fi t (x i )) t w 0i = Φ 1 (F t 0 i (x 0i )) t 0 David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 33

Basic Properties of Gaussian Copula for Stochastic Processes For given time t, the Gaussian copula defined for stochastic processes is the Gaussian copula for random variable vector X (t) with the correlation coefficient is Σ ρ. David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 34

Basic Properties of Gaussian Copula for Stochastic Processes For given time t, the Gaussian copula defined for stochastic processes is the Gaussian copula for random variable vector X (t) with the correlation coefficient is Σ ρ. If stochastic process vector X (t) satisfies the Gaussian copula condition, then X (t) is Markovian If stochastic process vector X (t) meets the Gaussian Copula condition, and the marginal distribution meets the following condition for any t > t 0 > 0, x i, x 0i Φ 1 (F t i (x i )) t = Φ 1 (F t t 0 i (x i x 0i )) t t 0 + Φ 1 (F t 0 i (x 0i )) t 0 Then, X (t) has independent, and homogeneous increments. 34 It is very preliminary and do not circulate without author s permission David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 35

Esscher Measure Changes for Stochastic Processes Y(t) = (Y 1 (t 1 ), Y 2 (t 2 ),..., Y n (t n )) T h = (h 1, h 2,..., h n ) For the moment generation vector we use [ ] M(h, t) = E e ht Y(t). Since we have independent and stationary decrements for the joint Brownian motions we still have the following Y(t) = [M(h, 1)] t, David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 36

Measure Changes for Risks specified by Normal Copula(continue) We use the positive martingale e ht Y(t) [M(h, 1)] t to define a new measure, the Esscher measure of parameter h for the stochastic processes Y(t). The risk neutral Esscher transform measure is the Esscher transform with parameter h such that, for each i = 1, 2,..., n, e rt V i (t) is a martingale. where e r = M(l i + h, 1) M(h, i = 1, 2,..., n, (11), 1) l i = (0,..., 0, 1, 0,..., 0), David X. where Li Shanghai theadvanced ith element Institute of Finance is 1, (SAIF) and Shanghai all other Theoretical Jiaotong elements Problems University(SJTU) in Credit are Portfolio zero. Modeling 37

Measure Changes for Risks specified by Normal Copula (continue) We have the following moment generating function for Y(t) Then, we have E M(h, t) = e t(ht µ+ 1 2 ht Σh). [ ] e zt Y(t) M(z + h, t) ; h = = e t(ht (µ+σh)+ 1 2 ht Σh). (12) M(h, t) David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 38

Joint Risk Neutral Distribution for Survival Times F (t 1,..., t n ) = Φ n (Φ 1 σ 2 Z [F 1 (t 1 )] + λ 1 β 1 t1 σ1 2,..., Φ 1 σ 2 ) Z [F n (t n )] + λ n β n tn σn 2 David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 39

Some general comments about the results It solves the problem of one period return, but multiperod survival or default It introduces volatility into the framework It is easy to compare different portfolio It has a good economic foundation David X. Li Shanghai Advanced Institute of Finance (SAIF) Shanghai Theoretical JiaotongProblems University(SJTU) in Credit Portfolio Modeling 40