A Gaussian-Poisson Conditional Model for Defaults

Transcription

1 1 / 26 A Gaussian-Poisson Conditional Model for Defaults Ambar N. Sengupta Department of Mathematics University of Connecticut 2016 High Frequency Conference

2 2 / 26 Initial remarks Risky Portfolios and Copula Models Idealized Model Default Times, Events, and Counts A Poisson model Conditioning

3 3 / 26 Initial remarks The purpose of this talk is to draw attention to the possible uses of combining different probabilistic models by means of conditioning.

4 3 / 26 Initial remarks The purpose of this talk is to draw attention to the possible uses of combining different probabilistic models by means of conditioning. The focus in this talk will be on the Gaussian copula model and the Poisson distribution, but this is meant to be a basic example rather than all there is to it.

5 3 / 26 Initial remarks The purpose of this talk is to draw attention to the possible uses of combining different probabilistic models by means of conditioning. The focus in this talk will be on the Gaussian copula model and the Poisson distribution, but this is meant to be a basic example rather than all there is to it. I will describe results obtained in joint work with Tyler Brannan.

6 4 / 26 Risky Portfolios Prior to the financial crisis there was great interest in modeling Credit Default Swaps and CDOs of various types.

7 4 / 26 Risky Portfolios Prior to the financial crisis there was great interest in modeling Credit Default Swaps and CDOs of various types. CDO:

8 4 / 26 Risky Portfolios Prior to the financial crisis there was great interest in modeling Credit Default Swaps and CDOs of various types. CDO: portfolio of assets that have default risk

9 4 / 26 Risky Portfolios Prior to the financial crisis there was great interest in modeling Credit Default Swaps and CDOs of various types. CDO: portfolio of assets that have default risk CDOs and other structured products haven t disappeared.

10 4 / 26 Risky Portfolios Prior to the financial crisis there was great interest in modeling Credit Default Swaps and CDOs of various types. CDO: portfolio of assets that have default risk CDOs and other structured products haven t disappeared. Securitization of debt products is still central to the financial system.

11 5 / 26 Copula Models Copula models describe correlated behavior of a family of random variables.

12 5 / 26 Copula Models Copula models describe correlated behavior of a family of random variables. Despite having suffered a severe blow to its reputation, the copula method for dependence modeling remains in widespread use

13 5 / 26 Copula Models Copula models describe correlated behavior of a family of random variables. Despite having suffered a severe blow to its reputation, the copula method for dependence modeling remains in widespread use from hydrology to actuarial science.

14 6 / 26 Idealized Model: default proxies As an illustration, consider a basket of M defaultable entities.

15 6 / 26 Idealized Model: default proxies As an illustration, consider a basket of M defaultable entities. The default behavior is governed by random variables X 1, X 2,..., X M.

16 6 / 26 Idealized Model: default proxies As an illustration, consider a basket of M defaultable entities. The default behavior is governed by random variables X 1, X 2,..., X M. When X j takes a value a threshold value c j we view this as default of the j-th constituent of a portfolio:

17 6 / 26 Idealized Model: default proxies As an illustration, consider a basket of M defaultable entities. The default behavior is governed by random variables X 1, X 2,..., X M. When X j takes a value a threshold value c j we view this as default of the j-th constituent of a portfolio: [X j c j ] = default of j-th asset

18 7 / 26 Idealized Model: Loss formula Default of name j results in a loss of amount l j.

19 7 / 26 Idealized Model: Loss formula Default of name j results in a loss of amount l j. Thus the loss resulting from name j is described by l j 1 [Xj c j ]

20 7 / 26 Idealized Model: Loss formula Default of name j results in a loss of amount l j. Thus the loss resulting from name j is described by l j 1 [Xj c j ] and the portfolio loss is L = M l j 1 [Xj c j ]. j=1

21 7 / 26 Idealized Model: Loss formula Default of name j results in a loss of amount l j. Thus the loss resulting from name j is described by l j 1 [Xj c j ] and the portfolio loss is L = M l j 1 [Xj c j ]. j=1 Note: In this analysis we ignore all issues of defaults occurring at different times...

22 8 / 26 Gaussian Copula The Gaussian Copula model posits that

23 8 / 26 Gaussian Copula The Gaussian Copula model posits that is Gaussian. (X 1,..., X M )

24 9 / 26 Other Copulas There are, of course, countless other choices instead of Gaussian.

25 10 / 26 Default times For our portfolio of assets, labeled 1,..., M, let τ j = default time of asset j

26 10 / 26 Default times For our portfolio of assets, labeled 1,..., M, let τ j = default time of asset j Let us observe the portfolio at discrete times: 0 = t 0 < t 1, <... < t n

27 11 / 26 Default events Let D j,k = [t k 1 < τ j t k ], the event that asset j defaults in the time window (t k 1, t k ].

28 12 / 26 Default counts Let ν k = M j=1 1 [τj t k ],

29 12 / 26 Default counts Let ν k = M j=1 1 [τj t k ], the number of events up to time t k. The number of events detected for the first time at time t k is ν k = M j=1 1 Dj,k

30 13 / 26 Poisson count model Consider now a counting process t N(t) {0, 1, 2, 3,...} with N(0) = 0, which is Poisson,

31 13 / 26 Poisson count model Consider now a counting process t N(t) {0, 1, 2, 3,...} with N(0) = 0, which is Poisson, at least when conditioned on some other events, such as specification of the interest rate process r( ).

32 14 / 26 Conditioning one model to the other We have on one hand the process t N(t) {0, 1, 2, 3,...} and on the other hand the count ν k of defaults by time t k.

33 14 / 26 Conditioning one model to the other We have on one hand the process t N(t) {0, 1, 2, 3,...} and on the other hand the count ν k of defaults by time t k. We introduce then the conditional probability P = Probability conditional on [ν 1 = N(t 1 ),..., ν n = N(t n )]

34 15 / 26 Conditioning one model to the other We have on one hand the process t N(t) {0, 1, 2, 3,...} and on the other hand the count ν k of defaults by time t k.

35 15 / 26 Conditioning one model to the other We have on one hand the process t N(t) {0, 1, 2, 3,...} and on the other hand the count ν k of defaults by time t k. We introduce then the conditional probability P = Probability conditional on [ν 1 = N(t 1 ),..., ν n = N(t n )]

36 16 / 26 How to actually define P precisely Often a main difficulty about a conditional probability measure is its precise definition, beyond just the idea.

37 16 / 26 How to actually define P precisely Often a main difficulty about a conditional probability measure is its precise definition, beyond just the idea. [ P P (A) = E 0 0 [ A [ν 1 = N(t 1 ),..., ν n = N(t n )] Ft r ]] n P [ ] 0 ν 1 = N(t 1 ),..., ν n = N(t n ) Ft r, n

38 16 / 26 How to actually define P precisely Often a main difficulty about a conditional probability measure is its precise definition, beyond just the idea. [ P P (A) = E 0 0 [ A [ν 1 = N(t 1 ),..., ν n = N(t n )] Ft r ]] n P [ ] 0 ν 1 = N(t 1 ),..., ν n = N(t n ) Ft r, n where F r t n is the sigma-algebra generated by the background (interest-rate) process r( ) up to time t n, and P 0 is the market pricing probability measure, or other relevant a priori measure.

39 17 / 26 The denominator We can check that the denominator is in fact positive:

40 17 / 26 The denominator We can check that the denominator is in fact positive: P 0 [ν 1 = N(t 1 ),..., ν n = N(t n ) Ft r n ] = n Λ a j P 0 [( ν 1,..., ν n ) = a] e Λ j j a a j!. (1) where Λ j is the mean of the Poisson variable N(t j ) N(t j 1 ), conditional on F r t n. j=1

41 18 / 26 The conditioned process r( ) Theorem The distribution of the process r on [0, t n ] is the same under the measure P as under P 0.

42 19 / 26 The conditioned process r( ) Proof. Consider now an event G F r t n. Then [ P P (G) = E 0 0 [ G [ν 1 = N(t 1 ),..., ν n = N(t n )] Ft r ]] n P [ ] 0 ν 1 = N(t 1 ),..., ν n = N(t n ) Ft r n

43 19 / 26 The conditioned process r( ) Proof. Consider now an event G F r t n. Then [ P P (G) = E 0 0 [ G [ν 1 = N(t 1 ),..., ν n = N(t n )] Ft r ]] n P [ ] 0 ν 1 = N(t 1 ),..., ν n = N(t n ) Ft r n [ E = E 0 0 [ 1 [ν1 =N(t 1 ),...,ν n=n(t n)]1 G Ft r ] ] n P [ ] 0 ν 1 = N(t 1 ),..., ν n = N(t n ) Ft r n

44 19 / 26 The conditioned process r( ) Proof. Consider now an event G F r t n. Then [ P P (G) = E 0 0 [ G [ν 1 = N(t 1 ),..., ν n = N(t n )] Ft r ]] n P [ ] 0 ν 1 = N(t 1 ),..., ν n = N(t n ) Ft r n [ E = E 0 0 [ 1 [ν1 =N(t 1 ),...,ν n=n(t n)]1 G Ft r ] ] n P [ ] 0 ν 1 = N(t 1 ),..., ν n = N(t n ) Ft r n = E 0 [1 G ] = P 0 [G]. Thus, the distribution of the process r on [0, t n ] is the same under the measure P as under P 0.

45 20 / 26 Default descriptor Y Let S = {1, 2,..., n, } M, (2) where we interpret a point y = (y 1,..., y M ) S to be a list of potential default times t yj.

46 20 / 26 Default descriptor Y Let S = {1, 2,..., n, } M, (2) where we interpret a point y = (y 1,..., y M ) S to be a list of potential default times t yj. Let Y = (Y 1,..., Y M ) be the S-valued random variable defined on (Ω, F) specified by Y j = k on D j,k. (3) Thus Y j = k if the default of asset j is observed first at time t k.

47 Default descriptor Y Let S = {1, 2,..., n, } M, (2) where we interpret a point y = (y 1,..., y M ) S to be a list of potential default times t yj. Let Y = (Y 1,..., Y M ) be the S-valued random variable defined on (Ω, F) specified by Y j = k on D j,k. (3) Thus Y j = k if the default of asset j is observed first at time t k. Moreover, for any y S, [Y = y] = M p=1 D p,y p. (4) The random variable Y gives a complete picture of which asset is observed to default at which time. 20 / 26

48 21 / 26 The distribution of default times The distribution of Y, under the probability measure P, is given by the values P[Y = y]=p[d 1,y1... D M,yM ]. (5) Theorem With framework as above, let F r t n be the sigma-algebra generated by {r(u) : 0 0 t n }. Then P [Y = y F r t n ] = P 0 [Y = y] n q=1 P 0 [Y = y] ȳ S Λ ν e q (y) Λq q ν q(y)! n q=1 Λ ν e q (y) Λq q ν q(y)!, with Λ q being the conditional expectation E 0 [Ñ(t q) F r t n ].

49 22 / 26 The distribution of default times The distribution of Y, under the probability measure P, is given by the values P[Y = y]=p[d 1,y1... D M,yM ]. (6) Theorem With framework as above, let F r t n be the sigma-algebra generated by {r(u) : 0 0 t n }. Then P [Y = y F r t n ] = P 0 [Y = y] n q=1 P 0 [Y = y] ȳ S Λ ν e q (y) Λq q ν q(y)! n q=1 Λ ν e q (y) Λq q ν q(y)!, with Λ q being the conditional expectation E 0 [Ñ(t q) F r t n ].

50 23 / 26 The Gaussian Case If we now assume that the default times τ 1,..., τ M are described by a Gaussian copula then more concrete formulas are obtained.

51 23 / 26 The Gaussian Case If we now assume that the default times τ 1,..., τ M are described by a Gaussian copula then more concrete formulas are obtained. In this case we are also able to compute exact formulas for sensitivities to various model parameters.

52 24 / 26 The Gaussian Case: Sensitivity For instance, we have found that for a single time horizon, ( E [lk s sgn ] ) ( ) Λ = sgn α α

53 24 / 26 The Gaussian Case: Sensitivity For instance, we have found that for a single time horizon, ( E [lk s sgn ] ) ( ) Λ = sgn α α where l s k is a senior tranche loss (number of defaults k),

54 24 / 26 The Gaussian Case: Sensitivity For instance, we have found that for a single time horizon, ( E [lk s sgn ] ) ( ) Λ = sgn α α where lk s is a senior tranche loss (number of defaults k), α is a model parameter, and Λ is the Poisson parameter.

55 25 / 26 Short version Conditional probabilities could give interesting/richer models.

56 Thank you! 26 / 26