Counterparty Risk Modeling: A Marked Default Time Perspective

Transcription

1 Counterparty Risk Modeling: A Marked Default Time Perspective Stéphane Crépey (numerics by Dong Li Wu and Hai Nam NGuyen) Université d'evry Val-d'Essonne, Laboratoire Analyse & Probabilités Advances in Financial Mathematics, 8 January 2013 This research presented in these slides beneted from the support of the Chair Markets in Transition under the aegis of Louis Bachelier laboratory, a joint initiative of École polytechnique, Université d'évry Val d'essonne and Fédération Bancaire Française 1 / 46

2 Outline 1 Bilateral Counterparty Risk under Funding Constraints / 46

3 Pricing and hedging counterparty risk (of defaulting counterparties) on.....(hundreds of) thousands of contracts...consistently in time and across all classes of assets Credit Valuation Adjustment (CVA) price of a giant hybrid option Large literature, in particular by Brigo et al. Related nonlinear funding issue under bilateral counterparty risk Crépey, S.: Bilateral Counterparty Risk under Funding Constraints - Part I: Pricing. Mathematical Finance online rst January 2013 Brigo, D., Pallavicini, A. and Perini, D.: Funding Valuation Adjustment consistent with TVA and DVA, wrong way risk, collateral, netting and re-hypotecation (working paper). Total Valuation Adjustment (TVA) 3 / 46

4 Netted portfolio with horizon T between the bank and its counterparty First party default time τ = τ b τ c Joint survival indicator process J t = 1 t<τ Eective time horizon τ = τ T Full pricing model ltration G Conditional expectation given (G t, Q) denoted by E t Bank's G τ -measurable exposure at default ξ(π) Bank's funding cost coecient g t (π) 4 / 46

5 Price and hedge: all-inclusive = mark-to-market TVA Clean price process (mark-to-market, ignoring counterparty risk) P t Risk-neutral expectation of promised cash ows discounted at the risk-free rate r t (set equal to 0 except in the numerics henceforth) Total valuation adjustment (TVA) dened (rather than derived for simplicity in these slides) by: τ ] Θ t = E t [1 τ<t ξ(p τ Θ τ ) + g s (P s Θ s )ds, t [0, τ] BSDE in integral form over the random time interval [0, τ] All-inclusive price (cost-of-the-hedge) Π = P Θ t A positive/negative Θ is deal adverse/friendly as it increases/decreases the bank's cost-of-the-hedge, hence it decreases/increases the bank's (buyer) price 5 / 46

6 Standard Reduced-Form Approach Modeling the counterparty default time τ through its intensity γ in a reference ltration F progressively enlarged by τ Crépey, S.: Bilateral Counterparty Risk under Funding Constraints - Part II: CVA. Mathematical Finance online rst January 2013 Computationally ecient Allows one to deal in an integrated setting with funding costs (also in a bilateral counterparty risk setup) But subject to an immersion hypothesis between a reference ltration and the ltration of the rst-to-default time τ of the two parties....and to the assumption that data do not jump at τ Assumptions ne in standard cases (e.g. for interest rate derivatives)..but too strong for cases of strong dependence between the underlying portfolio and the default of the two parties (e.g. for counterparty risk on credit derivatives) 6 / 46

7 Default Time with a Mark Dynamized copulas used for counterparty risk on credit derivatives Crépey, S., Jeanblanc, M. and Wu, D. L.: Informationally Dynamized Gaussian Copula, IJTAF, March Bielecki, T.R., Cousin, A., Crépey, S. and Herbertsson, A.: Dynamic Hedging of Portfolio Credit Risk in a Markov Copula Model, Journal of Optimization Theory and Applications (forthcoming). THIS WORK: unied perspective Modeling a default time τ with a mark Playing with the probability measure and/or the ltration Wrong-way and gap risks modeling 7 / 46

8 Reduced-form approaches possible in this perspective in the above dynamic copula setups Addressing the TVA high-dimensional nonlinear challenge of bilateral counterparty risk on credit derivatives under funding constraints Purely forward simulation schemes applicable (through expansions and/or particles) to the pre-default Markovian TVA equations over [0, T ].. Fujii, M. and Takahashi, A.: Perturbative expansion technique for non-linear FBSDEs with interacting particle method, arxiv Henry-Labordère, P.: Cutting CVA's complexity, Risk Magazine July as opposed to full model ltration TVA equations posed over the random interval [0, τ] 8 / 46

9 Outline 1 Bilateral Counterparty Risk under Funding Constraints / 46

10 τ = min e E τ e endowed with a mark e in a nite set E Stopping times τ e with Q(τ e τ e ) = 1 for e e Mark already implicitly present under bilateral counterparty risk in the standard approach: default of the bank alone, of the counterparty alone or of both simultaneously Hyp. ξ(π) = ξ e τ (π) on {τ = τ e } where ξ e t (π) is predictable Intensity γ t e of τ e Notation γ t f t = E γe t f t e for every f t = ft e Intensity γ t = e E γe t = γ t 1 E of τ Single step martingales for every semimartingale Y. ξ(y t )dj t + γ t ξ t (Y t )dt, 10 / 46

11 Reduced Pricing Model (F, P) Assumption (A). Let there be given, assumed to exist: A subltration F of G, A probability measure P Q such that an (F, P)-martingale stopped at τ is a (G, Q)-martingale. pseudo-stopping time τ in the immersion case P = Q... (A) the Azéma supermartingale of τ is FVP (has no martingale component)...but, via the mark ( e, richer ltration ) G than in the standard case where G t = F t (τ t) (τ < t) An F-adapted càdlàg process can jump at τ Conditional expectation given (F t, P) denoted by Ẽt 11 / 46

12 Reduced-Form TVA Modeling Proposition Assume an (F, P)-semimartingale Θ satises: Θ t = Ẽt T t fs ( Θ s )ds, t [0, T ] where f t (ϑ) is a pre-parties-default-f-representative of g t (P t ϑ) + γ t ξ t (P t ϑ) γ t ϑ and dene Θ = Θ on [0, τ) and Θ τ = 1 τ<t ξ(p τ Θ τ ). Then Θ satises the full model TVA (G, Q)-equation. Moreover, the (G, Q)-martingale component dµ t = dθ t + g t (P t Θ t )dt of Θ satises for t [0, τ]: ( dµ t = d µ t τ (ξ(p τ Θ τ ) Θ t )dj t + ( γ t ξ t (P t Θ ) t ) γ t Θ t )dt where d µ t = d Θ t + f t ( Θ t )dt is the (F, P)-martingale component of Θ. 12 / 46

13 Counterparty Risk on Credit Derivatives Let N = { 1, 0, 1,..., n}, N = {1,..., n}. Netted portfolio of credit derivatives on n underlying credit names (τ 1,..., τ n ) = (τ i ) i N Full credit dependence (wrong-way risk) model of (τ 1, τ 0, τ 1,..., τ n ) = (τ i ) i N with τ 1 τ b and τ 0 τ c In the following two sections we develop reduced-form TVA counterparty credit risk models for the (τ l ) l N 13 / 46

14 Outline 1 Bilateral Counterparty Risk under Funding Constraints / 46

15 References On the Dynamized Gaussian Copula Li, D.: On default correlation: A copula function approach, Journal of Fixed Income, Crépey, S., Jeanblanc, M. and Wu, D. L.: Informationally Dynamized Gaussian Copula. IJTAF, March Fermanian, J.-D. and Vigneron, O.: Pricing and hedging basket credit derivatives in the Gaussian copula. Risk Magazine, February 2010 (long version forthcoming in Quantitative Finance). El Karoui, N., Jeanblanc, M. and Jiao,Y.: What happens after a default: the conditional density approach. Stochastic Processes and their Applications, Jeanblanc, M. and Le Cam, Y.: Progressive enlargement of ltrations with initial times. Stochastic Processes and their Applications, / 46

16 Gaussian Copula Exponential-λ i default times τ i = λ 1 i log(φ(ε i )) ε i standard Gaussian r.v. with pairwise correlation ϱ 0 Φ Standard normal survival function The formula that killed Wall Street P(τ l > θ l, l N) = ( n R φ Standard normal density Φ( Φ 1 (P(τ l >θ l )) ϱy l=1 1 ϱ ) ) φ(y)dy > 0 Dynamized Gaussian Copula Informational dynamization ε i = + ς(t)db 0 t i ς( ) function with unit L 2 -norm B i standard BM with pairwise correlation ϱ Full model ltration G t = F B t ( (τi t) (τ i > t) ) i N 16 / 46

17 Conditional joint survival probability m t = (mt) l l N with mt l = t ς(u)db l 0 u θ t = (θt) l l N with θt l = τ l 1 τl t J t = supp(k t ) (resp. J t = N \ J t ) Random set of the obligors in default (resp. alive) at t Φ ρ,σ ((z j ) j J ) Survival function at (z j ) j J of a vector-gaussian distrib. with homogenous pairwise correlation ρ and stdev σ Proposition For every positive t, t l P(τ i > t i, i N G t ) = 1 {τi >t i, i J t} [ (ε j Φ ρt,σ t t(t t j ) )] [ ] Φ ρt,σ t (ε j t(t)) j Jt with for every s t ε j t(s) = h j(s) m j t σ(t) µ t for suitable ρ, σ, µ t = ρ, σ, µ (t, m t, θ t ) ρ t Contagion 17 / 46

18 Proof using A multidimensional counterpart of the key lemma of credit risk 1 {τ>t} E[X G t ] = 1 {τ>t} E(X 1 {τ>t} F B t ) P(τ > t F B t ) The stability of vector-gaussian distributions through conditioning of some components by the others Explicit update of the mean, volatility and correlation parameters 18 / 46

19 Markovian Structure (G, Q)-fundamental martingales dw i t = db i t γ i t dt, dm i t = d1 {τi t} λ i t dt for explicit processes γt, i λ i t γ i t = γ i (t, m t, θ t ), λ i t = λ i (t, m t, θ t ) explicit but combinatorially involved γt: i no immersion setup Process (m t, θ t ) is (G, Q)-Markov 19 / 46

20 Reduced-Form DGC TVA Model E = { 1, 0} Reduced ltration F t = F B t i N ( (τi t) (τ i > t) ) Equivalent measure P ( Q: no immersion setup) with fundamental (F, P)-martingales d W i t = db i t γ i dt, t i N, d M i t = d1 τi t λ i tdt, i N where γ i and λ i are pre-parties-default-f-representatives of γ i t and λi t W i t τ = W i t τ, M i t τ = M i t τ (A) satised by (F, P). 20 / 46

21 Unil. TVA on CDSs (MC with m = 10 4 runs) Unilateral TVA on one CDS computed by Fujii et al. expansions of increasing order/quality, for dierent levels of nonlinearity (unsecured borrowing basis λ); reference values computed by PHL branching scheme with 10 6 particles (ρ = 0.4). 21 / 46

22 Similar results regarding a portfolio of CDS contracts on ten dierent names. PHL reference values no longer available for variance reasons. 22 / 46

23 Unilateral TVA on one CDS computed by Fujii et al. best practice expansion, as a function of the DGC correlation parameter ϱ. 23 / 46

24 Similar results regarding a portfolio of CDS contracts on ten dierent names. 24 / 46

25 The errors (% relative standard deviation) of the dierent orders of the expansions don't explode with the number of names (ρ = 0.4) 25 / 46

26 The errors (% relative standard deviation) of the dierent orders of the expansions don't explode with the level of nonlinearity represented by the unsecured borrowing basis λ (ρ = 0.4) 26 / 46

27 Since the expansions are computed by purely forward MC schemes, their computation times are linear in the number of names 27 / 46

28 Outline 1 Bilateral Counterparty Risk under Funding Constraints / 46

29 References On the Dynamized Marshall-Olkin Copula Bielecki, T.R., Cousin, A., Crépey, S. and Herbertsson, A.: Dynamic Hedging of Portfolio Credit Risk in a Markov Copula Model, Journal of Optimization Theory and Applications (forthcoming). Elouerkhaoui, Y.: Pricing and Hedging in a Dynamic Credit Model. International Journal of Theoretical and Applied Finance, Brigo, D., Pallavicini, A., Torresetti, R.: Calibration of CDO Tranches with the Dynamical Generalized-Poisson Loss Model. Risk Magazine, May / 46

30 Static Marshall-Olkin Model Let I = {I 1,..., I m } denote a family of pre-specied subsets of N Sets of obligors susceptible to default simultaneously Defaults are the consequence of shocks aecting simultaneously pre-specied groups of obligors Dene, for Y Y = {{ 1}, {0}, {1},..., {n}} I, ane (as well as their sums) intensity processes Xt Y driven by independent BM under a pricing measure Q, and W Y t τ Y = inf{t > 0; t 0 X Y s ds ɛ Y } for independent standard exponential random variables ɛ Y Set, for every i N, τ i = Y Y; I i τ I = τ {i} I I; I i τ I 30 / 46

31 Example: n = 5 and Y = {{ 1}, {0}, {1}, {2}, {3}, {2, 3}, {0, 1, 2}, { 1, 0}} t 31 / 46

32 Dynamized Marshall-Olkin Informational dynamization through G t = F W t ( (τi t) (τ i > t) ) i N where W = (W Y t ) Y Y. Let also X = (X Y t ) Y Y. Proposition For any xed non-negative constants t, t 1,..., t n, we have: { Q (τ i > t i, i N G t ) = 1 {ti <τ i, i I t}e where t Y t = t max i Y Jt t i. For every obligor i and t i t, { Q(τ i > t i G t ) = 1 {τi >t}e exp ( ti t exp ( Y Y X i s ds) X i t t Y t t }, X Y s ds) Xt }, where X i = Y Y; i Y X Y (an ane process). 32 / 46

33 Markovian Structure (G, Q)-fundamental martingales W Y, Y Y and M Z, Z N where dm Z t = d1 τz t γ Z t dt τ Z Time of a joint default of names in set Z and only in Z γ Z t = Y Y; X Y Y t =Z t = γ Z (t, X t, H t ) Y t ) Y Y Set of survivors of set Y right before t X t = (X Y t H t = (Ht) i i N with H i t = 1 τi t Process (X t, H t ) is (G, Q)-Markov 33 / 46

34 Reduced-Form DMO TVA Model E = {Z N; 1 Z} {Z N; 0 Z} Let for every obligor i N τ i = min {Y Ỹ; i Y } τ Y where Ỹ = {{1},..., {n}} Ĩ, in which Ĩ consists of those I j in I which do not contain 1 or 0. Reduced ltration F such that for every t F t = F W t ( ( τi t) ( τ i > t) ) i N Markov copula DMO features fundamental (F, Q)-martingales W Y, Y Y and M Z, Z N (A) satised by (F, P = Q) 34 / 46

35 Unilateral TVA on a (mezzanine) CDO tranche computed by Fujii et al. expansions of order 1 to 3 (120 names) 35 / 46

36 The errors (% relative standard deviation) of the dierent orders of the expansions don't explode with the dimension ( λ = 0.01) 36 / 46

37 The errors (% relative standard deviation) of the dierent orders of the expansions don't explode with the level of nonlinearity represented by the unsecured borrowing basis (120 names) 37 / 46

38 Since the expansions are computed by purely forward MC schemes, their computation times don't explode with the number of names ( λ = 0.01) 38 / 46

39 Outline 1 Bilateral Counterparty Risk under Funding Constraints / 46

40 Cure period Time lag δ > 0 between the rst default time of a party and the close-out cash-ow No less than δ = two weeks for Basel III capital charges CVA computations > 5 years in the case of the New-York branch of Lehman Brothers (closing 31 dec 2013) Introduces a new layer of technicality since the terminal time of the problem is then given as the close-out (or liquidation) time (τ + δ), which, as opposed to the default time τ, is a predictable stopping time (as announced by τ), so that it has no intensity. Work in progress with Shiqi Song 40 / 46

41 For every time u, we write ū = u T, u δ = u + δ, ū δ = 1 {u<t } u δ + 1 {u T } T With a positive cure period δ, the relevant time horizon of the problem becomes τ δ and the exposure ξ δ (π) is G τδ -measurable. We dene (in particular, P t = P t for t τ). P t = P t + D t D τ, t [0, τ δ ] The full TVA BSDE with cure period is written, over [0, τ δ ], as: τδ ] Θ t = E t [1 τ<t ξ δ ( P τδ Θ τδ ) + g s ( P s Θ s )ds, 0 t τ δ t 41 / 46

42 Or equivalently in two parts, after and before τ : Θ t = E t [1 τ<t ξ δ ( P τδ Θ τδ ) + (1) τδ ] g s ( P s Θ s )ds, τ t τ δ t τ ] Θ t = E t [Θ τ + g s (P s Θ s )ds, 0 t τ. (2) t Here the idea is that a terminal condition to (2) is determined by a solution, assumed to exist, to (1). The BSDE (1) is radically dierent from (2) in that τ δ in (1) is predictable, as announced by τ, whereas τ in (2) is totally inaccessible, as endowed with an intensity. 42 / 46

43 Reduction of the cure period TVA BSDE (1) To get rid of the endogenous terminal condition in (1), we assume that the equation θ = 1 {τ<t } ξ δ ( P τδ θ), for a G τδ -measurable random variable θ, has a solution of the form θ = 1 {τ<t } ξ δ, for some suitable G τδ -measurable random variable ξ δ. Standing example: ξ δ (π) = ξ δ c 1 τ=τb (1 Rb )(π C) + for G τδ -measurable random variables ξ δ c, Rb and C : ξ δ c R b The CVA/DVA exposure A (0, 1]-valued recovery rate of the bank to its funder C The value of the accumulated collateral at time τ The above equation for a G τδ -measurable random variable θ is uniquely solved by θ = 1 τ<t ξ, δ where { R 1 ξ δ b = ξδ c + (1 R 1 )( P b τδ C) on {τ = τ b } {ξ δ c P τδ C} ξ δ otherwise 43 / 46

44 Assume ξ δ = ξ δ,τ where ξ δ,u For every time u, we write for t [ū, ū δ ] is G uδ -measurable for every time u P u t = P t + D t D u, f u t (ϑ) = g(p u t ϑ) The cure period TVA BSDE (1) is equivalent to the following BSDE for a (G, Q)-semimartingale Θ u over [u, ū δ ], conditional on τ = u: ( ūδ ) Θ u t = E t 1 {u<t } ξ δ,u + f u s (Θ u s )ds, t [ū, ū δ ] t 44 / 46

45 Reduction of the pre-default TVA BSDE (2) Assuming (1) solved, in order to solve the pre-default TVA BSDE (2), one can follow a reduced-form approach as in the no cure period case, with γ t Θt instead of γ t ξ t (π) in the denition of f t (π), where Θ is a G-predictable process (which exists) such that E(Θ τ τ G τ ) = Θ τ Accordingly let f δ t (ϑ) denote a pre-parties-default-f-representative of g t (P t ϑ) + γ t ( Θ t ϑ) 45 / 46

46 Proposition (TVA Modeling with ) Assume that, for every u (0, T ], a (G, Q)-semimartingale Θ u satises the following BSDE over [u, ū δ ] : ( ūδ ) Θ u t = Ẽt 1 {u<t } ξ δ,u + f u s (Θ u s )ds, t [u, ū δ ] and that an (F, P)-semimartingale Θ δ satises the following BSDE: Θ δ t = Ẽt T t t f δ s ( Θ δ s )ds, t [0, T ] Let Θ = Θ δ on [0, τ) and 1 τ<t Θ τ on [ τ, τ δ ]. Then the process Θ satises the full TVA equation with cure period on [0, τ δ ]. Moreover, on [0, τ], the (G, Q)-martingale component d µ t = dθ t + g t (P t Θ t )dt of Θ satises: ( dµ t = (Θ τ τ Θ ) t )dj t + γ t ( Θ t Θ t )dt, where d Θ δ t = d Θ δ t + f δ t ( Θ δ t )dt is the (F, P)-martingale component of Θ δ. The numerical computation of Θ involves resimulation for the Θ u. 46 / 46

47 Fujii and Takahashi's Expansions Θ t = Θ (0) (0) (1) (2) (3) Θ t + Θ t + Θ t + Θ t +... t = 0, [ Θ (1) T t = Ẽt [ Θ (2) T t = Ẽt [ Θ (3) T t = Ẽt t t t ( f s, X s, Θ (1) s ( Θ(2) s Θ Θ ) ] (0) Θ s ds, ( f s, X s, ( f s, X s, ) ] (0) Θ s ds,... ) (0) Θ s ( Θ(1) ) 2 2 ( s Θ f s, X s, 2 )) ] (0) Θ s ds. 47 / 46

48 Interacting particles implementation [ Θ (1) 0 = Ẽ 1 ζ1<t e λ1ζ1 λ 1 [ Θ (2) 0 = Ẽ e λ1ζ1 1 ζ1+ζ2<t e λ2ζ2 [ Θ (3,1) 0 = Ẽ ( f ζ 1, X )] ζ 0 (0) 1, Θ ζ 1, λ 1 ( f ζ 1, X ) ζ 0 (0) 1, Θ ζ 1 Θ ( f ζ 1 + ζ 2, X (0) ζ 1ζ1+ζ2, Θ ζ 1+ζ 2 )], λ 2 e λ1ζ1 1 ζ1+ζ2+ζ 3<T λ 1 e λ2ζ2 λ 2 e λ3ζ3 λ 3 Θ f ( f ζ 1, X ) ζ 0 (0) 1, Θ ζ 1 Θ ( ) ζ1 (0) f ζ 1 + ζ 2, X ζ 1+ζ 2, Θ ζ 1+ζ 2 ( )] ζ1+ζ2 (0) ζ 1 + ζ 2 + ζ 3, X ζ 1+ζ 2+ζ 3, Θ ζ 1+ζ 2+ζ 3 48 / 46

49 Θ (3,2) 0 = [1 Ẽ e λ1ζ1 1 ζ1+ζ2<t 1 ζ1+ζ 2 <T λ 1 2 e λ2ζ2 ( ) ζ1 (0) f ζ 1 + ζ 2, X ζ 1+ζ 2, Θ ζ 1+ζ 2 λ 2 e λ 2 ζ 2 f λ 2 2 Θ 2 f ( )] ζ 1 + ζ 2, ζ1 (0) X ζ 1+ζ, Θ 2 ζ 1+ζ. 2 ( ζ 1, X ) ζ 0 (0) 1, Θ ζ 1 Trick f (t, X t, Θ ) t = α(t) Θ t + ( f (t, X t, Θ ) ) t α(t) Θ t. 49 / 46