7 th General AMaMeF and Swissquote Conference 2015

Transcription

1 Linear Credit Damien Ackerer Damir Filipović Swiss Finance Institute École Polytechnique Fédérale de Lausanne 7 th General AMaMeF and Swissquote Conference 2015

2 Overview

3 Credit Risk(s) Default time τ Payments τ time Default likelihood variation P t+1 [τ T ] 5% 1% New regulatory requirements (Basel III, IFRS 9) Expected losses (12-month, lifetime), deterioration of credit quality, valuation adjustments (CVA, DVA), etc. Tractable models?

4 Ingredients F = (F t ) Doubly-stochastic default time risk-factors filtration (no default) S t = e t 0 λsds F-measurable survival process with the hazard-rate λ t 0 U Default time construction uniform r.v. on (0, 1) independent from F τ = inf {t 0 : S t U} H t = σ(1 {τ s} : s t) G t = (F t H t ) in which τ is a H stopping time all the observable information Survival process S t = P[τ > t F t ] = P[τ > t F ]

5 The Survival Process X t E R m risk-factor process and vector γ R m. Standard approach Model the hazard-rate process: λ t = γ X t 0. A new approach Model the survival process directly! t S t = P [τ > t F t ] = 1 0 which implies ds t = γ X t dt and such that γ X s ds > 0 (1) λ t = γ X t S t 0 (2) How to construct X t such that (1) and (2) are verified?

6 One Risk-Factor W.l.o.g. let γ = 1 such that ds t = X t dt. Assume that there exists a constant L > 0 such that 0 X t LS t then S t e Lt > 0 and L λ t 0. Assume that (X t, S t ) is a polynomial preserving diffusion, then dx t = (b + βx t + BS t )dt + σ X t ( LS t X t )dw t for some reals b, β, B and σ 0. Lemma The process (X t, S t ) is well-defined if and only if b = 0, B 0, and L 2 + βl + B 0.

7 Inward pointing condition The state space E is of the form S t (X 0, S 0 ) 1 One Risk-Factor II S u dx u ds u dx u 0 0 X u L X t

8 One Risk-Factor III Hazard-rate dynamics dλ t = (l 1 λ t )(λ t l 2 ) dt + σ λ t (L λ t ) dw t One risk-factor affine model (CIR process) dλ t = l 2 (λ t l 1 ) dt + σ λ t dw t Drift Diffusion 0 σ 3σ 0 l 1 L 0 σ 0 L

9 Multiple Risk-Factors W.l.o.g. let γ 1 = 1 and assume X t [0, LS t ] m such that 0 λ t L. ds t /dt = γ X t γ 1S t = LS t The dynamic of the multivariate process X t rewrites dx t = (b + βx t + BS t )dt + Σ(X t, S t )dw t Σ(X t, S t ) = diag(σ 1 X1t (LS t X 1t ),..., σ m Xmt (LS t X mt )) and with b, B R m, β R m m, and σ R m +. Lemma The process (X t, S t ) is well defined if and only if b = 0, B ( Lβ ij ) + and Lβ ii B i (L(γ i L + β ij )) + j i j i

10 Preliminaries Risk-neutral valuation There exists an equivalent risk-neutral martingale measure Q. The discount process is D t = e t 0 rsds with the short-rate r s. D t and S t have the same properties! Process moments (X t, S t ) polynomial preserving process with state space E Pol n (E) space of polynomials of order at most n on E H n vector of polynomials forming a basis of Pol n (E) G n matrix representation of A Poln(E) w.r.t. H n p vector representation of p(x, s) Pol n (E) w.r.t. H n Lemma (Cuchiero et al. (12), Filipović and Larsson (15)) E [p(x T, S T ) F t ] = H n (X t, S t ) e Gn(T t) p.

11 Defaultable bond Security B pays one if τ > T, zero otherwise. Assume for simplicity r t = 0, and consider the monomial basis H 1 (x, s) = {1, x 1,..., x m, s}. B(t, T ) = 1 {τ>t} E Q [ 1 {τ>t } G t ] = 1 {τ>t} E Q [ e T t λ udu F t ] = 1 {τ>t} E Q [ ST S t F t = 1 {τ>t} {1, X t, S t } e G 1(T t) {0, 0, 1/S t } ] where the matrix G 1 is given by ( 0 0 ) 0 G 1 = 0 β γ. 0 B 0 Only the first F t -conditional moment of S T is needed!

12 Contingent Cash-Flow Security C pays one at τ if and only if t < τ < T. Assume for simplicity that r t = r constant. C(t, T ) = 1 {τ>t} E Q [ 1 {t<τ<t } e r(τ t) G t ] = 1 {τ>t} T t = 1 {τ>t} T t e r(s t) E Q [ λ s e s t λudu F t ] ds e r(s t) E Q [ γ X s S t F t ] ds = 1 {τ>t} {1, X t, S t } (G 1 ) 1 ( e G 1 (T t) I ) {0, γ/s t, 0} with G 1 = G 1 diag(r) and using t 0 eas ds = A 1 (e At I ). No numerical integration over [t, T ]! Credit-Default-Swap sum of C and many Bs π cds t = {1, X t /S t } p cds (m=1) = {1, λ t } p cds

13 Modeling Choices Set r t = r constant over the estimation period. A cascading structure Consider m = 2 and dynamics of the form: ds t = X 1t dt dx 1t = κ 1 ( θ 1 X 2t dx 2t = κ 2 ( θ 2 S t X 1t ) dt + σ 1 X1t (LS t X 1t ) dw 1t X 2t ) dt + σ 2 X2t (LS t X 2t ) dw 2t Market price of risk dwt = dw t + Λ t dt Assume (X t, S t ) polynomial preserving process under P an Q Λ 1t = δ 1 X1t σ 1 LSt X 1t Λ 2t = δ 2 X 2t + δ 3 S t σ 2 X2t (LS t X 2t ) Stronger conditions on dynamics parameters! 10 parameters.

14 Estimation Results on AT&T 150 Market Fitted L 0 X 2t /S t X 1t /S t = λ t QML using UKF CDS spreads in basis points 1 to 10-year maturity L 8% RMSE < 5 b.p.

15 Estimation Results on Boeing 300 Market Fitted L 0 X 2t /S t X 1t /S t = λ t QML using UKF CDS spreads in basis points 1 to 10-year maturity L 12% RMSE < 4 b.p.

16 Single-name CDS Option European CDS option with maturity T, r s = 0 [ ( ) ] + πt opt = E Q 1 {τ>t } π cds Gt = 1 {τ>t} E Q S t T [ ( {X T, S T } p cds) + Ft ] Z = {X T, S T } p cds /S t [a, b] with known F t -cond moments. Payoff approximation Polynomial series p n (y) converging to (y) + on [a, b] such that E [p n (Z)] n πopt t with non-tight error upper bound p n (z) (z) + on [a, b].

17 CDS Option Example 0.15 n = 2 n = (z) + n = z 5-year CDS option on AT&T 1-year maturity 100 b.p. strike Chebyshev interpolation Z [ 0.058, 0.16] E[p n(z)] error bound n n

18 Conclusion New class of reduced-form models for credit-risk Survival process modeling S t = P[τ > t F t ] with PPP Analytical formulas for defaultable bonds and CDS prices Straightforward approximation of CDS options prices Promising directions: CVA, multi-name models,...