Integrated structural approach to Counterparty Credit Risk with dependent jumps

Transcription

1 1/29 Integrated structural approach to Counterparty Credit Risk with dependent jumps, Gianluca Fusai, Daniele Marazzina Cass Business School, Università Piemonte Orientale, Politecnico Milano September 2015

2 2/29 General context of the problem Firm 1 (S 1 ) - Counterparty; Firm 2 (S 2 ) - Investor Counterparty Credit Risk: risk of a party to a financial contract defaulting prior to/at the contract s expiration Credit Value Adjustment (): 1 = (1 R 1 )E ( 1 (τ1 min(τ 2,T ))Ψ + (τ 1 ; S 3, T ) ) Ψ ( ) - disc. value of OTC contract on S 3 (underlying asset) τ j = inf {t 0 : S j (t) K j }, j = 1, 2 R j - recovery rate Asset j, j = 1, 2 Motivation Regulatory framework - Bilateral vs Unilateral CCR not fully mitigated by collateral 65% losses from CCR due to during the financial crisis

3 3/29 Structural approach to credit risk for unified treatment pricing Right-Way-Risk/Wrong-Way-Risk Mitigating clauses - netting & collateral Calibration Multivariate Lévy processes Independent and stationary increments Brownian motion with drift + pure jump process Skewness and excess kurtosis Joint evolution of risk factors Improved calibration of credit spreads over short maturities Intermezzo 1 Efficient numerical schemes (exotic option pricing)

4 4/29 Structural approach to credit risk modelling Multivariate Lévy processes Construction Dependence features pricing calibration Correlation effect Gap risk s and work in progress

5 5/29 Structural approach to default Construction I Risk neutral dynamic S j (t) = S j (0)e (r q j ϕ j ( i))t+x j (t), X j (t) - Lévy process j = 1,..., n ϕ j ( i)t characteristic exponent of X j (t) r > 0 - risk free rate of interest q j > 0 - dividend yield of the j th asset Non defaultable underlying asset Dependence: factor representation (Ballotta and Bonfiglioli, 2014)

6 6/29 Multivariate Lévy processes Construction I Lévy processes Known characteristic function Invariant under linear transformation X j (t) = Y j (t) + a j Z(t) a j R, j = 1,..., n X j (t), Y j (t), Z(t) are Lévy processes Y j (t): idiosyncratic risk process Z(t): systematic risk process Y j (t) and Z(t) independent and distinct Correlation coefficient correctly represents dependence Var (Z(1)) ρ jl = a j a l Var (Xj (1))Var (X l (1)) Dependence structure can be isolated from marginal distributions

7 7/29 Pricing Pricing Contract Numerics Benchmark Long position in a contract on S 3 (equity index, commodity, currency rate,...) First passage time approach Default event: τ j = inf {t 0 : S j (t) K j }, j = 1, 2 First to default problem 1 = (1 R 1 )E ( 1 (τ1 T )1 (τ2 >τ 1 )Ψ + (τ 1 ; S 3, T ) ) Bucketing : default can only occur on time grid {t j : 0 j N} for t 0 = 0, t N = T 1 (1 R 1 ) N j=1 E (1 (tj 1 <τ 1 t j) 1 (τ 2 >t j) Ψ+ (t j ; S 3, T ) Conditioning on {Z(t), 0 < t T } 1 (1 R 1 ) N j=1 E [ P Z (t j 1 < τ 1 t j ) P Z (τ 2 > t j ) E Z ( Ψ + (t j ; S 3, T ) )] )

8 8/29 Exposure: Ψ + (t) = D(t)v + (t) Swap Pricing Contract Numerics Benchmark Payoff: 0 t T (T 1,, T NS = T : payment dates) ( v + (t) = i:ti (S >t 3 (t) e q 3 (T i t) K 3e r(t i t))) + ) = α(t, Z) (S 3(0)e (r ϕ Y ( i)t)+y 3 3 (t) + K(t, Z) α(t, Z) = e ϕ Z ( a 3 i)t+a 3 Z(t) i:t i >t e q 3 T i K(t, Z) = K 3 i:t i >t e r(t i t) /α(t, Z) Exposure: payoff of European vanilla call option Forward: set N S = 1 Standard Vanilla Option Pricing - COS method (Fang and Oosterlee, 2008)

9 9/29 Numerical implementation 1 (1 R 1 ) N i=1 E [ P Z (t i 1 < τ 1 t i ) P Z (τ 2 > t i ) E Z ( Ψ + (t i ; S 3, T ) )] { } τ j = inf t 0 : Y j (t) ln K j S j (0) (r q j ϕ j ( i))t a j Z(t) Stochastic barrier due to common factor Pricing Contract Numerics Benchmark

10 9/29 Numerical implementation Pricing Contract Numerics Benchmark 1 (1 R 1 ) N i=1 E [ P Z (t i 1 < τ 1 t i ) P Z (τ 2 > t i ) E Z ( Ψ + (t i ; S 3, T ) )] { } τ j = inf t 0 : Y j (t) ln K j S j (0) (r q j ϕ j ( i))t a j Z(t) Stochastic barrier due to common factor Monte Carlo joint with Transform techniques: MC+Hillbert(P) - P: n. grid points Monte Carlo: (M) trajectories of the commom component, Z Hilbert Transform: conditional probabilities (Feng and Linetsky, 2008) Benchmark for efficiency test: (nested) Monte Carlo FullMC(k) - k: n. of nested iterations COS method: conditional option price (for all strikes) (Fang and Oosterlee, 2008)

11 10/29 Results I: Benchmark and efficiency Pricing Contract Numerics Benchmark M: 10 5 ; COS: 2 9 points = P, L = 15 (trunc. range) Efficiency index: σ 2 MC t MC /(σ 2 H t H) 2.74 for k = 1 Monte Carlo nested iterations 6.45 for k = 10 3 Monte Carlo nested iterations NIG process

12 11/29 Market model: NIG Calibration Swap X j (t) - NIG process with parameters (θ j, σ j, k j ) X j (t) = θ j G j (t) + σ j W j (G j (t)) θ j R, σ j R ++ G j (t) unbiased subordinator IG(t/ k j, 1/ k j ), i.e. EG j (t) = t Var (G j (t)) = k j t Characteristic exponent ) ϕ j (u) = (1 t 1 2iuθ k j k + u 2 σ 2 j j k j

13 11/29 Market model: NIG Calibration Swap X j (t) - NIG process with parameters (θ j, σ j, k j ) X j (t) = θ j G j (t) + σ j W j (G j (t)) θ j R, σ j R ++ G j (t) unbiased subordinator IG(t/ k j, 1/ k j ), i.e. EG j (t) = t Var (G j (t)) = k j t Characteristic exponent ) ϕ j (u) = (1 t 1 2iuθ k j k + u 2 σ 2 j j k j θ j σ j k j RMSE Std. Dev. γ 1 γ 2 DB (S 1) E ENI (S 2) E BRENT (S 3) E

14 12/29 Calibration (θ j, σ j, k j ) for j = 1, 2, 3 NIG Calibration Non linear least square fit Default probabilities bootstrapped from CDS quotes (DB, ENI) Option prices (Brent) Market Data Term structure of interest rates bootstrapped using LIBOR and swap rates Swap

15 12/29 Calibration (θ j, σ j, k j ) for j = 1, 2, 3 NIG Calibration Non linear least square fit Default probabilities bootstrapped from CDS quotes (DB, ENI) Option prices (Brent) Market Data Term structure of interest rates bootstrapped using LIBOR and swap rates Swap Separation of margins from dependence Convolution Lack of liquid products suitable for correlation calibration Correlation Sensitivity analysis - Perturbation around sample correlation Intermezzo 2

16 13/29 Results II: Swap NIG Calibration Swap ρ 13 > 0: Right-Way-Risk ρ 13 < 0: Wrong-Way-Risk T =1 year; S 1(0) = S 2(0) = S 3(0) = 1 Weekly monitoring 10 6 Monte Carlo iterations, 2 10 grid points Multiple cash flows product ( amortization effect) Forward

17 14/29 Risk mitigation tool Notation I EE II CIIbil CvsWWR Amount posted when (uncollateralized) exposure exceeds prespecified threshold Cash amount - no investment Minimum Transfer Amount (MTA): Amount below which no margin transfer is made It reduces frequency of collateral exchanges Notation E(t): uncollateralized exposure H 1, H 2 : thresholds (uni/bilateral) M: Minimum Transfer Amount (MTA) C(t): collateral E C (t): collateralized exposure δt: margining period

18 15/29 Pricing I: unilateral case Notation I EE II CIIbil CvsWWR H 1 > 0 : threshold for collateral posted by counterparty in investor s favour C(t) = (E(t δt) H 1 ) + E C (t) = (E(t) C(t)) + = E(t) }{{} (E(t) E C (t)) 1 (C(t)>0) }{{} Uncoll. Exp. Risk Mitigation due to collateral Alternative representation E C (t) = v + (t)1 (v (t δt)<h1 ) + (v (t) v (t δt) + H 1 ) + 1 (v (t δt)>h1 ) }{{}}{{} Correlation Gap call Calendar Spread call Numerics: condition on v (t δt)

19 15/29 Pricing I: unilateral case Notation I EE II CIIbil CvsWWR H 1 > 0 : threshold for collateral posted by counterparty in investor s favour C(t) = (E(t δt) H 1 ) + 1 (E(t δt) H1>M) E C (t) = (E(t) C(t)) + = E(t) }{{} (E(t) E C (t)) 1 (C(t)>M) }{{} Uncoll. Exp. Risk Mitigation due to collateral Alternative representation E C (t) = v + (t)1 (v (t δt)<h1 +M) + (v (t) v (t δt) + H 1 ) + 1 (v (t δt)>h1 +M) }{{}}{{} Correlation Gap call Calendar Spread call Numerics: condition on v (t δt)

20 16/29 Results III: EE Swap with Notation I EE II CIIbil CvsWWR % EE reduction: Swap: 27% (H = 0.5); 22% (H = 1) 2 weeks lag; base case (ρ 13 = 0.22) Unilateral case

21 17/29 Pricing II: bilateral case Notation I EE II CIIbil CvsWWR H 2 < 0 : threshold for collateral posted by investor in counterparty s favour C(t) = (v (t δt) H 1 ) + + (v (t δt) H 2 ) }{{}}{{} E C (t) = v + (t) }{{} Uncoll. Exp. ( C (1) (t) C (2) (t) ) v (t) E (1) C (t) 1 (C (1) (t)>0) }{{} > 0 (Risk Mitigation) Alternative representation ( v (t) E (2) C (t) ) 1 (C (2) (t)<0) } {{ } < 0 (Credit Exposure) E C (t) = v + (t)1 (H2 <v (t δt)<h 1 ) + (v (t) v (t δt) + H 1 ) + 1 (v (t δt)>h1 ) }{{}}{{} Correlation Gap call Calendar Spread call + (v (t) v (t δt) + H 2 ) + 1 (v (t δt)<h2 ) }{{} Calendar Spread call MTA: similar to above

22 18/29 Results IV: bilateral case Notation I EE II CIIbil CvsWWR 2 weeks lag; base case (ρ 13 = 0.22)

23 19/29 Results V: vs W/RWR Notation I EE II CIIbil CvsWWR Right/Wrong-Way Risk effect dependent on the collateral agreement H 1 = 0.25; H 2 = 0; MTA = 0

24 20/29 Quantifying W : counterparty default between margining dates and relevant adverse change in the exposure for the investor P ( S 1(t) < K 1, S 2(t) > K 2, v (t) > 0 S 1(t ) > K 1, S 2(t ) > K 2, v (t ) < 0) = P ( X 1 (t) < ε 1, X 2 (t) > ε 2, X 3 (t) > ε 3 ) P ( a 1 Z(t) > ε 1, a 2 Z(t) > ε 2, a 3 Z(t) > ε 3 ) v (t): contract value ε j = (r q j ϕ j ( i)) t + j for j = 1, 2 ε 3 defined according to contract type (example - Forward: ε 3 = ϕ 3 ( i) t + 3 ) 1, 3, i.e. ε 1, ε 3

25 21/29 (ctd) ρ 13 < 0 (otherwise probability is zero) W a) ρ 12 > 0, ρ 23 < 0 ( { P max ε1 with P ( ε 2 < Z(t) < min a 2 }, ε 3 a 1 a 3 ) < Z(t) < ε 2 { a 2 }) ε 1, ε 3 a 1 a 3 { ε 2 > a 2 max ε1, ε 3 { a 1 } ε 2 > a 2 min ε 1, ε 3 a 1 a 3 a 3 } b) ρ 12 < 0, ρ 23 > 0 ( { }) P Z(t) > max ε1, ε 3 ( { a 1 a 3 }) P Z(t) < min ε 1, ε 3 a 1 a 3 a 1, a 2 < 0 < a 3 a 3 < 0 < a 1, a 2 a 1, a 2 < 0 < a 3 a 3 < 0 < a 1, a 2 a 1 < 0 < a 2, a 3 a 2, a 3 < 0 < a 1

26 22/29 Results VI: - 2 weeks W Scenarios

27 23/29 Upon default, losses are calculated at netted portfolio level (ϖ 1,..., ϖ N ) derivatives with discounted payoff Ψ i 1,W /0 = N b i ϖ i E [ ] 1 (τ1 min(τ 2,T ))Ψ + i 1,W = E [ 1 (τ1 min(τ 2,T ))Π +] Π = N b i ϖ i Ψ i Numerics Testing Reduction

28 23/29 Upon default, losses are calculated at netted portfolio level Numerics Testing Reduction (ϖ 1,..., ϖ N ) derivatives with discounted payoff Ψ i 1,W /0 = N b i ϖ i E [ ] 1 (τ1 min(τ 2,T ))Ψ + i 1,W = E [ 1 (τ1 min(τ 2,T ))Π +] A simple example Π = N b i ϖ i Ψ i N b swap contracts with maturity T and strike K j, j = 1,, N b ϖ j = 1/N b j Payoff with netting ( Y n l=3 w l K l (t) ) + Y = n l=3 ξ l ξ l = w l α l (t, Z)S l (0)e (r q l ϕ Y l ( i))t+y l (t) K(t) = K 3 j:t j >t e r(t j t) homogeneous copies of S 3

29 24/29 : Numerics Required: distribution of N b j=1 e ϕ Y j ( i)t+y j (t) Numerics Testing Reduction Y j (t): independent copies of the same (NIG) process Numerical methods Exact: via convolution Asymptotics: CLT implies ( ( )) Nb j=1 e ϕ Y ( i)t+y j j (t) Φ N b, N b e t(2ϕ Y ( i) ϕ j Yj ( 2i)) 1 Barakat (1976) approximation 1 2π e z 2 /2 ( 1 + γ 1 6N 1/2 h 3 (z) + γ 2 24N h 4(z) + γ2 1 72N h 6(z) γ 1, γ 2 : indices of skewness and excess kurtosis h k (z) = H k (x)φ(z), for φ(z) standard normal density (Edgeworth expansion) ) All info can be recovered from process Y (CF/pdf)

30 25/29 Results VII: Testing Numerics Testing Reduction Conv method becomes unstable for large N b Barakat approximation works well also for small N b

31 26/29 Results VIII: & Diversification Numerics Testing Reduction Tot. T =1 year; S j (0) = 1 for j = 1,, N b Weekly monitoring Base Case: ρ = 9.50%

32 27/29 s & Work in progress Multivariate structural default model with jumps and dependent components Unified treatment of,, Integrated numerical scheme for pricing, calibration and correlation fitting Sensitivity analysis with respect to relevant parameters Impact of dependence on relevant measurements Mathematically and computationally tractable framework Work in progress with and provisions (asymptotic results) Default risk of the reference name (compound option problem) - defaultable CDS, vulnerable options of more complex instruments (interest rate derivatives)...

33 Synopsis Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 27/29 CDS (DB, ENI) data source: Markit, June 26, 2014 Default probabilities computed using Markit calculator Option (BRENT) data source: CME, June 26, 2014 Settlement date: August 11, 2014 Back

34 Intermezzo 1: why jumps? Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 27/29

35 Intermezzo 1: why jumps? Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 27/29 Back to Back

36 Gap risk NIG Barrier: K 1 = Gaussian Barrier: K 1 = 0.9 S 1 S time Barrier: K 2 = S 1 S time Barrier: K 2 = time Exposure at t time Exposure at t 1 Synopsis Intermezzo 1 MC Construction II Convolution Fit time time WWR Forward Back SWPbreakdown EE MTA 27/29

37 At a glance Any 1-d model per component Full range of dependencies Correlation coefficient correctly represents dependence Var (Z(1)) ρ jl = a j a l Var (Xj (1))Var (X l (1)) Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 27/29 Dimension of parameter set: linear in n (n. assets) Unified approach for all classes of Lévy processes Subordinated Brownian motions: subordinator not required JD processes: law of jump sizes depends on nature underlying shock Dependence structure can be isolated from marginal distributions

38 Convolution X = Y + az in distribution Given X, choose Z parameters s.t. min 3 φ Xj (u) φ Yj (u)φ Z (a j u) 2 du, j = 1, 2, 3 j=1 Choose a to fit given correlation matrix Obtain Y parameters s.t. c X j k = c Y j k + ak j c k Z, j = 1, 2, 3; k = 1,, 4 Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 27/29

39 Convolution Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 27/29 X = Y + az in distribution Given X, choose Z parameters s.t. min 3 φ Xj (u) φ Yj (u)φ Z (a j u) 2 du, j = 1, 2, 3 j=1 Choose a to fit given correlation matrix Obtain Y parameters s.t. c X j k = c Y j k + ak j c k Z, j = 1, 2, 3; k = 1,, 4 Correlation matrix: sample correlation (2 years daily observations) DB 1 ENI BRENT

40 Convolution: Fitting error Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 27/29 Back

41 Intermezzo 2 Right-way risk: the exposure tends to decrease when the counteparty credit quality deteriorates Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 27/29 Firm 1 credit quality worsens, call option on S 3 moves out of the money ρ X 13 > 0 Wrong-way risk: the exposure tends to increase when the counteparty credit quality deteriorates Firm 1 credit quality worsens, call option on S 3 moves in the money ρ X 13 < 0 Back

42 : Forward Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 27/29 ρ 13 > 0: Right-Way-Risk T =1 year; S 1(0) = S 2(0) = S 3(0) = 1 Weekly monitoring 10 6 Monte Carlo iterations, 2 10 grid points Single cash flow product ( diffusion effect) ρ 13 < 0: Wrong-Way-Risk Back

43 Impact of collateral (unilateral) Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 28/29 Swap contract; 2 weeks lag; base case (ρ 13 = 0.22)

44 and MTA: EE Swap 0.7 Calendar Spread package Swap 1 Correlation Swap Option price time 1.5 EE C Swap Option price time Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 29/29 Expected Exposure time Uncoll. EE Coll. EE H = 1; M = 0 Coll. EE H = 1; M = 1 Coll. EE H = 1; M = 1.5 % EE reduction: Swap: 22% (H = 1, M = 0); 6.6% (H = 1, M = 1); 2% (H = 1, M = 1.5); 2 weeks lag; base case (ρ 13 = 0.22) Unilateral case

45 Expected Exposure with collateral I Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 29/29 Forward contract; 2 weeks lag; base case (ρ 13 = 0.22)

46 Pricing II: bilateral case Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 29/29 H 2 < 0 : collateral posted by investor in counterparty s favour C(t) = (v (t δt) H 1 ) + 1 (v (t δt) H1 >M)+ (v (t δt) H 2 ) 1 (v (t δt) H2 < M) }{{}}{{} E C (t) = C (1) (t) v + (t) }{{} Uncoll. Exp. ( ) v (t) E (1) C (t) 1 (C (1) (t)>0) }{{} > 0 (Risk Mitigation) Alternative representation C (2) (t) ( v (t) E (2) C (t) ) 1 (C (2) (t)<0) } {{ } < 0 (Credit Exposure) E C (t) = v + (t)1 (H2 M<v (t δt)<h 1 +M) + (v (t) v (t δt) + H 1) + 1 (v (t δt)>h1 +M) }{{}}{{} Correlation Gap call Calendar Spread call + (v (t) v (t δt) + H 2) + 1 (v (t δt)<h2 M) }{{} Calendar Spread call

47 NIG vs Brownian motion: the tails Synopsis Intermezzo 1 MC Construction II Convolution Fit WWR Forward SWPbreakdown EE MTA 29/29 NIG distribution Upper tail P (X > x) 2C e λ U x 2C ( ) λ U π erfc λu x x Lower tail P (X < x) 2C e λ Lx 2C ( ) λ L π erfc λl x x Gaussian distribution Upper tail P(X > x) = 1 ( ) x µ 2 erfc σ 2 Lower tail P(X x) = ( ) x µ 2 erf σ 2 x > 0 x > 0