Gianluca Fusai CASS & UPO Integrated structural approach to Introduction Counterparty Credit Risk with Contribution Agenda dependent jumps Model CVA Example Collateral Laura Ballotta, Gianluca Fusai, Daniele Marazzina Gap Risk Netting Cass Business School, DISEI-Universit` a Piemonte Orientale, Politecnico Conclusion Milano Appendix EM-LYON QUANTITATIVE APPROACHES IN MANAGEMENT AND ECONOMICS Friday, the 27th of November, 2015 1/31
Introduction • General context of the problem • Firm 1 ( S 1 ) - Counterparty; Firm 2 ( S 2 ) - Investor Gianluca Fusai CASS & UPO • Counterparty Credit Risk : risk of a party to a financial Introduction contract defaulting prior to/at the contract’s expiration Contribution • Credit Value Adjustment (CVA): Agenda � � 1 ( τ 1 ≤ min ( τ 2 , T )) Ψ + ( τ 1 ; S 3 , T ) CVA 1 = ( 1 − R 1 ) E Model CVA • Ψ ( · ) - disc. value of OTC contract on S 3 (underlying asset) Example • τ j = inf { t ≥ 0 : S j ( t ) ≤ K j } , j = 1 , 2 Collateral • R j - recovery rate Asset j , j = 1 , 2 Gap Risk Netting Conclusion • Motivation Appendix • Regulatory framework - Bilateral vs Unilateral • CCR not fully mitigated by collateral • ≈ 65% losses from CCR due to CVA during the financial crisis 2/31
Contribution • Structural approach to credit risk for unified treatment • CVA pricing Gianluca Fusai CASS & UPO • Right-Way-Risk/Wrong-Way-Risk • Mitigating clauses - netting & collateral Introduction Contribution • Gap Risk Agenda • Model Calibration Model CVA • Multivariate L´ evy processes Example • Independent and stationary increments Collateral Gap Risk • Brownian motion with drift + pure jump process Netting • Skewness and excess kurtosis Conclusion • Joint evolution of risk factors Appendix • Improved calibration of credit spreads over short maturities Intermezzo 1 • Efficient numerical schemes (exotic option pricing) • Case Study in the oil market 3/31
Agenda • Structural approach to credit risk modelling Gianluca Fusai • Multivariate L´ evy processes CASS & UPO • Construction Introduction • Dependence features Contribution Agenda • CVA pricing Model • Model calibration CVA Example • Correlation effect Collateral • Collateral Gap Risk • Gap risk Netting Conclusion • Netting Appendix • Conclusions and work in progress 4/31
Structural approach to default • Risk neutral dynamic Gianluca Fusai CASS & UPO S j ( t ) = S j ( 0 ) e ( r − q j − ϕ j ( − i )) t + X j ( t ) , ∀ j = 1 , ..., n Introduction • X j ( t ) - L´ evy process Contribution Agenda • ϕ j ( − i ) t characteristic exponent of X j ( t ) Model Construction I CVA • r > 0 - risk free rate of interest Example Collateral • q j > 0 - dividend yield of the j th asset Gap Risk Netting • “Non defaultable” underlying asset Conclusion Appendix • Dependence: factor representation (Ballotta and Bonfiglioli, 2014) 5/31
The default event in the first-hitting time structural approach Gianluca Fusai CASS & UPO Introduction Contribution Agenda Model Construction I CVA Example Collateral Gap Risk Netting Conclusion Appendix Figure: Simulated paths of the log-firm value. Paths crossing the barrier determine the default event before maturity 6/31
Multivariate L´ evy processes • L´ evy processes • Known characteristic function Gianluca Fusai CASS & UPO • Invariant under linear transformation Introduction Contribution • X j ( t ) = Y j ( t ) + a j Z ( t ) a j ∈ R , ∀ j = 1 , ..., n Agenda • X j ( t ) , Y j ( t ) , Z ( t ) are L´ evy processes Model Construction I • Y j ( t ) : idiosyncratic risk process CVA Example • Z ( t ) : systematic risk process Collateral Gap Risk • Y j ( t ) and Z ( t ) independent and distinct Netting • Correlation coefficient correctly represents dependence Conclusion Appendix V ar ( Z ( 1 )) ρ jl = a j a l � V ar ( X j ( 1 )) V ar ( X l ( 1 )) • Dependence structure can be isolated from marginal distributions 7/31
CVA Pricing • Long position in a contract on S 3 (equity index, commodity, currency rate,...) Gianluca Fusai CASS & UPO • First passage time approach Introduction Default event: τ j = inf { t ≥ 0 : S j ( t ) ≤ K j } , j = 1 , 2 Contribution Agenda • First to default problem Model � � 1 ( τ 1 ≤ T ) 1 ( τ 2 >τ 1 ) Ψ + ( τ 1 ; S 3 , T ) CVA CVA 1 = ( 1 − R 1 ) E Pricing Contract Numerics • ‘Bucketing’: default can only occur on time grid Benchmark { t j : 0 ≤ j ≤ N } for t 0 = 0 , t N = T Example � � CVA 1 ≈ ( 1 − R 1 ) � N 1 ( t j − 1 <τ 1 ≤ t j ) 1 ( τ 2 > t j )Ψ + ( t j ; S 3 , T ) Collateral j = 1 E Gap Risk Netting • Conditioning on { Z ( t ) , 0 < t ≤ T } Conclusion CVA 1 ≈ Appendix � � �� ( 1 − R 1 ) � N Ψ + ( t j ; S 3 , T ) j = 1 E P Z ( t j − 1 < τ 1 ≤ t j ) P Z ( τ 2 > t j ) E Z 8/31
Exposure: Ψ + ( t ) = D ( t ) v + ( t ) • Swap Gianluca Fusai CASS & UPO Payoff: 0 ≤ t ≤ T ( T 1 , · · · , T N S = T : payment dates) Introduction �� � S 3 ( t ) e − q 3 ( Ti − t ) − K 3 e − r ( Ti − t ) �� + v + ( t ) = Contribution i : Ti > t Agenda � � + ( r − ϕ Y 3 ( − i ) t )+ Y 3 ( t ) − K ( t , Z ) = α ( t , Z ) S 3 ( 0 ) e Model CVA α ( t , Z ) = e − ϕ Z ( − a 3 i ) t + a 3 Z ( t ) � i : Ti > t e − q 3 Ti Pricing Contract � Numerics i : Ti > t e − r ( Ti − t ) /α ( t , Z ) K ( t , Z ) = K 3 Benchmark Example • Exposure: payoff of European vanilla call option Collateral Gap Risk Netting • Forward: set N S = 1 Conclusion Appendix • Standard Vanilla Option Pricing - COS method (Fang and Oosterlee, 2008) 9/31
Numerical implementation CVA 1 � � �� ≈ ( 1 − R 1 ) � N Ψ + ( t i ; S 3 , T ) i = 1 E P Z ( t i − 1 < τ 1 ≤ t i ) P Z ( τ 2 > t i ) E Z Gianluca Fusai CASS & UPO � � Introduction K j • τ j = inf t ≥ 0 : Y j ( t ) ≤ ln S j ( 0 ) − ( r − q j − ϕ j ( − i )) t − a j Z ( t ) Contribution • Stochastic barrier due to common factor Agenda Model CVA Pricing Contract Numerics Benchmark Example Collateral Gap Risk Netting Conclusion Appendix 10/31
Numerical implementation CVA 1 � � �� ≈ ( 1 − R 1 ) � N Ψ + ( t i ; S 3 , T ) i = 1 E P Z ( t i − 1 < τ 1 ≤ t i ) P Z ( τ 2 > t i ) E Z Gianluca Fusai CASS & UPO � � Introduction K j • τ j = inf t ≥ 0 : Y j ( t ) ≤ ln S j ( 0 ) − ( r − q j − ϕ j ( − i )) t − a j Z ( t ) Contribution • Stochastic barrier due to common factor Agenda Model • Monte Carlo joint with Transform techniques: CVA Pricing MC+Hillbert( P ) - P : n. grid points Contract Numerics • Monte Carlo: (M) trajectories of the commom component, Z Benchmark Example • Hilbert Transform: conditional probabilities (Feng and Collateral Linetsky, 2008) Gap Risk Netting • Benchmark for efficiency test: (nested) Monte Carlo Conclusion FullMC( k ) - k : n. of nested iterations Appendix • COS method: conditional option price (for all strikes) (Fang and Oosterlee, 2008) 10/31
Results I: Benchmark and efficiency Gianluca Fusai CASS & UPO Introduction Contribution Agenda Model CVA Pricing Contract Numerics Benchmark Example Collateral Gap Risk • M : 10 5 ; COS: 2 9 points = P , L = 15 (trunc. range) Netting • Efficiency index: σ 2 MC t MC / ( σ 2 Conclusion H t H ) Appendix • 2.74 for k = 1 Monte Carlo nested iterations • 6.45 for k = 10 3 Monte Carlo nested iterations • NIG process 11/31
Market model: Example • X j ( t ) - NIG process with parameters ( θ j , σ j , k j ) Gianluca Fusai CASS & UPO • X j ( t ) = θ j G j ( t ) + σ j W j ( G j ( t )) θ j ∈ R , σ j ∈ R ++ Introduction � � • G j ( t ) unbiased subordinator IG ( t / k j , 1 / k j ) , i.e. Contribution Agenda E G j ( t ) = t V ar ( G j ( t )) = k j t Model CVA • Characteristic exponent Example � � � NIG t 1 − 1 − 2 iu θ j k + u 2 σ 2 ϕ j ( u ) = j k j Calibration k j CVA Swap Collateral Gap Risk Netting Conclusion Appendix 12/31
Market model: Example • X j ( t ) - NIG process with parameters ( θ j , σ j , k j ) Gianluca Fusai CASS & UPO • X j ( t ) = θ j G j ( t ) + σ j W j ( G j ( t )) θ j ∈ R , σ j ∈ R ++ Introduction � � • G j ( t ) unbiased subordinator IG ( t / k j , 1 / k j ) , i.e. Contribution Agenda E G j ( t ) = t V ar ( G j ( t )) = k j t Model CVA • Characteristic exponent Example � � � NIG t 1 − 1 − 2 iu θ j k + u 2 σ 2 ϕ j ( u ) = j k j Calibration k j CVA Swap Collateral Gap Risk θ j σ j k j RMSE Std. Dev. γ 1 γ 2 Netting Conclusion DB ( S 1 ) -0.22 0.25 0.55 1.33E-03 0.29 -1.09 2.48 Appendix ENI ( S 2 ) -0.18 0.10 0.34 1.29E-03 0.20 -0.87 1.14 BRENT ( S 3 ) 0.07 0.19 0.08 2.18E-03 0.19 0.09 0.25 12/31
Calibration • ( θ j , σ j , k j ) for j = 1 , 2 , 3 • Non linear least square fit Gianluca Fusai CASS & UPO • Default probabilities bootstrapped from CDS quotes Introduction (DB, ENI) Contribution • Option prices (Brent) Agenda Market Data Model CVA • Term structure of interest rates bootstrapped using Example LIBOR and swap rates NIG Calibration CVA Swap Collateral Gap Risk Netting Conclusion Appendix 13/31
Recommend
More recommend
