Toward a benchmark GPU platform to simulate XVA

Transcription

1 Diallo - Lokman (INRIA) MonteCarlo16 1 / 22 Toward a benchmark GPU platform to simulate XVA Babacar Diallo a joint work with Lokman Abbas-Turki INRIA 6 July 2016

2 Diallo - Lokman (INRIA) MonteCarlo16 2 / 22 Plan Introduction Simulation algorithms Without funding constraints With funding constraints Some simulation results Common-Shock Model Conclusion

3 Introduction Diallo - Lokman (INRIA) MonteCarlo16 3 / 22 Plan Introduction Simulation algorithms Without funding constraints With funding constraints Some simulation results Common-Shock Model Conclusion

4 Introduction Diallo - Lokman (INRIA) MonteCarlo16 4 / 22 Credit Valuation Adjustment In a nancial transaction between a party C that has to pay another party B some amount V, the CVA value is the price of the insurance contract that covers the default of party C to pay the whole sum V. CVA t,t = (1 R)E t ( V + τ 1 t<τ T ) (1) R is the recovery to make if the counterparty defaults (Assume R = 0), τ is the random default time of the counterparty, T is the protection time horizon. Numerical simulation N 1 ( ) CVA 0,T E V t + k 1 τ (tk,t k+1 ], (2) k=0 N the number of time steps used for SDEs discretization. Importance Hold sucient amount of liquid assets to face the counterparty default. Basel III includes the calculation of the CVA (Credit Valuation Adjustment) as an important part of the prudential rules.

5 Introduction Diallo - Lokman (INRIA) MonteCarlo16 5 / 22 TVA denition Total valuation adjustment (>CVA+DVA+FVA), it covers: Both defaults: τ = τ c τ b, CVA and DVA. Funding our risk and the risk of the counterparty: Nonlinear BSDE part, FVA. S. Crépey(2012) Ignoring the external funding and denoting β t = e t 0 r udu where r is the risk-free short rate process, Θ satises the following BSDE on [0, τ T ] TVA BSDE simulation [ τ T β tθ t = E β τ 1 τ<t (V τ R τ ) + β sg s(v s Θ ] s)ds Gt t where G is the extension of F by the natural ltration generated by τ c and by τ b. R is the total close-out cash-ow specied thanks to CSA (Credit Support Annex) and g is the funding coecient. Only for European contracts. Requires a good approximation of the exposure V. Practitioners usually use rough approximations. No trustable procedure in the general case. (3)

6 Simulation algorithms Diallo - Lokman (INRIA) MonteCarlo16 6 / 22 Plan Introduction Simulation algorithms Without funding constraints With funding constraints Some simulation results Common-Shock Model Conclusion

7 Simulation algorithms Without funding constraints With funding constraints CVA0,T = N X1 k=0 + E Pk+ 1 1τ (kh,(k+1)h], Θk = Ek (Θk+1 + hg (k + 1, Pk+1, Θk+1 )), h= T. N ΘN = 0. An example of a two stage simulation with M0 = 2, M6 = 8 and M8 = 4 Diallo - Lokman (INRIA) MonteCarlo16 7 / 22

8 Diallo - Lokman (INRIA) MonteCarlo16 8 / 22 Simulation algorithms CVA 0,T N 1 approximation ĈVA 0,T = 1 M0 ( Fk+1 2 P1 (S1 i M ),..., P ) k+1 (Sk+1 i ) k=0 0 i=1 With { F 1 k+1 (x 1,..., x k+1 ) = E ( 1 τ (kh,(k+1)h] P 1 = x 1,..., P k+1 = x k+1 ), F 2 k+1 (x 1,..., x k+1 ) = (x k+1 ) + F 1 k+1 (x 1,..., x k+1 ). (4) (5) Θ k approximation For k = 1,..., N 1 Θ k (x)= t ψ(x)ψ 1 1 M0 ( k ψ(s j k M ) Θ k+1 (S j k+1 ) + 1 ( 0 N g k +1, Θ k+1 (S j k+1 ), P ) k+1 (S j k+1 ) j=1 and Θ N (x) = 0, Θ0 (S 0 ) = 1 M0 ( Θ 1 (S j1 M ) + 1N ( g 1, Θ 1 (S j 1 ), P ) 1 (S j 1 ). 0 j=1 Where Ψ k = T 1 M0 ψ( S k i M )t ψ( S k i ) with: {S i } i {1,...,M0 } and { S i } i {1,...,M0 } are two 0 i=0 independent simulations of the underlying asset S, ψ is a basis of monomial functions where K is its cardinal and T is an operator that must satisfy some desired properties. (6)

9 Simulation algorithms Without funding constraints Diallo - Lokman (INRIA) MonteCarlo16 9 / 22 Theorem (Lokman Abbas-Turki - Mohamed Mikou: TVA on American Derivatives) [ ) ] 2 E (ĈVA0,T CVA 0,T N2 ( ( max M Var Fk+1 2 P1 (S1 i ),..., P )) k+1 (Sk+1 i ) 0 k {0,...,N 1} N N + 4M 2 j=1 j N 1 + 4NM 2 j=1 j N 1 + 4NM 2 j=1 j ( [ ]) 2 E V j (Sj i )f j (P j (Sj i ))F j 3 (P 1(S1 i ),..., P j (Sj i )) E V j (Sj i )f j 2 N 1 (P j (Sj i )) Fk+1 4 (P 1(S1 i ),..., P j (Sj i ), Si j ) ( [ ] ) 2 E V j (Sj i )F j 1 (P 1(S1 i ),..., P j (Sj i )) P j (Sj i ) = 0 N ϕ j (0) + N (N j + 1) 2 O ( Mj ( )) Where ϕ j is the density of P j (Sj i ), V j (x) = Var Pj (x) P j (x). k=j j=1 ( 1 M 4 j ) Good choices For example If ϕ j (0) is big then take M j M 0, otherwise M j M 0 /N. In both cases, N must be small when compared to M 0. M j = N j N 1 M M0 1 with either M 1 = N or M 1 = M 0. (7)

10 Simulation algorithms With funding constraints Diallo - Lokman (INRIA) MonteCarlo16 10 / 22 Theorem (Lokman Abbas-Turki - Mohamed Mikou: TVA on American Derivatives) As long as {Θ i (x)} 0 i N 1 are of class C s on the support of S R d, there exists a positive constant C such that for each 0 k N 1 ( ) [ ) ] 2 E ( Θk (Sk i ) Θ k(sk i ) CK N 1 E V l+1(s j l+1 ) 2 P g l + 1, Θ l+1 (S j l+1 ), P l+1(s j l+1 ) 2 N 2 2M l=k l ( ) K +O + K 2 M 0 N 2 + K M 0 N 4 Ml 2 + K 1 2s/d N 2 + K 2s/d. Good choice Take M l M 0 /N, N must be suciently small N 10. For example M l = N l N 1 M M0 1 with M 1 = N. (8)

11 Some simulation results Diallo - Lokman (INRIA) MonteCarlo16 11 / 22 Plan Introduction Simulation algorithms Without funding constraints With funding constraints Some simulation results Common-Shock Model Conclusion

12 Some simulation results Diallo - Lokman (INRIA) MonteCarlo16 12 / 22 Within less than 1 minute simulation on GPU: M 0 = 131K, N = 10, Neds = 50 European Path-dependent option Φ(S T ) = ( S 1 T 2 + S2 T 2 S3 T ) + M 1 Θ 0 Θ 0 std CVA 0,T CVA 0,T std M N M N M

13 Some simulation results Diallo - Lokman (INRIA) MonteCarlo16 13 / 22 Within less than 1 minute simulation on GPU: M 0 = 131K, N = 10, Neds = 50 European Path-dependent option Φ(S T ) = ( 3S 1 T S2 T 10 S3 T ) + ( 7S 1 T S2 T 10 S3 T ) + M 1 Θ 0 Θ 0 std CVA 0,T CVA 0,T std M N M N M N M

14 Common-Shock Model Diallo - Lokman (INRIA) MonteCarlo16 14 / 22 Plan Introduction Simulation algorithms Without funding constraints With funding constraints Some simulation results Common-Shock Model Conclusion

15 Common-Shock Model Diallo - Lokman (INRIA) MonteCarlo16 15 / 22 Dynamic Marshall-Olkin (DMO) Default time model n + 2 credit names { 1, 0,..., n}. Y = {{ 1}, {0},..., {n}, I 1, I 2,..., I m}, (9) I j contains at least two obligors, I = {I 1, I 2,..., I m}. λ Y (t, X t) shock intensities, where X t = ( Xt Y ) Markov factor process { η Y = inf t > 0 : t 0 Y Y λ Y (s, X s)ds > E Y }, (10) E Y random variables i.i.d and exponentially distributed with parameter 1. If Y 1 Y 2, Q(η Y 1 = η Y 2) = 0. For each obligor i we dene τ i = min η Y, (11) Y Y,i Y as the default time of obligor i in common-shock model, τ b = τ 1, τ c = τ 0. H i t = 1 τi t. (12)

16 Common-Shock Model Diallo - Lokman (INRIA) MonteCarlo16 16 / 22 Conditionnal survival probability CDS clean price (Bielecki; Cousin; Crépey: A Bottom-Up Dynamic Model of Portfolio Credit Risk. Part I: Markov Copula Perspective ) The conditionnal survival probability function of every obligor is given by, for every t i t, Q (τ i > t i F t(w, H)) = Q(τ i > t i H t, X t) ti λ Y (s, X Y s )ds = (1 Ht i )E t Y Y,i Y e Xt i s ( ) r u + λ Y (u, X Ti u Y ) du Pt i = 1 Ht i t Y Y,i Y E e t (1 R i ) Y Y,i Y λ Y (s, Xs Y ) S i ds X t, R i are the recovery rates; S i are the contractual spreads.

17 Common-Shock Model Model example Individual default intensities X i are independent homogenous CIR processes, dxt i = a i (b i Xt i )dt + c i Xt i dw i. The individual default intensities are function of X t where κ i is a constant. λ i t = κ i + X i t, (13) Systemic shock intensities λ I (t, X t) = α I inf i I λi t, (14) (α I ) I I are nonnegative constants, I I α I 1. Idiosyncratic intensities λ i (t, X t) = λ i t i I λ I (t, X t) 0. (15) Diallo - Lokman (INRIA) MonteCarlo16 17 / 22

18 Diallo - Lokman (INRIA) MonteCarlo16 18 / 22 CVA 0,T = E k=0 Common-Shock Model P τ0 + 1 {τ0 <T }β τ0 (1 R 0 ) N 1 E 1 {τ0 (t k,t k+1 ]} βt k+1 (1 R 0) P tk+1 + i pay i rec i pay i rec 1 {τi =τ<t i }(1 R i ) 1 {τi (t k,t k+1 ]} (1 R i ) +, (16) +, (17) where τ c = τ 0. CDS portfolio clean price P = P i. (18) i pay i rec P i (t k, H i,l t k, X l t k ) (1 H i,l t k ) 1 M k (1 R i ) t q M k T i r tp + λ Y (t p, X Y,l,j t p ) (t p t p 1 ) t e p=t k Y Y,i Y j=1 t q=t k Y Y,i Y λ Y (t q, X Y,l,j t q ) S i (t q t q 1 ).

19 Conclusion Diallo - Lokman (INRIA) MonteCarlo16 19 / 22 Plan Introduction Simulation algorithms Without funding constraints With funding constraints Some simulation results Common-Shock Model Conclusion

20 Conclusion Diallo - Lokman (INRIA) MonteCarlo16 20 / 22 Mathematical and computing work suited to GPUs More to come Architecture evolution Market factor TVA simulation within a minute. Credit factor CVA simulation with 100 CDS within few minutes. Better use of GPU cache memory (shared). Compute XVA on any general bank portfolio in less than one hour: simulation of all prices +XVA. High bandwidth memory (HBM) on GPU to increase throughput. 3d XPoint non volatile memory to store the huge number of prices.

21 Conclusion Diallo - Lokman (INRIA) MonteCarlo16 21 / 22 References L.A. Abbas-Turki and M.A. Mikou. TVA on American Derivatives: Stéphane Crépey and Tomasz R. Bielecki. Counterparty Risk and Funding A Tale of Two Puzzles. Chapman and HALL Tomasz R. Bielecki; Areski Cousin; Stéphane Crépey. A Bottom-Up Dynamic Model of Portfolio Credit Risk. Part I: Markov Copula Perspective. Recent Advances in Financial Engineering 2012, Takahashi and Y. Muromachi and T. Shibata (Eds.), pp and 51-74, 2014.

22 The End Diallo - Lokman (INRIA) MonteCarlo16 22 / 22 Thank you Questions?