From CVA to the Resolution of a Large Number of Small Random Systems - - PowerPoint PPT Presentation

SLIDE 1

From CVA to the Resolution of a Large Number of Small Random Systems

Lokman Abbas-Turki

First part from a joint work with M. A. Mikou Last part from a joint work with S. Graillat

UPMC, LPMA

6 April 2016

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 1 / 38

SLIDE 2

Plan

Introduction From linear/linear to linear/nonlinear From nonlinear/linear to nonlinear/nonlinear Simulation algorithms Without funding constraints With funding constraints Adaptation issues to American options The difference with the references LDLt Householder tridiagonalization + PCR Divide and conquer for eigenproblem Some simulation results Conclusion

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 2 / 38

SLIDE 3

Introduction

Plan

Introduction From linear/linear to linear/nonlinear From nonlinear/linear to nonlinear/nonlinear Simulation algorithms Without funding constraints With funding constraints Adaptation issues to American options The difference with the references LDLt Householder tridiagonalization + PCR Divide and conquer for eigenproblem Some simulation results Conclusion

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 3 / 38

SLIDE 4

Introduction From linear/linear to linear/nonlinear

Credit Valuation Adjustment

In a financial 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 . CVAt,T = (1 − R)Et

• V +

τ 1t<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

CVA0,T ≈

N−1

• k=0

E

• V +

tk 1τ∈(tk ,tk+1]

• ,

(2) N ≤ the number of time steps used for SDEs discretization.

Importance

◮

Hold sufficient 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.

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 4 / 38

SLIDE 5

Introduction From linear/linear to linear/nonlinear

Various kind of contracts

Simulating assets St = (S1

t , ..., Sd t ) trajectories then contracts trajectories

to get Vt as a sum: Vt =

• ie

φexp

ie (St) +

• ii

φeui

ii (St) +

• id

φeud

id,t(St) +

• ia

φam

ia,t(St),

(3) where ie, ii, id and ia are the exposure indices and: φexp explicit function, for example: φexp(Stk ) = S1

tk − S2 tk .

φeui is a path-independent European contract, φeui(Stk ) = E(f eui(ST )|Stk ). (4) φeud

t

is a path-dependent European contract, φeud

tk (Stk ) = E(f eud tk

(Stk+1)|Stk ), (5) for example: f eud

tk

(Stk+1) = ( max

i=0,..,k S1 ti ∨ S1 tk+1 − S2 tk+1)+.

φam

t

is an American contract, involving an optimal stopping problem φam

tk (Stk ) = f (Stk ) ∨ E(φam tk+1(Stk+1)|Stk )

(6) with f an explicit payoff that generally does not depend on the asset path.

Common problem

ϕ(x) = E(f (Stk+1)|Stk = x).

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 5 / 38

SLIDE 6

Introduction From nonlinear/linear to nonlinear/nonlinear

TVA definition

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 βt = e−

t

0 rudu where r is the

risk-free short rate process, Θ satisfies the following BSDE on [0, τ ∧ T] βtΘt = E

• βτ1τ<T (Vτ − Rτ) +

τ∧T

t

βsgs(Vs − Θs)ds

• Gt
• (7)

where G is the extension of F by the natural filtration generated by τ c and by τ b. R is the total close-out cash-flow specified thanks to CSA (Credit Support Annex) and g is the funding coefficient.

TVA BSDE simulation

◮

Only for European contracts.

◮

Requires a good approximation of the exposure V .

◮

Practitioners usually use rough approximations.

◮

No trustable procedure in the general case.

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 6 / 38

SLIDE 7

Introduction From nonlinear/linear to nonlinear/nonlinear

WOLOG: V = P

For one American contract

◮

Pt is the American derivative exposition.

◮

Let τ ∗ ∈ [0, T] be the optimal stopping time associated to Pt.

Theorem

Θ = ˜ ΘJ + (1 − J)ξτ, where Jt = 1{τ>t} and the pre-default TVA ˜ Θ satisfies the following BSDE on [0, τ ∗] ˜ βt ˜ Θt = Et τ∗

t

˜ βs ˜ gs(Ps − ˜ Θs)ds

• , ˜

gt(Pt − ˜ Θt) = gt(Pt − ˜ Θt) + γt ˜ ξt. (8)

◮

˜ βt = e−

t

0(γu+ru)du. ◮

˜ ξt :=

1 Et(1{τ>t}) Et(ξt1{τ>t}) with ξt := Pt − E(R1{t=¯ τ}|Gt).

Extension

Possible to extend (8) on a portfolio of various American options.

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 7 / 38

SLIDE 8

Simulation algorithms

Plan

Introduction From linear/linear to linear/nonlinear From nonlinear/linear to nonlinear/nonlinear Simulation algorithms Without funding constraints With funding constraints Adaptation issues to American options The difference with the references LDLt Householder tridiagonalization + PCR Divide and conquer for eigenproblem Some simulation results Conclusion

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 8 / 38

SLIDE 9

Simulation algorithms

Without funding constraints

CVA0,T =

N−1

• k=0

E

• P+

k+11τ∈(kh,(k+1)h]

• ,

h = T N .

With funding constraints

Θk = Ek (Θk+1 + hg(k + 1, Pk+1, Θk+1)) , ΘN = 0.

The exposure

Pk(x) = E

• Φk,τk (Sτk )|Sk = x
• with

τN = N, ∀k ∈ {N − 1, ..., 0}, τk = k1Γk + τk+11Γc

k ,

(9) where Γk =

• Φk+1,k(Sk) > E
• Pk+1(Sk+1)
• Sk
• . The conditional

expectation involved in Γk is approximated using a regression on a basis

• f monomial functions where Kk is its cardinal.

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 9 / 38

SLIDE 10

Simulation algorithms

Without funding constraints

CVA0,T =

N−1

• k=0

E

• P+

k+11τ∈(kh,(k+1)h]

• ,

h = T N .

With funding constraints

Θk = Ek (Θk+1 + hg(k + 1, Pk+1, Θk+1)) , ΘN = 0.

An example of a two stage simulation with M0 = 2, M6 = 8 and M8 = 4

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 10 / 38

SLIDE 11

Simulation algorithms

CVA0,T approximation

• CVA0,T =

N−1

• k=0

1 M0

M0

• i=1

F 2

k+1

• P1(Si

1), ...,

Pk+1(Si

k+1)

• (10)

With F 1

k+1(x1, ..., xk+1) = E

• 1τ∈(kh,(k+1)h]|P1 = x1, ..., Pk+1 = xk+1
• ,

F 2

k+1(x1, ..., xk+1) = (xk+1)+F 1 k+1(x1, ..., xk+1).

(11)

Θk approximation

                   For k = 1, ..., N − 1

• Θk(x)= tψ(x)Ψ−1

k

  1 M0

M0

• j=1

ψ(Sj

k)

• Θk+1(Sj

k+1) + 1

N g

• k+1,

Θk+1(Sj

k+1),

Pk+1(Sj

k+1)

• 

 and ΘN(x) = 0,

• Θ0(S0) =

1 M0

M0

• j=1
• Θ1(Sj

1) + 1

N g

• 1,

Θ1(Sj

1),

P1(Sj

1)

• .

(12) Where Ψk = T   1 M0

M0

• i=0

ψ( Si

k)tψ(

Si

k)

  with: {Si}i∈{1,...,M0} and { Si}i∈{1,...,M0} are two 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.

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 11 / 38

SLIDE 12

Simulation algorithms Without funding constraints

Theorem

E

• CVA0,T − CVA0,T

2 ≤ N2 M0 max

k∈{0,...,N−1} Var

• F 2

k+1

• P1(Si

1), ...,

Pk+1(Si

k+1)

• +

N

• j=1

1 4NM2

j

• E
• Vj(Si

j )f ′′ j (Pj(Si j ))F 3 j (P1(Si 1), ..., Pj(Si j ))

2 +

N

• j=1

1 4NM2

j

  E   Vj(Si

j )f ′′ j (Pj(Si j )) N−1

• k=j

F 4

k+1(P1(Si 1), ..., Pj(Si j ), Si j )

   

2

+

N

• j=1

N 4M2

j

• E
• Vj(Si

j )F 1 j (P1(Si 1), ..., Pj(Si j ))|Pj(Si j ) = 0

• ϕj(0)

2 + N

N

• j=1

(N − j + 1)2O

• 1

M4

j

• Where

ϕj is the density of Pj(Si

j ), Vj(x) = Var

Mj

• Pj(x) − Pj(x)
• .

Good choices

If ϕj(0) is big then take Mj ∼ √M0, otherwise Mj ∼ √M0/N. In both cases, N must be small when compared to √M0. When American options are involved, make sure that K 3

j /Mj is small enough.

For example

Mj = N − j N − 1 M1 with either M1 = √M0 N

• r M1 =
• M0.

(13)

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 12 / 38

SLIDE 13

Simulation algorithms With funding constraints

Theorem

As long as {Θi(x)}0≤i≤N−1 are of class Cs on the support of S ∈ Rd, there exists a positive constant C such that for each 0 ≤ k ≤ N − 1 E

• Θk(Si

k) − Θk(Si k)

2 ≤ CK N2

N−1

• l=k

  E   Vl+1(Sj

l+1)∂2 Pg

• l + 1, Θl+1(Sj

l+1), Pl+1(Sj l+1)

• 2Ml

   

2

+O

• K

M0 + K 2 N2M0 + K N4M2

l

+ K 1−2s/d N2 + K −2s/d

• .

Good choice

Take Ml ∼

• M0/N, N must be sufficiently small N ∼ 10. When

American options are involved, make sure that K 3

l /Ml is small enough.

For example

Ml = N − l N − 1 M1 with M1 =

• M0

N . (14)

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 13 / 38

SLIDE 14

Adaptation issues to American options

Plan

Introduction From linear/linear to linear/nonlinear From nonlinear/linear to nonlinear/nonlinear Simulation algorithms Without funding constraints With funding constraints Adaptation issues to American options The difference with the references LDLt Householder tridiagonalization + PCR Divide and conquer for eigenproblem Some simulation results Conclusion

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 14 / 38

SLIDE 15

Adaptation issues to American options

An example of a two stage simulation with M0 = 2, M6 = 8 and M8 = 4 Inner dynamic programming

τN = N, ∀k ∈ {N − 1, ..., 0}, τk = k1Γk + τk+11Γc

k ,

(15) where Γk =

• Φk+1,k(Sk) > Rk · ψ(Sk)
• . The vector Rk minimizes the

quadratic error ||Pk+1(Sk+1) − Rk · ψ(Sk)||L2 (16) and denoting Ak = E (ψ(Sk)tψ(Sk)), we get Rk = A−1

k

E (ψ(Sk)Pk+1(Sk+1)) . (17)

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 15 / 38

SLIDE 16

Adaptation issues to American options The difference with the references

Three main methods for symmetric big matrices

Cholesky factorization

◮

• V. Volkov and J. Demmel. LU, QR and Cholesky Factorizations using

Vector Capabilities of GPUs. Berkeley Technical Report. 2008.

◮

• G. Ballard, J. Demmel, O. Holtz and O. Schwartz,

Communication-Optimal Parallel and Sequential Cholesky Decomposition. SIAM J. SCI. COMPUT. 32(6), 3495–3523. 2010.

Tridiagonal form + cyclic reduction

◮

• Y. Zhang , J. Cohen and J. D. Owens. 15th ACM SIGPLAN Symposium
• n Principles and Practice of Parallel Programming, 127–136. 2010.

◮

• D. Goddeke and R. Strzodka. Cyclic Reduction Tridiagonal Solvers on

GPUs Applied to Mixed Precision Multigrid. Parallel and Distributed Systems, IEEE Trans. 22(1), 22–32. 2010.

Tridiagonal form + eigenproblem

◮

• J. W. Demmel, O. A. Marques, B. N. Parlett and C. Vomel. Performance

and Accuracy of Lapack’s Symmetric Tridiagonal Eigensolvers. SIAM J.

• SCI. COMPUT. 30(3), 1508–1526. 2008.

◮

• C. Vomel, S. Tomov and J. Dongarra. Divide & Conquer on Hybrid

Gpu-Accelerated Multicore Systems. SIAM J. SCI. COMPUT. 34(2), 70–82. 2012.

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 16 / 38

SLIDE 17

Adaptation issues to American options The difference with the references

None of the previous works can be used directly

The reason

◮

Large number of small random linear systems: The size does not exceed 64 and the communication is reduced.

◮

Some of these random systems could be ill-conditioned.

• Ak,l =

1 Mk

Mk

• j=1

ψl(S(j)

tk )ψl(S(j) tk )t

(18)

Typical condition numbers for linear regression n = 30 in the Black & Scholes model

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 17 / 38

SLIDE 18

Adaptation issues to American options The difference with the references

Must we systematically use Householder tridiagonalization with divide & conquer when we suspect the random linear systems to be ill-conditioned?

Our answer

◮

Perform Householder tridiagonalization O(4n3/3) and solve the linear systems cheaply using parallel cyclic reduction O(n log2(n)).

◮

Take a decision according to the value of the residue error:

* If the residue error is small then we already have good

solutions.

* Otherwise, we must perform divide & conquer O(4n3/3)

diagonalizations and discard the smallest eigenvalues.

◮

The next time we solve this same kind of linear systems:

* If they used to be well-conditioned then we just process LDLt

O(n3/6).

* Otherwise we execute directly the combination of

Householder tridiagonalization and divide & conquer diagonalization.

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 18 / 38

SLIDE 19

Adaptation issues to American options LDLt

Shared occupation n(n + 1)/2 + n and complexity O(n3/6)

A = LDLt, Dj,j = Aj,j −

j−1

• k=1

L2

j,kDk,k,

Li,j = 1 Dj,j  Ai,j −

j−1

• k=1

Li,kLj,kDk,k   if i > j. (19)

Standard LDLt parallel strategy

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 19 / 38

SLIDE 20

Adaptation issues to American options LDLt

Three different versions studied

1. An SIMD version that requires only independent threads, one for each linear system. 2. A collaborative version that involves n collaborative threads for each linear system with n unknowns. 3. An optimal hybrid solution that involves n∗ (n∗ < n) collaborative threads for each linear system with n unknowns.

The speedup of the collaborative and the hybrid versions when compared to the SIMD implementation.

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 20 / 38

SLIDE 21

Adaptation issues to American options LDLt

(a) Optimal number of collaborative threads (b) Number of systems solved within a second

(a) (b)

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 21 / 38

SLIDE 22

Adaptation issues to American options Householder tridiagonalization + PCR

Householder tridiagonalization: Shared occupation n2 + 2n and complexity O(4n3/3)

1. An SIMD version that requires only independent threads, one for each linear system. 2. A collaborative version that involves n collaborative threads for each linear system with n unknowns.

For symmetric A

U = Ht

3...Ht nAHn...H3 =

             d1 c1 c1 d2 c2 c2 d3 ... ... ... ... ... ... cn−1 cn−1 dn              , (20) with each Householder matrix H given by H = I − uut/b, b = utu/2. (21)

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 22 / 38

SLIDE 23

Adaptation issues to American options Householder tridiagonalization + PCR

From CR: Shared occupation 3n and complexity O(nlog2(n))

CR when n = 8

e1 e7 e5 z2 z8 z4 z6

Stepg1:gForwardgreductiongtog ag4-unknowngsystemginvolvingg z2,gz4,gz6gandgz8 Stepg2:gForwardgreductiongto ag2-unknowngsystemginvolvingg z4gandgz8 Stepg3:gSolveg2-unknowngsystem Stepg4:gBackwardgsubstitutiongto solvegthegrestg2gunknowns Stepg5:gBackwardgsubstitutiongto solvegthegrestg4gunknowns

A A A A A A A A A A A A A A A A A A A A A A A A A A A A

A A A A A A

A

A

e3 z3 z1 z5 z7 e1 e2 e3 e4 e5 e6 e7 e8 e'8 e'6 e'4 e'2 e''4 e''8 e'6 e'2 z4 z8 z2 z4 z6 z8

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 23 / 38

SLIDE 24

Adaptation issues to American options Householder tridiagonalization + PCR

To a new version of PCR: Shared occupation 4n and complexity O(nlog2(n))

PCR when n = 8

Step 1: Reduced to 2 systems

• f 4 unknowns

Step 2: Reduced to 4 systems

• f 2 unknowns

Step 3: Solve

A A A A A A A A A A A A A A A A A A A A A A e'7 e'5 e'3 e'1 A A A A A A A A A A A A A A A A A A A A e''6 e''2 e''5 e''1 e''8 e''4 e''7 e''3 e'7 e'5 e'3 e'1 e'2 e'4 e'6 e'8 A A A A A A A A A A A A A A A A z4 z2 z3 z1 z8 z6 z7 z5 e''7 e''3 e''5 e''1 e''2 e''6 e''4 e''8 e'8 e'6 e'4 e'2 e4 e3 e2 e1 e5 e6 e7 e8

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 24 / 38

SLIDE 25

Adaptation issues to American options Householder tridiagonalization + PCR

         d1 c1 c1 d2 c2 c2 d3 c3 c3 d4 c4 c4 d5 c5 c5 d6 c6 c6 d7          (R) − →          d′

1

c′

2

d′

2

c′

3

c′

2

d′

3

c′

4

c′

3

d′

4

c′

5

c′

4

d′

5

c′

6

c′

5

d′

6

c′

6

d′

7

         (P) − →          d′

1

c′

2

c′

2

d′

3

c′

4

c′

4

d′

5

c′

6

c′

6

d′

7

d′

2

c′

3

c′

3

d′

4

c′

5

c′

5

d′

6

         (R) − →          d′′

1

c′′

2

d′′

3

c′′

4

c′′

2

d′′

5

c′

4

d′

7

d′′

2

c′′

3

d′′

4

c′′

3

d′′

6

         (P) − →          d′′

1

c′′

2

c′′

2

d′′

5

d′′

3

c′′

4

c′′

4

d′

7

d′′

2

c′′

3

c′′

3

d′′

6

d′′

4

        

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 25 / 38

SLIDE 26

Adaptation issues to American options Householder tridiagonalization + PCR

(a) Number of PCRs + two matrix/vector multiplications performed per second. (b) LDLt vs. tridiagonal + PCR

(a) (b)

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 26 / 38

SLIDE 27

Adaptation issues to American options Divide and conquer for eigenproblem

For ill-conditioned systems

Standard procedure

◮

Tridiagonal Householder decomposition A = QUQt where Q is orthogonal and U is symmetric tridiagonal.

◮

Divide & conquer algorithm for symmetric tridiagonal eigenproblems to establish U = ODOt where O is orthogonal and D is diagonal.

◮

Discard the smallest eigenvalues of D that provide a condition number larger than 105. U =                   d1 c1 c1 ... ... ... ... cm−1 cm−1 dm − cm dm+1 − cm cm+1 cm+1 ... ... ... ... cn−1 cn−1 dn                   + cm1m,m+11t

m,m+1

= U1 U2

• + cm1m,m+11t

m,m+1

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 27 / 38

SLIDE 28

Adaptation issues to American options Divide and conquer for eigenproblem

Shared occupation 2n(n + 2) + 21+⌊log2(n−1)⌋ and complexity O(4n3/3)

Steps

1. U = O1 O2 D1 D2

• + cmuut

Ot

1

Ot

2

• where u =

Ot

1

Ot

2

• 1m,m+1 =

last column of Ot

1

first column of Ot

2

• .

2. Let Λ = {λ1, ..., λn}, ordered family of eigenvalues of D1 D2

• . If

cm = 0 and the eigenvalue λ of U satisfies λ / ∈ Λ, then its value is

• btained as a solution of

n

• i=1

u2

i

λi − λ + 1 cm = 0. (23) 3. From u and the solutions of (23), Löwner’s Theorem provides vector u that is used to compute the eigenvector Vλ of D1 D2

• + cmuut

4. Let W = (Vλ)λ eigenvalue of U, we get the eigenvectors of U thanks to the multiplication Q1 Q2

• W .

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 28 / 38

SLIDE 29

Adaptation issues to American options Divide and conquer for eigenproblem

Additional details on steps 1. and 2.

Step 1.

Advantage: Pure divide and conquer algorithm, it prevents to have eigenvalues of multiplicity bigger than two at each conquering step.

Step 2.

Using Gragg’s scheme (based on Newton’s method): Choose hk such that hk(λ) = xk,0 + xk,1/(λk − λ) + xk,2/(λk+1 − λ) matches

n

• i=1

u2

i

λi − λ + 1 cm at its root ∈ (λk, λk+1) up to the second derivative. Advantage: Cubic monotonic convergence.

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 29 / 38

SLIDE 30

Adaptation issues to American options Divide and conquer for eigenproblem

Comparison with Householder tridiagonalization

Execution time comparison: TDC/THouseholder

10 20 30 40 50 60 70 2 3 4 5 6 7 8 9 10 System size

Commments

◮

Small matrices.

◮

Iterative algorithm to solve the secular equation.

◮

Divergence produced by deflation.

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 30 / 38

SLIDE 31

Some simulation results

Plan

Introduction From linear/linear to linear/nonlinear From nonlinear/linear to nonlinear/nonlinear Simulation algorithms Without funding constraints With funding constraints Adaptation issues to American options The difference with the references LDLt Householder tridiagonalization + PCR Divide and conquer for eigenproblem Some simulation results Conclusion

Lokman (UPMC, LPMA) GPU Tech. Conf. 2016 31 / 38

SLIDE 32

Some simulation results

Within less than 1 minute simulation on GPU: M0 = 131K, N = 10, Neds = 50

European Path-dependent

• ption

Φ(ST ) =

• S1

T

2 + S2

T

2 − S

3 T

• +

M1 Θ0 Θ0 std CVA0,T CVA0,T std √M0 N 0.01364 4 ∗ 10−5 0.0296 2 ∗ 10−4 √M0 √ N 0.01307 4 ∗ 10−5 0.0294 2 ∗ 10−4 √M0 0.01265 3 ∗ 10−5 0.0291 2 ∗ 10−4

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SLIDE 33

Some simulation results

Within less than 1 minute simulation on GPU: M0 = 131K, N = 10, Neds = 50

European Path-dependent

• ption

Φ(ST ) =

• 3S1

T

10 + 7S2

T

10 − S

3 T

• +

−

• 7S1

T

10 + 3S2

T

10 − S

3 T

• +

M1 Θ0 Θ0 std CVA0,T CVA0,T std √M0 N 2.72 ∗ 10−3 10−5 0.0365 8 ∗ 10−4 √M0 √ N 2.44 ∗ 10−3 10−5 0.0453 8 ∗ 10−4 √M0 2.28 ∗ 10−3 10−5 0.0520 8 ∗ 10−4 √ N√M0 2.24 × 10−3 10−5 0.0528 8 × 10−4

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SLIDE 34

Some simulation results

Within less than 1 minute simulation on GPU: M0 = 131K, N = 10, Neds = 50 and n = 10

American option

Φ(ST ) =

• K − S1

T

3 − S2

T

3 − S3

T

3

• +

M1 Θ0 Θ0 std CVA0,T CVA0,T std √M0 √ N 0.0242 10−4 0.0356 2 ∗ 10−4 √M0 0.0229 10−4 0.0351 2 ∗ 10−4

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SLIDE 35

Conclusion

Plan

Introduction From linear/linear to linear/nonlinear From nonlinear/linear to nonlinear/nonlinear Simulation algorithms Without funding constraints With funding constraints Adaptation issues to American options The difference with the references LDLt Householder tridiagonalization + PCR Divide and conquer for eigenproblem Some simulation results Conclusion

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SLIDE 36

Conclusion

Mathematical and computing work suited to GPUs

Mathematical part

◮

Extending CVA (TVA) on American options.

◮

Judicious choice of the number of inner and outer trajectories.

Computing part

◮

CUDA source code of: LDLt, Householder reduction, parallel cyclic reduction that is not necessary a power of two and divide and conquer for eigenproblem.

◮

Execution time comparison of the different methods mentioned above.

◮

Original method to further optimize the adaptation of LDLt to our context.

◮

Original parallel cyclic reduction that can be used for any vector size and not only a power of two.

◮

Precise answer to the following question: Must we systematically use Householder tridiagonalization with divide & conquer when we suspect the random linear systems to be ill-conditioned?

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SLIDE 37

Conclusion

Future work

◮

Further developments on the use of Nested Monte Carlo on GPUs for BSDEs.

◮

Studying the rounding errors and error propagation.

◮

Use CADNA library to test each procedure: http://www-pequan.lip6.fr/cadna/

Source code

◮

http://www.proba.jussieu.fr/~abbasturki/soft.htm

◮

Premia library: https://www.rocq.inria.fr/mathfi/Premia/index.html

References

◮

L.A. Abbas-Turki and Stef Graillat. Resolution of a large number of small random symmetric linear systems in single precision arithmetic on GPUs: https://hal.archives-ouvertes.fr/hal-01295549

◮

L.A. Abbas-Turki and M.A. Mikou. TVA on American Derivatives: https://hal.archives-ouvertes.fr/hal-01142874

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SLIDE 38

The End

Thank you Questions?

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