Numerical scheme for quadratic BSDEs and Dynamic Risk Measures

Transcription

1 Numerical scheme for quadratic BSDEs, Numerics and Finance, Oxford Man Institute Julien Azzaz I.S.F.A., Lyon, FRANCE 4 July 2012 Joint Work In Progress with Anis Matoussi, Le Mans, FRANCE

2 Outline Framework BSDE Extended framework for numerical scheme Dynamic Risk Measures Axioms A famous example Duality Numerical Scheme Classical approximation Dual Scheme Numerical experiments Some figures Optimizations

3 Outline Framework BSDE Extended framework for numerical scheme Dynamic Risk Measures Axioms A famous example Duality Numerical Scheme Classical approximation Dual Scheme Numerical experiments Some figures Optimizations

4 BSDE T T Y t = ξ T + g(z s )ds Z s dw s. t t (qbsde[g, ξ T ] for short) Hypothesis on driver and terminal value (H) : g : (t, z) [0, T ] R d g(t, z) R; C > 0 s.t. P as, g(t, z) C(1 + z 2 ), t, z; P as, t, z g(t, z) is continuous ; g is convex in variable z; g is (F t ) B(R d )-measurable; ξ T L (F T ). Existence and maximality of a solution (Kobylanski 2000)

5 Extended framework for numerical scheme FBSDE ds t = b(s t )dt + σ(s t )dw t S 0 = x dy t = g(t, Z t )dt Z t dw t Y T = ξ T ξ T = ϕ(s T ) or ξ T = ϕ((s t ) t [0,T ] ).

6 Outline Framework BSDE Extended framework for numerical scheme Dynamic Risk Measures Axioms A famous example Duality Numerical Scheme Classical approximation Dual Scheme Numerical experiments Some figures Optimizations

7 Axioms Theorem (g-conditional risk measure) (Convexity) R g S (λξ1 T + (1 λ)ξ2 T ) λrg S (ξ1 T ) + (1 λ)rg S (ξ2 T ) (Monotonic) R g S (ξ1 T ) Rg S (ξ2 T ) (Translation invariance) R g S (ξ T + η) = R g S (ξ T ) η (Arbitrage-free) R g S (ξ1 T ) = Rg S (ξ2 T ) on A t = {S < T } ξt 1 = ξ2 T a.s. on A S.

8 Axioms Theorem (Time-consistent) R g S (ξ U) = R g S ( Rg T (ξ U)) (Almost Conditionally invariant) ( t [0, T ], g(t, 0) = 0) ( R g S (1{B}ξ T ) = 1{B}R g S (ξ T ) a.s. ). (Almost Homogeneous) ( t [0, T ] g(t,.) is homogeneous) ( R g S (λ Sξ T ) = λ S R g S (ξ T ) a.s

9 A famous example Dynamic entropic risk measure g(t, z) = z2 t 2γ. e γ t (ξ T ) = 1 γ log E [exp ( γξ T ) /F t ], t [0, T ]. It will be used as a benchmark for simulations.

10 Duality Definition (Legendre-Fenchel Transform) Let g : z [0, T ] g(z) R be a convex function. Its Legendre-Fenchel transform G is the polar process associated to g, defined by : G(µ) = sup { µ, z g(z)} = sup { µ, z g(z)}. z R d z Dom(g) Proposition (Subdifferential) g(z) = {µ/g(z) µ, z G(µ)}.

11 Duality Theorem µ g(z) µ is optimal for : g(z) = inf µ {G(µ) µ, z } (i.e. g(z) = G( µ) µ, z ).

12 Duality Dual representation BMO-processes BMO(P) {ϕ H 2d s.t. C > 0, t, E [ T Proposition t ] } ϕ s 2 ds/f t C, P a.s. µ = (µ t ) t [0,T ] BMO(P) (Γ µ t ) t [0,T ] is a u.i. martingale. ( Γ µ t t = exp 0 µ sdw s 1 ) t 2 0 µ s 2 ds dqµ dp F t = Γ µ t

13 Duality Theorem (Dual representation; El Karoui and Barrieu 2009) Let g and ξ T verify hypothesis (H). It follows, using previous convex analysis results, that G, g and µ are well defined as above. Then, the solution of qbsde[g, ξ T ] (Y t, Z t ) t [0,T ] exists. We denote (R g t (ξ T )) t [0,T ] := (Y t ) t [0,T ]. Moreover, this solution process can be written as : R g t (ξ T ) = E Q µ [ T ] ξ T G(s, µ)ds/f t, t [0, T ]. t Here above, µ is a process defined by : s [0, T ], µ s g(z s ) and it belongs in BMO(P). The equivalent probability measure Q µ is defined as previously.

14 Outline Framework BSDE Extended framework for numerical scheme Dynamic Risk Measures Axioms A famous example Duality Numerical Scheme Classical approximation Dual Scheme Numerical experiments Some figures Optimizations

15 Classical approximation S N t k+1 = S N t k + b(s tk ) k t + σ(s tk ) k W. How to use dual representation? Yt N N = ξ T (1a) Zt N k = 1 [ ] k t E t k k W Yt N k+1 (1b) [ ] Yt N k = E tk Yt N k+1 + g tk (Zt N k ) k t (1c)

16 Dual Scheme Zt N k = 1 [ ] k t E t k k W Yt N k+1, (2a) Find µ tk s.t. g(zt N k ) = G( µ tk ) < µ tk, Zt N k >, (2b) ( T Evaluate Ê k exp µ s dw s 1 T ) µ 2 s ds, (2c) t k 2 t k [( T ) ] Evaluate Yt N k E P k ξ T G(s, µs )ds Ê k. (2d) Convergence is a W.I.P. Conjecture Convergence will be as the one of Gobet, Lemor and Warin (2005) Want to make the same for unbounded terminal condition (with jumps): Barrieu et El Karoui (2010), El Karoui, Matoussi and Ngoupeyou (2012). t k

17 Outline Framework BSDE Extended framework for numerical scheme Dynamic Risk Measures Axioms A famous example Duality Numerical Scheme Classical approximation Dual Scheme Numerical experiments Some figures Optimizations

18 Some figures ξ T = sin(s T ), γ = 1, T = 1; LSM : [ 1, x, x 2, x 3] ; Tolerance of order 10 4 for optimization tools; M = N Y e e e e e-01 Error

19 Some figures A few statistics on trajectories of Y Figure: N=40, M=5000

20 Optimizations In our simulations, use of Matlab and specific Matlab functions to make optimizations (cf. Legendre-Fenchel Transform and µ). Example : Find the minimizer of f (x) Tolerance on f Tolerance on x Influence on rate of convergence of the global scheme. Example : ( tn ) =T Ê k = exp µ s s W µ2 s 2 s. s=t k Example : To find µ, we use some function with default tolerance in X and Y of If we reduce it to 10 4, we gain 40% in execution time and lose 10% of the precision. Joint work with Denis Clot (MCF at I.S.F.A.)

21 Optimizations THANK YOU