Nonlinear Monte Carlo Methods. From American Options to Fully Nonlinear PDEs

Transcription

1 : From Amercan Optons to Fully Nonlnear PDEs Ecole Polytechnque Pars PDEs and Fnance Workshop KTH, Stockholm, August 20-23, 2007

2 Outlne 1 Monte Carlo Methods for Amercan Optons 2 3 4

3 Outlne 1 Monte Carlo Methods for Amercan Optons 2 3 4

4 Prcng Amercan Optons n Complete Markets In the context of a complete market wth a nonrsky asset S 0 : S 0 t = e rt, t 0, and a rsky securty wth prce process {S t, t 0}... the no-arbtrage prce of the Amercan put opton wth strke K > 0 and maturty T > 0 : P 0 = sup E [ e rτ (K S τ ) +] = E [e rτ (K S τ ) +] τ T T where T T = {stoppng tmes wth values n [0, T ]} and τ = mn { t 0 : P t = (K S t ) +}

5 Dscrete-tme Approxmaton Let t n = h n, = 1,..., n, and h n = T n Defne the so-called Snell envelope : Y n T = (K S T ) + and Y n t n = max { (K St n ) +, Et n Then, an approxmaton of the Amercan put prce s : Y n 0 P 0, the error s known to be of order n 1/2,.e. lm sup n n (Y n 0 P 0 ) < and an approxmaton of the optmal stoppng polcy s : { τn := nf t n : Yt n n = ( ) } K S t n [ ]} e rhn Yt n +1 n

6 Approxmaton of Condtonal Expectatons Man observaton : n the present context all condtonal expectatons are regressons,.e. [ ] [ ] E t n Yt n +1 n = E Yt n +1 n S t n = Classcal methods from statstcs : Kernel regresson <Carrère> Projecton on subspaces of L 2 (P) <Longstaff-Schwartz, Gobet-Lemor-Warn AAP05> from numercal probablstc methods quantzaton... <Bally-Pagès SPA03> Integraton by parts <Bouchard-Ekeland-Touz FS04>

7 Approxmaton of the Replcatng Strategy Put prce s P t = P(t, S t ) a determnstc functon of (t, S t ) The replcatng strategy of the Amercan put s : t = P s (t, S t), t < τ An approxmaton of the replcatng strategy wthn a Monte Carlo estmaton of the put prce s : [ ] n t n = E t n Yt n W t n n ; W t n +1 = W t n +1 W t n h n <Broade-Glasserman, Fourné-Lasry-Lebuchoux-Lons-Touz > Fnally, the Monte Carlo scheme s : YT n = (K S T ) + and { (K Ŷt n ) [ ]} + n = max St n, e rh n Ê t n Y n t+1 n [ ] ˆ n t n = Ê t n Yt n W t n n h n

8 From Amercan Optons to Fully Nonlnear PDEs Objectve : Monte Carlo technque for the approxmaton of the Amercan opton prce and hedge extends to solutons of Fully nonlnear PDEs. Fully Nonlnear PDEs are encountered n many areas of appled mathematcs. In partcular, stochastc control problems can be characterzed n terms of the Bellman (dynamc programmng) equaton 0 = v { 1 t sup b(x, u) Dv + 1 u U 2 2 Tr [ σσ T (x, u)d 2 v ] +f (x, u)v k(x, u) 1 } 2 optmal stoppng problems can also be characterzed n terms of the correspondng Bellman equaton (free boundary)

9 Outlne 1 Monte Carlo Methods for Amercan Optons 2 3 4

10 Backward SDE : Defnton Fnd an F W adapted (Y, Z) satsfyng : T T Y t = ξ + F r (Y r, Z r )dr Z r dw r t t.e. dy t = F t (Y t, Z t )dt + Z t dw t and Y T = ξ where the generator F : Ω [0, T ] R R d R, and {F t (y, z), t [0, T ]} s F W adapted If F s Lpschtz n (y, z) unformly n (ω, t), and ξ L 2 (P), then there s a unque soluton satsfyng E sup Y t 2 + E t T T 0 Z t 2 dt <

11 Markov BSDE s Let X. be defned by the (forward) SDE dx t = b(t, X t )dt + σ(t, X t )dw t and F t (y, z) = f (t, X t, y, z), f : [0, T ] R d R R d R ξ = g(x T ) L 2 (P), g : R d R If f contnuous, Lpschtz n (x, y, z) unformly n t, then there s a unque soluton to the BSDE dy t = f (t, X t, Y t, Z t )dt + Z t σ(x t )dw t, Y T = g (X T ) Moreover, there exsts a measurable functon V : Y t = V (t, X t ), 0 t T

12 BSDE s and semlnear PDE s By defnton, Y s Y t = V (s, X s ) V (t, X t ) = s t f (X r, Y r, Z r )dr + s t Z r σ(x r )dw r If V (t, x) s smooth, t follows from Itô s lemma that : s t LV (r, X r )dr + s t = DV (r, X r ) σ(x r )dw r s t f (X r, Y r, Z r )dr + s where L s the Dynkn operator assocated to X : LV = V t + b DV Tr[σσT D 2 V ] t Z r σ(x r )dw r

13 Stochastc representaton of solutons of a semlnear PDE Under some condtons, the semlnear PDE V LV (t, x) f (x, V (t, x), DV (t, x)) = 0 t V (T, x) = g(x) has a unque soluton whch can be represented as where Y t,x solves the BSDE V (t, x) = Y t,x t Y s = g(x T ) + T s f (X r, Y r, Z r )dr T s Z r σ(x r )dw r, t s T and X t = x, dx s = b(x s )ds + σ(x s )dw s, t s T

14 Extenson of Feynman-Kac s formula Let f 0, then V (t, x) = Y t,x t = g ( X t,x ) T T t Z r σ ( Xr t,x ) dwr = take condtonal expectatons V (t, x) = E [ g(x t,x T )] wth : Xt t,x = x and dxr t,x = b ( Xr t,x ) ( ) dr + σ X t,x r dwr = Numercal soluton by Monte Carlo : ˆV (t, x) := 1 N N g =1 ( ) ˆX () T V (t, x) a.s. (LLN) and N ( ) ˆV (t, x) V (t, x) = N (0, V [g(x T )]) (CLT)

15 Dscrete-tme approxmaton <Bally-Pagès SPA03, Zhang AAP04, Bouchard-Touz SPA04> Numercal soluton of a sem-lnear PDE by smulatng the assocated backward SDE by means of Monte Carlo methods Start from Euler dscretzaton : Yt n n = g ( Xt n ) n s gven, and E n [ W t+1 Yt n +1 Yt n = f ( Xt n, Yt n, Zt n ) t +Zt n σ ( Xt n ) Wt+1 = Dscrete-tme approxmaton : Y n t n Y n t = E n [ Y n t +1 ] Z n t = ( t ) 1 E n = g ( X n t n ) and + f ( Xt n, Yt n, Zt n ) t +..., 0 n 1, [ ] Yt n +1 W t+1 = Smlar to numercal computaton of Amercan optons

16 Dscrete-tme approxmaton <Bally-Pagès SPA03, Zhang AAP04, Bouchard-Touz SPA04> Numercal soluton of a sem-lnear PDE by smulatng the assocated backward SDE by means of Monte Carlo methods Start from Euler dscretzaton : Yt n n = g ( Xt n ) n s gven, and E n [ W t+1 Yt n +1 Yt n = f ( Xt n, Yt n, Zt n ) t +Zt n σ ( Xt n ) Wt+1 = Dscrete-tme approxmaton : Y n t n Y n t = E n [ Y n t +1 ] Z n t = ( t ) 1 E n = g ( X n t n ) and + f ( Xt n, Yt n, Zt n ) t +..., 0 n 1, [ ] Yt n +1 W t+1 = Smlar to numercal computaton of Amercan optons

17 Dscrete-tme approxmaton <Bally-Pagès SPA03, Zhang AAP04, Bouchard-Touz SPA04> Numercal soluton of a sem-lnear PDE by smulatng the assocated backward SDE by means of Monte Carlo methods Start from Euler dscretzaton : Yt n n = g ( Xt n ) n s gven, and E n [ W t+1 Y n t +1 Y n t = f ( X n t, Y n t, Z n t ) t +Z n t σ ( X n t ) Wt+1 = Dscrete-tme approxmaton : Y n t n Y n t = E n [ Y n t +1 ] Z n t = ( t ) 1 E n = g ( X n t n ) and + f ( Xt n, Yt n, Zt n ) t +..., 0 n 1 [ ] Yt n +1 W t+1 = Smlar to numercal computaton of Amercan optons

18 Dscrete-tme approxmaton <Bally-Pagès SPA03, Zhang AAP04, Bouchard-Touz SPA04> Numercal soluton of a sem-lnear PDE by smulatng the assocated backward SDE by means of Monte Carlo methods Start from Euler dscretzaton : Yt n n = g ( Xt n ) n s gven, and E n [ W t+1 Y n t +1 Y n t = f ( X n t, Y n t, Z n t ) t +Z n t σ ( X n t ) Wt+1 = Dscrete-tme approxmaton : Y n t n Y n t = E n [ Y n t +1 ] Z n t = ( t ) 1 E n = g ( X n t n ) and + f ( Xt n, Yt n, Zt n ) t +..., 0 n 1 [ ] Yt n +1 W t+1 = Smlar to numercal computaton of Amercan optons

19 Dscrete-tme approxmaton, contnued Theorem Assume f and g are Lpschtz. Then : { [ 1 ]} lm sup n sup E Yt n Y t 2 + E Zt n Z t 2 dt n 0 0 t 1 0 < Same rate of convergence as for the smulaton of (forward) SDEs n the present context all condtonal expectatons are regressons,.e. [ ] E Yt n +1 F t [ ] E Yt n +1 W t+1 F t [ = E = E ] Yt n +1 X t [ ] Yt n +1 W t+1 X t = can be approxmated as n the case of Amercan optons...

20 Smulaton of Backward SDE s 1. Smulate trajectores of the forward process X (well understood) 2. Backward algorthm : Ŷt n n = g ( X n ) t n ] ( ) Ŷ t n 1 = Ên t 1 [Ŷ n t + f Xt n 1, Ŷt n 1, Ẑt n 1 t, 1 n, ] Ẑt n 1 = Ên t 1 [Ŷ n W t t t (truncaton of Ŷ n and Ẑ n needed n order to control the L p error)

21 Smulaton of BSDEs : bound on the rate of convergence Theorem For p > 1 : lm sup n max 0 n n 1 d/(4p) N 1/2p Ŷt n Yt n L p < For the tme step 1, and lmt case p = 1 : n rate of convergence of 1 n f and only f n 1 d 4 N 1/2 = n 1/2,.e. N = n 3+ d 2

22 Outlne 1 Monte Carlo Methods for Amercan Optons 2 3 4

23 Man purpose Enlarge the class of BSDE s n order to obtan a stochastc representaton of Fully Nonlnear PDE s (In partcular, representaton of general stochastc control problems) Gradent s related to the representaton of a random varable as a stochastc ntegral (up to the drver) In order to obtan a fully nonlnear PDE, one needs to nclude the Hessan n the drver... = Requres understandng local behavor of double stochastc ntegrals...

24 : Defnton ˆf (x, y, z, γ) := f (x, y, z, γ) Tr[σσT (x)γ] non-decreasng n γ Consder the 2nd order BSDE : dx t = σ(x t )dw t dy t = f (t, X t, Y t, Z t, Γ t )dt + Z t σ(x t )dw t, Y T = g(x T ) dz t = α t dt + Γ t σ(x t )dw t A soluton of (2BSDE) s a process (Y, Z, α, Γ) wth values n R R n R n S n Queston : exstence? unqueness? n whch class? <Cherdto, Soner, Touz and Vctor CPAM 2007>

25 Second order BSDE : Exstence and unqueness Feynman-Kac formula for fully-nonlnear PDE Theorem Let Assumpton (f) hold, and suppose g has lnear growth. Suppose further that (E) satsfes the comparson Assumpton Com and has a smooth soluton v : [0, T ] R d R satsfyng max { Dv(t, x), D 2 v(t, x), LDv(t, x) } m (1 + x p ) D 2 v(t, x) D 2 v(s, y) m (1 + x p + y p ) ( t s + x y ) Then the process ( Ȳ, Z, ᾱ, Γ ) defned by Ȳ t := v(t, X t ), Z t := Dv(t, X t ), ᾱ t := LDv(t, X t ), Γ t := V xx (t, X t ) s the unque soluton of (2BSDE) wth Z A 0,x

26 Second order BSDE : Class of solutons Let A m t,x be the class of all processes Z of the form Z s = z + s t α r dr + s t Γ r dx t,x r, s [t, T ] where z R d, α and Γ are respectvely R d and S d (R d ) progressvely measurable processes wth max { Z s, α s, Γ s } m ( 1 + X s t,x p ), Γ r Γ s m ( 1 + X r t,x p + X t,x s p ) ( r s + Xr t,x Our unqueness result s restrcted to solutons (Y, Z, α, Γ) of (2BSDE) such that Z A t,x := m 0 A m t,x Xs t,x )

27 Idea of proof of unqueness : from fnance Defne the stochastc target problems { V (t, x) := nf y : Y t,y,z T g ( X t,x ) } T a.s. for some Z At,x (Seller super-replcaton cost n fnance), and { U(t, x) := sup y : Y t,y,z T g ( X t,x ) } T a.s. for some Z At,x (Buyer super-replcaton cost n fnance) By defnton : V (t, X t ) Y t U(t, X t ) for every soluton (Y, Z, α, Γ) of (2BSDE) wth Z A 0,x Man techncal result : V s a (dscontnuous) vscosty super-soluton of the nonlnear PDE (E) = U s a (dscontnuous) vscosty subsoluton of (E) Assumpton Com = V U

28 A probablstc numercal scheme for fully nonlnear PDEs By analogy wth BSDE, we ntroduce the followng dscretzaton for 2BSDEs : Y n t n = g ( Xt n ) n, Yt n 1 = E n [ ] ( ) 1 Y n t + f Xt n 1, Yt n 1, Zt n 1, Γ n t 1 t, 1 n, [ ] Zt n 1 = E n 1 Yt n W t t [ Γ n t 1 = E n 1 Yt n ( W t ) 2 ] t ( t ) 2

29 Intuton From Greeks Calculaton Frst, use the approxmaton f (x) h=0 E[f (x + W h )] Then, ntegraton by parts shows that f (x) f (x + y) e y 2 /(2h) dy 2π = f (x + y) y e y [ 2 /2 dy = E f (x + W h ) W ] h h 2π h = f (x + y) y 2 h e y 2/2 [ ( W 2 )] h 2 dy = E f (x + W h ) h h 2π h 2 Connecton wth Fnte Dfferences : W h ( 1 h 2 δ ) 2 δ 1 [ E ψ(x + W h ) W ] h h ψ(x + h) ψ(x h) 2h Centered FD!

30 The Convergence Result <Fahm, Soner and Touz 2007> Theorem Suppose that f s Lpschtz and f γ L σ. Then Y n 0 v(t, x) where v s the unque vscosty soluton of the nonlnear PDE. Bounds on the approxmaton error are avalable Ths convergence result s weaker than that of (frst order) Backward SDEs...

31 Outlne 1 Monte Carlo Methods for Amercan Optons 2 3 4

32 Comments on the 2BSDE algorthm n BSDEs the drft coeffcent µ of the forward SDE can be changed arbtrarly by Grsanov theorem (mportance samplng...) n 2BSDEs both µ and σ can be changed. Numercal results (together wth above theorem) however recommend prudence... The heat equaton v t + v xx = 0 correspond to a BSDE wth zero drver. Splttng the Laplacan n two peces, t can also be vewed as a 2BSDE wth drver f (γ) = 1 2 γ numercal experments show that the 2BSDE algorthm performs better than the pure fnte dfferences scheme

33 Numercal example : portfolo optmzaton (X. Warn) Wth U(x) = e ηx, want to solve : [ ( T )] V (t, x) := sup E U x + θ u σ(λdu + dw u ) θ t An explct soluton s avalable V s the characterzed by the fully nonlnear PDE V t λ2 (V x) 2 V xx = 0 and V (T,.) = U

34 Fg.: Relatve Error (Regresson), dmenson 1

35 Fg.: Relatve Error (Regresson), dmenson 2

36 Varyng the drft of the FSDE Drft FSDE Relatve error (Regresson) -1 0, ,8 0, ,6 0, ,4 0, ,2 0, , ,2 0, ,4 0, ,6 0, ,8 0,

37 Varyng the volatlty of the FSDE Volatlty FSDE Relatve error Relatve error (Regresson) (Quantzaton) 0,2 0, , ,4 0, , ,6 0, , ,8 0, , , , ,2 0, , ,4 0, , ,6 0, ,