[PPT] - A probabillistic Numerical Method for Fully Nonlinear PDEs Nizar PowerPoint Presentation

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Motivation : American options The Probabilistic Scheme Numerical results

A probabillistic Numerical Method for Fully Nonlinear PDEs

Nizar TOUZI

Ecole Polytechnique Paris Joint work with Xavier WARIN and Arash FAHIM

Computational Methos in Finance Fields Institute, Toronto March 22-24, 2010

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 2

Motivation : American options The Probabilistic Scheme Numerical results

Outline

1 Motivation : American options 2 The Probabilistic Scheme

A natural MC-FD scheme The semilinear case The fully nonlinear case

3 Numerical results

On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 3

Motivation : American options The Probabilistic Scheme Numerical results

Pricing and Hedging US Options

(Ω, F, P), P : risk-neutral measure, complete market
Consider an American option defined by the payoff process

{Gt, t ≥ 0}

Then, pricing and hedging reduce to :

V0 = sup {E [Gτ] : τ stopping time ≤ T} which can be approximated by the (discrete-time) Snell envelop(tk := kT/n) : V n

T := GT

and V n

tk := max

Gtk, E
V n

tk+1|Ftk

Standard numerical scheme !

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SLIDE 4

Motivation : American options The Probabilistic Scheme Numerical results

Approximation of conditional expectations

Main observation : the latter conditional expectations are regressions : E

Y n

ti+1|Fti

=

E

Y n

ti+1|Xti

=

⇒ Classical methods from statistics :

Kernel regression <Carrière>
Projection on subspaces of L2(P) <Longstaff-Schwarz,

Gobet-Lemor-Warin AAP05> from numerical probabilistic methods

quantization... <Bally-Pagès SPA03>

Stochastic mesh <Broadie-Glasserman> Integration by parts <Lions-Reigner 00, Bouchard-T. SPA04>

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SLIDE 5

Motivation : American options The Probabilistic Scheme Numerical results

Objective : from US option to nonlinear PDEs

Suggest a Monte Carlo type of scheme for nonlinear PDEs
Numerical complexity reduces to the same problem as US
ptions...
Nonlinear PDEs appear in many problems in finance

Continuous-time portfolio optimization Algorithmic trading under market impact Hedging in illiquid markets ...

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 6

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

Outline

1 Motivation : American options 2 The Probabilistic Scheme

A natural MC-FD scheme The semilinear case The fully nonlinear case

3 Numerical results

On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 7

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

The Monte Carlo component

Consider the fully nonlinear PDE :

= −vt(t, x) − F (t, x, Dv(t, x)) , Dv := (v, Dv, D2v)

Isolate a diffusion part in the equation :

= −vt(t, x) − 1 21∆v(t, x) − f (t, x, Dv(t, x))

Let Xs = x + 1Ws−t+h, s ≥ t − h, evaluate at (s, Xs), and take

expectations : = E t

t−h

−(vt + 1 2∆v)(s, Xs)ds − t

t−h

f (., Dv) (s, Xs)ds

=

v(t − h, x) − E

v(t, Xt) +

t

t−h

f (., Dv) (s, Xs)ds

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Nonlinear Monte Carlo

SLIDE 8

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

The Finite-Differences component

From the previous slide :

ˆ v(t − h, x) = E [ˆ v(t, Xt)] + h f (., E[Dˆ v(t, Xt)])

Need to avoid the calculation of Dˆ

v and D2ˆ v at each time step = ⇒ Integration by parts E[Dˆ v(t, Xt)] = E

v(t, Xt)Wh

h

, E[D2ˆ

v(t, Xt)] = E

v(t, Xt)W 2

h − h

h2

yields the numerical scheme :

ˆ v(t − h, x) = E [ˆ v(t, Xt)] + h f

x, E [ˆ

v(t, Xt)] , E

ˆ

v(t, Xt)Wh

h

, E
ˆ

v(t, Xt)W 2

h −h

h2

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Nonlinear Monte Carlo

SLIDE 9

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

The Finite-Differences component

From the previous slide :

ˆ v(t − h, x) = E [ˆ v(t, Xt)] + h f (., E[Dˆ v(t, Xt)])

Need to avoid the calculation of Dˆ

v and D2ˆ v at each time step = ⇒ Integration by parts E[Dˆ v(t, Xt)] = E

v(t, Xt)Wh

h

, E[D2ˆ

v(t, Xt)] = E

v(t, Xt)W 2

h − h

h2

yields the numerical scheme :

ˆ v(t − h, x) = E [ˆ v(t, Xt)] + h f

x, E [ˆ

v(t, Xt)] , E

ˆ

v(t, Xt)Wh

h

, E
ˆ

v(t, Xt)W 2

h −h

h2

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Nonlinear Monte Carlo

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Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

Intuition From Greeks Calculation

Using the approximation f ′(x) ∼h=0 E[f ′(x + Wh)] :

f ′(x) ∼

f ′(x + y)e−y2/(2h)

√ 2π dy =

f (x + y)y

h e−y2/(2h) √ 2π dy = E

f (x + Wh)Wh

h

Similarly, by an additional integration by parts :

f ′′(x) =

f (x + y)y2 − h

h2 e−y2/(2h) √ 2π dy = E

f (x + Wh)

W 2

h − h

h2

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Nonlinear Monte Carlo

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Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

A probabilistic numerical scheme for fully nonlinear PDEs

This suggests the following discretization : Y n

tn

= g

X n

tn

,

Y n

ti−1

= En

i−1

Y n

ti

+ f
X n

ti−1, Y n ti−1, Z n ti−1, Γn ti−1

∆ti , 1 ≤ i ≤ n ,

Z n

ti−1

= En

i−1

Y n

ti

∆Wti ∆ti

Γn

ti−1

= En

i−1

Y n

ti

|∆Wti|2 − ∆ti |∆ti|2

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Nonlinear Monte Carlo

SLIDE 12

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

Connection with Finite Differences : Xh := x + Wh

Consider the binomial approximation of the Brownian motion

“Wh ∼ √ h 1 2δ{1} + 1 2δ{−1} ′′ Then : E

ψ′(Xh)
= E
ψ(Xh)Wh

h

∼ ψ(x +

√ h) − ψ(x − √ h) 2 √ h

With the trinomial approximation of the Brownian motion

“Wh ∼ √ 3h 1 6δ{1} + 2 3δ{0} + 1 6δ{−1} ′′ Then : E

ψ′′(Xh)
= E
ψ(Xh)W 2

h −h

h2

∼ ψ(x+

√ 3h) − 2ψ(x) + ψ(x− √ 3h) 3h

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 13

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

Description of the scheme

1. Simulate trajectories of the forward process X (well understood)
2. Backward algorithm :
ˆ

Y n

tn

= g

X n

tn

ˆ

Y n

ti−1

=

En

ti−1

ˆ

Y n

ti

+ f
X n

ti−1, ˆ

Y n

ti−1, ˆ

Z n

ti−1

∆ti , 1 ≤ i ≤ n ,

ˆ Z n

ti−1

=

En

ti−1

ˆ

Y n

ti

∆Wti ∆ti

Question : what kind of objet are we simulating ?

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 14

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

Backward SDE : Definition

Find an FW −adapted (Y , Z) satisfying : Yt = G + T

t

Fr(Yr, Zr)dr − T

t

Zr · dWr i.e. dYt = −Ft(Yt, Zt)dt + Zt · dWt and YT = G where the generator F : Ω × [0, T] × R × Rd − → R, and {Ft(y, z), t ∈ [0, T]} is FW − adapted If F is Lipschitz in (y, z) uniformly in (ω, t), and G ∈ L2(P), then there is a unique solution satisfying E sup

t≤T

|Yt|2 + E T |Zt|2dt < ∞

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 15

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

Markov BSDE’s

Let X. be defined by the (forward) SDE dXt = b(t, Xt)dt + σ(t, Xt)dWt and Ft(y, z) = f (t, Xt, y, z) , f : [0, T] × Rd × R × Rd − → R G = g(XT) ∈ L2(P) , g : Rd − → R If f continuous, Lipschitz in (x, y, z) uniformly in t, then there is a unique solution to the BSDE dYt = −f (t, Xt, Yt, Zt)dt + Zt · σ(t, Xt)dWt , YT = g (XT) Moreover, there exists a measurable function V : Yt = V (t, Xt) , 0 ≤ t ≤ T

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 16

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

BSDE’s and semilinear PDE’s

By definition,

Yt+h − Yt = V (t + h, Xt+h) − V (t, Xt) = − t+h

t

f (Xr, Yr, Zr)dr + t+h

t

Zr · σ(Xr)dWr

If V (t, x) is smooth, it follows from Itô’s formula that :

t+h

t

LV (r, Xr)dr + t+h

t

DV (r, Xr) · σ(Xr)dWr = − t+h

t

f (Xr, Yr, Zr)dr + t+h

t

Zr · σ(Xr)dWr where L is the Dynkin operator associated to X : LV = Vt + b · DV + 1 2Tr[σσTD2V ]

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 17

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

Discrete-time approximation of BSDEs

<Bally-Pagès SPA03, Zhang AAP04, Bouchard-T. SPA04> Monte Carlo methods Start from Euler discretization : Y n

tn = g

X n

tn

is given, and

En

i [∆Wti+1 → Y n ti+1−Y n ti = −f

X n

ti , Y n ti , Z n ti

∆ti+Z n

ti·σ

X n

ti

∆Wti+1

= ⇒ Discrete-time approximation : Y n

tn = g

X n

tn

and

Y n

ti

= En

i

Y n

ti+1

+ f
X n

ti , Y n ti , Z n ti

∆ti+... , 0 ≤ i ≤ n − 1

Z n

ti

= En

i

Y n

ti+1

∆Wti+1 ∆ti

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Nonlinear Monte Carlo

SLIDE 18

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

Convergence of the discrete-time approximation of BSDEs

0 = t0 < t1 < . . . < tn = T, ti := i T

n

Theorem (Zhang 04, Bouchard-T. 04) Assume f and g are

Lipschitz. Then :

lim sup

n→∞ n1/2

sup

0≤t≤1

Y n

t − YtL2 + Z n − ZH2

< ∞

Theorem (Gobet-Labart 06) Under additional conditions : lim sup

n→∞ nY n 0 − Y0L2

< ∞ Weak error...

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 19

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

Simulation of BSDEs : bound on the rate of convergence

Error estimate for the Malliavin-based algorithm Theorem For p > 1 : lim sup

n→∞

max

0≤i≤n n−1−d/(4p)N1/2p

ˆ

Y n

ti − Y n ti

Lp

< ∞ For the time step 1 n, and limit case p = 1 : rate of convergence of

1 √n

if and only if n−1− d

4 N1/2 = n1/2,

i.e. N = n3+ d

2

Recent developments in Crisan, Manolarakis and T.

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SLIDE 20

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

Recall the probabilistic numerical scheme

Compared to the semilinear case, the fully nonlinear case exhibits

ne more regression but the ibp weight is more singular...

Y n

tn

= g

X n

tn

,

Y n

ti−1

= En

i−1

Y n

ti

+ f
X n

ti−1, Y n ti−1, Z n ti−1, Γn ti−1

∆ti , 1 ≤ i ≤ n ,

Z n

ti−1

= En

i−1

Y n

ti

∆Wti ∆ti

Γn

ti−1

= En

i−1

Y n

ti

|∆Wti|2 − ∆ti |∆ti|2

No results on the convergence of the process... Second Order

BSDEs <Soner, T., Zhang 2010>

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SLIDE 21

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

Convergence of the Discretization

Assumption The PDE satisfies comparison of bounded super and subsolutions in the sense of viscosity solutions Theorem (i) Suppose that f is Lipschitz uniformly in x, µ, σ bounded, σ invertible, and 0 ≤ ∇γf ≤ σσT. Then Y n

0 (t, x) −

→ v(t, x) uniformly on compacts where v is the unique viscosity solution of the nonlinear PDE. (ii) If f is either convex or concave in (y, z, γ), i.e. HJB operator, −Ch1/10 ≤ v − vh ≤ Ch1/4

Proof : stability, consistency, monotonicity (Barles-Souganidis 91)
Bounds on the approximation error (Krylov, Barles-Jakobsen)
Compare with Bonnans and Zidani...

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SLIDE 22

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

An Implementable MC-FD Scheme

In order to define an implementable scheme, we consider an approximation EN of E, and : ˆ Y n

tn

= g

X n

tn

,

ˆ Y n

ti−1

= ˆ En

i−1

ˆ

Y n

ti

+ f
X n

ti−1, ˆ

Y n

ti−1, ˆ

Z n

ti−1, ˆ

Γn

ti−1

∆ti , 1 ≤ i ≤ n ,

ˆ Z n

ti−1

= ˆ En

i−1

ˆ

Y n

ti

∆Wti ∆ti

ˆ

Γn

ti−1

= ˆ En

i−1

ˆ

Y n

ti

|∆Wti|2 − ∆ti |∆ti|2

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Nonlinear Monte Carlo

SLIDE 23

Motivation : American options The Probabilistic Scheme Numerical results A natural MC-FD scheme The semilinear case The fully nonlinear case

Convergence of the MC-FD Scheme

Assumption There exist constants C, λ, ν > 0 such that for any bounded function ψ :

(ˆ

EN − E)ψ(Wh)Hi(Wh)

p

≤ C h−λN−ν where H0(w) = 1, H1(w) = w, H2(w) = wTw − I Theorem (i) Let the conditions of the previous theorem hold, and limh→0 hλ+2Nν

h = ∞. Then

ˆ Y n

0 (t, x) −

→ v(t, x) uniformly on compacts where v is the unique viscosity solution of the nonlinear PDE. (ii) If f is either convex or concave in (y, z, γ), i.e. HJB operator, and limh→0 hλ+ 21

10 Nν

h > 0

−Ch1/10 ≤ v − vh ≤ Ch1/4

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 24

Motivation : American options The Probabilistic Scheme Numerical results On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables

Outline

1 Motivation : American options 2 The Probabilistic Scheme

A natural MC-FD scheme The semilinear case The fully nonlinear case

3 Numerical results

On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 25

Motivation : American options The Probabilistic Scheme Numerical results On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables

Comments on the 2BSDE algorithm

in BSDEs the drift coefficient µ of the forward SDE can be

changed arbitrarily by Girsanov theorem : importance sampling... Well-understood in the linear case, open question for the semilinear case...

in 2BSDEs both µ and σ can be changed : previous theorem and

numerical results however recommend prudence...

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 26

Motivation : American options The Probabilistic Scheme Numerical results On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables

Comments on the 2BSDE algorithm

in BSDEs the drift coefficient µ of the forward SDE can be

changed arbitrarily by Girsanov theorem : importance sampling... Well-understood in the linear case, open question for the semilinear case...

in 2BSDEs both µ and σ can be changed : previous theorem and

numerical results however recommend prudence...

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 27

Motivation : American options The Probabilistic Scheme Numerical results On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables

Portfolio optimization, dimension 1

With U(y) = −e−ηy, want to solve : V (t, y) := sup

Z

E [U (YT)] where dYu = ZudSu and dSu = Suσ(dWu + λdu)

An explicit solution is available
V is the characterized by the fully nonlinear PDE

−Vt + 1 2λ2 (Vy)2 Vyy = 0 and V (T, .) = U

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SLIDE 28

Motivation : American options The Probabilistic Scheme Numerical results On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables

Varying the drift of the FSDE

Drift FSDE Relative error (Regression)

1

0,0648429

0,8

0,0676044

0,6

0,0346846

0,4

0,0243774

0,2

0,0172359 0,0124126 0,2 0,00880041 0,4 0,00656142 0,6 0,00568952 0,8 0,00637239

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SLIDE 29

Motivation : American options The Probabilistic Scheme Numerical results On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables

Varying the volatility of the FSDE

Diffusion FSDE Relative error Relative error (Regression) (Quantization) 0,2 0,581541 0,526552 0,4 0,42106 0,134675 0,6 0,0165435 0,0258884 0,8 0,0170161 0,00637319 1 0, 0124126 0,0109905 1,2 0,0211604 0,0209174 1,4 0,0360543 0,0362259 1,6 0,0656076 0,0624566

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 30

Motivation : American options The Probabilistic Scheme Numerical results On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables

Optimal investment under stochastic volatility

Zero interest rate, and 1 stock : dSt = µStdt +

YtStdW (1)

t

dYt = k(m − Yt)dt + c

Yt
ρdW (1)

t

+

1 − ρ2dW (2)

t

,

Optimal investment with exponential expected utility = ⇒ v(T, x, y) = −e−ηx, −vt−k(m−y)vy−1 2c2yvyy+(µvx + ρcyvxy)2 2yvxx = 0 Quasi explicit solution from Zariphopoulou : v(t, x, y) = −e−ηx

exp
−1

2 T

t

µ2 ˜ Ys ds

L1−ρ2

where the process ˜ Y is defined by d ˜ Yt = (k(m − ˜ Yt) − µcρ)dt + c

˜

YtdWt

Nizar TOUZI Nonlinear Monte Carlo

SLIDE 31

Motivation : American options The Probabilistic Scheme Numerical results On the choice of µ and σ Portfolio optimization, 2 state variables Portfolio optimization, 5 state variables