[PPT] - Ninomiya-Victoir scheme: strong convergence, antithetic version and PowerPoint Presentation

SLIDE 1

Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators

CERMICS ´ Ecole des Ponts ParisTech project team ENPC-INRIA-UPEM Mathrisk

April 18, 2016 Joint work with Benjamin Jourdain and Emmanuelle Cl´ ement

SLIDE 2

Outline

1

Introduction

2

Monte Carlo Methods The Standard Monte Carlo Method The Multilevel Monte Carlo

3

Numerical Schemes The Ninomiya-Victoir Scheme An antithetic version of the Ninomiya-Victoir scheme

4

Application to the Heston Model The Heston model The Ninomiya-Victoir scheme in the Heston model

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Goal

We are interested in the computation, by Monte Carlo methods, of the expectation Y = E [f (XT)], where X = (Xt)t∈[0,T] is the solution to a multidimensional stochastic differential equation (SDE) and f : Rn → R a given function such that E

f (XT)2

< +∞. We will focus on minimizing the computational complexity subject to a given target error ǫ ∈ R∗

+.

To measure the accuracy of an estimator ˆ Y , we will consider the root mean squared error: RMSE

ˆ

Y ; Y

= E

1 2

Y − ˆ

Y

2

. (1)

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Itˆ

-type SDE

We consider a general Itˆ

-type SDE of the form

   dXt = b(Xt)dt +

d

j=1

σj(Xt)dW j

t

X0 = x (2) where: x ∈ Rn, (Xt)t∈[0,T] is a n−dimensional stochastic process, W =

W 1, . . . , W d

is a d−dimensional standard Brownian motion, b, σ1, . . . , σd : Rn → Rn are Lipschitz continuous.

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Outline

1

Introduction

2

Monte Carlo Methods The Standard Monte Carlo Method The Multilevel Monte Carlo

3

Numerical Schemes The Ninomiya-Victoir Scheme An antithetic version of the Ninomiya-Victoir scheme

4

Application to the Heston Model The Heston model The Ninomiya-Victoir scheme in the Heston model

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Standard Monte Carlo Method

The standard Monte Carlo method consists in: discretizing the SDE, using a numerical scheme X N, with N ∈ N∗ steps, approximating the expectation using M ∈ N∗ independent path simulations. To be clear, the crude Monte Carlo estimator is given by ˆ YCMC = 1 M

M

k=1

f

X N,k

T

(3)

where X N,k are independent copies of a numerical scheme X N.

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Complexity analysis

Bias

B

ˆ

YCMC; Y

= E
ˆ

YCMC

− Y = E
f
X N

T

− E [f (XT)] .

(4) The bias is related to the weak error of the scheme: E

f
X N

T

− f (XT)
= c1

Nα + o 1 Nα

.

(5)

Variance

V

ˆ

YCMC

= 1

M V

f
X N

T

.

(6)

Cost

CCMC = C × M × N = O

ǫ−(2+ 1

α)

. (7)

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Outline

1

Introduction

2

Monte Carlo Methods The Standard Monte Carlo Method The Multilevel Monte Carlo

3

Numerical Schemes The Ninomiya-Victoir Scheme An antithetic version of the Ninomiya-Victoir scheme

4

Application to the Heston Model The Heston model The Ninomiya-Victoir scheme in the Heston model

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The Multilevel Monte Carlo

The main idea of this technique is to use the following telescopic summation to control the bias: E

f
X 2L

T

= E
f
X 1

T

+

L

l=1

E

f
X 2l

T

− f
X 2l−1

T

.

Then, a generalized multilevel Monte Carlo estimator is built as follows: ˆ YMLMC =

L

l=0

1 Ml

Ml

k=1

Z l

k

(8) where

Z l

k

0≤l≤L,1≤k≤Ml are independent random variables such that:

E

Z 0

= E

f
X 1

T

(9)

and: ∀l ∈ {1, . . . , L} , E

Z l

= E

f
X 2l

T

− f
X 2l−1

T

.

(10)

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Bias and variance

Bias

B

ˆ

YMLMC; Y

= E
ˆ

YMLMC

− Y = E
f
X 2L

T

− E [f (XT)] .

(11) The bias is related to the weak error of the scheme: E

f
X 2L

T

− f (XT)
= c1

2αL + o 1 2αL

.

(12)

Variance

V

ˆ

YMLMC

=

L

l=0

1 Ml V

Z l

. (13)

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Cost and canonical exemple

Cost

For a given discretization level l ∈ {0, . . . , L}, the computational cost of simulating one sample Z l is Cλl2l, where: C ∈ R+ is a constant, depending only on the discretization scheme, ∀l ∈ N, λl ∈ Q∗

+ is a weight, depending only on l,

CMLMC = C

L

l=0

Mlλl2l. (14)

Natural choice for Z l, l ∈ {0, . . . , L}

Z 0 = f

X 1

T

(15)

Z l = f

X 2l

T

− f
X 2l−1

T

, ∀l ∈ {1, . . . , L} .

(16) For this canonical choice, it is natural to take λ0 = 1 and λl = 3

2.

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Optimal complexity

Theorem (Complexity theorem (Giles))

Assume that ∃ (α, c1) ∈ R∗

+ × R∗ and ∃ (β, c2) ∈

R∗

+

2 such that ∀l ∈ N: E

f
X 2l

T

− Y = c1

2αl + o 1 2αl

(17)

and V

Z l

= c2 2βl + o 1 2βl

.

(18) Then, the optimal complexity is given by:              C∗

MLMC = O

ǫ−2

if β > 1, C∗

MLMC = O

ǫ−2
log

1 ǫ 2 if β = 1, C∗

MLMC = O

ǫ−2+ β−1

α

if β < 1.

(19)

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Optimal parameters

L∗ =     log2 √

2|c1| ǫ

α

    (20) M∗

0 =

    2 ǫ2

V [Z 0]

λ0

λ0V [Z 0] +

L∗

l=1
c2λl2l(1−β)

    (21) ∀l ∈ {1, . . . , L∗} , M∗

l =

2

ǫ2

c2

λl2l(β+1)

λ0V [Z 0] +

L∗

l=1
c2λl2l(1−β)
.

(22)

Regression

One can estimate (α, β, c1, c2) by using a regression: V

Z l

∼ c2 2βl (23) E

Z l

∼ c1 (1 − 2α) 2αl . (24)

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Outline

1

Introduction

2

Monte Carlo Methods The Standard Monte Carlo Method The Multilevel Monte Carlo

3

Numerical Schemes The Ninomiya-Victoir Scheme An antithetic version of the Ninomiya-Victoir scheme

4

Application to the Heston Model The Heston model The Ninomiya-Victoir scheme in the Heston model

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Stratonovich form

Assuming C1 regularity for diffusion coefficients σ1, . . . , σd, the Itˆ

-type

SDE can be written in Stratonovich form:    dXt = σ0(Xt)dt +

d

j=1

σj(Xt) ◦ dW j

t

X0 = x (25) where σ0 = b − 1

2 d

j=1

∂σjσj and ∂σj is the Jacobian matrix of σj defined as follows ∂σj =

∂σj

ik

i,k∈[

[1;n] ] =

∂xkσij

i,k∈[ [1;n] ] .

(26)

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The Ninomiya-Victoir scheme

Notations

tk = k T

N

k∈[

[0;N] ] is the subdivision of [0, T].

ηN = (η1, . . . , ηN) is a sequence of independent, identically distributed Rademacher random variables independent of W . ∀j ∈ {1, . . . , d} , ∆W j

tk+1 = W j tk+1 − W j tk.

For j ∈ {0, . . . , d} and x0 ∈ Rd, let (exp(tσj)x0)t∈R solve the ODE dx(t)

dt

= σj (x (t)) x (0) = x0.

Scheme

If ηk+1 = 1 X NV ,N,ηN

tk+1

= exp T 2N σ0

exp
∆W d

tk+1σd

. . . exp

∆W 1

tk+1σ1

exp T 2N σ0

X NV ,N,ηN

tk

and if ηk+1 = −1 X NV ,N,ηN

tk+1

= exp T 2N σ0

exp
∆W 1

tk+1σd

. . . exp

∆W d

tk+1σ1

exp T 2N σ0

X NV ,N,ηN

tk

.

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Order 2 of weak convergence

Denoting by (X x

t )t≥0 the solution to the SDE starting from X x 0 = x ∈ Rn,

for f : Rn → Rn smooth, u(t, x) = E [f (X x

t )] solves the Feynman-Kac

PDE

∂u

∂t (t, x) = Lu(t, x), (t, x) ∈ [0, ∞) × Rn

u(0, x) = f (x), x ∈ Rn with L = b.∇x + 1

2Tr

(σ1, . . . , σd)(σ1, . . . , σd)∗∇2

x

= σ0 + 1

2

d

j=1(σj)2

the infinitesimal generator. ∂2u ∂t2 = ∂ ∂t Lu = L ∂ ∂t u = L2u and u(t1, x) = f (x) + t1Lf (x) + t2

1

2 L2f (x) + O(t3

1).

Ninomiya and Victoir have designed their scheme so that E[f (X NV ,N,ηN

t1

)] = f (x) + t1Lf (x) + t2

1

2 L2f (x) + O(t3

1).

One step error O( 1

N3 ) Nsteps

− → O( 1

N2 ) global error.

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Order 1/2 of strong convergence

Theorem (Strong convergence)

Assume that the vector fields, b, ∀j ∈ {1, . . . , d} , σj and ∂σjσj are Lipschitz continuous functions. Then: ∀p ≥ 1, ∃CNV ∈ R∗

+, ∀N ∈ N∗

E

max

0≤k≤N

Xtk − X NV ,N,ηN

tk

2p
η
≤ CNV

Np . (27)

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Outline

1

Introduction

2

Monte Carlo Methods The Standard Monte Carlo Method The Multilevel Monte Carlo

3

Numerical Schemes The Ninomiya-Victoir Scheme An antithetic version of the Ninomiya-Victoir scheme

4

Application to the Heston Model The Heston model The Ninomiya-Victoir scheme in the Heston model

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The Ninomiya-Victoir scheme: antithetic version

We consider two grids: a coarse grid with time step hl−1 =

T 2l−1 , a fine grid with time step

hl = T

2l and we introduce some notations:

∀k ∈

0, . . . , 2l−1

, tk = khl−1, ∀k ∈

0, . . . , 2l−1 − 1
, tk+ 1

2 =

k + 1

2

hl−1,

η2l = (η1, . . . , η2l), ∆W c

tk+1 = Wtk+1 − Wtk, ∆W f tk+ 1

2

= Wtk+ 1

2 − Wtk and ∆W f

tk+1 = Wtk+1 − Wtk+ 1

2 .

On the coarsest grid, X NV ,2l−1,η2l is defined inductively by: η2k+1 = 1: X NV ,2l−1,η2l

tk+1

= exp hl−1 2 σ0

exp
∆W d,c

tk+1σd

. . . exp

∆W 1,c

tk+1σ1

exp hl−1 2 σ0

X NV ,2l−1,η2l

tk

, (28) and if η2k+1 = −1: X NV ,2l−1,η2l

tk+1

= exp hl−1 2 σ0

exp
∆W 1,c

tk+1σd

. . . exp

∆W d,c

tk+1σ1

exp hl−1 2 σ0

X NV ,2l−1,η2l

tk

. (29)

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The Ninomiya-Victoir: antithetic version

Similarly, on the finest grid: η2k+1 = 1: X NV ,2l,η2l

tk+ 1

2

= exp hl 2 σ0

exp
∆W d,f

tk+ 1

2

σd

. . . exp
∆W 1,f

tk+ 1

2

σ1

exp

hl 2 σ0

X NV ,2l,η2l

tk

, (30) and if η2k+1 = −1: X NV ,2l,η2l

tk+ 1

2

= exp hl 2 σ0

exp
∆W 1,f

tk+ 1

2

σd

. . . exp
∆W d,f

tk+ 1

2

σ1

exp

hl 2 σ0

X NV ,2l,η2l

tk

, (31) if η2k+2 = 1: X NV ,2l,η2l

tk+1

= exp hl 2 σ0

exp
∆W d,f

tk+1σd

. . . exp

∆W 1,f

tk+1σ1

exp hl 2 σ0

X NV ,2l,η2l

tk+ 1

2

, (32) and if η2k+2 = −1: X NV ,2l,η2l

tk+1

= exp hl 2 σ0

exp
∆W 1,f

tk+1σd

. . . exp

∆W d,f

tk+1σ1

exp hl 2 σ0

X NV ,2l,η2l

tk+ 1

2

. (33) The antithetic scheme ˜ X NV ,2l,η2l is defined by the same discretization, except that the Brownian increment ∆W f

tk+ 1

2

and ∆W f

tk+1 are swapped.

SLIDE 22

Strong coupling with order one between successive levels

Considering: Z l

NV = 1

4

f
˜

X NV ,2l,η2l

T

+ f
˜

X NV ,2l,−η2l

T

+ f
X NV ,2l,η2l

T

+ f
X NV ,2l,−η2l

T

− 1

2

f
X NV ,2l−1,η2l

T

+ f
X NV ,2l−1,−η2l

T

,

(34) we have a first order of convergence.

Theorem

Assume that f ∈ C2 (Rn, R) and b ∈ C2 (Rn, Rn) with bounded first and second order derivatives, and, ∀j ∈ {1, . . . , d} , σj ∈ C3 (Rn, Rn) with bounded first and second order derivatives and with polynomially growing third order derivatives. Then: ∀p ≥ 1, ∃c ∈ R∗

+, ∀l ∈ N∗, E

Z l

NV

2p

≤ c 22pl . (35)

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Derived MLMC estimator

The antithetic MLMC estimator, ˆ Y NV

MLMC , with the Ninomiya-Victoir

scheme is defined as follows ˆ Y NV

MLMC = L∗

l=0

1 M∗

l M∗

l

k=1

Z l,k

NV

where Z 0

NV = f

X NV ,1,η

T

r Z 0

NV = 1 2

f
X NV ,1,η

T

+ f
X NV ,1,−η

T

,

and for l ∈ {0, . . . , L∗},Z l,k

NV are independent copies of Z l NV .

Practical procedure

Step 1: Estimate α, β, c1, c2 and V

Z 0

NV

.

Step 2: Compute L∗ and (M∗

l )0≤l≤L∗.

Step 3: Compute ˆ Y NV

MLMC.

SLIDE 24

Outline

1

Introduction

2

Monte Carlo Methods The Standard Monte Carlo Method The Multilevel Monte Carlo

3

Numerical Schemes The Ninomiya-Victoir Scheme An antithetic version of the Ninomiya-Victoir scheme

4

Application to the Heston Model The Heston model The Ninomiya-Victoir scheme in the Heston model

SLIDE 25

The Heston model

Heston model

     dUt = (r − δ − 1 2Vt)dt +

VtdW 1

t

dVt = κ(θ − Vt)dt + σ

Vt
ρdW 1

t +

1 − ρ2dW 2

t

,

(36) where the asset price S is given by St = exp(Ut) and θ ∈ R∗

+ is the long implied variance, or long run average price

variance; as t tends to infinity, the expected value of Vt tends to θ, κ ∈ R∗

+ is the rate at which Vt reverts to θ,

σ ∈ R∗

+ is the volatility of the implied volatility and determines the

variance of Vt, r ∈ R the annualized risk-free interest rate, continuously compounded, δ ∈ R∗

+ is the annualized continuous yield dividend,

ρ ∈] − 1, 1[ is the correlation between the two Brownian motion (ie stock price and implied volatility).

SLIDE 26

The Heston model

In this 2−dimensional model, the Brownian vector fields are given by σ1 u v

=

√v ρσ√v

,

σ2 u v

=
σ
1 − ρ2√v
.

The drift coefficient is b u v

=

r − δ − 1

2v

κ (θ − v)

.

The Stratonovich drift is given by σ0 = b − 1

2

∂σ1σ1 + ∂σ2σ2

: σ0 u v

=

r − δ − 1

2v − 1 4ρσ

κ (θ − v) − σ2

4

.

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Outline

1

Introduction

2

Monte Carlo Methods The Standard Monte Carlo Method The Multilevel Monte Carlo

3

Numerical Schemes The Ninomiya-Victoir Scheme An antithetic version of the Ninomiya-Victoir scheme

4

Application to the Heston Model The Heston model The Ninomiya-Victoir scheme in the Heston model

SLIDE 28

The Ninomiya-Victoir scheme

The Ninomiya-Victoir scheme is well defined fo setting ξ = θ − σ2

4κ ≥ 0, for a given uniform

grid, is inductively defined by: 1st step: ¯ U0

tk+1 = UNV ,η tk

+ 1 2

r − δ − 1

2ρσ − 1 2ξ

t1 + 1

2κ (v − ξ)

exp
−1

2κt1

− 1
,

¯ V 0

tk+1 =

V NV ,η

tk

− ξ exp

−1

2κt1

− 1
+ ξ.

2nd step: If ηk+1 = 1: ¯ U1,η

tk+1 = ¯

U0

tk+1 +

¯

V 0

tk+1∆W 1 tk+1 + 1

4ρσ

∆W 1

tk+1

2 , ¯ V 1,η

tk+1 =

¯

V 0

tk+1 + 1

2σρ∆W 1

tk+1

2 , ¯ U2,η

tk+1 = ¯

U1,η

tk+1,

¯ V 2,η

tk+1 =

¯

V 1,η

tk+1 + 1

2σ

1 − ρ2∆W 2

tk+1

2 .

SLIDE 29

The Ninomiya-Victoir scheme

If ηk+1 = −1: ¯ U1,η

tk+1 = ¯

U0,η

tk+1,

¯ V 1,η

tk+1 =

¯

V 0

tk+1 + 1

2σ

1 − ρ2∆W 2

tk+1

2 , ¯ U2,η

tk+1 = ¯

U1,η

tk+1 +

¯

V 1,η

tk+1∆W 1 tk+1 + 1

4ρσ

∆W 1

tk+1

2 , ¯ V 2,η

tk+1 =

¯

V 1,η

tk+1 + 1

2σρ∆W 1

tk+1

2 . 3rd step: UNV ,η

tk+1

= ¯ U2,η

tk+1 + 1

2

r − δ − 1

2ρσ − 1 2ξ

t1 + 1

2κ (v − ξ)

exp
−1

2κt1

− 1
,

V NV ,η

tk+1

=

¯

V 2,η

tk+1 − ξ

exp

−1

2κt1

− 1
+ ξ.

SLIDE 30

The Ninomiya-Victoir scheme

If ηk+1 = −1: ¯ U1,η

tk+1 = ¯

U0,η

tk+1,

¯ V 1,η

tk+1 =

¯

V 0

tk+1 + 1

2σ

1 − ρ2∆W 2

tk+1

2 , ¯ U2,η

tk+1 = ¯

U1,η

tk+1 +

¯

V 1,η

tk+1∆W 1 tk+1 + 1

4ρσ

∆W 1

tk+1

2 , ¯ V 2,η

tk+1 =

¯

V 1,η

tk+1 + 1

2σρ∆W 1

tk+1

2 . 3rd step: UNV ,η

tk+1

= ¯ U2,η

tk+1 + 1

2

r − δ − 1

2ρσ − 1 2ξ

t1 + 1

2κ (v − ξ)

exp
−1

2κt1

− 1
,

V NV ,η

tk+1

=

¯

V 2,η

tk+1 − ξ

exp

−1

2κt1

− 1
+ ξ.

SLIDE 31