Numerical Solution of Stochastic Differential Equations ... · Numerical Solution of Stochastic...

Numerical Solution of Stochastic DifferentialEquations with Jumps in Finance

Eckhard PlatenSchool of Finance and Economics and School of Mathematical Sciences

University of Technology, Sydney

Kloeden, P.E.& Pl, E.: Numerical Solution of Stochastic Differential Equations

Springer, Applications of Mathematics23 (1992,1995,1999).

Pl, E.& Heath, D.: A Benchmark Approach to Quantitative Finance,Springer Finance (2010).

Pl, E.& Bruti-Niberati, N.: Numerical Solution of SDEs with Jumps in Finance,

Springer, Stochastic Modelling and Applied Probability64 (2010).

Jump-Diffusion Multi-Factor Models

Bjork, Kabanov & Runggaldier (1997)

Øksendal & Sulem (2005)

• Markovian

• explicit transition densities in special cases

• benchmark framework

• discrete time approximations

• suitable for simulation

• Markov chain approximations

c© Copyright E. Platen SDE Jump MC 1

Pathwise Approximations:

• scenario simulation of entire markets

• testing statistical techniques on simulated trajectories

• filtering hidden state variablesPl. & Runggaldier (2005, 2007)

• hedge simulation

• dynamic financial analysis

• extreme value simulation

• stress testing

=⇒ higher order strong schemespredictor-corrector methods


Strong Convergence

• Applications: scenario analysis, filtering and hedge simulation

• strong order γ if

εs(∆) =

√

E(

∣

∣XT − Y ∆N

∣

∣

2)

≤ K∆γ


Probability Approximations:

• derivative prices

• sensitivities

• expected utilities

• portfolio selection

• risk measures

• long term risk management

=⇒ Monte Carlo simulation, higher order weak schemes,

predictor-corrector,

variance reduction, Quasi Monte Carlo,

or Markov chain approximations, lattice methods


Weak Convergence

• Applications: derivative pricing, utilities, risk measures

• weak order β if

εw(∆) = |E(g(XT )) − E(g(Y ∆N ))| ≤ K∆β


Essential Requirements:

• parsimonious models

• long time horizons

• respect no-arbitrage in discrete time approximation

• numerically stable methods

• efficient methods for high-dimensional models

• higher order schemes


Continuous and Event Driven Risk

• Wiener processes W k, k ∈ {1, 2, . . .,m}

• counting processes pk

intensityhk

jump martingaleqk

dWm+kt = dqkt =

(

dpkt − hkt dt

) (

hkt

)− 12

k ∈ {1, 2, . . . , d−m}

W t = (W 1t , . . . ,W

mt , q

1t , . . . , q

d−mt )⊤


Primary Security Accounts

dSjt = S

jt−

(

ajt dt+

d∑

k=1

bj,kt dW k

t

)

Assumption 1

bj,kt ≥ −

√

hk−mt

k ∈ {m+ 1, . . . , d}.

Assumption 2

Generalized volatility matrixbt = [bj,kt ]dj,k=1 invertible.


• market price of risk

θt = (θ1t , . . . , θdt )

⊤ = b−1t [at − rt 1]

• primary security account

dSjt = S

jt−

(

rt dt+

d∑

k=1

bj,kt (θkt dt+ dW k

t )

)

• portfolio

dSδt =

d∑

j=0

δjt dS

jt


• fraction

πjδ,t = δ

jt

Sjt

Sδt

• portfolio

dSδt = Sδ

t−

{

rt dt+ π⊤δ,t− bt (θt dt+ dW t)

}


Assumption 3√

hk−mt > θkt

=⇒ numeraire portfolio (NP) exists

• generalized NP volatility

ckt =

θkt for k ∈ {1, 2, . . . ,m}θk

t

1−θk

t(h

k−m

t)−

12

for k ∈ {m+ 1, . . . , d}

• NP fractions

πδ∗,t = (π1δ∗,t, . . . , πd

δ∗,t)⊤ =

(

c⊤t b−1t

)⊤


• Numeraire Portfolio - Benchmark

dSδ∗

t = Sδ∗

t−

(

rt dt+ c⊤t (θt dt+ dW t))

• optimal growth rate

gδ∗

t = rt +1

2

m∑

k=1

(θkt )2

−d∑

k=m+1

hk−mt

ln

1 +θkt

√

hk−mt − θkt

+θkt

√

hk−mt


• benchmarked portfolio

Sδt =

Sδt

Sδ∗

t

Theorem 4 Any nonnegative benchmarked portfolioSδ is an

(A, P )-supermartingale, that is,

Sδt ≥ Et(S

δT )

0 ≤ t ≤ T < ∞=⇒ no strong arbitrage

Pl. & Heath (2010)


Multi-Factor Models

model under benchmark approach mainly:

• benchmarked primary security accounts

Sjt =

Sjt

Sδ∗

t

j ∈ {0, 1, . . . , d}

supermartingales, often SDE driftless,

(local martingales, sometimes martingales)


model savings account

S0t = exp

{∫ t

0

rs ds

}

=⇒ NP

Sδ∗

t =S0t

S0t

=⇒ stock

Sjt = S

jt S

δ∗

t

model additionally dividend rates

and foreign interest rates


Example

• benchmarked security:

dSt = St−

(

√

VtdWSt + dqt

)

• squared volatility:

• Bates model

dVt = ξ(η − Vt) dt+ q√

Vt dWVt

•32

model

d1

Vt

= ξ

(

η − 1

Vt

)

dt+ q

√

1

Vt

dWVt


0

0.5

1

1.5

2

2.5

3

0 5 10 15 20

time

Figure 1: Simulated benchmarked primary security accounts.


0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20

time

Figure 2: Simulated primary security accounts.


0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 5 10 15 20

time

GOPEWI

Figure 3: Naive Diversification: NP and EWI ford = 50.


0

10

20

30

40

50

60

0 5 9 14 18 23 27 32

Figure 4: Benchmarked primary security accounts MMM.


0

50

100

150

200

250

300

350

400

450

0 5 9 14 18 23 27 32

Figure 5: Primary security accounts under the MMM.


0

10

20

30

40

50

60

70

80

90

100

0 5 9 14 18 23 27 32

EWI

GOP

Figure 6: NP and EWI.


• fair security

benchmarked security martingale⇐⇒ fair

• minimal replicating portfolio

fair nonnegative portfolioSδ with Sδτ = Hτ

=⇒ minimal price

• real world pricing formula

VHτ(t) = Sδ∗

t Et

(

Hτ

Sδ∗

τ

)

No need for equivalent risk neutral probability measure!


Fair Hedging

• benchmarked fair portfolio

Sδt = Et

(

Hτ

Sδ∗

τ

)

=⇒ minimal price

• martingale representation

Hτ

Sδ∗

τ

= Et

(

Hτ

Sδ∗

τ

)

+

d∑

k=1

∫ τ

t

xkHτ

(s) dW ks +MHτ

(t)

MHτ- martingale (when pooled⇒ vanishing)

MHτandW k orthogonal

Follmer & Schweizer (1991), Du& Pl. (2011)


Numerical Solution of SDEs

Kloeden& Pl. (1999)

Milstein (1995)

Kloeden, Pl.& Schurz (2003)

Jackel (2002)

Glasserman (2004)

Pl. & Bruti-Liberati (2010)

• major problem:

propagation of errors during simulation


Simulation of SDEs with Jumps

• strong schemes(paths)exact simulationTaylor (Wagner-Pl. expansion)explicitderivative-freepredictor-correctorimplicit, balanced implicit

• weak schemes(probabilities)Taylor (Wagner-Pl. expansion)simplifiedexplicitderivative-freepredictor-corrector, implicit


Jump-Adapted Time Discretization

t0 t1 t2 t3 = T

regular

τ1

r

τ2

r jump times

t0 t1 t2 t3 t4 t5 = T

r r jump-adapted


• intensity of jump process

– regular schemes =⇒ high intensity

– jump-adapted schemes=⇒ low intensity


SDE with Jumps

dXt = a(t,Xt)dt+ b(t,Xt)dWt + c(t−, Xt−) dpt

X0 ∈ ℜd

• pt = Nt: Poisson process, intensityλ < ∞• pt =

∑Nt

i=1(ξi − 1): compound Poisson,ξi i.i.d r.v.

• Poisson random measure


• time discretization

tn = n∆

• discrete time approximation

Y ∆n+1 = Y ∆

n + a(Y ∆n )∆ + b(Y ∆

n )∆Wn + c(Y ∆n )∆pn


Literature on Strong Schemes with Jumps

• Pl (1982), Mikulevicius& Pl (1988)

=⇒ γ ∈ {0.5, 1, . . .} Taylor schemes and jump-adapted

• Maghsoodi (1996, 1998) =⇒ strong schemesγ ≤ 1.5

• Jacod & Protter (1998) =⇒ Euler scheme for semimartingales

• Gardon (2004) =⇒ γ ∈ {0.5, 1, . . .} strong schemes

• Higham & Kloeden (2005) =⇒ implicit Euler scheme

• Bruti-Liberati& Pl (2007) =⇒ γ ∈ {0.5, 1, . . .}explicit, implicit, derivative-free, predictor-corrector


Euler Scheme

• Euler scheme

Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn + c(Yn)∆pn

where

∆Wn ∼ N (0,∆) and ∆pn = Ntn+1−Ntn ∼ Poiss(λ∆)

• strong orderγ = 0.5


Strong Taylor Scheme

Wagner-Platen expansion (strong orderγ = 1.0) =⇒

Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn + c(Yn)∆pn + b(Yn)b′(Yn) I(1,1)

+ b(Yn) c′(Yn) I(1,−1) + {b(Yn + c(Yn)) − b(Yn)} I(−1,1)

+ {c (Yn + c(Yn)) − c(Yn)} I(−1,−1)

with

I(1,1) = 12{(∆Wn)

2 − ∆}, I(−1,−1) =1

2{(∆pn)

2 − ∆pn}

I(1,−1) =∑N(tn+1)

i=N(tn)+1 Wτi− ∆pn Wtn , I(−1,1) = ∆pn ∆Wn − I(1,−1)

• simulation jump timesτi : Wτi=⇒ I(1,−1) andI(−1,1)

• Computational effort heavily dependent on intensityλ


Derivative-Free Strong Schemes

avoid computation of derivatives

⇓

order1.0 derivative-free strong scheme


• implicit methods

Talay (1982)

Klauder & Petersen (1985)

Milstein (1988)

Hernandez & Spigler(1992, 1993)

Saito & Mitsui(1993a, 1993b)

Kloeden& Pl. (1992, 1999)

Milstein, Pl.& Schurz (1998)

Higham (2000)

Alcock & Burrage (2006)

• solve algebraic equation


• ad hoc attempts lead to explosions

• balanced implicit methods


Alcock & Burrage (2006)


Implicit Strong Schemes

wide stability regions

⇓

implicit Euler scheme

order1.0 implicit strong Taylor scheme


Predictor-Corrector Euler Scheme

• corrector

Yn+1 = Yn +(

θ aη(Yn+1) + (1 − θ) aη(Yn))

∆n

+(

η b(Yn+1) + (1 − η) b(Yn))

∆Wn +

p(tn+1)∑

i=p(tn)+1

c (ξi)

aη = a− η b b′

• predictor

Yn+1 = Yn + a(Yn)∆n + b(Yn)∆Wn +

p(tn+1)∑

i=p(tn)+1

c (ξi)

θ, η ∈ [0, 1] degree of implicitness


Jump-Adapted Strong Approximations

jump-adapted time discretisation⇓

jump times included in time discretisation

• jump-adapted Euler scheme

Ytn+1− = Ytn + a(Ytn)∆tn + b(Ytn)∆Wtn

and

Ytn+1= Ytn+1− + c(Ytn+1−)∆pn

• strong orderγ = 0.5


Merton SDE :µ = 0.05, σ = 0.2,ψ = −0.2, λ = 10,X0 = 1, T = 1

0 0.2 0.4 0.6 0.8 1T

0

0.2

0.4

0.6

0.8

1X

Figure 7: Plot of a jump-diffusion path.


0 0.2 0.4 0.6 0.8 1T

-0.00125

-0.001

-0.00075

-0.0005

-0.00025

0

0.00025

0.0005Error

Figure 8: Plot of the strong error for Euler(red) and1.0 Taylor(blue) scheme.


Merton SDE :µ = −0.05, σ = 0.1, λ = 1,X0 = 1, T = 0.5

-10 -8 -6 -4 -2 0Log2dt

-25

-20

-15

-10

Log 2Error

15TaylorJA

1TaylorJA

1Taylor

EulerJA

Euler

Figure 9: Log-log plot of strong error versus time step size.


Literature on Weak Schemes with Jumps

• Mikulevicius& Pl (1991)

=⇒ jump-adapted orderβ ∈ {1, 2 . . .} weak schemes

• Liu & Li (2000) =⇒ orderβ ∈ {1, 2 . . .} weak Taylor, extrapo-

lation and simplified schemes

• Kubilius& Pl (2002) and Glasserman & Merener (2003)

=⇒ jump-adapted Euler with weaker assumptions on coefficients

• Bruti-Liberati&Pl (2006) =⇒ jump-adapted orderβ ∈ {1, 2 . . .}derivative-free, implicit and predictor-corrector schemes


Simplified Euler Scheme

• Euler scheme =⇒ weak orderβ = 1

• simplified Euler scheme

Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn + c(Yn) (ξn − 1)∆pn

• if ∆Wn and∆pn match the first 3 moments of∆Wn and∆pn up to

anO(∆2) error =⇒ weak orderβ = 1

•

P (∆Wn = ±√∆) =

1

2


Jump-Adapted Taylor Approximations

• jump-adapted Euler scheme =⇒ weak orderβ = 1

• jump-adapted order 2 weak Taylor scheme

Ytn+1− = Ytn + a∆tn + b∆Wtn +b b′

2

(

(∆Wtn)2 − ∆tn

)

+ a′ b∆Ztn

+1

2

(

a a′ +1

2a′ ′b2

)

∆2tn

+

(

a b′ +1

2b′′ b2

)

{∆Wtn ∆tn− ∆Ztn}

and

Ytn+1= Ytn+1− + c(Ytn+1−)∆pn

• weak orderβ = 2 (can be simplified and made derivative free)


Predictor-Corrector Schemes

• predictor-corrector =⇒ numerical stability and efficiency

• jump-adapted predictor-corrector Euler scheme

Ytn+1− = Ytn +1

2

{

a(Ytn+1−) + a}

∆tn + b∆Wtn

with predictor

Ytn+1− = Ytn + a∆tn + b∆Wtn

• weak orderβ = 1


-5 -4 -3 -2 -1Log2dt

-2

-1

0

1

2

3

Log2Error

PredCorrJA

ImplEulerJA

EulerJA

Figure 10: Log-log plot of weak error versus time step size.


Regular Approximations

• higher order schemes : time, Wiener and Poisson multiple integrals

• random jump size difficult to handle

• higher order schemes: computational effort dependent on intensity


Numerical Stability

• roundoff and truncation errors

• propagation of errors

• numerical stability priority over higher order


• specially designed test equations

Hernandez & Spigler (1992, 1993)

Milstein (1995)

Kloeden& Pl. (1999)

Saito & Mitsui(1993a, 1993b, 1996)

Hofmann& Pl. (1994, 1996)

Higham (2000)


• linear test dynamics

Xt = X0 exp{

(1 − α)λ t+√

α |λ|Wt

}

α, λ ∈ ℜ

=⇒

P(

limt→∞

Xt = 0)

= 1 ⇐⇒ (1 − α)λ < 0


• linear It o SDE

dXt =

(

1 − 3

2α

)

λXt dt+√

α |λ|Xt dWt

• corresponding Stratonovich SDE

dXt = (1 − α)λXt dt+√

α |λ|Xt ◦ dWt

• α = 0 no randomness

• α = 23

Ito SDE no drift =⇒ martingale

• α = 1 Stratonovich SDE no drift


Definition 5 Y = {Yt, t ≥ 0} is calledasymptotically stableif

P(

limt→∞

|Yt| = 0)

= 1.

impact of perturbations declines asymptotically over time


• stability region Γ

those pairs(λ∆, α) ∈ (−∞, 0)× [0, 1) for which approximationY

asymptotically stable


• transfer function

∣

∣

∣

∣

Yn+1

Yn

∣

∣

∣

∣

= Gn+1(λ∆, α)

Y asymptotically stable ⇐⇒

E(ln(Gn+1(λ∆, α))) < 0

Higham (2000)


• Euler scheme

Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn

Gn+1(λ∆, α) =

∣

∣

∣

∣

1 +

(

1 − 3

2α

)

λ∆ +√

|αλ|∆Wn

∣

∣

∣

∣

∆Wn ∼ N (0,∆)


Figure 11: A-stability region for the Euler scheme


• semi-drift-implicit predictor-corrector Euler method

Yn+1 = Yn +1

2

(

a(Yn+1) + a(Yn))

∆ + b(Yn)∆Wn

Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn

Gn+1(λ∆, α) =

∣

∣

∣

∣

1 + λ∆

(

1 − 3

2α

){

1 +1

2

(

λ∆

(

1 − 3

2α

)

+√

−αλ∆Wn

)}

+√

−αλ∆Wn

∣

∣

∣

∣


Figure 12: -stability region for semi-drift-implicit predictor-corrector Euler

methodc© Copyright E. Platen SDE Jump MC 59

Figure 13: A-stability region for the predictor-correctorEuler method with

θ = 0 andη = 12


Figure 14: A-stability region for the symmetric predictor-corrector Euler


p-Stability

Pl. & Shi (2008)

Definition 6 For p > 0 a processY = {Yt, t > 0} is calledp-stable

if

limt→∞

E(|Yt|p) = 0.

Forα ∈ [0, 11+p/2

) andλ < 0 test SDE isp-stable.


• Stability region those triplets(λ∆, α, p) for which Y is p-stable.


For λ∆ < 0,α ∈ [0, 1) andp > 0 Y p-stable ⇐⇒

E((Gn+1(λ∆, α))p) < 1

• for p > 0

=⇒

E(ln(Gn+1(λ∆, α))) ≤ 1

pE((Gn+1(λ∆, α))p − 1) < 0

=⇒ asymptotically stable


00.2

0.40.6

0.81

Α

-10-8-6-4-2

0

ΛD

0

0.5

1

1.5

2

p

Figure 15: Stability region for the Euler scheme


00.2

0.40.6

0.81

Α

-10-8-6-4-2

0

ΛD

0

0.5

1

1.5

2

p

Figure 16: Stability region for semi-drift-implicit predictor-corrector Euler


00.2

0.40.6

0.81

Α

-10-8-6-4-2

0

ΛD

0

0.5

1

1.5

2

p

Figure 17: Stability region for the predictor-corrector Euler method with

θ = 0 andη = 12


Stability of Some Implicit Methods

• semi-drift implicit Euler scheme

Yn+1 = Yn +1

2(a(Yn+1) + a(Yn))∆ + b(Yn)∆Wn

• full-drift implicit Euler scheme

Yn+1 = Yn + a(Yn+1)∆ + b(Yn)∆Wn

solve algebraic equation


00.2

0.40.6

0.81

Α

-10-8-6-4-2

0

ΛD

0

0.5

1

1.5

2

p

Figure 18: Stability region for semi-drift implicit Euler method


00.2

0.40.6

0.81

Α

-10-8-6-4-2

0

ΛD

0

0.5

1

1.5

2

p

Figure 19: Stability region for full-drift implicit Euler method


• balanced implicit Euler method


Yn+1 = Yn+

(

1 − 3

2α

)

λYn∆+√

α|λ|Yn∆Wn+c|∆Wn|(Yn−Yn+1)


00.2

0.40.6

0.81

Α

-10-8-6-4-2

0

ΛD

0

0.5

1

1.5

2

p

Figure 20: Stability region for a balanced implicit Euler method


00.2

0.40.6

0.81

Α

-10-8-6-4-2

0

ΛD

0

0.5

1

1.5

2

p

Figure 21: Stability region for the simplified symmetric Euler method


00.2

0.40.6

0.81

Α

-10-8-6-4-2

0

ΛD

0

0.5

1

1.5

2

p

Figure 22: Stability region for the simplified symmetric implicit Euler

Schemec© Copyright E. Platen SDE Jump MC 74

00.2

0.40.6

0.81

Α

-10-8-6-4-2

0

ΛD

0

0.5

1

1.5

2

p

Figure 23: Stability region for the simplified fully implicit Euler Scheme


Variance Reduction via Integral Representations

Heath & Platen (2002)

The HP Variance Reduced Estimator

• SDE

dXs,xt = a(t,Xs,x

t ) dt+m∑

j=1

bj(t,Xs,xt ) dW j

t


L0 f(t, x) =∂f(t, x)

∂t+

d∑

i=1

ai(t, x)∂f(t, x)

∂xi

+1

2

d∑

i,k=1

m∑

j=1

bi,j(t, x) bk,j(t, x)∂2f(t, x)

∂xi ∂xk


u(0, x) = E(

h(τ,X0,xτ )

)

= E(

u(τ,X0,xτ )

)

= u(0, x) + E

(∫ τ

0

L0 u(t,X0,xt ) dt

)

= u(0, x) +

∫ T

0

E(

1{t<τ}L0 u(t,X0,x

t ))

dt

• unbiased estimator for u(0, x)

Zτ = u(0, x) +

∫ τ

0

L0 u(t,X0,xt ) dt

HP estimator


0 0.1 0.2 0.3 0.4 0.5

10

20

30

40

50

Figure 24: Simulated outcomes for the intrinsic value(S0,x1

t −K)+, t ∈[0, T ].


0 0.1 0.2 0.3 0.4 0.5

6.6

6.65

6.7

6.75

Figure 25: Simulated outcomes for the estimatorZt, t ∈ [0, T ].


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