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NUMERICAL METHODS OF FINANCE
Eckhard PlatenSchool of Finance and Economics and Department of Mathematical Sciences
University of Technology, Sydney
Platen, E.& Bruti-Liberati, N.: Numerical Solution of SDEs with Jumps i n FinanceSpringer, Applications of Mathematics (2010).
Platen, E.& Heath, D.: A Benchmark Approach to Quantitative FinanceSpringer Finance, 700 pp., 199 illus., Hardcover, ISBN-10 3-540-26212-1 (2010).
Kloeden, P.& Platen, E.: Numerical Solution of Stochastic DifferentialEquationsSpringer, Applications of Mathematics, 600 pp., Hardcover, ISBN-3-540-54062-8 (1999).
Kloeden, P., Platen, E.& Schurz, H.: Numerical Solution of SDEs throughComputer Experiments, Springer, Universitext (2003).
4 Stochastic Expansions
Stochastic Taylor Expansions
Deterministic Taylor Formula
• ordinary differential equation
d
dtXt = a(Xt)
• integral form
Xt = Xt0 +
∫ t
t0
a(Xs) ds
c© Copyright E. Platen NS of SDEs Chap. 4 107
deterministic chain rule
=⇒d
dtf(Xt) = a(Xt)
∂
∂xf(Xt)
• operator
L = a∂
∂x
c© Copyright E. Platen NS of SDEs Chap. 4 108
=⇒integral equation
f(Xt) = f(Xt0) +
∫ t
t0
Lf(Xs) ds
• special case
f(x)≡ x
Lf = a,LLf = (L)2f = La, . . .
• apply tof = a
c© Copyright E. Platen NS of SDEs Chap. 4 109
=⇒
Xt = Xt0 +
∫ t
t0
(
a(Xt0) +
∫ s
t0
La(Xz) dz
)
ds
= Xt0 + a(Xt0)
∫ t
t0
ds+
∫ t
t0
∫ s
t0
La(Xz) dz ds
=⇒
nontrivial Taylor expansion
c© Copyright E. Platen NS of SDEs Chap. 4 110
• applyf = La
=⇒
Xt = Xt0 + a(Xt0)
∫ t
t0
ds+ La(Xt0)
∫ t
t0
∫ s
t0
dz ds+R1
remainder term
R1 =
∫ t
t0
∫ s
t0
∫ z
t0
(L)2a(Xu) du dz ds
c© Copyright E. Platen NS of SDEs Chap. 4 111
• continuing
=⇒
classical deterministic Taylor formula
f(Xt) = f(Xt0) +
r∑
l=1
(t− t0)ℓ
l!(L)ℓf(Xt0)
+
∫ t
t0
· · ·∫ s2
t0
(L)r+1f(Xs1) ds1 . . . dsr+1
for t ∈ [t0, T ] andr ∈ N
c© Copyright E. Platen NS of SDEs Chap. 4 112
Wagner-Platen Expansion
Wagner & Platen (1978), Platen (1982b),
Platen & Wagner (1982) and Kloeden & Platen (1992)
• SDE
Xt = Xt0 +
∫ t
t0
a(Xs) ds+
∫ t
t0
b(Xs) dWs
c© Copyright E. Platen NS of SDEs Chap. 4 113
• Ito formula
f(Xt) = f(Xt0)
+
∫ t
t0
(
a(Xs)∂
∂xf(Xs) +
1
2b2(Xs)
∂2
∂x2f(Xs)
)
ds
+
∫ t
t0
b(Xs)∂
∂xf(Xs) dWs
= f(Xt0) +
∫ t
t0
L0f(Xs) ds+
∫ t
t0
L1f(Xs) dWs
c© Copyright E. Platen NS of SDEs Chap. 4 114
operators
L0 = a∂
∂x+
1
2b2∂2
∂x2
and
L1 = b∂
∂x
f(x)≡ x =⇒ L0f = a andL1f = b
c© Copyright E. Platen NS of SDEs Chap. 4 115
• apply Ito formula tof = a andf = b
Xt = Xt0
+
∫ t
t0
(
a(Xt0) +
∫ s
t0
L0a(Xz) dz +
∫ s
t0
L1a(Xz) dWz
)
ds
+
∫ t
t0
(
b(Xt0) +
∫ s
t0
L0b(Xz) dz +
∫ s
t0
L1b(Xz) dWz
)
dWs
= Xt0 + a(Xt0)
∫ t
t0
ds+ b(Xt0)
∫ t
t0
dWs +R2
c© Copyright E. Platen NS of SDEs Chap. 4 116
remainder term
R2 =
∫ t
t0
∫ s
t0
L0a(Xz) dz ds+
∫ t
t0
∫ s
t0
L1a(Xz) dWz ds
+
∫ t
t0
∫ s
t0
L0b(Xz) dz dWs +
∫ t
t0
∫ s
t0
L1b(Xz) dWz dWs
c© Copyright E. Platen NS of SDEs Chap. 4 117
• apply Ito formula tof = L1b
=⇒
Xt = Xt0 + a(Xt0)
∫ t
t0
ds+ b(Xt0)
∫ t
t0
dWs
+L1b(Xt0)
∫ t
t0
∫ s
t0
dWz dWs +R3
c© Copyright E. Platen NS of SDEs Chap. 4 118
remainder term
R3 =
∫ t
t0
∫ s
t0
L0a(Xz) dz ds+
∫ t
t0
∫ s
t0
L1a(Xz) dWz ds
+
∫ t
t0
∫ s
t0
L0b(Xz) dz dWs
+
∫ t
t0
∫ s
t0
∫ z
t0
L0L1b(Xu) du dWz dWs
+
∫ t
t0
∫ s
t0
∫ z
t0
L1L1b(Xu) dWu dWz dWs
example for Wagner-Platen expansion
c© Copyright E. Platen NS of SDEs Chap. 4 119
• multiple Ito integrals
∫ t
t0
ds = t− t0,
∫ t
t0
dWs = Wt −Wt0 ,
∫ t
t0
∫ s
t0
dWz dWs =1
2
(
(Wt −Wt0)2 − (t− t0)
)
• remainder termR3
consisting of next following multiple Ito integrals
with nonconstant integrands
c© Copyright E. Platen NS of SDEs Chap. 4 120
Generalized Wagner-Platen Expansion
• expanding with respect to processX = Xt, t ∈ [0, T ]
• Ito formula
f(t,Xt) = f(t0, Xt0) +
∫ t
t0
∂
∂tf(s,Xs) ds
+
∫ t
t0
∂
∂xf(s,Xs) dXs +
1
2
∫ t
t0
∂2
∂x2f(s,Xs) d[X]s
quadratic variation
[X]t = [X]t0 +
∫ t
t0
b2 (Xs) ds
c© Copyright E. Platen NS of SDEs Chap. 4 121
• expand further
f(t,Xt) = f(t0, Xt0)
+
∫ t
t0
∂
∂tf(t0, Xt0) +
∫ s
t0
∂2
∂t2f(z,Xz) dz
+
∫ s
t0
∂2
∂x ∂tf(z,Xz) dXz +
∫ s
t0
1
2
∂2
∂x2
∂
∂tf(z,Xz) d[X]z
ds
c© Copyright E. Platen NS of SDEs Chap. 4 122
+
∫ t
t0
∂
∂xf(t0, Xt0) +
∫ s
t0
∂2
∂t ∂xf(z,Xz) dz
+
∫ s
t0
∂2
∂x2f(z,Xz) dXz +
∫ s
t0
1
2
∂3
∂x3f(z,Xz) d[X]z
dXs
+1
2
∫ t
t0
∂2
∂x2f(t0, Xt0) +
∫ s
t0
∂3
∂t ∂x2f(z,Xz) dz
+
∫ s
t0
∂3
∂x3f(z,Xz) dXz +
∫ s
t0
1
2
∂4
∂x4f(z,Xz) d[X]z
d[X]s
= f(t0, Xt0) +∂
∂tf(t0, Xt0) (t− t0) +
∂
∂xf(t0, Xt0) (Xt −Xt0)
+1
2
∂2
∂x2f(t0, Xt0) ([X]t − [X]t0) +Rf(t0, t)
c© Copyright E. Platen NS of SDEs Chap. 4 123
f(t,Xt) = f(t0, Xt0) +∂
∂tf(t0, Xt0) (t− t0)
+∂
∂xf(t0, Xt0) (Xt −Xt0)
+1
2
∂2
∂x2f(t0, Xt0) ([X]t − [X]t0)
+∂2
∂x ∂tf(t0, Xt0)
∫ t
t0
∫ s
t0
dXz ds
+1
2
∂2
∂x2
∂
∂tf(t0, Xt0)
∫ t
t0
∫ s
t0
d[X]z ds
+∂2
∂t ∂xf(t0, Xt0)
∫ t
t0
∫ s
t0
dz dXs
c© Copyright E. Platen NS of SDEs Chap. 4 124
+∂2
∂x2f(t0, Xt0)
∫ t
t0
∫ s
t0
dXz dXs
+1
2
∂3
∂x3f(t0, Xt0)
∫ t
t0
∫ s
t0
d[X]z dXs
+1
2
∂4
∂t ∂x3f(t0, Xt0)
∫ t
t0
∫ s
t0
dz d[X]s
+1
2
∂3
∂x3f(t0, Xt0)
∫ t
t0
∫ s
t0
dXz d[X]s
+1
4
∂4
∂x4f(t0, Xt0)
∫ t
t0
∫ s
t0
d[X]z d[X]s + Rf(t0, t)
c© Copyright E. Platen NS of SDEs Chap. 4 125
Expansions with Jumps
• counting processN = Nt, t ∈ [0, T ]
• for right-continuous processZ
jump size
∆Zt = Zt − Zt−
=⇒Nt =
∑
s∈(0,t]
∆Ns
c© Copyright E. Platen NS of SDEs Chap. 4 126
• measurable function
f : ℜ → ℜ
=⇒f(Nt) = f(N0) +
∑
s∈(0,t]
∆ f(Ns)
consistent with Ito formula
=⇒
f(Nt) = f(N0) +
∫
(0,t]
(
f(Ns− + 1) − f(Ns−))
dNs
c© Copyright E. Platen NS of SDEs Chap. 4 127
• measurable function
∆ f(N) = f(N + 1) − f(N)
=⇒
f(Nt) = f(N0) +
∫
(0,t]
∆ f(Ns−) dNs
= f(N0) +
∫
(0,t]
∆ f(N0) dNs
+
∫
(0,t]
∫
(0,s2)
∆(
∆ f(Ns1−))
dNs1 dNs2
=⇒
multiple stochastic integrals with respect toN
c© Copyright E. Platen NS of SDEs Chap. 4 128
Engel (1982)
Platen (1984)
Studer (2001)
∫
(0,t]
dNs = Nt
∫
(0,t]
∫
(0,s1)
dNs1 dNs2 =1
2 !Nt (Nt − 1),
∫
(0,t]
∫
(0,s1)
∫
(0,s2)
dNs1 dNs2 dNs3 =1
3 !Nt (Nt − 1) (Nt − 2),
∫
(0,t]
∫
(0,s1)
. . .
∫
(0,sn)
dNs1 . . . dNsn−1dNsn =
(
Nt
n
)
for Nt ≥ n
0 otherwise
c© Copyright E. Platen NS of SDEs Chap. 4 129
for t ∈ [0, T ], where
(
i
n
)
=i (i− 1) (i− 2) . . . (i− n+ 1)
1 · 2 · . . . · n =i !
n ! (i− n) !
for i ≥ n
c© Copyright E. Platen NS of SDEs Chap. 4 130
• stochastic Taylor expansion
f(Nt) = f(N0) +
∫
(0,t]
∆ f(N0) dNs
+
∫
(0,t]
∫
(0,s2)
∆(
∆ f(N0))
dNs1 dNs2 + R3(t)
with
R3(t) =
∫
(0,t]
∫
(0,s2)
∫
(0,s3)
∆(
∆(
∆ f(Ns1−)))
dNs1 dNs2 dNs3
=⇒
f(Nt) = f(N0) + ∆ f(N0)
(
Nt
1
)
+ ∆(
∆ f(N0))
(
Nt
2
)
+ R3(t)
c© Copyright E. Platen NS of SDEs Chap. 4 131
∆ f(N0) = ∆ f(0) = f(1) − f(0)
∆(
∆ f(N0))
= f(2) − 2 f(1) + f(0)
=⇒
f(Nt) = f(0) + (f(1) − f(0))Nt
+(f(2) − 2 f(1) + f(0))1
2Nt (Nt − 1) + R3(t)
forNt ≤ 3 R3(t) equals zero
c© Copyright E. Platen NS of SDEs Chap. 4 132
Measuring Market Risk for Linear Portfolios
Primary Securities
St =(
S(0)t , . . . , S
(d)t
)⊤
dS(j)t = S
(j)t
rt dt+
d∑
k=1
bj,kt
(
θkt dt+ dW kt
)
j ∈ 0, 1, . . . , d
c© Copyright E. Platen NS of SDEs Chap. 4 133
invertible volatility matrix
bt = [bj,kt ]dj,k=1
market price for risk
θt = (θ1t , θ2t , . . . , θ
dt )
⊤ = b−1t (at − rt 1)
savings account
S(0)t = exp
∫ t
0
rs ds
c© Copyright E. Platen NS of SDEs Chap. 4 134
Strategies and Portfolios
• strategy
δ = δt = (δ0t , δ1t , . . . , δ
dt )
⊤, t ∈ [0, T ]
• portfolio
Sδt =
d∑
j=0
δjt S(j)t = δ⊤t St
• self-financing
dSδt =
d∑
j=0
δjt dS(j)t = δ⊤t dSt
c© Copyright E. Platen NS of SDEs Chap. 4 135
• jth fraction
πjδ(t) = δjt
S(j)t
Sδt
,
d∑
j=0
πjδ(t) = 1
c© Copyright E. Platen NS of SDEs Chap. 4 136
• portfolio SDE
dSδt = Sδ
t
(
rt dt+
d∑
k=1
βkδ (t) (θ
kt dt+ dW k
t )
)
• portfolio volatility
βkδ (t) =
d∑
j=0
πjδ(t) b
j,kt
=⇒d ln
(
Sδt
)
= (rt + gδ(t)) dt+d∑
k=1
βkδ (t) dW
kt
c© Copyright E. Platen NS of SDEs Chap. 4 137
• portfolio net growth rate
gδ(t) =d∑
k=1
βkδ (t)
(
θkt − 1
2βk
δ (t)
)
• growth optimal portfolio
dSδ∗t = Sδ∗
t
(
rt dt+
d∑
k=1
θkt (θkt dt+ dW kt )
)
c© Copyright E. Platen NS of SDEs Chap. 4 138
General and Specific Market Risk
• each institution obliged to put aside sufficient capital as abuffer for large losses
=⇒ regulatory capital
investment bank, managed fund, insurance company or similar business
• market risk - risk of losing money from adverse movements of financial mar-
kets
=⇒ specificandgeneral market risk
Basle(1996a, 1996b)
Platen & Stahl (2003)
c© Copyright E. Platen NS of SDEs Chap. 4 139
• general market risk
risk exposure of the portfolio against the equity market as awhole
• specific market risk
risk of holding an individual security within a portfolio which is not covered by
general market risk
c© Copyright E. Platen NS of SDEs Chap. 4 140
• broadly based index
required by regulations for measuring general market risk
take world stock index =⇒ general market risk
• particular portfolio - Sδt
correspondingbenchmarked portfolio
Sδt =
Sδt
Sδ∗t
=⇒ specific risk
efficient and natural separation of market risk
into general and specific market risk
c© Copyright E. Platen NS of SDEs Chap. 4 141
Coherent Risk Measures
Artzner, Delbaen, Eber & Heath (1997)
Follmer & Schiedt (2002)
A mapping is acoherent risk measureif
(i) If X ≥ 0, then(X) ≤ 0;
(ii) (X1 +X2) ≤ (X1) + (X2);
(ii) (λX) = λ (X) for λ ≥ 0;
(iv) (a+X) = (X) − a for each constanta ∈ ℜ.
c© Copyright E. Platen NS of SDEs Chap. 4 142
(i) - natural negative sign for positive movements
(ii) - subadditivity
(iii) - positive homogeneity
(iv) - translation invariance
• there exist also convex risk measures
c© Copyright E. Platen NS of SDEs Chap. 4 143
Value at Risk
Basle (1996a, 1996b)
• strictly increasing distribution functionFX
• quantile function
F−1X (α) = infx ∈ ℜ : FX(x) > α
• α-quantile
qα = F−1X (α)
c© Copyright E. Platen NS of SDEs Chap. 4 144
Value at Risk of a random return
X =Sδ
t+∆ − Sδt
Sδt
for a portfolioSδt and a given levelα ∈ (0, 1) is
VaRα(X) = −F−1X (α)Sδ
t
=⇒VaRα(X) = −qα Sδ
t
Use Wagner-Platen expansion to approximateX
c© Copyright E. Platen NS of SDEs Chap. 4 145
• Gaussian return
meanµ = 0
variancev
qα denotes theα-quantile of a standard Gaussian random variable
=⇒VaRα(X) = −qα
√v Sδ
t
q0.05 = −1.645, q0.01 = −2.326
c© Copyright E. Platen NS of SDEs Chap. 4 146
-4 -2 2 40
0.2
0.4
0.6
0.8
1
Figure 4.1: Gaussian distribution function.
c© Copyright E. Platen NS of SDEs Chap. 4 147
• VaR is not a coherent risk measure
under general circumstances
Artzner, Delbaen, Eber & Heath (1997)
• random return linear combination
of underlying risk factorsX1, . . . , Xd
elliptical joint distribution
density is constant on ellipsoids
=⇒
VaRα(X) is acoherent risk measure
Embrechts, McNeal & Straumann (1999)
c© Copyright E. Platen NS of SDEs Chap. 4 148
• in general,subadditivity can be violated
risk capital assigned to a diversified position may be
bigger than the sum of the risk capitals allocated
• for optionVaRα(Sδ(T )) can be misleading
• in practice if portfolio is diversified
and market model is realistic
subadditivity is likely
=⇒ VaRα coherent risk measure
c© Copyright E. Platen NS of SDEs Chap. 4 149
VaR for Conditionally Gaussian Random Variables
• conditionally Gaussian random variable
stochastic but independent variancev
=⇒ mixture of normals
• if 1v
is χ2-distributed withν degrees of freedom
=⇒ X is Studentt with ν degrees of freedom
• if v is gamma distributed
=⇒ X is variance gamma
c© Copyright E. Platen NS of SDEs Chap. 4 150
• if X is a linear combination of returns
X =
d∑
i=1
πiδ Ri
with
Ri = Zi
√v
Zi - standard Gaussian
v - independent random variance
=⇒ R1, . . . , Rd is elliptical
=⇒ coherent risk measure
c© Copyright E. Platen NS of SDEs Chap. 4 151
• correlated conditionally Gaussianreturns
covariance matrix
D =[
dj,ℓ]d
j,ℓ=1
D = b b⊤
Cholesky decomposition
b =[
bj,ℓ]d
j,ℓ=1
invertible matrix
c© Copyright E. Platen NS of SDEs Chap. 4 152
• return vector
R = (R1, . . . , Rd)⊤ =
√v bZ
correlated according toD
R1, . . . , Rd is elliptical
=⇒ representation
X = π⊤δ R = |π⊤
δ b| ξ
weight vector
πδ =(
π1δ , . . . , π
dδ
)⊤∈ ℜd
ξ normal mixture distributed scalar random variable
c© Copyright E. Platen NS of SDEs Chap. 4 153
with variancec2 ∆
For Studentt distributed multivariate log-returns
ξ is Studentt distributed
c© Copyright E. Platen NS of SDEs Chap. 4 154
• VaRα(X) is in this case acoherent risk measure
=⇒ internal risk measure for capital allocation
significantly simplifies the VaR calculation
VaRα(X) ≈ −√
π⊤δ Dπδ c
√∆ tαS
δt
tα - α-quantile of standardized normal mixture
distribution
Sδt - present face value of the portfolio
qα - standard Gaussianα-quantile
c© Copyright E. Platen NS of SDEs Chap. 4 155
• event factor
ϕα =tαqα
Platen & Stahl (2003)
=⇒
VaRα(X) = −√
π⊤δ Dπδ c
√∆ qα ϕα S
δt
c© Copyright E. Platen NS of SDEs Chap. 4 156
• event risk
Basle(1996a, 1996b)
considerStudent t distributed log-returns
with ν ∈ 2, 3, 4, 5, 10
=⇒ event factorϕα
ν ∞ 10 5 4 3 2
ϕ0.01 1 1.06 1.11 1.12 1.14 1.16
Table 1: Event factorϕα in dependence on degrees of freedomν.
c© Copyright E. Platen NS of SDEs Chap. 4 157
• extensivehistorical simulations
Gibson (2001)
=⇒ average event factor ϕ0.01 ≈ 1.12
typical portfolios of US institutions
coincides exactly with the event factor that is obtained fortheStudent t distri-
bution with four degreesof freedom
=⇒ supports alternative market model
c© Copyright E. Platen NS of SDEs Chap. 4 158
Internal Models
• internal modelsfor calculatingregulatory capital
Basle(1996a, 1996b)
reduces the requiredregulatory capital
• if the portfolio of an institution is not linear
but forms adiversified portfolio
=⇒ approximately GOP
Fergusson & Platen (2006)
Studentt distributed
c© Copyright E. Platen NS of SDEs Chap. 4 159
Wagner-Platen Expansion for a BS Portfolio
Studer (2001)
Giannelli & Primbs (2001)
Black-Scholes dynamics ofS(1)
∆S(δ)t =
∫ t+∆
t
δ(0)s dS(0)s +
∫ t+∆
t
δ(1)s dS(1)s
=
∫ t+∆
t
δ(0)s S(0)s rs ds+
∫ t+∆
t
δ(1)s S(1)s
×(
(
rs + b1,1s θ1s)
ds+ b1,1s dW 1s
)
c© Copyright E. Platen NS of SDEs Chap. 4 160
≈(
δ(0)t S
(0)t rt + δ
(1)t S
(1)t
(
rt + b1,1t θ1t)
)
∆
+ δ(1)t S
(1)t b1,1t ∆W 1
t
+∂
∂S(1)
[
δ(1)t S
(1)t b1,1t
]
δ(1)t S
(1)t b1,1t
1
2
[
(∆W 1t )
2 − ∆]
= ∆S(δ)t
∆W 1t = W 1
t+∆ −W 1t
c© Copyright E. Platen NS of SDEs Chap. 4 161
Multiple Stochastic Integrals
• d-dimensional Ito equation
Xt = Xt0 +
∫ t
t0
a(s,Xs) ds+
m∑
j=1
∫ t
t0
bj(s,Xs) dWjs
• equivalent Stratonovich equation
Xt = Xt0 +
∫ t
t0
a(s,Xs) ds+
m∑
j=1
∫ t
t0
bj(s,Xs) dW js
ai = ai − 1
2
m∑
j=1
d∑
k=1
bk,j∂bi,j
∂xk
c© Copyright E. Platen NS of SDEs Chap. 4 162
Multi-indices
α = (j1, j2, . . . , jℓ)
where
ji ∈ 0, 1, . . . ,m
for i ∈ 1, 2, . . . , ℓ andm ∈ N is amulti-indexof length
ℓ = ℓ(α) ∈ N
c© Copyright E. Platen NS of SDEs Chap. 4 163
• m - number of components of Wiener process
v - multi-index of length zero
ℓ(v) = 0
ℓ((0, 1)) = 2, ℓ((0, 1, 0)) = 3
• n(α) - number of components of a multi-indexα which equal0
n((1, 0, 1)) = 1, n((0, 1, 0)) = 2, n((0, 0)) = 2
c© Copyright E. Platen NS of SDEs Chap. 4 164
• set of all multi-indices
Mm =
(j1, j2, . . . , jℓ) : ji ∈ 0, 1, . . . ,m,
i ∈ 1, 2, . . . , ℓ, for ℓ ∈ N
∪ v
• Givenα ∈ Mm with ℓ(α) ≥ 1,
−α orα− obtained by deleting
the first or the last component ofα, respectively.
−(1, 0) = (0), (1, 0)− = (1) and
−(0, 1, 1) = (1, 1), (0, 1, 1)− = (0, 1)
c© Copyright E. Platen NS of SDEs Chap. 4 165
• for α= (j1, j2, . . ., jk), α= (j1, j2, . . ., jℓ)
concatenation operation∗
α ∗ α = (j1, j2, . . . , jk, j1, j2, . . . , jℓ)
for α= (0, 1, 2) andα= (1, 3)
α ∗ α = (0, 1, 2, 1, 3) andα ∗ α = (1, 3, 0, 1, 2)
c© Copyright E. Platen NS of SDEs Chap. 4 166
Multiple It o Integrals
Let andτ be two stopping times
0 ≤ ≤ τ ≤ T
multi-indexα= (j1, j2, . . ., jℓ) ∈ Mm
f ∈ Hα
c© Copyright E. Platen NS of SDEs Chap. 4 167
• multiple It o integral
Iα[f·],τ =
fτ for ℓ = 0
∫ τ
Iα−[f·],s ds for ℓ ≥ 1 and jℓ = 0
∫ τ
Iα−[f·],s dW
jℓs for ℓ ≥ 1 and jℓ ≥ 1
• in the caseft = 1
abbreviate Iα,τ by Iα
c© Copyright E. Platen NS of SDEs Chap. 4 168
Relationships between Multiple Ito Integrals
• write
Iα,t = Iα[1]0,t
W 0t = t
• relationship
for α = (j1, . . . , ji, ji+1, . . . , jℓ)
W jt Iα,t =
ℓ∑
i=0
I(j1,...,ji,j,ji+1,...,jℓ),t
+
ℓ∑
i=1
1ji=j 6=0 I(j1,...,ji−1,0,ji+1,...,jℓ),t
c© Copyright E. Platen NS of SDEs Chap. 4 169
Hermite Polynomials and Multiple It o Integrals
Kloeden & Platen (1999)
• Hermite polynomials
Hn(x) = (−1)n ex2 dn
dxnexp−x2
generating function
exp2 z x− z2 =
∞∑
n=0
Hn(x)zn
n !
c© Copyright E. Platen NS of SDEs Chap. 4 170
• monic Hermite polynomials
Hn(t, x) =
(
t
2
)n2
Hn
(
x√2t
)
• scaled monic Hermite polynomials
hn(t, x) =1
n !Hn(t, x)
for n ∈ 0, 1, . . ., x ∈ ℜ, t ∈ (0,∞)
c© Copyright E. Platen NS of SDEs Chap. 4 171
=⇒ Hermite polynomials
Hn(x) = n !
[n2]
∑
j=0
(−1)j (2x)n−2j
j ! (n− 2j) !
for n ∈ 0, 1, . . . andx ∈ ℜ
[y] - largest integer not greater thany
=⇒ scaled monic Hermite polynomials
hn(t, x) =
[n2]
∑
j=0
(−1)j xn−2j
j ! (n− 2j) !
(
t
2
)j
c© Copyright E. Platen NS of SDEs Chap. 4 172
with properties∂
∂xhn(t, x) = hn−1(t, x)
and∂
∂thn(t, x) = −1
2
∂2
∂x2hn(t, x)
for x ∈ ℜ, t ∈ (0,∞) andn ∈ 1, 2, . . .
c© Copyright E. Platen NS of SDEs Chap. 4 173
• Forα = (j1, . . . , jℓ) = (0, . . . , 0)
with ji = 0 for i ∈ 1, 2, . . . , ℓ
Iα,t =
∫ t
0
. . .
∫ s2
0
ds1 . . . dsℓ =tℓ
ℓ !
• Forα = (j1, . . . , jℓ) = (j, . . . , j)
with ji = j ∈ 1, 2, . . . ,m for i ∈ 1, 2, . . . , ℓ
Iα,t =
∫ t
0
. . .
∫ s2
0
dW js1 . . . dW
jsℓ = hℓ(t,W
jt )
c© Copyright E. Platen NS of SDEs Chap. 4 174
I(j),t = W jt
I(j,j),t =1
2
(
(
W jt
)2
− t
)
I(j,j,j),t =1
3 !
(
(
W jt
)3
− 3 tW jt
)
I(j,j,j,j),t =1
4 !
(
(
W jt
)4
− 6 t (W jt )
2 + 3 t2)
I(j,j,j,j,j),t =1
5 !
(
(
W jt
)5
− 10 t (W jt )
3 + 15 t2W jt
)
for t ∈ [0,∞), j ∈ 1, 2, . . . ,m
c© Copyright E. Platen NS of SDEs Chap. 4 175
Multiple Stratonovich Integrals
stopping times
0 ≤ ≤ τ ≤ T
Jα[g(·, X·)],τ =
g(τ,Xτ ) for ℓ = 0
∫ τ
Jα−[g(·, X·)],s ds for ℓ ≥ 1, jℓ = 0
∫ τ
Jα−[g(·, X·)],s dW jℓ
s for ℓ ≥ 1, jℓ ≥ 1
c© Copyright E. Platen NS of SDEs Chap. 4 176
Relationships between Multiple Stratonovich Integrals
Jα,t = Jα[1]0,t
W 0t = t
Let j1, . . ., jℓ ∈ 0, 1, . . . ,m andα = (j1, . . ., jℓ) ∈ Mm whereℓ ∈ N .
Then
W jt Jα,t =
ℓ∑
i=0
J(j1,...,ji,j,ji+1,...,jℓ),t
for all t ∈ [0, T ].
c© Copyright E. Platen NS of SDEs Chap. 4 177
Coefficient Functions
• operators
L0 =∂
∂t+
d∑
k=1
ak ∂
∂xk+
1
2
d∑
k,l=1
m∑
j=1
bk,jbl,j∂2
∂xk∂xℓ
Lj =
d∑
k=1
bk,j∂
∂xk
j ∈ 1, 2, . . . ,m
c© Copyright E. Platen NS of SDEs Chap. 4 178
• It o coefficient function
fα =
f for l = 0
Lj1f−α for l ≥ 1
multi-indexα= (j1,. . ., jℓ)
functionf : [0, T ] × ℜd → ℜ
for f(t, x) = x
f(0) = a, f(1) = b, f(1,1) = b b′ andf(0,1) = a b′ + 12b2b′′
c© Copyright E. Platen NS of SDEs Chap. 4 179
Stratonovich Coefficient Functions
• operators
L0 =∂
∂t+
d∑
k=1
ak ∂
∂xk
Lj = Lj =
d∑
k=1
bk,j∂
∂xk
for j ∈ 1, 2, . . . ,m
a = a− 1
2
m∑
j=1
Ljbj
c© Copyright E. Platen NS of SDEs Chap. 4 180
• Stratonovich coefficient function
fα
=
f for l = 0
Lj1f−αfor l ≥ 1
for α= (j1, . . ., jℓ)
c© Copyright E. Platen NS of SDEs Chap. 4 181
• Examples for Stratonovich coefficient functions
f(0)
= a, f(j1)
= bj1 , f(0,0)
= a a′,
f(0,j1)
= a bj1 ′, f(j1,0)
= a′bj1 , f(j1,j2)
= bj1bj2 ′,
f(0,0,0)
= a(
a a′′ + (a′)2)
, f(0,0,j1)
= a(
a bj1 ′′ + a′bj1 ′)
,
f(0,j1,0)
= a(
a′′bj1 + a′bj1 ′)
, f(j1,0,0)
= bj1(
a a′′ + (a′)2)
,
c© Copyright E. Platen NS of SDEs Chap. 4 182
f(0,j1,j2)
= a(
bj1bj2 ′′ + bj1 ′bj2 ′)
, f(j1,0,j2)
= bj1(
a bj1 ′′ + a′bj1 ′)
,
f(j1,j2,0)
= bj1(
a′′bj2 + a′bj2 ′)
, f(j1,j2,j3)
= bj1(
bj2bj3 ′′ + bj2 ′bj3 ′)
,
wherej1, j2, j3 ∈ 1, 2, . . . ,m
c© Copyright E. Platen NS of SDEs Chap. 4 183
Hierarchical Sets
We call a subsetA⊂ Mm ahierarchical setif
A 6= ∅
supα∈A
ℓ(α) < ∞
and
−α ∈ A for each α ∈ A \ v.
Examples:
v, v, (0), (1), v, (0), (1), (1, 1) are hierarchical sets
c© Copyright E. Platen NS of SDEs Chap. 4 184
Remainder Sets
For a given hierarchical setA
B(A) = α ∈ Mm \A : −α ∈ A.
Remainder setconsists of all of the next following multi-indices with respect to the
given hierarchical set.
Examples:
Whenm= 1 then
B (v) = (0), (1)and
B (v, (0), (1)) = (0, 0), (0, 1), (1, 0), (1, 1)
c© Copyright E. Platen NS of SDEs Chap. 4 185
General Stochastic Taylor Expansion
Wagner-Platen Expansion
Theorem 4.1 (Wagner-Platen)
Let andτ be two stopping times with
0 ≤ (ω) ≤ τ(ω) ≤ T,
a.s.,A⊂ Mm a hierarchical set
andf : [0, T ]×ℜd → ℜ a sufficiently smooth function, then we have
f(τ,Xτ ) =∑
α∈A
Iα [fα(,X)],τ +∑
α∈B(A)
Iα [fα(·, X·)],τ .
c© Copyright E. Platen NS of SDEs Chap. 4 186
Example
d=m= 1
for functionf(t, x)≡ x,
times= 0, τ = t
hierarchical set A = α ∈ Mm : ℓ(α) ≤ 3
drift a(t, x) = a(x)
diffusion coefficient b(t, x) = b(x)
dXt = a (Xt) dt+ b(Xt) dWt
=⇒
Wagner-Platen expansion in the form:
c© Copyright E. Platen NS of SDEs Chap. 4 187
Xt = X0 + a I(0) + b I(1) +
(
a a′ +1
2b2a′′
)
I(0,0)
+
(
a b′ +1
2b2b′′
)
I(0,1) + b a′I(1,0) + b b′I(1,1)
+
[
a
(
a a′′ + (a′)2 + b b′a′′ +1
2b2a′′′
)
+1
2b2(
a a′′′ + 3 a′a′′
+(
(b′)2 + b b′′)
a′′ + 2 b b′a′′′)+1
4b4a(4)
]
I(0,0,0)
+
[
a
(
a′b′ + a b′′ + b b′b′′ +1
2b2b′′′
)
+1
2b2(
a′′b′ + 2 a′b′′
+a b′′′ +(
(b′)2 + b b′′)
b′′ + 2 b b′b′′′ +1
2b2b(4)
)
]
I(0,0,1)
c© Copyright E. Platen NS of SDEs Chap. 4 188
+
[
a(
b′a′ + b a′′)+1
2b2(
b′′a′ + 2 b′a′′ + b a′′′)]
I(0,1,0)
+
[
a(
(b′)2 + b b′′)
+1
2b2(
b′′b′ + 2 b b′′ + b b′′′)
]
I(0,1,1)
+ b
(
a a′′ + (a′)2 + b b′a′′ +1
2b2a′′′
)
I(1,0,0)
+ b
(
a b′′ + a′b′ + b b′b′′ +1
2b2b′′′
)
I(1,0,1)
+ b(
a′b′ + a′′b)
I(1,1,0) + b(
(b′)2 + b b′′)
I(1,1,1) +R6
c© Copyright E. Platen NS of SDEs Chap. 4 189
Examples for Wagner Platen Expansions
• Vasicek interest rate model
drt = γ (r − rt) dt+ β dWt
for t ∈ [0, T ], r0 ≥ 0
c© Copyright E. Platen NS of SDEs Chap. 4 190
• hierarchical set
A = α ∈ M1 : ℓ(α) ≤ 3
rt = r0 + γ (r − r0) t+ βWt − γ2 (r − r0)t2
2
−β γ
∫ t
0
Ws ds+ γ3 (r − r0)t3
6
+β γ2
∫ t
0
∫ s2
0
Ws1 ds1 ds2 +R6
c© Copyright E. Platen NS of SDEs Chap. 4 191
• Black-Scholes dynamics
dSt = St (a dt+ σ dWt)
for t ∈ [0, T ], S0 ≥ 0
• Wagner-Platen expansion
hierarchical set
A = α ∈ M1 : ℓ(α) ≤ 3
c© Copyright E. Platen NS of SDEs Chap. 4 192
is given by
St = S0
(
1 + a t+ σWt + a2 t2
2+ aσ (I(0,1) + I(1,0))
+σ2 1
2
(
(Wt)2 − t
)
+ a3 t3
6
+ a2 σ (I(0,0,1) + I(0,1,0) + I(1,0,0))
+ aσ2 (I(0,1,1) + I(1,0,1) + I(1,1,0))
+σ3 1
6
(
(Wt)3 − 3 tWt
)
)
+R6
c© Copyright E. Platen NS of SDEs Chap. 4 193
• relationship
tWt = I(0) I(1) = I(0,1) + I(1,0)
Wtt2
2= I(1) I(0,0) = I(0,0,1) + I(0,1,0) + I(1,0,0)
t1
2
(
(Wt)2 − t
)
= I(0) I(1,1) = I(0,1,1) + I(1,0,1) + I(1,1,0)
c© Copyright E. Platen NS of SDEs Chap. 4 194
• Wagner-Platen expansion
St = S0
(
1 + a t+ σWt + a2 t2
2+ aσ tWt
+σ2
2
(
(Wt)2 − t
)
+ a3 t3
6+ a2σWt
t2
2
+ aσ2 t
2
(
(Wt)2 − t
)
+ σ3 1
6
(
(Wt)3 − 3 tWt
)
)
+R6
c© Copyright E. Platen NS of SDEs Chap. 4 195
• squared Bessel process
dXt = ν dt+ 2√Xt dWt
for t ∈ [0, T ] withX0 > 0
hierarchical set
A = α ∈ M1 : ℓ(α) ≤ 2
c© Copyright E. Platen NS of SDEs Chap. 4 196
• Wagner-Platen expansion
Xt = X0 + (ν − 1) t+ 2√X0Wt
+ν − 1√X0
∫ t
0
s dWs + (Wt)2 + R
c© Copyright E. Platen NS of SDEs Chap. 4 197
Stratonovich-Taylor Expansion
(Kloeden-Platen) andτ stopping times
0 ≤ ≤ τ ≤ T , f : [0, T ] × ℜd → ℜ and
A⊂ Mm hierarchical set, then
f (τ,Xτ ) =∑
α∈A
Jα
[
fα(,X)
]
,τ+
∑
α∈B(A)
Jα
[
fα(·, X·)
]
,τ
c© Copyright E. Platen NS of SDEs Chap. 4 198
• example Stratonovich-Taylor expansion
Xt = X0 + a J(0) + b J(1) + a a′ J(0,0) + a b′ J(0,1) + b a′J(1,0)
+ b b′J(1,1) + a(
a a′′ + (a′)2)
J(0,0,0) + a(
a b′′ + a′b′)
J(0,0,1)
+ a(
a′′b+ a′b′)
J(0,1,0) + b(
a a′′ + (a′)2)
J(1,0,0)
+ a(
b b′′ + (b′)2)
J(0,1,1) + b(
a b′′ + a′b′)
J(1,0,1)
+ b(
a′′b+ a′b′)
J(1,1,0) + b(
b b′′ + (b′)2)
J(1,1,1) +R7
c© Copyright E. Platen NS of SDEs Chap. 4 199
Moments of Multiple It o Integrals
First Moments
α ∈ Mm \ v with ℓ(α) 6= n(α)
E(
Iα [f·],τ
∣
∣
∣A
)
= 0
c© Copyright E. Platen NS of SDEs Chap. 4 200
Estimates of Higher Moments
α ∈ Mm
(
E
(
∣
∣
∣Iα [g·],τ
∣
∣
∣
2q ∣∣
∣A
)) 1q
≤(
2 (2 q − 1) eT)ℓ(α)−n(α)
(τ − )ℓ(α)+n(α)R8,
where
R8 =
(
E
(
sup≤s≤τ
|gs|2q∣
∣
∣A
)) 1q
c© Copyright E. Platen NS of SDEs Chap. 4 201
• truncated Wagner-Platen expansions
Xk(t) =∑
α∈Λk
Iα [fα(0, X0)]0,t
hierarchical set
Λk = α ∈ Mm : ℓ(α) + n(α) ≤ k
under appropriate assumptions
Xta.s.= lim
k→∞Xk(t)
a.s.=
∑
α∈Mm
Iα [fα(0, X0)]0,t
c© Copyright E. Platen NS of SDEs Chap. 4 202
Exercises of Chapter 4
4.1 Use the Wagner-Platen expansion with time incrementh > 0 and
Wiener process incrementWt0+h −Wt0 in the expansion part to ex-
pand the incrementXt0+h −Xt0 of a geometric Brownian motion at
time t0, where
dXt = aXt dt+ bXt dWt.
4.2 Expand the geometric Brownian motion from Exercise 4.1 such that all
double integrals appear in the expansion part.
4.3 For multi-indicesα = (0, 0, 0, 0), (1, 0, 2), (0, 2, 0, 1) determine
−α,α−, ℓ(α) andn(α).
4.4 Write out in full the multiple Ito stochastic integralsI(0,0),t, I(1,0),t,
I(1,1),t andI(1,2),t.
c© Copyright E. Platen NS of SDEs Chap. 4 203
4.5 Express the multiple Ito integralI(0,1) in terms ofI(1), I(0) andI(1,0).
4.6 Verify thatI(1,0),∆ is Gaussian distributed with
E(I(1,0),∆) = 0, E(
(I(1,0),∆)2)
=∆3
3,
E(I(1,0),∆ I(1),∆) =∆2
2.
4.7 For the cased = m = 1 determine the Ito coefficient functionsf(1,0)andf(1,1,1).
4.8 Which of the following sets of multi-indices are not hierarchical sets:
∅, (1), v, (1), v, (0), (0, 1), v, (0), (1), (0, 1) ?
4.9 Determine the remainder sets that correspond to the hierarchical sets
v, (1) andv, (0), (1), (0, 1).
c© Copyright E. Platen NS of SDEs Chap. 4 204
4.10 Determine the truncated Wagner-Platen expansion at time t = 0 for the
solution of the Ito SDE
dXt = a(t,Xt) dt+ b(t,Xt) dWt
using the hierarchical setA = v, (0), (1), (1, 1).
4.11 In the notation of Sect.4.2 where the component−1 denotes in a multi-
indexα a jump term, determine forα = (1, 0,−1) its lengthℓ(α),
its number of zeros and its number of time integrations.
4.12 For the multi-indexα = (1, 0,−1) derive in the notation of Sect.4.2
−α.
c© Copyright E. Platen NS of SDEs Chap. 4 205
5 Introduction to Scenario Simulation
Discrete Time Approximation
0 = τ0 < τ1 < · · · < τn < · · · < τN = T
one-dimensional SDE
dXt = a(t,Xt) dt+ b(t,Xt) dWt
c© Copyright E. Platen NS of SDEs Chap. 5 206
• Euler scheme
Euler-Maruyama scheme
Maruyama (1955)
Yn+1 = Yn + a(τn, Yn) (τn+1 − τn) + b(τn, Yn)(
Wτn+1−Wτn
)
Y0 = X0,
Yn = Yτn
∆n = τn+1 − τn
c© Copyright E. Platen NS of SDEs Chap. 5 207
maximum step size
∆ = maxn∈0,1,...,N−1
∆n
• equidistant time discretization
τn = n∆
∆n ≡ ∆ = TN
c© Copyright E. Platen NS of SDEs Chap. 5 208
• Euler approximationrecursively computed
• random increments
∆Wn = Wτn+1−Wτn
for n ∈ 0, 1, . . . , N−1
Wiener process
W = Wt t ∈ [0, T ]
Gaussian distributed
mean
E (∆Wn) = 0
variance
E(
(∆Wn)2) = ∆n
c© Copyright E. Platen NS of SDEs Chap. 5 209
Gaussian pseudo-random numbers generated
• abbreviation
f = f(τn, Yn)
Euler scheme
Yn+1 = Yn + a∆n + b∆Wn,
discrete time approximations
c© Copyright E. Platen NS of SDEs Chap. 5 210
Interpolation
• discrete time approximation
stochastic process on[0, T ]
piecewise constant interpolation
Yt = Ynt
for t ∈ [0, T ]
nt = maxn ∈ 0, 1, . . . , N : τn ≤ t
largest integern for whichτn does not exceedt
c© Copyright E. Platen NS of SDEs Chap. 5 211
• linear interpolation
Yt = Ynt +t− τnt
τnt+1 − τnt
(Ynt+1 − Ynt)
c© Copyright E. Platen NS of SDEs Chap. 5 212
Simulating Geometric Brownian Motion
• illustrate scenario simulation
standard market model as geometric Brownian motions
dXt = aXt dt+ bXt dWt
for t ∈ [0, T ],X0 > 0
• drift coefficient
a(t, x) = ax
• diffusion coefficient
b(t, x) = b x
c© Copyright E. Platen NS of SDEs Chap. 5 213
• appreciation rate a
• volatility b 6= 0
• explicit solution
Xt = X0 exp
((
a− 1
2b2)
t+ bWt
)
• Wiener process
W = Wt, t ∈ [0, T ]
c© Copyright E. Platen NS of SDEs Chap. 5 214
Scenario Simulation
• simulate trajectory of Euler approximation
1. initial valueY0 =X0
2. proceed recursively
Yn+1 = Yn + aYn ∆n + b Yn ∆Wn
for n ∈ 0, 1, . . . , N−1
∆Wn = Wτn+1−Wτn
c© Copyright E. Platen NS of SDEs Chap. 5 215
• comparison with explicit solution at timeτn
Xτn = X0 exp
(
(
a− 1
2b2)
τn + b
n∑
i=1
∆Wi−1
)
for n ∈ 0, 1, . . . , N−1Euler approximation
mathematically another new object
different
negative ?
alternative ways ?
c© Copyright E. Platen NS of SDEs Chap. 5 216
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8 1
Figure 5.1: Euler approximations for∆ = 0.25 and∆ = 0.0625 and
exact solution for Black-Scholes SDE.c© Copyright E. Platen NS of SDEs Chap. 5 217
Strong Approximation
• not specified acriterion for classification
• scenario simulation
approximates paths
testing ofcalibration methods
statisticalestimators
filtering
c© Copyright E. Platen NS of SDEs Chap. 5 218
• Monte-Carlo simulation
approximatesprobabilities
functionals
simulatesexpectations
moments
pricesof contingent claims
or risk measuresas Value at Risk
c© Copyright E. Platen NS of SDEs Chap. 5 219
Order of Strong Convergence
• absolute error criterion
ε(∆) = E(∣
∣
∣XT − Y ∆T
∣
∣
∣
)
A discrete time approximationY ∆ converges strongly with orderγ > 0 at time
T if there exists a positive constantC, which does not depend on∆, and aδ0> 0 such that
ε(∆) = E(∣
∣
∣XT − Y ∆T
∣
∣
∣
)
≤ C∆γ
for each∆ ∈ (0, δ0).
c© Copyright E. Platen NS of SDEs Chap. 5 220
Strong Taylor Schemes
• useWagner-Platen expansions
appropriate truncation
• operators
L0 =∂
∂t+
d∑
k=1
ak ∂
∂xk+
1
2
d∑
k,ℓ=1
m∑
j=1
bk,j bℓ,j∂
∂xk ∂xℓ
L0 =∂
∂t+
d∑
k=1
ak ∂
∂xk
c© Copyright E. Platen NS of SDEs Chap. 5 221
and
Lj = Lj =d∑
k=1
bk,j∂
∂xk
for j ∈ 1, 2, . . . ,m, where
ak = ak − 1
2
m∑
j=1
Lj bk,j
c© Copyright E. Platen NS of SDEs Chap. 5 222
• multiple It o integrals
I(j1,...,jℓ) =
∫ τn+1
τn
. . .
∫ s2
τn
dW j1s1 . . . dW
jℓsℓ
• multiple Stratonovich integrals
J(j1,...,jℓ) =
∫ τn+1
τn
. . .
∫ s2
τn
dW j1s1 . . . dW jℓ
sℓ
for j1, . . . , jℓ ∈ 0, 1, . . . ,m, ℓ ∈ 1, 2, . . .andn ∈ 0, 1, . . .
W 0t = t
for all t ∈ [0, T ]
c© Copyright E. Platen NS of SDEs Chap. 5 223
• It o SDE
dXt = a(t,Xt) dt+m∑
j=1
bj(t,Xt) dWjt
• equivalentStratonovich SDE
dXt = a(t,Xt) dt+
m∑
j=1
bj(t,Xt) dW jt
c© Copyright E. Platen NS of SDEs Chap. 5 224
Euler Scheme
simplest strong Taylor approximation
order of strong convergenceγ = 0.5
• d=m= 1
Euler scheme
Yn+1 = Yn + a∆ + b∆W
∆ = τn+1 − τn
∆W = ∆Wn = Wτn+1−Wτn
N(0,∆) independent Gaussian distributed
c© Copyright E. Platen NS of SDEs Chap. 5 225
• multi-dimensional Euler scheme
m = 1 andd ∈ 1, 2, . . .
kth component
Y kn+1 = Y k
n + ak ∆ + bk ∆W
for k ∈ 1, 2, . . . , d
a= (a1, . . ., ad)⊤ andb= (b1, . . ., bd)⊤
c© Copyright E. Platen NS of SDEs Chap. 5 226
• general multi-dimensional Euler scheme
d,m ∈ 1, 2, . . .
Y kn+1 = Y k
n + ak ∆ +
m∑
j=1
bk,j ∆W j
∆W j = W jτn+1
−W jτn
N(0,∆) independent Gaussian distributed
∆W j1 and∆W j2 independent forj1 6= j2
b= [bk,j]d,mk,j=1 d×m-matrix
truncated Wagner-Platen expansion
c© Copyright E. Platen NS of SDEs Chap. 5 227
Theorem 5.1 Suppose that we have initial valuesX0 andY0 = Y ∆0 such that
E(|X0|2
)
< ∞
and
E
(
∣
∣
∣X0 − Y ∆0
∣
∣
∣
2) 1
2
≤ K1 ∆12 .
Furthermore, assume the Lipschitz condition
|a(t, x) − a(t, y)| + |b(t, x) − b(t, y)| ≤ K2 |x− y|,
the linear growth condition
|a(t, x)| + |b(t, x)| ≤ K3 (1 + |x| )
c© Copyright E. Platen NS of SDEs Chap. 5 228
and
|a(s, x) − a(t, x)| + |b(s, x) − b(t, x)| ≤ K4 (1 + |x| ) |s− t| 12
for all s, t∈ [0, T ] andx, y ∈ℜd, where the constantsK1, . . .,K4 do not depend
on∆. Then the Euler approximationY ∆ converges with strong orderγ = 0.5, that
is we have the estimate
E(∣
∣
∣XT − Y ∆T
∣
∣
∣
)
≤ K5 ∆12 ,
where the constantK5 does not depend on∆.
c© Copyright E. Platen NS of SDEs Chap. 5 229
A Simulation Example
• Black-Scholes SDE
dXt = µXt dt+ σXt dWt
• exact solution
XT = X0 exp
(
µ− σ2
2
)
T + σWT
c© Copyright E. Platen NS of SDEs Chap. 5 230
absolute error
ε(∆) = E(|XT − Y ∆N |)
default parameters:
X0 = 1, µ = 0.06, σ = 0.2 andT = 1
5000 simulations
fitted line
ln(ε(∆)) = −3.86933 + 0.46739 ln(∆)
c© Copyright E. Platen NS of SDEs Chap. 5 231
-4 -3 -2 -1 0
Log -dt
-6
-5.5
-5
-4.5
-4Log-Strong
Error
Figure 5.2: Log-log plot of the absolute errorε(∆) for an Euler Scheme
againstln(∆).
c© Copyright E. Platen NS of SDEs Chap. 5 232
Milstein Scheme
Milstein (1974)
• Milstein scheme
d = m = 1
Yn+1 = Yn + a∆ + b∆W +1
2b b′
(∆W )2 − ∆
orderγ = 1.0 of strong convergence
c© Copyright E. Platen NS of SDEs Chap. 5 233
• multi-dimensional Milstein scheme
m= 1 andd ∈ 1, 2, . . .
Y kn+1 = Y k
n + ak ∆ + bk ∆W +
(
d∑
ℓ=1
bℓ∂bk
∂xℓ
)
1
2
(∆W )2 − ∆
c© Copyright E. Platen NS of SDEs Chap. 5 234
• general multi-dimensional Milstein scheme
d,m ∈ 1, 2, . . .
kth component
Y kn+1 = Y k
n + ak ∆ +m∑
j=1
bk,j∆W j +m∑
j1,j2=1
Lj1bk,j2I(j1,j2)
Ito integralsI(j1,j2)
• alternatively
Y kn+1 = Y k
n + ak ∆ +
m∑
j=1
bk,j∆W j +
m∑
j1,j2=1
Lj1bk,j2J(j1,j2)
c© Copyright E. Platen NS of SDEs Chap. 5 235
Stratonovich integralsJ(j1,j2)
j1 6= j2 j1, j2 ∈ 1, 2, . . . ,m
J(j1,j2) = I(j1,j2) =
∫ τn+1
τn
∫ s1
τn
dW j1s2 dW
j2s1
cannot be simply expressed by∆W j1 and∆W j2
I(j1,j1) =1
2
(∆W j1)2 − ∆
and J(j1,j1) =1
2
(
∆W j1)2
for j1 ∈ 1, 2, . . . ,m
c© Copyright E. Platen NS of SDEs Chap. 5 236
Commutative Noise
• commutativity condition
Lj1bk,j2 = Lj2bk,j1
for all j1, j2 ∈ 1, 2, . . . ,m, k ∈ 1, 2, . . . , d and
(t, x) ∈ [0, T ]×ℜd
• satisfied for Black-Scholes SDEs
additive noise
single Wiener process
c© Copyright E. Platen NS of SDEs Chap. 5 237
• Milstein scheme under commutative noise
Y kn+1 = Y k
n +ak ∆+
m∑
j=1
bk,j∆W j +1
2
m∑
j1,j2=1
Lj1bk,j2∆W j1∆W j2
for k ∈ 1, 2, . . . , d
no double Wiener integrals
c© Copyright E. Platen NS of SDEs Chap. 5 238
A Black-Scholes Example
• correlated Black-Scholes dynamics
d = m = 2
• first risky security
dX1t = X1
t
[
r dt+ θ1(θ1 dt+ dW 1t ) + θ2(θ2 dt+ dW 2
t )]
= X1t
[(
r +1
2
(
(θ1)2 + (θ2)2)
)
dt
+ θ1 dW 1t + θ2 dW 2
t
]
c© Copyright E. Platen NS of SDEs Chap. 5 239
• second risky security
dX2t = X2
t
[
r dt+ (θ1 − σ1,1) (θ1 dt+ dW 1t )]
= X2t
[(
r + (θ1 − σ1,1)
(
1
2θ1 +
1
2σ1,1
))
dt
+(θ1 − σ1,1) dW 1t
]
c© Copyright E. Platen NS of SDEs Chap. 5 240
=⇒
L1 b1,2 =2∑
k=1
bk,1∂
∂xkb1,2 = θ1X1
t θ2
=
2∑
k=1
bk,2∂
∂xkb1,1 = L2 b1,1
and
L1 b2,2 =
2∑
k=1
bk,1∂
∂xkb2,2 = 0 =
2∑
k=1
bk,2∂
∂xkb2,1 = L2 b2,1
=⇒ commutativity condition satisfied
=⇒ Milstein scheme :
c© Copyright E. Platen NS of SDEs Chap. 5 241
Y 1n+1 = Y 1
n + Y 1n
(
r +1
2
(
(θ1)2 + (θ2)2)
∆ + θ1 ∆W 1 + θ2 ∆W 2
+1
2(θ1)2(∆W 1)2 +
1
2(θ2)2(∆W 2)2 + θ1 θ2 ∆W 1 ∆W 2
)
Y 2n+1 = Y 2
n + Y 2n
((
r +1
2(θ1 − σ1,1) (θ1 + σ1,1)
)
∆
+(θ1 − σ1,1)∆W 1 +1
2(θ1 − σ1,1)2 (∆W 1)2
)
( may produce negative trajectories )
c© Copyright E. Platen NS of SDEs Chap. 5 242
A Square Root Process Example
• square root process of dimensionν
dXt =ν
4η
(
1
η−Xt
)
dt+√Xt dW
1t
X0 = 1η
for ν > 2
• Milstein
Yn+1 = Yn +ν
4η
(
1
η− Yn
)
∆ +√Yn ∆W +
1
4
(
∆W 2 − ∆)
c© Copyright E. Platen NS of SDEs Chap. 5 243
0
5
10
15
20
25
30
35
40
0 20 40 60 80 100
Figure 5.3: Square root process simulated by the Milstein scheme.
c© Copyright E. Platen NS of SDEs Chap. 5 244
Approximate Double Wiener Integrals
Forj1 6= j2 with j1, j2 ∈ 1, 2, . . . ,m approximate
J(j1,j2) = I(j1,j2) by
Jp(j1,j2)
= ∆
(
1
2ξj1ξj2 +
√p (µj1,p ξj2 − µj2,p ξj1)
)
+∆
2π
p∑
r=1
1
r
(
ζj1,r(√
2 ξj2 + ηj2,r
)
− ζj2,r(√
2 ξj1 + ηj1,r
))
where
p =1
12− 1
2π2
p∑
r=1
1
r2
c© Copyright E. Platen NS of SDEs Chap. 5 245
ξj , µj,p, ηj,r andζj,r
independentN(0, 1)
ξj =1√∆
∆W j
for j ∈ 1, 2, . . . ,m, r ∈ 1, . . . p andp ∈ 1, 2, . . . if
p = p(∆) ≥ K
∆
=⇒γ = 1.0
c© Copyright E. Platen NS of SDEs Chap. 5 246
Convergence Theorem
E(|X0|2
)
< ∞
E
(
∣
∣
∣X0 − Y ∆0
∣
∣
∣
2) 1
2
≤ K1 ∆12
c© Copyright E. Platen NS of SDEs Chap. 5 247
|a(t, x) − a(t, y)| ≤ K2 |x− y|∣
∣
∣bj1(t, x) − bj1(t, y)
∣
∣
∣ ≤ K2 |x− y|∣
∣
∣Lj1bj2(t, x) − Lj1bj2(t, y)
∣
∣
∣ ≤ K2 |x− y|
|a(t, x)| +∣
∣
∣Lja(t, x)
∣
∣
∣ ≤ K3 (1 + |x|)∣
∣
∣bj1(t, x)
∣
∣
∣+∣
∣
∣Ljbj2(t, x)
∣
∣
∣ ≤ K3 (1 + |x|)∣
∣
∣LjLj1bj2(t, x)
∣
∣
∣ ≤ K3 (1 + |x|)
c© Copyright E. Platen NS of SDEs Chap. 5 248
and
|a(s, x) − a(t, x)| ≤ K4 (1 + |x| ) |s− t| 12∣
∣
∣bj1(s, x) − bj1(t, x)
∣
∣
∣ ≤ K4 (1 + |x| ) |s− t| 12∣
∣
∣Lj1bj2(s, x) − Lj1bj2(t, x)
∣
∣
∣ ≤ K4 (1 + |x| ) |s− t| 12
for all s, t ∈ [0, T ], x, y ∈ ℜd, j ∈ 0, . . .,m andj1, j2 ∈ 1, 2, . . . ,m.
Then the Milstein scheme converges with strong orderγ = 1.0
E(∣
∣
∣XT − Y ∆T
∣
∣
∣
)
≤ K5 ∆.
Kloeden & Platen (1999).
c© Copyright E. Platen NS of SDEs Chap. 5 249
A Simulation Study
-4 -3 -2 -1 0
Log -dt
-9
-8
-7
-6
-5Log-Strong
Error
Figure 5.4: Log-log plot of the absolute error against log-step size for a
Milstein scheme.c© Copyright E. Platen NS of SDEs Chap. 5 250
• linear regression
ln(ε(∆)) = −4.91021 + 0.95 ln(∆)
c© Copyright E. Platen NS of SDEs Chap. 5 251
Order 1.5 Strong Taylor Scheme
• simulation tasks that require more accurate schemes
extreme log-returns
• including further multiple stochastic integrals
• order 1.5 strong Taylor scheme
d = m = 1
c© Copyright E. Platen NS of SDEs Chap. 5 252
Yn+1 = Yn + a∆ + b∆W +1
2b b′
(∆W )2 − ∆
+ a′ b∆Z +1
2
(
a a′ +1
2b2 a′′
)
∆2
+
(
a b′ +1
2b2 b′′
)
∆W ∆ − ∆Z
+1
2b(
b b′′ + (b′)2)
1
3(∆W )2 − ∆
∆W
c© Copyright E. Platen NS of SDEs Chap. 5 253
• double integral
∆Z = I(1,0) =
∫ τn+1
τn
∫ s2
τn
dWs1 ds2
Gaussian
E(∆Z) = 0
E((∆Z)2) =1
3∆3
E(∆Z∆W ) =1
2∆2
c© Copyright E. Platen NS of SDEs Chap. 5 254
• two independentN(0, 1)
U1 andU2
=⇒∆W = U1
√∆, ∆Z =
1
2∆
32
(
U1 +1√3U2
)
• triple Wiener integral
I(1,1,1) =1
2
1
3(∆W 1)2 − ∆
∆W 1
scaled monic Hermite polynomial
c© Copyright E. Platen NS of SDEs Chap. 5 255
• multi-dimensional order 1.5 strong Taylor scheme
d ∈ 1, 2, . . . andm = 1
Y kn+1 = Y k
n + ak ∆ + bk ∆W
+1
2L1 bk (∆W )2 − ∆ + L1 ak ∆Z
+L0 bk ∆W ∆ − ∆Z +1
2L0 ak ∆2
+1
2L1 L1 bk
1
3(∆W )2 − ∆
∆W
c© Copyright E. Platen NS of SDEs Chap. 5 256
• general multi-dimensional order 1.5 strong Taylor scheme
d,m ∈ 1, 2, . . .
Y kn+1 = Y k
n + ak ∆ +1
2L0 ak ∆2
+
m∑
j=1
(bk,j∆W j + L0 bk,j I(0,j) + Lj ak I(j,0))
+
m∑
j1,j2=1
Lj1 bk,j2 I(j1,j2) +
m∑
j1,j2,j3=1
Lj1 Lj2 bk,j3 I(j1,j2,j3)
in case of additive noise simplifies considerably
strong orderγ = 1.5
c© Copyright E. Platen NS of SDEs Chap. 5 257
Approximate Multiple Stochastic Integrals
Let ξj , ζj,1, . . . , ζj,p, ηj,1, . . . , ηj,p, µj,p andφj,p
be independentN(0, 1)
for j, j1, j2, j3 ∈ 1, 2, . . . ,m and somep ∈ 1, 2, . . .
set
I(j) = ∆W j =√∆ ξj, I(j,0) =
1
2∆(√
∆ ξj + aj,0
)
with
aj,0 = −√2∆
π
p∑
r=1
1
rζj,r − 2
√
∆ p µj,p
c© Copyright E. Platen NS of SDEs Chap. 5 258
where
p =1
12− 1
2π2
p∑
r=1
1
r2
I(0,j) = ∆W j ∆ − I(j,0), I(j,j) =1
2
(∆W j)2 − ∆
I(j,j,j) =1
2
1
3(∆W j)2 − ∆
∆W j
Ip(j1,j2) =1
2∆ ξj1 ξj2 − 1
2
√∆(ξj1 aj2,0 − ξj2 aj1,0) +Ap
j1,j2∆
c© Copyright E. Platen NS of SDEs Chap. 5 259
Order 2.0 Strong Taylor Scheme
• use Stratonovich-Taylor expansion
d=m= 1
Yn+1 = Yn + a∆ + b∆W +1
2!b b′(∆W )2 + b a′ ∆Z
+1
2a a′ ∆2 + a b′∆W ∆ − ∆Z
+1
3!b(
b b′)′
(∆W )3 +1
4!b(
b(
b b′)′)′
(∆W )4
+ a(
b b′)′J(0,1,1) + b
(
a b′)′J(1,0,1) + b
(
b a′)′ J(1,1,0)
c© Copyright E. Platen NS of SDEs Chap. 5 260
Approximate Multiple Stratonovich Integrals
∆W = Jp(1) =
√∆ ζ1
∆Z = Jp(1,0) =
1
2∆(√
∆ ζ1 + a1,0
)
Jp(1,0,1) =
1
3!∆2ζ21 − 1
4∆ a2
1,0 +1
π∆
32 ζ1 b1 − ∆2Bp
1,1
Jp(0,1,1) =
1
3!∆2ζ21 − 1
2π∆
32 ζ1 b1 +∆2Bp
1,1 −1
4∆
32 a1,0 ζ1 +∆2 Cp
1,1
Jp(1,1,0) =
1
3!∆2ζ21 +
1
4∆ a2
1,0− 1
2π∆
32 ζ1 b1+
1
4∆
32 a1,0 ζ1−∆2 Cp
1,1
c© Copyright E. Platen NS of SDEs Chap. 5 261
with
a1,0 = − 1
π
√2∆
p∑
r=1
1
rξ1,r − 2
√
∆p µ1,p
p =1
12− 1
2π2
p∑
r=1
1
r2
b1 =
√
∆
2
p∑
r=1
1
r2η1,r +
√
∆αp φ1,p
αp =π2
180− 1
2π2
p∑
r=1
1
r4
Bp1,1 =
1
4π2
p∑
r=1
1
r2(
ξ21,r + η21,r
)
c© Copyright E. Platen NS of SDEs Chap. 5 262
and
Cp1,1 = − 1
2π2
p∑
r,l=1r 6=l
r
r2 − l2
(
1
lξ1,r ξ1,ℓ − l
rη1,r η1,ℓ
)
ζ1, ξ1,r η1,r, µ1,p andφ1,p ∼ N(0, 1) i.i.d.
c© Copyright E. Platen NS of SDEs Chap. 5 263
Multi-dimensional Order 2.0 Strong Taylor Scheme
m= 1
Y kn+1 = Y k
n + ak ∆ + bk ∆W +1
2!L1 bk (∆W )2 + L1 ak ∆Z
+1
2L0ak ∆2 + L0bk ∆W ∆ − ∆Z
+1
3!L1L1bk (∆W )3 +
1
4!L1L1L1bk (∆W )4
+L0L1bk J(0,1,1) + L1L0bk J(1,0,1) + L1L1ak J(1,1,0)
c© Copyright E. Platen NS of SDEs Chap. 5 264
• general multi-dimensional order 2.0 strong Taylor scheme
Y kn+1 = Y k
n + ak ∆ +1
2L0ak ∆2
+
m∑
j=1
(
bk,j ∆W j + L0bk,jJ(0,j) + Ljak J(j,0)
)
+
m∑
j1,j2=1
(
Lj1bk,j2J(j1,j2) + L0Lj1bk,j2J(0,j1,j2)
+Lj1L0bk,j2J(j1,0,j2) + Lj1Lj2ak J(j1,j2,0)
)
+
m∑
j1,j2,j3=1
Lj1Lj2bk,j3J(j1,j2,j3)
+
m∑
j1,j2,j3,j4=1
Lj1Lj2Lj3bk,j4J(j1,j2,j3,j4)
c© Copyright E. Platen NS of SDEs Chap. 5 265
Convergence Theorem
• hierarchical set
Aγ =
α ∈ M : ℓ(α) + n(α) ≤ 2 γ or ℓ(α) = n(α) = γ +1
2
c© Copyright E. Platen NS of SDEs Chap. 5 266
• order γ strong Taylor approximation
Y ∆t = Y ∆
nt+
∑
α∈Aγ\vIα[
fα(
τnt , Y∆nt
)]
τnt,t
=∑
α∈Aγ
Iα[
fα(
τnt , Y∆nt
)]
τnt,t
for γ ∈ 0.5, 1.0, 1.5, 2.0, . . .
stochastic interpolation
c© Copyright E. Platen NS of SDEs Chap. 5 267
Theorem
Y ∆ = Y ∆t , t ∈ [0, T ] orderγ strong Taylor approximation
γ ∈ 0.5, 1.0, 1.5, 2.0, . . .
|fα(t, x) − fα(t, y)| ≤ K1 |x− y|
for all α ∈ Aγ , t ∈ [0, T ] andx, y ∈ ℜd,
f−α ∈ C1,2 and fα ∈ Hα
for all α ∈ Aγ ∪ B (Aγ) and
|fα(t, x)| ≤ K2 (1 + |x|)
c© Copyright E. Platen NS of SDEs Chap. 5 268
for all α ∈ Aγ ∪ B (Aγ), t ∈ [0, T ] andx ∈ ℜd. Then
E
(
sup0≤t≤T
∣
∣
∣Xt − Y ∆t
∣
∣
∣
2 ∣∣A0
)
≤ K3
(
1 + |X0|2)
∆2γ +K4
∣
∣
∣X0 − Y ∆0
∣
∣
∣
2
,
whereK1,K2,K3, andK4 do not depend on∆.
c© Copyright E. Platen NS of SDEs Chap. 5 269
Derivative Free Strong Schemes
• similar to Runge-Kutta schemes for ODEs
Explicit Order 1.0 Strong Schemes
Platen (1984)
d=m= 1
Yn+1 = Yn + a∆ + b∆W +1
2√∆
b(τn, Υn) − b
(∆W )2 − ∆
with supporting value
Υn = Yn + a∆ + b√∆
c© Copyright E. Platen NS of SDEs Chap. 5 270
• multi-dimensional Platen scheme
m = 1
Y kn+1 = Y k
n + ak ∆ + bk ∆W
+1
2√∆
(
bk(τn, Υn) − bk)
(
(∆W )2 − ∆)
with the vector supporting value
Υn = Yn + a∆ + b√∆
c© Copyright E. Platen NS of SDEs Chap. 5 271
• general multi-dimensional case
Y kn+1 = Y k
n + ak ∆ +
m∑
j=1
bk,j ∆W j
+1√∆
m∑
j1,j2=1
(
bk,j2(
τn, Υj1n
)
− bk,j2)
I(j1,j2)
with
Υjn = Yn + a∆ + bj
√∆
for j ∈ 1, 2, . . .
c© Copyright E. Platen NS of SDEs Chap. 5 272
• commutative noise
Y kn+1 = Y k
n + ak ∆ +1
2
m∑
j=1
(
bk,j(
τn, Υn
)
+ bk,j)
∆W j
with
Υn = Yn + a∆ +m∑
j=1
bj ∆W j
strong orderγ = 1.0
c© Copyright E. Platen NS of SDEs Chap. 5 273
ln(ε(∆)) = −4.71638 + 0.946112 ln(∆)
-4 -3 -2 -1 0
Log -dt
-9
-8
-7
-6
-5Log-Strong
Error
Figure 5.5: Log-log plot of the absolute error for a Platen scheme.
c© Copyright E. Platen NS of SDEs Chap. 5 274
• two stage Runge-Kutta methodγ = 1.0
m = d = 1
Burrage (1998)
Yn+1 = Yn +(
a(Yn) + 3 a(Yn)) ∆
4
+1
4
(
b(Yn) + 3 b(Yn))
∆W
with
Yn = Yn +2
3(a(Yn)∆ + b(Yn)∆W )
c© Copyright E. Platen NS of SDEs Chap. 5 275
Explicit Order 1.5 Strong Schemes
Platen (1984)
d = m = 1
with
Υ± = Yn + a∆ ± b√∆
and
Φ± = Υ+ ± b(Υ+)√∆
c© Copyright E. Platen NS of SDEs Chap. 5 276
Yn+1 = Yn + b∆W +1
2√∆
(
a(Υ+) − a(Υ−))
∆Z
+1
4
(
a(Υ+) + 2a+ a(Υ−))
∆
+1
4√∆
(
b(Υ+) − b(Υ−)) (
(∆W )2 − ∆)
+1
2∆
(
b(Υ+) − 2b+ b(Υ−)) (
∆W∆ − ∆Z)
+1
4∆
(
b(Φ+) − b(Φ−) − b(Υ+) + b(Υ−))
×(
1
3(∆W )2 − ∆
)
∆W
c© Copyright E. Platen NS of SDEs Chap. 5 277
• order1.5 Platen scheme
Y kn+1 = Y k
n + ak ∆ +
m∑
j=1
bk,j ∆W j
+1
2√∆
m∑
j2=0
m∑
j1=1
(
bk,j2(
Υj1+
)
− bk,j2(
Υj1−
) )
I(j1,j2)
+1
2∆
m∑
j2=0
m∑
j1=1
(
bk,j2(
Υj1+
)
− 2bk,j2 + bk,j2(
Υj1−
))
I(0,j2)
+1
2∆
m∑
j1,j2,j3=1
(
bk,j3(
Φj1,j2+
)
− bk,j3(
Φj1,j2−
)
− bk,j3(
Υj1+
)
+ bk,j3(
Υj1−
))
I(j1,j2,j3)
c© Copyright E. Platen NS of SDEs Chap. 5 278
with
Υj± = Yn +
1
ma∆ ± bj
√∆
and
Φj1,j2± = Υj1
+ ± bj2(
Υj1+
) √∆
where we interpretbk,0 asak
c© Copyright E. Platen NS of SDEs Chap. 5 279
A Simulation Study
-4 -3 -2 -1 0
Log -dt
-12
-10
-8
-6
-4
Log-Strong
Error
1.5 Taylor
R -K
Milstein
Euler
Figure 5.6: Log-log plot of the absolute error for various strong schemes.
c© Copyright E. Platen NS of SDEs Chap. 5 280
Method CPU Time
Euler 3.5 Seconds
Milstein 4.1 Seconds
1.0 Strong Platen 4.1 Seconds
1.5 Strong Taylor 4.4 Seconds
Table 2: Computational times for various strong methods.
500000 time steps
c© Copyright E. Platen NS of SDEs Chap. 5 281
Numerical Stability
• ability of a scheme to control the propagation
of initial and roundoff errors
• numerical stability has higher priority than
a potentially high strong order
c© Copyright E. Platen NS of SDEs Chap. 5 282
DeterministicA-Stability
• one-step method
Yn+1 = Yn + Ψ(τn, Yn, Yn+1,∆)∆
ordinary differential equation
dx
dt= a(t, x)
a(t, x) satisfies Lipschitz condition
c© Copyright E. Platen NS of SDEs Chap. 5 283
• numerically stable
if there exist positive constants∆0 andM such that
|Yn − Yn| ≤ M |Y0 − Y0|
n ∈ 0, 1, . . . , N, ∆ < ∆0
and any two solutionsY , Y
corresponding to the initial valuesY0 Y0, respectively
• asymptotically numerically stable
if there exist∆0 andM such that
limn→∞
|Yn − Yn| ≤ M |Y0 − Y0|
for any twoY , Y
c© Copyright E. Platen NS of SDEs Chap. 5 284
• test equation
dxt
dt= λxt
with λ ∈ ℜ
c© Copyright E. Platen NS of SDEs Chap. 5 285
numerical scheme in recursive form
Yn+1 = G(λ∆)Yn
• A-stability region:
thoseλ∆ ∈ ℜ for which
|G(λ∆)| < 1
c© Copyright E. Platen NS of SDEs Chap. 5 286
• explicit Euler scheme
Yn+1 = Yn + a(tn, Yn)∆
Yn+1 = (1 + λ∆)Yn
A-stability region
open unit interval centered at−1 and ending at 0 since
|G(λ∆)| = |1 + λ∆|
c© Copyright E. Platen NS of SDEs Chap. 5 287
-2 -1 0
-1
-0.5
0.5
1
Figure 5.7: Region ofA-stability for the deterministic Euler scheme.
c© Copyright E. Platen NS of SDEs Chap. 5 288
• implicit Euler scheme
Yn+1 = Yn + a(tn+1, Yn+1)∆
Yn+1 = Yn + λ∆Yn+1
(1 − λ∆)Yn+1 = Yn
G(λ∆) =1
1 − λ∆
|G(λ∆)| = 1
|1 − λ∆|exterior of an interval with center at1 beginning at 0
• scheme is calledA-stable if it covers at least the left half axis
c© Copyright E. Platen NS of SDEs Chap. 5 289
StochasticA-Stability
• test equation
dXt = λXt dt+ dWt
λ ∈ ℜ
c© Copyright E. Platen NS of SDEs Chap. 5 290
• discrete time approximation
Yn+1 = GA(λ∆)Yn + Zn,
Zn does not depend onY0, Y1, . . . , Yn, Yn+1
c© Copyright E. Platen NS of SDEs Chap. 5 291
• A-stability region
real numbersλ∆ for which
|GA(λ∆)| < 1
If A-stability region covers left half of the real axis
=⇒ thenA-stable
c© Copyright E. Platen NS of SDEs Chap. 5 292
Drift Implicit Euler Scheme
• drift implicit Euler scheme
strong orderγ = 0.5
d = m = 1
Yn+1 = Yn + a (τn+1, Yn+1) ∆ + b∆W
c© Copyright E. Platen NS of SDEs Chap. 5 293
• family of drift implicit Euler schemes
Yn+1 = Yn + αa (τn+1, Yn+1) + (1 − α) a ∆ + b∆W
degree of implicitness α ∈ [0, 1]
for α= 0 explicit Euler scheme
for α= 1 drift implicit Euler scheme
c© Copyright E. Platen NS of SDEs Chap. 5 294
• general multi-dimensional
family of drift implicit Euler schemes
Y kn+1 = Y k
n +(
αkak (τn+1, Yn+1) + (1 − αk) a
k)
∆
+
m∑
j=1
bk,j ∆W j
αk ∈ [0, 1], k ∈ 1, 2, . . . , d
c© Copyright E. Platen NS of SDEs Chap. 5 295
• for test equation
Yn+1 = Yn +(
αλYn+1 + (1 − α)λYn
)
∆ + ∆Wn
Yn+1 = GA(λ∆)Yn + ∆Wn (1 − αλ∆)−1
transfer function
GA(λ∆) = (1 − αλ∆)−1 (1 + (1 − α)λ∆)
c© Copyright E. Platen NS of SDEs Chap. 5 296
Yn corresponding discrete time approximation that starts at initial valueY0
Yn+1 − Yn+1 = GA(λ∆) (Yn − Yn)
= (GA(λ∆))n (Y0 − Y0)
as long as
|GA(λ∆)| < 1
the initial error(Y0 − Y0) is decreased
=⇒ λ∆ ∈(
− 2
1 − 2α, 0
)
for α ≥ 12
A-stable
c© Copyright E. Platen NS of SDEs Chap. 5 297
Drift Implicit Milstein Scheme
d = m = 1
• It o version
Yn+1 = Yn + a (τn+1, Yn+1) ∆ + b∆W +1
2b b′(
(∆W )2 − ∆)
c© Copyright E. Platen NS of SDEs Chap. 5 298
• Stratonovich version
Yn+1 = Yn + a (τn+1, Yn+1) ∆ + b∆W +1
2b b′(∆W )2
adjusted Stratonovich drift a= a− 12b b′
both schemes are different
c© Copyright E. Platen NS of SDEs Chap. 5 299
• multi-dimensional case withd=m ∈ 1, 2, . . . andcommutative noise
Y kn+1 = Y k
n +
αk ak (τn+1, Yn+1) + (1 − αk) a
k
∆
+
m∑
j=1
bk,j ∆W j
+1
2
m∑
j1,j2=1
Lj1bk,j2
∆W j1∆W j2 − 1j1=j2∆
and
Y kn+1 = Y k
n +
αk ak (τn+1, Yn+1) + (1 − αk) a
k
∆
+
m∑
j=1
bk,j ∆W j +1
2
m∑
j1,j2=1
Lj1bk,j2∆W j1∆W j2
c© Copyright E. Platen NS of SDEs Chap. 5 300
• general drift implicit Milstein scheme
Ito version
Y kn+1 = Y k
n +(
αk ak (τn+1, Yn+1) + (1 − αk) a
k)
∆
+
m∑
j=1
bk,j ∆W j +
m∑
j1,j2=1
Lj1bk,j2I(j1,j2)
c© Copyright E. Platen NS of SDEs Chap. 5 301
Stratonovich version
Y kn+1 = Y k
n +(
αk ak (τn+1, Yn+1) + (1 − αk) a
k)
∆
+
m∑
j=1
bk,j ∆W j +
m∑
j1,j2=1
Lj1bk,j2J(j1,j2)
αk ∈ [0, 1] for k ∈ 1,. . ., d
multiple stochastic integralsI(j1,j2) andJ(j1,j2)
approximated
c© Copyright E. Platen NS of SDEs Chap. 5 302
Drift Implicit Order 1.0 Strong Runge-Kutta Scheme
d=m= 1
Yn+1 = Yn + a (τn+1, Yn+1) ∆ + b∆W
+1
2√∆
(
b(
τn, Υn
)− b) (
(∆W )2 − ∆)
with supporting value
Υ = Yn + a∆ + b√∆
c© Copyright E. Platen NS of SDEs Chap. 5 303
• family of drift implicit order 1.0 strong Runge-Kutta schem es
Y kn+1 = Y k
n +(
αk a (τn+1, Yn+1) + (1 − αk) ak)
∆
+m∑
j=1
bk,j ∆W j
+1√∆
m∑
j1,j2=1
(
bk,j2(
τn, Υj1n
)
− bk,j2)
I(j1,j2)
with
Υjn = Yn + a∆ + bj
√∆
for j ∈ 1, 2, . . . ,mαk ∈ [0, 1] for k ∈ 1, 2, . . . , d
c© Copyright E. Platen NS of SDEs Chap. 5 304
• for commutative noise
Y kn+1 = Y k
n +(
αk ak (τn+1, Yn+1) + (1 − αk) a
k)
∆
+1
2
m∑
j=1
(
bk,j(
τn, Ψn
)
+ bk,j)
∆W j
with
Ψn = Yn + a∆ +
m∑
j=1
bj ∆W j
αk ∈ [0, 1] for k ∈ 1, 2, . . . , d
c© Copyright E. Platen NS of SDEs Chap. 5 305
Alternative Implicit Methods
• implicit methods are important
• overcome a range of numerical instabilities
• the above strong schemes do not provide implicit diffusion terms
=⇒ important limitation
• drift implicit methods are well adapted for small noise and additive noise
for relatively large multiplicative noise
implicit diffusion terms seem unavoidable
c© Copyright E. Platen NS of SDEs Chap. 5 306
• illustration with multiplicative noise
dXt = σXt dWt
• explicit strong methods have large errors for not too small time step sizes
• very small time step size may require unrealistic computational time
• cannot apply drift implicit schemes
c© Copyright E. Platen NS of SDEs Chap. 5 307
• balanced implicit methods
Milstein, Platen & Schurz (1998)
m = d = 1
Yn+1 = Yn + a∆ + b∆W + (Yn − Yn+1)Cn
where
Cn = c0(Yn)∆ + c1(Yn) |∆W |
c0, c1 positive, real valued uniformly bounded functions
strong orderγ = 0.5
c© Copyright E. Platen NS of SDEs Chap. 5 308
• low order strong convergence
price to pay for numerical stability
• family of specific methods providing a balance between approximating diffusion
terms
c© Copyright E. Platen NS of SDEs Chap. 5 309
Simulation Study for the Balanced Method
• Euler scheme
Yn+1 = Yn + σ Yn ∆W
• no simple stochastic counterpart of deterministic implicit Euler method since
Yn+1 = Yn + σ Yn+1 ∆W
Yn+1 =Yn
1 − σ∆W
fails because
E|(1 − σ∆W )−1| = +∞
c© Copyright E. Platen NS of SDEs Chap. 5 310
• partially implicit scheme
Yn+1 = Yn +
(
σ∆W +σ2
2(∆W )2
)
Yn − σ2
2Yn+1 ∆
• balanced implicit method
Yn+1 = Yn + σ Yn ∆W + σ (Yn − Yn+1) |∆W |
=⇒ implicitness also in diffusion term
c© Copyright E. Platen NS of SDEs Chap. 5 311
• explicit solution
XT = exp
σWT − σ2
2T
X0
• absolute error
εT (∆) = E(|XT − YN |)
c© Copyright E. Platen NS of SDEs Chap. 5 312
Figure 5.8: Estimated absolute errorεT (∆) of the Euler method at timeT
for time step sizes∆ = 2−4 and2−5.
c© Copyright E. Platen NS of SDEs Chap. 5 313
Figure 5.9: Absolute error for the partially implicit method.
c© Copyright E. Platen NS of SDEs Chap. 5 314
Figure 5.10: Absolute error for the balanced implicit method.
c© Copyright E. Platen NS of SDEs Chap. 5 315
The General Balanced Method and its Convergence
• d-dimensional SDE
dXt = a(t,Xt) dt+
m∑
j=1
bj(t,Xt) dWjt
• family of balanced implicit methods
Yn+1 = Yn+a(τn, Yn)∆+
m∑
j=1
bj(τn, Yn)∆Wjn+Cn(Yn−Yn+1)
c© Copyright E. Platen NS of SDEs Chap. 5 316
where
Cn = c0(τn, Yn)∆ +
m∑
j=1
cj(τn, Yn) |∆W jn|
c0, c1, . . . , cm
d× d-matrix-valued uniformly bounded functions
c© Copyright E. Platen NS of SDEs Chap. 5 317
• assume that for any sequence of real(αi) with α0 ∈ [0, α], α1 ≥ 0, . . .,
αm ≥ 0, whereα ≥ ∆
matrix
M(t, x) = I + α0 c0(t, x) +
m∑
j=1
aj cj(t, x)
has an inverse
|(M(t, x))−1| ≤ K < ∞
c© Copyright E. Platen NS of SDEs Chap. 5 318
Theorem (Milstein-Platen-Schurz)
The balanced implicit method converges with strong orderγ = 0.5, that is for all
k ∈ 0, 1, . . . , N and step size∆ = TN
,N ∈ 1, 2, . . . one has
E(|Xτk − Yk|
∣
∣A0
) ≤ (
E(|Xτk − Yk|2
∣
∣A0
))12
≤ K(
1 + |X0|2)
12 ∆
12 ,
whereK does not depend on∆.
c© Copyright E. Platen NS of SDEs Chap. 5 319
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
aη = a− η b b′
• predictor
Yn+1 = Yn + a(Yn)∆n + b(Yn)∆Wn
θ, η ∈ [0, 1] degree of implicitness
c© Copyright E. Platen NS of SDEs Chap. 5 320
Strong Approximation of SDEs with Jumps
Some Jump Diffusions in Finance
• Merton model Merton (1976)
dSt = St− (a dt+ σ dWt + dYt)
• compound Poisson process
Yt =
Nt∑
k=1
ξk
with
dYt = ξNt dNt
c© Copyright E. Platen NS of SDEs Chap. 5 321
• Poisson process
N = Nt, t ∈ [0, T ] with intensityλ > 0
jump sizes
i.i.d. random variablesξ1, ξ2, . . .
• explicit solution
St = S0 exp
(
a− 1
2σ2
)
t+ σWt
Nt∏
k=1
(ξk + 1)
c© Copyright E. Platen NS of SDEs Chap. 5 322
• compound Poisson process
jump size
∆Yτk = Yτk − Yτk− = ξk
• asset price jump size
∆Sτk = Sτk − Sτk− = Sτk− ξk
=⇒Sτk = Sτk− (ξk + 1)
=⇒ extension of the Black-Scholes model
c© Copyright E. Platen NS of SDEs Chap. 5 323
• for ξ1 = −1
default of the stock
• for ξk = δ − 1, k ∈ 1, 2, . . . with δ ∈ [0, 1]
δ recovery rate of a stock
c© Copyright E. Platen NS of SDEs Chap. 5 324
• scalar jump diffusion
dXt = a(t,Xt) dt+ b(t,Xt) dWt + c(t−, Xt−) dNt
at a jump timeτ
Xτ = Xτ− + c(τ−, Xτ−)
• multi-dimensional jump diffusion
interacting factorsX1t , . . . , X
dt
independent standard Wiener processesW 1, . . . ,Wm
c© Copyright E. Platen NS of SDEs Chap. 5 325
n Poisson processes with intensitiesλ1(t,Xt), . . . , λn(t,Xt)
SDE
dXit = ai(t,Xt) dt+
m∑
k=1
bi,k(t,Xt) dWkt
+
n∑
ℓ=1
ci,ℓ(t−, Xt−) dN ℓt
i ∈ 1, 2, . . . , d
credit risk, operational risk and insurance modeling
c© Copyright E. Platen NS of SDEs Chap. 5 326
Random Jump Size
• Poisson jump measurepℓϕℓ
(·, ·)
• mark set E = ℜ\0
ϕℓ(E) < ∞
ℓ ∈ 1, 2, . . . , d
c© Copyright E. Platen NS of SDEs Chap. 5 327
jumps arrive at timeτ1 < τ2 < . . . with marksv1, v2, . . .
Nℓt
∑
k=1
ci,ℓ(vk) as∫ t
0
∫
Eci,ℓ(v) pℓ
ϕℓ(dv, dt)
∫ t
0
∫
Eci,ℓ(v)
(
pℓϕℓ
(dv, dt) − ϕℓ(dv) dt)
(A, P )-martingale
c© Copyright E. Platen NS of SDEs Chap. 5 328
• jump diffusion SDE
dXit = ai(t,Xt) dt+
m∑
k=1
bi,k(t,Xt) dWkt
+n∑
ℓ=1
∫
Eci,ℓ(v, t−, Xt−) pℓ
ϕℓ(dv, dt)
Xiτ = Xi
τ− + ci,ℓ(v, τ−, Xτ−).
(A, P )-martingale measure
qℓϕℓ(dv, dt) = pℓ
ϕℓ(dv, dt) − ϕℓ(dv) dt
c© Copyright E. Platen NS of SDEs Chap. 5 329
• Levy process models
Barndorff-Nielsen (1998)
Madan & Seneta (1990)
• Affine jump-diffusions
Duffie, Pan & Singleton (2000)
• interest rate term structure
Bjork, Kabanov & Runggaldier (1997)
Glasserman & Merener (2003)
c© Copyright E. Platen NS of SDEs Chap. 5 330
Discrete Time Approximation with Jump Times
• discrete time approximation of jump diffusions
Platen (1982a, 1984)
Platen & Rebolledo (1985)
Maghsoodi & Harris (1987)
Mikulevicius & Platen (1988)
Maghsoodi (1996, 1998)
Kubilius & Platen (2002)
Glasserman & Merener (2003)
Bruti-Liberati & Platen (2007)
c© Copyright E. Platen NS of SDEs Chap. 5 331
• jump adapted time discretization
ti with t0 < t1 < . . .
superposition of the random jump timesτ1, τ2, . . . of the Poisson processes
pℓϕℓ
(E, [0, ·]) and a deterministic grid
can be precomputed
• maximum step size is∆ > 0
ti+1 − ti ≤ ∆ a.s
c© Copyright E. Platen NS of SDEs Chap. 5 332
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
c© Copyright E. Platen NS of SDEs Chap. 5 333
Jump-Adapted Strong Approximations
jump-adapted time discretization⇓
jump times included in time discretization
• jump-adapted Euler scheme
Ytn+1− = Ytn + a(Ytn)∆tn + b(Ytn)∆Wtn
and
Ytn+1= Ytn+1− + c(Ytn+1−)∆pn
• γ = 0.5
c© Copyright E. Platen NS of SDEs Chap. 5 334
Euler Scheme
d = m = n = 1
Yti+1− = Yti + a (ti+1 − ti) + b(
Wti+1−Wti
)
jump part
Yti+1= Yti+1− +
∫
Ec(v, ti+1−, Yti+1−) pϕ(dv, ti+1)
• multi-dimensional
Euler scheme
Y iti+1− = Y i
ti + ai (ti+1 − ti) +
m∑
k=1
bi,k(
W kti+1
−W kti
)
c© Copyright E. Platen NS of SDEs Chap. 5 335
with
Y iti+1
= Y iti+1− +
n∑
ℓ=1
∫
Eci,ℓ(v, ti+1−, Yti+1−) pℓ
ϕℓ(dv, ti+1)
for i ∈ 1, 2, . . . , d
Platen (1982a)
γ = 0.5
c© Copyright E. Platen NS of SDEs Chap. 5 336
Order 1.0 Taylor Scheme
Y iti+1− = Y i
ti + ai (ti+1 − ti) +
m∑
k=1
bi,k(
W kti+1
−W kti
)
+
m∑
j1,j2=1
Lj1 bi,j2 I(j1,j2)ti,ti+1
with
Y iti+1
= Y iti+1− +
n∑
ℓ=1
∫
Eci,ℓ(v, ti+1−, Yti+1−) pℓ
ϕℓ(dv, ti+1)
for i ∈ 1, 2, . . . , d
c© Copyright E. Platen NS of SDEs Chap. 5 337
• if noise is commutative =⇒ scheme simplifies
Platen (1982a)
γ = 1.0
c© Copyright E. Platen NS of SDEs Chap. 5 338
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 5.11: Plot of a jump-diffusion path.
c© Copyright E. Platen NS of SDEs Chap. 5 339
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 5.12: Plot of the strong error for Euler(red) and1.0 Taylor(blue)
scheme.
c© Copyright E. Platen NS of SDEs Chap. 5 340
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 5.13: Log-log plot of strong error versus time step size.
c© Copyright E. Platen NS of SDEs Chap. 5 341
Exercises of Chapter 5, 6 and 7
5.1 Write down for the linear SDE
dXt = (µXt + η) dt+ γ Xt dWt
withX0 = 1 the Euler scheme and the Milstein scheme.
5.2 Determine for the Vasicek short rate model
drt = γ(r − rt) dt+ β dWt
the Euler and Milstein schemes. What are the differences between the resulting
two schemes?
5.3 Write down for the SDE in Exercise 5.1 the order 1.5 strongTaylor scheme.
5.4 Derive for the Black-Scholes SDE
dXt = µXt dt+ σXt dWt
withX0 = 1 the explicit order 1.0 strong scheme.
c© Copyright E. Platen NS of SDEs Chap. 5 342
11 Monte Carlo Simulation of SDEs
Introduction to Monte Carlo Simulation
• classical Monte Carlo methods
Hammersley & Handscomb (1964)
Fishman (1996)
• Monte Carlo methods for SDEs
Kloeden & Platen (1999)
Kloeden, Platen & Schurz (2003)
Milstein (1995), Glasserman (2004)
c© Copyright E. Platen NS of SDEs Chap. 11 465
Weak Convergence Criterion
• CP (ℜd,ℜ) set of all polynomialsg : ℜd → ℜ
• SDE
dXt = a(t,Xt) dt+
m∑
j=1
bj(t,Xt) dWjt
• a discrete time approximationY ∆ converges with weak orderβ > 0 toX at
time T as∆ → 0 if for eachg ∈ CP (ℜd,ℜ) there exists a constantCg,
which does not depend on∆ and∆0 ∈ [0, 1] such that
µ(∆) = |E(g(XT )) − E(g(Y ∆T ))| ≤ Cg ∆β
for each∆ ∈ (0,∆0)
• absolute weak error criterion
c© Copyright E. Platen NS of SDEs Chap. 11 466
Systematic and Statistical Error
• functional
u = E(g(XT ))
• weak approximationsY ∆
• raw Monte Carlo estimate
uN,∆ =1
N
N∑
k=1
g(Y ∆T (ωk))
N independent simulated realizations
c© Copyright E. Platen NS of SDEs Chap. 11 467
Y ∆T (ω1), Y
∆T (ω2), . . . , Y
∆T (ωN)
ωk ∈ Ω for k ∈ 1, 2, . . . , N
• discrete time weak approximationY ∆T
c© Copyright E. Platen NS of SDEs Chap. 11 468
• weak error
µN,∆ = uN,∆ − E(g(XT ))
• systematic error µsys
• statistical error µstat
µN,∆ = µsys+ µstat
c© Copyright E. Platen NS of SDEs Chap. 11 469
• systematic error
µsys = E(µN,∆)
= E
(
1
N
N∑
k=1
g(Y ∆T (ωk))
)
− E(g(XT ))
= E(g(Y ∆T )) − E(g(XT ))
µ(∆) = |µsys|
c© Copyright E. Platen NS of SDEs Chap. 11 470
• statistical error
Central Limit Theorem
asymptotically Gaussian with mean zero and variance
Var(µstat) = Var(µN,∆) =1
NVar(g(Y ∆
T ))
deviation
Dev(µstat) =√
Var(µstat) =1√N
√
Var(g(Y ∆T ))
decreases at slow rate1√N
asN → ∞
may need an extremely large numberN of sample paths
c© Copyright E. Platen NS of SDEs Chap. 11 471
Confidence Intervals
• statistical error is halved by a fourfold increase inN
• Monte Carlo approach is very general
• high-dimensional functionals
• do usually not know the variance
of raw Monte Carlo estimates
c© Copyright E. Platen NS of SDEs Chap. 11 472
• form batches
average of each batch
approximately Gaussian
Studentt confidence intervals
• length of a confidence interval
proportional to the square root of the variance
reformulate the random variable
same mean but a much smaller variance
• variance reduction techniques
c© Copyright E. Platen NS of SDEs Chap. 11 473
Example of Raw Monte Carlo Simulation
u = E(g(X))
X ∼ N(0, 1)
g(X) =(
exp
r∆ + σ√∆X
)2
r = 0.05, σ = 0.2, ∆ = 1
u = E(
exp
2(
r∆ + σ√∆X
))
= exp(
r + σ2) 2∆ ≈ 1.197
c© Copyright E. Platen NS of SDEs Chap. 11 474
• Raw Monte Carlo estimators
uN =1
N
N∑
i=1
exp
2(
r∆ + σ√∆X(ωi)
)
N ∈ 1, 2, . . . , 2000
asymptotically normal
c© Copyright E. Platen NS of SDEs Chap. 11 475
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
0 200 400 600 800 1000 1200 1400 1600 1800 2000
^u_Nu
Figure 11.1: Raw Monte Carlo estimates in dependence on the number of
simulations.c© Copyright E. Platen NS of SDEs Chap. 11 476
Normalized Monte Carlo Error
ZN =(uN − u)
√
Var(g(X))
√N ∼ N (0, 1)
CLT
Var(g(X)) = exp4∆ (r + 2σ2) (1 − exp−4∆σ2) ≈ 0.25.
c© Copyright E. Platen NS of SDEs Chap. 11 477
-4
-3
-2
-1
0
1
2
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Figure 11.2: Normalized raw Monte Carlo error.
c© Copyright E. Platen NS of SDEs Chap. 11 478
-4
-3
-2
-1
0
1
2
3
9000 9200 9400 9600 9800 10000
Figure 11.3: Independent realizations of normalized Monte Carlo errors.
c© Copyright E. Platen NS of SDEs Chap. 11 479
Weak Taylor Schemes
Euler and Simplified Weak Euler Scheme
• Euler scheme
Y kn+1 = Y k
n + ak ∆ +
m∑
j=1
bk,j ∆W jn
∆W jn = W j
τn+1−W j
τn
truncated Wagner-Platen expansion
weak convergenceβ = 1.0
=⇒ weak orderβ = 1.0 Taylor scheme
c© Copyright E. Platen NS of SDEs Chap. 11 480
• simplified weak Euler scheme
Y kn+1 = Y k
n + ak ∆ +
M∑
j=1
bk,j ∆W jn
∆W jn independentAτn+1
-measurable random variables with
∣
∣
∣E(
∆W jn
)∣
∣
∣+
∣
∣
∣
∣
E
(
(
∆W jn
)3)∣
∣
∣
∣
+
∣
∣
∣
∣
E
(
(
∆W jn
)2)
− ∆
∣
∣
∣
∣
≤ K∆2
=⇒ β = 1.0
• two-point distributed random variable
P(
∆W jn = ±
√∆)
=1
2
c© Copyright E. Platen NS of SDEs Chap. 11 481
• Simulation Example
SDE
dXt = aXt dt+ bXt dWt
a = 1.5, b = 0.01 andT = 1
test functiong(X) = x
N = 16, 000, 000
=⇒
confidence intervals of negligible length
• absolute weak errors
µ(∆) = |E(XT ) − E(YN)|
c© Copyright E. Platen NS of SDEs Chap. 11 482
-5 -4 -3 -2 -1 0Log2-dt
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Log2-WeakErrors
Weak Error
Simp Euler
Euler
Figure 11.4: Log-log plot of the weak error for an SDE with multiplicative
noise for the Euler and simplified Euler schemes.
c© Copyright E. Platen NS of SDEs Chap. 11 483
Weak Order 2.0 Taylor Scheme
further multiple stochastic integrals
• weak order 2.0 Taylor schemeone-dimensional cased = m = 1
Yn+1 = Yn + a∆ + b∆Wn +1
2b b′(
(∆Wn)2 − ∆
)
+ a′ b∆Zn +1
2
(
a a′ +1
2a′′b2
)
∆2
+
(
a b′ +1
2b′′b2
)
(
∆Wn ∆ − ∆Zn
)
∆Zn = I(1,0) =
∫ τn+1
τn
∫ s
τn
dWz ds
generate pair of correlated Gaussian random variables∆Wn and∆Zn
c© Copyright E. Platen NS of SDEs Chap. 11 484
• simplified weak order 2.0 Taylor scheme
Yn+1 = Yn + a∆ + b∆W +1
2b b′
(
(
∆W)2
− ∆
)
+1
2
(
a′ b+ a b′ +1
2b′′b2
)
∆W ∆
+1
2
(
a a′ +1
2a′′b2
)
∆2
c© Copyright E. Platen NS of SDEs Chap. 11 485
∣
∣
∣E(
∆W)∣
∣
∣+
∣
∣
∣
∣
E
(
(
∆W)3)∣
∣
∣
∣
+
∣
∣
∣
∣
E
(
(
∆W)5)∣
∣
∣
∣
+
∣
∣
∣
∣
E
(
(
∆W)2)
− ∆
∣
∣
∣
∣
+
∣
∣
∣
∣
E
(
(
∆W)4)
− 3∆2
∣
∣
∣
∣
≤ K∆3
• three-point distributed random variable
P(
∆W = ±√3∆
)
=1
6, P
(
∆W = 0)
=2
3
c© Copyright E. Platen NS of SDEs Chap. 11 486
• weak order 2.0 Taylor scheme with scalar noise
d ∈ 1, 2, . . . withm = 1
Y kn+1 = Y k
n + ak ∆ + bk ∆W +1
2L1bk
(
(∆W )2 − ∆)
+1
2L0ak ∆2 + L0bk
(
∆W ∆ − ∆Z)
+ L1ak ∆Z
c© Copyright E. Platen NS of SDEs Chap. 11 487
• weak order 2.0 Taylor scheme
d,m ∈ 1, 2, . . .
Y kn+1 = Y k
n + ak ∆ +1
2L0ak ∆2
+m∑
j=1
(
bk,j∆W j + L0bk,j I(0,j) + Ljak I(j,0)
)
+
m∑
j1,j2=1
Lj1bk,j2 I(j1,j2)
c© Copyright E. Platen NS of SDEs Chap. 11 488
• simplified weak order 2.0 Taylor scheme
Y kn+1 = Y k
n + ak ∆ +1
2L0ak ∆2
+
m∑
j=1
(
bk,j +1
2∆(
L0bk,j + Ljak)
)
∆W j
+1
2
m∑
j1,j2=1
Lj1bk,j2(
∆W j1∆W j2 + Vj1,j2
)
c© Copyright E. Platen NS of SDEs Chap. 11 489
• two-point distributed random variables
P (Vj1,j2 = ±∆) =1
2
for j2 ∈ 1, . . ., j1 − 1,
Vj1,j1 = −∆
and
Vj1,j2 = −Vj2,j1
for j2 ∈ j1 + 1, . . .,m andj1 ∈ 1, 2, . . . ,m
β = 2.0
c© Copyright E. Platen NS of SDEs Chap. 11 490
Weak Order 3.0 Taylor Scheme
• weak order 3.0 Taylor scheme
d,m ∈ 1, 2, . . .
Y kn+1 = Y k
n + ak ∆ +
m∑
j=1
bk,j∆W j +
m∑
j=0
Ljak I(j,0)
+
m∑
j1=0
m∑
j2=1
Lj1bk,j2 I(j1,j2) +
m∑
j1,j2=0
Lj1Lj2ak I(j1,j2,0)
+
m∑
j1,j2=0
m∑
j3=1
Lj1Lj2bk,j3 I(j1,j2,j3)
c© Copyright E. Platen NS of SDEs Chap. 11 491
• simplified weak order 3.0 Taylor scheme
m = 1, d = 1
Yn+1 = Yn + a∆ + b∆W +1
2L1b
(
(
∆W)2
− ∆
)
+L1a∆Z +1
2L0a∆2 + L0b
(
∆W ∆ − ∆Z)
+1
6
(
L0L0b+ L0L1a+ L1L0a)
∆W ∆2
+1
6
(
L1L1a+ L1L0b+ L0L1b)
(
(
∆W)2
− ∆
)
∆
+1
6L0L0a∆3 +
1
6L1L1b
(
(
∆W)2
− 3∆
)
∆W
c© Copyright E. Platen NS of SDEs Chap. 11 492
∆W ∼ N(0,∆), ∆Z ∼ N
(
0,1
3∆3
)
E(
∆W ∆Z)
=1
2∆2
c© Copyright E. Platen NS of SDEs Chap. 11 493
Forβ = 3.0 we can set approximately
I(1) = ∆W 1, I(1,0) ≈ ∆Z, I(0,1) ≈ ∆∆W − ∆Z
I(1,1) ≈ 1
2
(
(
∆W)2
− ∆
)
I(0,0,1) ≈ I(0,1,0) ≈ I(1,0,0) ≈ 1
6∆2∆W
I(1,1,0) ≈ I(1,0,1) ≈ I(0,1,1) ≈ 1
6∆
(
(
∆W)2
− ∆
)
I(1,1,1) ≈ 1
6∆W
(
(
∆W)2
− 3∆
)
where∆W and∆Z
are correlated Gaussian random variables
c© Copyright E. Platen NS of SDEs Chap. 11 494
Hofmann (1994)
=⇒ additive noise,β = 3.0, m ∈ 1, 2, . . .
I(0) = ∆, I(j) = ξj√∆, I(0,0) =
∆2
2, I(0,0,0) =
∆3
6
I(j,0) ≈ 1
2
(
ξj + ϕj1√3∆
)
∆32 , I(j,0,0) ≈ I(0,j,0) ≈ ∆
52
6ξj
I(j1,j2,0) ≈ ∆2
6
(
ξj1 ξj2 +Vj1,j2
∆
)
independent four point distributed random variables
ξ1, . . . , ξm
c© Copyright E. Platen NS of SDEs Chap. 11 495
P
(
ξj = ±√
3 +√6
)
=1
12 + 4√6
P
(
ξj = ±√
3 −√6
)
=1
12 − 4√6
independent three point distributed random variables
ϕ1, . . . , ϕm
independent two-point distributed random variables
Vj1,j2
c© Copyright E. Platen NS of SDEs Chap. 11 496
• multi-dimensional case
d,m ∈ 1, 2, . . .
additive noise
weak order3.0 Taylor scheme
c© Copyright E. Platen NS of SDEs Chap. 11 497
Y kn+1 = Y k
n + ak ∆ +
m∑
j=1
bk,j ∆W j +1
2L0ak ∆2 +
1
6L0L0ak ∆3
+
m∑
j=1
(
Ljak ∆Zj + L0bk,j(
∆W j∆ − ∆Zj)
+1
6
(
L0L0bk,j + L0Ljak + LjL0ak)
∆W j ∆2
)
+1
6
m∑
j1,j2=1
Lj1Lj2ak(
∆W j1∆W j2 − Ij1=j2 ∆)
∆
c© Copyright E. Platen NS of SDEs Chap. 11 498
∆W j and∆Zj
independent pairs of
correlated Gaussian random variables
c© Copyright E. Platen NS of SDEs Chap. 11 499
Weak Order 4.0 Taylor Scheme
Wagner-Platen expansion=⇒
• weak order4.0 Taylor scheme
d,m ∈ 1, 2, . . .
Y kn+1 = Y k
n +
4∑
ℓ=1
m∑
j1,...,jℓ=0
Lj1 · · ·Ljℓ−1 bk,jℓ I(j1,...,jℓ)
β = 4.0
Platen (1984)
c© Copyright E. Platen NS of SDEs Chap. 11 500
• simplified weak order 4.0 Taylor scheme for additive noise
Yn+1 = Yn + a∆ + b∆W +1
2L0a∆2 + L1a∆Z
+L0b(
∆W ∆ − ∆Z)
+1
3!
(
L0L0b+ L0L1a)
∆W ∆2
+L1L1a
(
2∆W ∆Z − 5
6
(
∆W)2
∆ − 1
6∆2
)
+1
3!L0L0a∆3 +
1
4!L0L0L0a∆4
c© Copyright E. Platen NS of SDEs Chap. 11 501
+1
4!
(
L1L0L0a+ L0L1L0a+ L0L0L1a + L0L0L0b)
∆W ∆3
+1
4!
(
L1L1L0a+ L0L1L1a+ L1L0L1a)
(
(
∆W)2
− ∆
)
∆2
+1
4!L1L1L1a∆W
(
(
∆W)2
− 3∆
)
∆
∆W ∼ N(0,∆), ∆Z ∼ N(0,1
3∆3)
E(∆W ∆Z) =1
2∆2
c© Copyright E. Platen NS of SDEs Chap. 11 502
A Simulation Study
• SDE with additive noise
dXt = aXt dt+ b dWt
X0 = 1, a = 1.5, b = 0.01 andT = 1
g(X) = x
c© Copyright E. Platen NS of SDEs Chap. 11 503
-5 -4 -3 -2 -1 0
Log2-dt
-20
-15
-10
-5
0
Log2-WeakErrors
Weak Error
4Taylor
3Taylor
2Taylor
Euler
Figure 11.5: Log-log plot of the weak error for an SDE with additive noise
using weak Taylor schemes.
c© Copyright E. Platen NS of SDEs Chap. 11 504
• SDE with multiplicative noise
dXt = aXt dt+ bXt dWt
g(X) = x
c© Copyright E. Platen NS of SDEs Chap. 11 505
-5 -4 -3 -2 -1 0
Log2-dt
-15
-10
-5
0Log2-WeakErrors
Weak Error
4Taylor
3Taylor
2Taylor
Euler
Figure 11.6: Log-log plot of the weak error for an SDE with multiplicative
noise using weak Taylor schemes.
c© Copyright E. Platen NS of SDEs Chap. 11 506
Convergence Theorem
• Wagner-Platen expansion
• It o coefficient functions
f(t, x) ≡ x
fα(t, x) = Lj1 · · ·Ljℓ−1 bjℓ(x)
α = (j1, . . . , jℓ) ∈ Mm, m ∈ 1, 2, . . .
c© Copyright E. Platen NS of SDEs Chap. 11 507
• multiple It o integral
Iα,τn,τn+1=
∫ τn+1
τn
∫ sℓ
τn
· · ·∫ s2
τn
dW j1s1 . . . dW
jℓ−1sℓ−1
dW jℓsℓ
dW 0s = ds
c© Copyright E. Platen NS of SDEs Chap. 11 508
• hierarchical set
Γβ = α ∈ Mm : l(α) ≤ β
• time discretization
0 = τ0 < τ1 < . . . τn < . . .
• weak Taylor scheme of orderβ
Yn+1 = Yn +∑
α∈Γβ\vfα (τn, Yn) Iα,τn,τn+1
=∑
α∈Γβ
fα (τn, Yn) Iα,τn,τn+1
Y0 = X0
c© Copyright E. Platen NS of SDEs Chap. 11 509
Theorem
For someβ ∈ 1, 2, . . . and autonomousX letY ∆ be a weak Taylor scheme of
orderβ. Suppose thata andb are Lipschitz continuous with componentsak, bk,j
∈ C2(β+1)P
(ℜd,ℜ) for all k ∈ 1, 2, . . . , d andj ∈ 0, 1, . . . ,m, and that
thefα satisfy a linear growth bound
|fα(t, x)| ≤ K (1 + |x|) ,
for all α ∈ Γβ, x ∈ ℜd and t ∈ [0, T ], whereK < ∞. Then for eachg ∈C2(β+1)P
(ℜd,ℜ) there exists a constantCg, which does not depend on∆, such
that
µ(∆) =∣
∣
∣E (g (XT )) − E(
g(
Y ∆T
))∣
∣
∣ ≤ Cg ∆β,
that isY ∆ converges with weak orderβ toX at timeT as∆ → 0.
c© Copyright E. Platen NS of SDEs Chap. 11 510
Derivative Free Weak Approximations
Explicit Weak Order 2.0 Scheme
• explicit weak order 2.0 scheme
m = 1 andd ∈ 1, 2, . . .
Yn+1 = Yn +1
2
(
a(
Υ)
+ a)
∆
+1
4
(
b(
Υ+)
+ b(
Υ−)
+ 2b)
∆W
+1
4
(
b(
Υ+)
− b(
Υ−))
(
(
∆W)2
− ∆
)
∆12
c© Copyright E. Platen NS of SDEs Chap. 11 511
with supporting values
Υ = Yn + a∆ + b∆W
and
Υ± = Yn + a∆ ± b√∆
∆W Gaussian or three-point distributed
P(
∆W = ±√3∆)
=1
6and P
(
∆W = 0)
=2
3
c© Copyright E. Platen NS of SDEs Chap. 11 512
• multi-dimensional explicit weak order 2.0 scheme
d,m ∈ 1, 2, . . .
Yn+1 = Yn +1
2
(
a(
Υ)
+ a)
∆
+1
4
m∑
j=1
[
(
bj(
Rj+
)
+ bj(
Rj−
)
+ 2bj)
∆W j
+
m∑
r=1r 6=j
(
bj(
Ur+
)
+ bj(
Ur−)− 2bj
)
∆W j ∆− 12
]
c© Copyright E. Platen NS of SDEs Chap. 11 513
+1
4
m∑
j=1
[
(
bj(
Rj+
)
− bj(
Rj−
)
)(
(
∆W j)2
− ∆
)
+
m∑
r=1r 6=j
(
bj(
Ur+
)− bj(
Ur−)
)
(
∆W j∆W r + Vr,j
)
]
∆− 12
c© Copyright E. Platen NS of SDEs Chap. 11 514
with supporting values
Υ = Yn + a∆ +
m∑
j=1
bj ∆W j, Rj± = Yn + a∆ ± bj
√∆
and
U j± = Yn ± bj
√∆
∆W j andVr,j as before
c© Copyright E. Platen NS of SDEs Chap. 11 515
• additive noise
=⇒
Yn+1 = Yn +1
2
a
(
Yn + a∆ +
m∑
j=1
bj ∆W j
)
+ a
∆
+m∑
j=1
bj ∆W j
c© Copyright E. Platen NS of SDEs Chap. 11 516
Explicit Weak Order 3.0 Schemes
• explicit weak order 3.0 scheme
d ∈ 1, 2, . . . withm = 1
Yn+1 = Yn + a∆ + b∆W +1
2
(
a+ζ + a−
ζ − 3
2a− 1
4
(
a+ζ + a−
ζ
)
)
∆
+
√
2
∆
(
1√2
(
a+ζ − a−
ζ
)
− 1
4
(
a+ζ − a−
ζ
)
)
ζ∆Z
+1
6
[
a(
Yn +(
a+ a+ζ
)
∆ + (ζ + ) b√∆)
− a+ζ − a+
+ a]
×[
(ζ + ) ∆W√∆ + ∆ + ζ
(
(
∆W)2
− ∆
)]
c© Copyright E. Platen NS of SDEs Chap. 11 517
with
a±φ = a
(
Yn + a∆ ± b√∆φ
)
and
a±φ = a
(
Yn + 2a∆ ± b√2∆φ
)
∆W ∼N(0,∆) and ∆Z ∼N(0, 13∆3)
E(∆W∆Z) =1
2∆2
P (ζ = ±1) = P ( = ±1) =1
2
c© Copyright E. Platen NS of SDEs Chap. 11 518
• explicit weak order 3.0 scheme for scalar noise
Yn+1 = Yn + a∆ + b∆W +1
2Ha ∆ +
1
∆Hb ∆Z
+
√
2
∆Ga ζ∆Z +
1√2∆
Gb ζ
(
(
∆W)2
− ∆
)
+1
6F++
a
(
∆ + (ζ + )√∆∆W + ζ
(
(
∆W)2
− ∆
))
+1
24
(
F++b + F−+
b + F+−b + F−−
b
)
∆W
c© Copyright E. Platen NS of SDEs Chap. 11 519
+1
24√∆
(
F++b − F−+
b + F+−b − F−−
b
)
(
(
∆W)2
− ∆
)
ζ
+1
24∆
(
F++b + F−−
b − F−+b − F+−
b
)
(
(
∆W)2
− 3
)
∆W ζ
+1
24√∆
(
F++b + F−+
b − F+−b − F−−
b
)
(
(
∆W)2
− ∆
)
c© Copyright E. Platen NS of SDEs Chap. 11 520
with
Hg = g+ + g− − 3
2g − 1
4
(
g+ + g−)
,
Gg =1√2
(
g+ − g−)
− 1
4
(
g+ − g−)
,
F+±g = g
(
Yn +(
a+ a+)
∆ + b ζ√∆ ± b+
√∆)
− g+
− g(
Yn + a∆ ± b √∆)
+ g
F−±g = g
(
Yn +(
a+ a−)
∆ − b ζ√∆ ± b−
√∆)
− g−
−g(
Yn + a∆ ± b √∆)
+ g
c© Copyright E. Platen NS of SDEs Chap. 11 521
where
g± = g(
Yn + a∆ ± b ζ√∆)
and
g± = g(
Yn + 2 a∆ ±√2 b ζ
√∆)
with g being equal to eithera or b.
c© Copyright E. Platen NS of SDEs Chap. 11 522
Extrapolation Methods
Weak Order 2.0 Extrapolation
• equidistant time discretizations
• simulate functional
E(
g(
Y ∆T
))
using Euler scheme
with step size∆
• repeat with the double step size2∆
=⇒E(
g(
Y 2∆T
))
c© Copyright E. Platen NS of SDEs Chap. 11 523
• combine these two functionals
=⇒• weak order 2.0 extrapolation
V ∆g,2(T ) = 2E
(
g(
Y ∆T
))
− E(
g(
Y 2∆T
))
Talay & Tubaro (1990)
Richardson extrapolation
c© Copyright E. Platen NS of SDEs Chap. 11 524
• example
geometric Brownian motion
dXt = aXt dt+ bXt dWt
X0 = 0.1, a = 1.5 andb = 0.01
Richardson extrapolationV ∆g,2(T )
g(x) = x
=⇒ absolute weak error
µ(∆) =∣
∣
∣V∆g,2(T ) − E (g (XT ))
∣
∣
∣
β = 2.0
c© Copyright E. Platen NS of SDEs Chap. 11 525
-12
-11
-10
-9
-8
-7
-6
-6 -5.5 -5 -4.5 -4 -3.5 -3
time
Figure 11.7: Log-log plot for absolute weak error of Richardson extrapola-
tion against step size.
c© Copyright E. Platen NS of SDEs Chap. 11 526
Higher Weak Order Extrapolation
• weak order 4.0 extrapolation
V ∆g,4(T ) =
1
21
[
32E(
g(
Y ∆T
))
− 12E(
g(
Y 2∆T
))
+ E(
g(
Y 4∆T
))
]
c© Copyright E. Platen NS of SDEs Chap. 11 527
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-4 -3.5 -3 -2.5 -2
time
Figure 11.8: Log-log plot for absolute weak error of weak order 4.0 extra-
polation.c© Copyright E. Platen NS of SDEs Chap. 11 528
Implicit and Predictor Corrector Methods
• numerical stability has highest priority
Drift Implicit Euler Scheme
d,m ∈ 1, 2, . . .
Yn+1 = Yn + a (τn+1, Yn+1)∆ +
m∑
j=1
bj (τn, Yn)∆Wj
P(
∆W j = ±√∆)
=1
2
c© Copyright E. Platen NS of SDEs Chap. 11 529
• family of drift implicit simplified Euler schemes
Yn+1 = Yn +(
(1 − α) a (τn, Yn) + αa (τn+1, Yn+1))
∆
+
m∑
j=1
bj (τn, Yn)∆Wj
α degree of drift implicitness
A-stable forα ∈ [0.5, 1]
region ofA-stability
circle of radiusr = (1 − 2α)−1 centered at−r
c© Copyright E. Platen NS of SDEs Chap. 11 530
Fully Implicit Euler Scheme
simplified schemes allow
implicit diffusion coefficient term
• fully implicit weak Euler scheme
Yn+1 = Yn + a (Yn+1)∆ + b (Yn+1)∆W
a = a− b b′
c© Copyright E. Platen NS of SDEs Chap. 11 531
• family of implicit weak Euler schemes
Yn+1 = Yn +(
α aη (τn+1, Yn+1) + (1 − α) aη (τn, Yn))
∆
+m∑
j=1
(
η bj (τn+1, Yn+1) + (1 − η) bj (τn, Yn))
∆W j
aη = a− η
m∑
j1,j2=1
d∑
k=1
bk,j1∂bj2
∂xk
for α, η ∈ [0, 1]
c© Copyright E. Platen NS of SDEs Chap. 11 532
Implicit Weak Order 2.0 Taylor Scheme
Yn+1 = Yn + a (Yn+1)∆ + b∆W
− 1
2
(
a (Yn+1) a′ (Yn+1) +
1
2b2 (Yn+1) a
′′ (Yn+1)
)
∆2
+1
2b b′
(
(
∆W)2
− ∆
)
+1
2
(
−a′ b+ a b′ +1
2b′′b2
)
∆W ∆
P(
∆W = ±√3∆)
=1
6and P
(
∆W = 0)
=2
3
Milstein (1995)
c© Copyright E. Platen NS of SDEs Chap. 11 533
• family of implicit weak order 2.0 Taylor schemes
Yn+1 = Yn +(
αa (τn+1, Yn+1) + (1 − α) a)
∆
+1
2
m∑
j1,j2=1
Lj1bj2(
∆W j1∆W j2 + Vj1,j2
)
+
m∑
j=1
(
bj +1
2
(
L0bj + (1 − 2α)Lja)
∆
)
∆W j
+1
2(1 − 2α)
(
β L0a+ (1 − β)L0a (τn+1, Yn+1))
∆2
α = 0.5 =⇒
c© Copyright E. Platen NS of SDEs Chap. 11 534
Yn+1 = Yn +1
2
(
a (τn+1, Yn+1) + a)
∆
+
m∑
j=1
bj ∆W j +1
2
m∑
j=1
L0bj ∆W j ∆
+1
2
m∑
j1,j2=1
Lj1bj2(
∆W j1∆W j2 + Vj1,j2
)
c© Copyright E. Platen NS of SDEs Chap. 11 535
Implicit Weak Order 2.0 Scheme
m = 1
Platen (1995)
• implicit weak order 2.0 scheme
Yn+1 = Yn +1
2(a+ a (Yn+1))∆
+1
4
(
b(
Υ+)
+ b(
Υ−)
+ 2 b)
∆W
+1
4
(
b(
Υ+)
− b(
Υ−))
(
(
∆W)2
− ∆
)
∆− 12
c© Copyright E. Platen NS of SDEs Chap. 11 536
with supporting values
Υ± = Yn + a∆ ± b√∆
∆W can be chosen as before
c© Copyright E. Platen NS of SDEs Chap. 11 537
• implicit weak order 2.0 scheme
d,m ∈ 1, 2, . . .
c© Copyright E. Platen NS of SDEs Chap. 11 538
Yn+1 = Yn +1
2(a+ a (Yn+1))∆
+1
4
m∑
j=1
[
bj(
Rj+
)
+ bj(
Rj−
)
+ 2bj
+
m∑
r=1r 6=j
(
bj(
Ur+
)
+ bj(
Ur−)− 2bj
)
∆− 12
]
∆W j
+1
4
m∑
j=1
[
(
bj(
Rj+
)
− bj(
Rj−
)
) (
(
∆W j)2
− ∆
)
+
m∑
r=1r 6=j
(
bj(
Ur+
)− bj(
Ur−)
)
(
∆W j∆W r + Vr,j
)
]
∆− 12
c© Copyright E. Platen NS of SDEs Chap. 11 539
supporting values
Rj± = Yn + a∆ ± bj
√∆
and
U j± = Yn ± bj
√∆
A-stable
β = 2.0
c© Copyright E. Platen NS of SDEs Chap. 11 540
Weak Order 1.0 Predictor-Corrector Methods
• modified trapezoidal method of weak orderβ = 1.0
corrector
Yn+1 = Yn +1
2
(
a(
Yn+1
)
+ a)
∆ + b∆W
predictor
Yn+1 = Yn + a∆ + b∆W
P(
∆W = ±√∆)
=1
2
c© Copyright E. Platen NS of SDEs Chap. 11 541
• family of weak order 1.0 predictor-corrector methods
corrector
Yn+1 = Yn +(
α aη
(
τn+1, Yn+1
)
+ (1 − α) aη (τn, Yn))
∆
+
m∑
j=1
(
η bj(
τn+1, Yn+1
)
+ (1 − η) bj (τn, Yn))
∆W j
for α, η ∈ [0, 1], where
aη = a− η
m∑
j1,j2=1
d∑
k=1
bk,j1∂bj2
∂xk
c© Copyright E. Platen NS of SDEs Chap. 11 542
and predictor
Yn+1 = Yn + a∆ +
m∑
j=1
bj ∆W j
c© Copyright E. Platen NS of SDEs Chap. 11 543
Weak Order 2.0 Predictor-Corrector Methods
• weak order 2.0 predictor-corrector method
corrector
Yn+1 = Yn +1
2
(
a(
Yn+1
)
+ a)
∆ + Ψn
with
Ψn = b∆W +1
2b b′
(
(
∆W)2
− ∆
)
+1
2
(
a b′ +1
2b2 b′′
)
∆W ∆
c© Copyright E. Platen NS of SDEs Chap. 11 544
and as predictor
Yn+1 = Yn + a∆ + Ψn
+1
2a′ b∆W ∆ +
1
2
(
a a′ +1
2a′′b2
)
∆2
P(
∆W = ±√3∆)
=1
6and P
(
∆W = 0)
=2
3
c© Copyright E. Platen NS of SDEs Chap. 11 545
• general multi-dimensional case
corrector
Yn+1 = Yn +1
2
(
a(
τn+1, Yn+1
)
+ a)
∆ + Ψn
where
Ψn =
m∑
j=1
(
bj +1
2L0bj∆
)
∆W j
+1
2
m∑
j1,j2=1
Lj1bj2(
∆W j1 ∆W j2 + Vj1,j2
)
and predictor
Yn+1 = Yn + a∆ + Ψn +1
2L0a∆2 +
1
2
m∑
j=1
Lja∆W j ∆
c© Copyright E. Platen NS of SDEs Chap. 11 546
• derivative free weak order 2.0 predictor-corrector method
corrector
Yn+1 = Yn +1
2
(
a(
Yn+1
)
+ a)
∆ + φn
where
φn =1
4
(
b(
Υ+)
+ b(
Υ−)
+ 2 b)
∆W
+1
4
(
b(
Υ+)
− b(
Υ−))
(
(
∆W)2
− ∆
)
∆− 12
with supporting values
Υ± = Yn + a∆ ± b√∆
c© Copyright E. Platen NS of SDEs Chap. 11 547
and with predictor
Yn+1 = Yn +1
2
(
a(
Υ)
+ a)
∆ + φn
with the supporting value
Υ = Yn + a∆ + b∆W
c© Copyright E. Platen NS of SDEs Chap. 11 548
• multi-dimensional generalization
corrector
Yn+1 = Yn +1
2
(
a(
Yn+1
)
+ a)
∆ + φn
c© Copyright E. Platen NS of SDEs Chap. 11 549
where
φn =1
4
m∑
j=1
[
bj(
Rj+
)
+ b(
Rj−
)
+ 2 bj
+m∑
r=1r 6=j
(
bj(
Ur+
)
+ b(
Ur−)− 2 bj
)
∆− 12
]
∆W j
+1
4
m∑
j=1
[
(
bj(
Rj+
)
− b(
Rj−
))
(
(
∆W)2
− ∆
)
+
m∑
r=1r 6=j
(
bj(
Ur+
)− b(
Ur−)
)(
∆W j ∆W r + Vr,j
)
]
∆− 12
c© Copyright E. Platen NS of SDEs Chap. 11 550
supporting values
Rj± = Yn + a∆ ± bj
√∆ and U j
± = Yn ± bj√∆
c© Copyright E. Platen NS of SDEs Chap. 11 551
predictor
Yn+1 = Yn +1
2
(
a(
Υ)
+ a)
∆ + φn
supporting value
Υ = Yn + a∆ +
m∑
j=1
bj ∆W j
local error
Zn+1 = Yn+1 − Yn+1
c© Copyright E. Platen NS of SDEs Chap. 11 552
Weak Approximation with Jumps
Platen (1982a)
Mikulevicius & Platen (1988)
Kubilius & Platen (2002)
Bruti-Liberati & Platen (2006)
Jump Diffusion
• SDE
Xt = x+
∫ t
0
a(Xs) ds+
∫ t
0
b(Xs) dWs
+
∫ t
0
∫
Ec(v,Xs) qϕ(dv, ds)
c© Copyright E. Platen NS of SDEs Chap. 11 553
mark set E = ℜ\0
• jump martingale measure
qϕ(dv, ds) = pϕ(dv, ds) − ϕ(dv) ds
intensity measureϕ(dv)ds
ϕ(E) < ∞
c© Copyright E. Platen NS of SDEs Chap. 11 554
• simulation of functionals
u(s, y) = E(
g(Xs,yT )
∣
∣As
)
c© Copyright E. Platen NS of SDEs Chap. 11 555
• Kolmogorov backward equation
∂
∂su(s, y) +
d∑
i=1
ai(y)∂
∂yiu(s, y)
+1
2
d∑
i,r=1
m∑
j=1
bi,j(y) br,j(y)∂2
∂yi ∂yru(s, y)
+
∫
E
(
u(s, y + c(v, y)) − u(s, y)
−d∑
i=1
ci(v, y)∂
∂yiu(s, y)
)
ϕ(dv) = 0
c© Copyright E. Platen NS of SDEs Chap. 11 556
for all (s, y) ∈ (0, T ) × ℜd with
u(T, y) = g(y)
for y ∈ ℜd
c© Copyright E. Platen NS of SDEs Chap. 11 557
Jump Adapted Time Discretization
• absolute weak error criterion
µ(∆) =∣
∣E(g(X0,yT )) − E(g(YT ))
∣
∣ ≤ K∆β
• Poisson process
pϕ = pϕ(E, [0, t]), t ∈ [0, T ]
finite intensity ϕ(E) < ∞
generates a sequence of jump times
c© Copyright E. Platen NS of SDEs Chap. 11 558
• jump adapted time discretization with maximum
step size∆ > 0
0 = τ0 < τ1 < . . . < τnT = T
including all jump times ofpϕ
nt = maxn ∈ 0, 1, . . . : tn ≤ t
maxn∈1,...,nT
(τn − τn−1) ≤ ∆
c© Copyright E. Platen NS of SDEs Chap. 11 559
Jump Adapted Weak Euler Scheme
Yτn+1− = Yτn + a(Yτn) (τn+1 − τn) + b(Yτn) (Wτn+1−Wτn)
−∫
Ec(v, Yτn)ϕ(du) (τn+1 − τn),
Yτn+1= Yτn+1− +
∫
Ec(v, Yτn+1−) pϕ(dv, τn+1)
for n ∈ 0, 1, . . . , nT − 1 andY0 = x
β = 1
simplified weak Euler scheme can be used
to approximate diffusion part
c© Copyright E. Platen NS of SDEs Chap. 11 560
Second Order Weak Schemes
d = m = 1
Yτn+1− = Yτn + a(Yτn) (τn+1 − τn) + b(Yτn)∆W
+ b(Yτn) b′(Yτn)
1
2
(
(∆W )2 − (τn+1 − τn))
+ b(Yτn) a′(Yτn)∆Z
+
(
a(Yτn) b′(Yτn) +
1
2b(Yτn)
2 b′′(Yτn)
)
× (∆W (τn+1 − τn) − ∆Z)
+
(
a(Yτn) a′(Yτn) +
1
2b(Yτn)
2 a′′(Yτn)
)
(τn+1 − τn)2
2
Yτn+1= Yτn+1− +
∫
Ec(v, Yτn+1−) pϕ(dv, τn+1)
c© Copyright E. Platen NS of SDEs Chap. 11 561
for n ∈ 0, 1, . . . , nT − 1
with Y0 = x, a = a− ∫
E c dϕ
∆W =
∫ τn+1
τn
dWs ∼ N(0, τn+1 − τn)
∆Z =
∫ τn+1
τn
∫ s2
τn
dWs1 ds2 ∼ N
(
0,(τn+1 − τn)
3
3
)
E(∆W ∆Z) =(τn+1 − τn)
2
2
c© Copyright E. Platen NS of SDEs Chap. 11 562
Jump Adapted Weak Taylor Approximations
• Wagner-Platen expansion
hierarchical set
Aβ = α ∈ Mm : l(α) ≤ β
c© Copyright E. Platen NS of SDEs Chap. 11 563
• jump adapted weak Taylor approximation of order β
Yτn+1− =∑
α∈Aβ
Iα(fα(Yτn))τn,τn+1
Yτn+1= Yτn+1− +
∫
Ec(v, Yτn+1−) pϕ(dv, τn+1)
n ∈ 0, 1, . . . , nT − 1, with Y0 = x
f(x) = x
c© Copyright E. Platen NS of SDEs Chap. 11 564
Theorem (Mikulevicius-Platen)
Let be given a functiong ∈ C2(β+1)P (ℜd,ℜ), a time discretizationτnn∈0,1,...
with maximum step size∆ > 0 and a corresponding jump adapted weak Taylor ap-
proximationY of orderβ.
(i) a, b andc are2(β+1)-times continuously differentiable, where the derivatives
are uniformly bounded.
(ii) For all α ∈ Mm with l(α) ≤ β andy ∈ ℜd it holds|fα(y)| ≤ K (1 +
|y|).Then the jump adapted weak Taylor approximationY of order β converges with
weak orderβ, which means that
|E(g(XT )) − E(g(YT ))| ≤ C∆β,
whereC is a constant not depending on∆.
c© Copyright E. Platen NS of SDEs Chap. 11 565
Exercises of Chapter 11
11.1 Verify that the two point distributed random variable∆W with
P (∆W = ±√∆) =
1
2
satisfies the moment conditions∣
∣
∣E(
∆W)∣
∣
∣+
∣
∣
∣
∣
E
(
(
∆W)3)∣
∣
∣
∣
+
∣
∣
∣
∣
E
(
(
∆W)2)
− ∆
∣
∣
∣
∣
≤ K∆2.
11.2 Prove that the three point distributed random variable∆W with
P(
∆W = ±√3∆
)
=1
6and P
(
∆W = 0)
=2
3
satisfies the moment conditions∣
∣
∣E(
∆W)∣
∣
∣+
∣
∣
∣
∣
E
(
(
∆W)3)∣
∣
∣
∣
+
∣
∣
∣
∣
E
(
(
∆W)5)∣
∣
∣
∣
+
+
∣
∣
∣
∣
E
(
(
∆W)2)
− ∆
∣
∣
∣
∣
+
∣
∣
∣
∣
E
(
(
∆W)4)
− 3∆2
∣
∣
∣
∣
≤ K∆3.
c© Copyright E. Platen NS of SDEs Chap. 11 566
11.3 For the Black-Scholes model with SDE
dXt = µXt dt+ σXt dWt
for t ∈ [0, T ] with X0 = 0 andW a standard Wiener process write down
a simplified Euler scheme. Which weak order of convergence does this scheme
achieve ?
11.4 For the BS model in Exercise 11.3 describe a simplified weak order 2.0 method.
11.5 For the BS model in Exercise 11.3 construct a drift implicit Euler scheme.
11.6 For the BS model in Exercise 11.3 describe a fully implicit Euler scheme.
c© Copyright E. Platen NS of SDEs Chap. 11 567
14 Numerical Stability
• roundoff and truncation errors
• propagation of errors
• numerical stability priority over higher order
c© Copyright E. Platen NS of SDEs Chap. 14 568
• specially designed test equations
Hernandez & Spigler (1992, 1993)
Milstein (1995)
Kloeden& Pl. (1999)
Saito & Mitsui(1993a, 1993b, 1996)
Hofmann& Pl. (1994, 1996)
Higham (2000)
c© Copyright E. Platen NS of SDEs Chap. 14 569
• 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
c© Copyright E. Platen NS of SDEs Chap. 14 570
• 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
c© Copyright E. Platen NS of SDEs Chap. 14 571
Definition 14.1 Y = Yt, t ≥ 0 is calledasymptotically stableif
P(
limt→∞
|Yt| = 0)
= 1.
impact of perturbations declines asymptotically over time
c© Copyright E. Platen NS of SDEs Chap. 14 572
• stability region Γ
those pairs(λ∆, α) ∈ (−∞, 0) × [0, 1) for which approximationY
asymptotically stable
c© Copyright E. Platen NS of SDEs Chap. 14 573
• transfer function
∣
∣
∣
∣
Yn+1
Yn
∣
∣
∣
∣
= Gn+1(λ∆, α)
Y asymptotically stable ⇐⇒
E(ln(Gn+1(λ∆, α))) < 0
Higham (2000)
c© Copyright E. Platen NS of SDEs Chap. 14 574
• Euler scheme
Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn
Gn+1(λ∆, α) =
∣
∣
∣
∣
1 +
(
1 − 3
2α
)
λ∆ +√
|αλ|∆Wn
∣
∣
∣
∣
∆Wn ∼ N (0,∆)
c© Copyright E. Platen NS of SDEs Chap. 14 575
Figure 14.1: A-stability region for the Euler scheme.
c© Copyright E. Platen NS of SDEs Chap. 14 576
• 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
∣
∣
∣
∣
c© Copyright E. Platen NS of SDEs Chap. 14 577
Figure 14.2: A-stability region for semi-drift-implicit predictor-corrector Eu-
ler method.
c© Copyright E. Platen NS of SDEs Chap. 14 578
• drift-implicit predictor-corrector Euler method
Yn+1 = Yn + a(Yn+1)∆ + b(Yn)∆Wn
Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn
Gn+1(λ∆, α) =
∣
∣
∣
∣
1 + λ∆
(
1 − 3
2α
)
1 + λ∆
(
1 − 3
2α
)
+√
−αλ∆Wn
+√
−αλ∆Wn
∣
∣
∣
∣
c© Copyright E. Platen NS of SDEs Chap. 14 579
Figure 14.3: A-stability region for drift-implicit predictor-corrector Euler
method.
c© Copyright E. Platen NS of SDEs Chap. 14 580
• semi-implicit diffusion predictor-corrector Euler metho d
Yn+1 = Yn + a 12(Yn)∆ +
1
2
(
b(Yn+1) + b(Yn))
∆Wn
Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn
Gn+1(λ∆, α) =
∣
∣
∣
∣
1 + λ∆(1 − α) +√
−αλ∆Wn
×
1 +1
2
(
λ∆
(
1 − 3
2α
)
+√
−αλ∆Wn
)∣
∣
∣
∣
c© Copyright E. Platen NS of SDEs Chap. 14 581
Figure 14.4: A-stability region for the predictor-corrector Euler method with
θ = 0 andη = 12
.
c© Copyright E. Platen NS of SDEs Chap. 14 582
• symmetric predictor-corrector Euler method
Yn+1 = Yn+1
2
(
a 12(Yn+1) + a 1
2(Yn)
)
∆+1
2
(
b(Yn+1) + b(Yn))
∆Wn
Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn
Gn+1(λ∆, α) =
∣
∣
∣
∣
1 + λ∆(1 − α)
1 +1
2
(
λ∆
(
1 − 3
2α
)
+√
−αλ∆Wn
)
+√
−αλ∆Wn
1 +1
2
(
λ∆
(
1 − 3
2α
)
+√
−αλ∆Wn
)∣
∣
∣
∣
c© Copyright E. Platen NS of SDEs Chap. 14 583
Figure 14.5: A-stability region for the symmetric predictor-corrector Euler
method.
c© Copyright E. Platen NS of SDEs Chap. 14 584
Gn+1(λ∆, α) =
∣
∣
∣
∣
1 + λ∆
(
1 − 1
2α
)
1 + λ∆
(
1 − 3
2α
)
+√
−αλ∆Wn
+√
−αλ∆Wn
1 + λ∆
(
1 − 3
2α
)
+√
−αλ∆Wn
∣
∣
∣
∣
=
∣
∣
∣
∣
1 +
1 + λ∆
(
1 − 3
2α
)
+√
−αλ∆Wn
×
λ∆
(
1 − 1
2α
)
+√
−αλ∆Wn
∣
∣
∣
∣
c© Copyright E. Platen NS of SDEs Chap. 14 585
Figure 14.6: A-stability region for fully implicit predictor-corrector Euler
method.
c© Copyright E. Platen NS of SDEs Chap. 14 586
0
5
10
15
20
25
30
35
40
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
time
exact solutionpredictor corrector
Figure 14.7: Exact solution and approximate solution generated by the sym-
metric predictor-corrector Euler scheme.c© Copyright E. Platen NS of SDEs Chap. 14 587
p-Stability
Pl. & Shi (2008)
Definition 14.2 For p > 0 a processY = Yt, t > 0 is calledp-stableif
limt→∞
E(|Yt|p) = 0.
Forα ∈ [0, 11+p/2
) andλ < 0 test SDE isp-stable.
c© Copyright E. Platen NS of SDEs Chap. 14 588
• Stability region those triplets(λ∆, α, p) for which Y is p-stable.
c© Copyright E. Platen NS of SDEs Chap. 14 589
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
c© Copyright E. Platen NS of SDEs Chap. 14 590
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 14.8: Stability region for the Euler scheme.
c© Copyright E. Platen NS of SDEs Chap. 14 591
Figure 14.9: Paths of exact solution, Euler scheme with∆ = 0.2 and Euler
scheme with∆ = 5.
c© Copyright E. Platen NS of SDEs Chap. 14 592
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 14.10: Stability region for semi-drift-implicit predictor-corrector Eu-
ler method.
c© Copyright E. Platen NS of SDEs Chap. 14 593
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 14.11: Stability region for drift-implicit predictor-corrector Euler
method.
c© Copyright E. Platen NS of SDEs Chap. 14 594
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 14.12: Stability region for the predictor-correctorEuler method with
θ = 0 andη = 12
.
c© Copyright E. Platen NS of SDEs Chap. 14 595
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 14.13: Stability region for the symmetric predictor-corrector Euler
method.
c© Copyright E. Platen NS of SDEs Chap. 14 596
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 14.14: Stability region for fully implicit predictor-corrector Euler
method.
c© Copyright E. Platen NS of SDEs Chap. 14 597
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
c© Copyright E. Platen NS of SDEs Chap. 14 598
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 14.15: Stability region for semi-drift implicit Euler method.
c© Copyright E. Platen NS of SDEs Chap. 14 599
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 14.16: Stability region for full-drift implicit Euler method.
c© Copyright E. Platen NS of SDEs Chap. 14 600
• balanced implicit Euler method
Milstein, Pl.& Schurz (1998)
Yn+1 = Yn+
(
1 − 3
2α
)
λYn∆+√
α|λ|Yn∆Wn+c|∆Wn|(Yn−Yn+1)
c© Copyright E. Platen NS of SDEs Chap. 14 601
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 14.17: Stability region for a balanced implicit Eulermethod.
c© Copyright E. Platen NS of SDEs Chap. 14 602
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 14.18: Stability region for the simplified symmetric Euler method.
c© Copyright E. Platen NS of SDEs Chap. 14 603
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 14.19: Stability region for the simplified symmetric implicit Euler
Scheme.
c© Copyright E. Platen NS of SDEs Chap. 14 604
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 14.20: Stability region for the simplified fully implicit Euler Scheme.
c© Copyright E. Platen NS of SDEs Chap. 14 605
16 Variance Reduction Techniques
Various Variance Reduction Methods
• classical
Hammersley & Handscomb (1964)
Ermakov (1975), Boyle (1977)
Maltz & Hitzl (1979), Rubinstein (1981)
Ermakov & Mikhailov (1982), Ripley (1983)
Kalos & Whitlock (1986), Bratley, Fox & Schrage (1987)
Chang (1987),Wagner (1987)
Law & Kelton (1991), Ross (1990)
c© Copyright E. Platen NS of SDEs Chap. 16 632
• stochastic differential equations
Boyle (1977)
Boyle, Broadie & Glasserman (1997)
Broadie & Glasserman (1997b), Fu (1995)
Grant, Vora & Weeks (1997), Joy, Boyle & Tan (1996)
Glasserman (2004)
Longstaff & Schwartz (2001)
Milstein (1988), Kloeden & Platen (1999)
Hofmann, Platen & Schweizer (1992), Heath (1995)
Goldman, Heath, Kentwell & Platen (1995)
Fournie, Lasry & Touzi (1997)
Newton (1994)
c© Copyright E. Platen NS of SDEs Chap. 16 633
Antithetic Variates
• probability space
(Ω,AT ,A, P ) whereΩ = C([t0, T ],ℜ2)
ω(t) = (ω1(t), ω2(t))⊤ ∈ ℜ2 for t ∈ [t0, T ]
• coordinate mappings
W 1t (ω) = ω1(t) andW 2
t (ω) = ω2(t)
dXt0,xt = a
(
t,Xt0,xt
)
dt+ b(
t,Xt0,xt
)
dWt
c© Copyright E. Platen NS of SDEs Chap. 16 634
Forω ∈ Ω, defineω ∈ Ω by ω(t) = (−ω1(t),−ω2(t))⊤, t ∈ [t0, T ]
h(
Xt0,xT
)
(ω) = h(
Xt0,xT
)
(ω)
• unbiased estimator
h(
Xt0,xT
)
=1
2
(
h(
Xt0,xT
)
+ h(
Xt0,xT
))
=⇒
Var(
h(
Xt0,xT
))
=1
4
(
Var(
h(
Xt0,xT
))
+ Var(
h(
Xt0,xT
))
+ 2 Cov(
h(
Xt0,xT
)
, h(
Xt0,xT
)))
c© Copyright E. Platen NS of SDEs Chap. 16 635
Stratified Sampling
• set of events
Ai ⊆ AT , i ∈ 1, 2, . . . , N
N⋃
i=1
Ai = Ω, Ai ∩Aj = ∅
for i, j ∈ 1, 2, . . . , N, P (Ai) = 1N
A = σ Ai, i ∈ 1, 2, . . . , N
c© Copyright E. Platen NS of SDEs Chap. 16 636
• random variable
Z : Ω → ℜ
• restriction ofZ toAi
ZAi(ω) = Z(ω) for ω ∈ Ai
c© Copyright E. Platen NS of SDEs Chap. 16 637
• unbiased estimator forE(Z)
Z =1
N
N∑
i=1
ZAi
whereZAi independent
E(Z) =1
N
N∑
i=1
E(ZAi) =
N∑
i=1
∫
Ai
Z dP =
∫
Ω
Z dP = E(Z)
independence
=⇒
c© Copyright E. Platen NS of SDEs Chap. 16 638
Var(Z) =
N∑
i=1
Var(ZAi)
N2
=1
NE(Var(Z
∣
∣A))
≤ 1
NVar(Z)
c© Copyright E. Platen NS of SDEs Chap. 16 639
Example: simplified weak Euler approximationY ∆
∆Wk two-point distributed
two-point variates withN time steps
underlying sample space
=⇒ΩN = −1, 10,1,...,N−1
‘path’ for Wk
given by∆Wk(ω) = ωk
√∆
probabilitiesPN(ω) = 12N
only a tiny fraction of these paths can be sampled
c© Copyright E. Platen NS of SDEs Chap. 16 640
A stratified Monte Carlo estimation could consist of exhausting all pathsω ∈ ΩN
up to timetN′ and then sampling randomly;
reduces duplicate traversals of the early nodes of the lattice.
c© Copyright E. Platen NS of SDEs Chap. 16 641
Measure Transformation Method
see Kloeden & Platen (1999)
• d-dimensional diffusion
dXs,xt = a(t,Xs,x
t ) dt+
m∑
j=1
bj(t,Xs,xt ) dW j
t
on(Ω,AT ,A, P )
• approximate the functional
u(s, x) = E(
g(Xs,xT )
∣
∣As
)
c© Copyright E. Platen NS of SDEs Chap. 16 642
• Kolmogorov backward equation
L0 u(s, x) = 0
for (s, x) ∈ (0, T ) × ℜd with
u(T, y) = g(y)
L0 =∂
∂s+
d∑
k=1
ak ∂
∂xk+
1
2
d∑
k,ℓ=1
m∑
j=1
bk,j bℓ,j∂2
∂xk ∂xℓ
c© Copyright E. Platen NS of SDEs Chap. 16 643
• Girsanov transformation
W jt = W j
t −∫ t
0
dj(
z, X0,xz
)
dz
dj denotes given, rather flexible real-valued function
• Radon-Nikodym derivative
dP
dP=
Θt
Θ0
c© Copyright E. Platen NS of SDEs Chap. 16 644
• SDE
dX0,xt = a
(
t, X0,xt
)
dt+
m∑
j=1
bj(
t, X0,xt
)
dW jt
=
a(
t, X0,xt
)
−m∑
j=1
bj(
t, X0,xt
)
dj(
t, X0,xt
)
dt
+m∑
j=1
bj(
t, X0,xt
)
dW jt
• Radon-Nikodym derivative process
Θt = Θ0 +
m∑
j=1
∫ t
0
Θz dj(
z, X0,xz
)
dW jz
(A, P )-martingale
c© Copyright E. Platen NS of SDEs Chap. 16 645
• diffusion processX0,x with respect toP
same drift and diffusion coefficients asXs,x
=⇒
E(
g(
X0,xT
))
=
∫
Ω
g(
X0,xT
)
dP
=
∫
Ω
g(
X0,xT
)
dP
=
∫
Ω
g(
X0,xT
) ΘT
Θ0dP
= E
(
g(
X0,xT
) ΘT
Θ0
)
c© Copyright E. Platen NS of SDEs Chap. 16 646
• unbiased estimator
g(
X0,xT
) ΘT
Θ0
no particular choice ofdj made so far
c© Copyright E. Platen NS of SDEs Chap. 16 647
• ideally choosedj in the form
dj(t, x) = − 1
u(t, x)
d∑
k=1
bk,j(t, x)∂u(t, x)
∂xk
then it can be shown that
u(
t, X0,xt
)
Θt = u(0, x)Θ0
for all t ∈ [0, T ]
=⇒u(0, x) = g
(
X0,xT
) ΘT
Θ0
c© Copyright E. Platen NS of SDEs Chap. 16 648
=⇒
g(
X0,xT
) ΘT
Θ0
is not random
c© Copyright E. Platen NS of SDEs Chap. 16 649
• guess a functionu
dj(t, x) = − 1
u(t, x)
d∑
k=1
bk,j(t, x)∂u(t, x)
∂xk
• used unbiased estimator
g(
X0,xT
) ΘT
Θ0
E
(
g(
X0,xT
) ΘT
Θ0
)
= E(
g(
X0,xT
))
c© Copyright E. Platen NS of SDEs Chap. 16 650
Control Variates and Integral Representations
Clewlow & Carverhill (1992, 1994)
Basic Control Variate Method
• valuation martingale
Mt = u(
t,Xt0,xt
)
= E(
h(
Xt0,xT
) ∣
∣
∣ At
)
construct an accurate and fast estimate of
E(h(Xt0,xT )) = u(t0, x)
c© Copyright E. Platen NS of SDEs Chap. 16 651
• control variate
findY with known meanE(Y )
• unbiased estimator
Z = h(Xt0,xT ) − α(Y − E(Y ))
α ∈ ℜ
E(Z) = E(h(Xt0,xT ))
c© Copyright E. Platen NS of SDEs Chap. 16 652
• known valuation function
u(
t, Xt0,xt
)
= E(
h(
Xt0,xT
) ∣
∣
∣ At
)
Xt0,x approximatesXt0,xT
• unbiased estimator
ZT = h(
Xt0,xT
)
− α(
h(
Xt0,xT
)
− E(
h(
Xt0,xT
)))
= u(
T,Xt0,xT
)
− α(
u(
T, Xt0,xT
)
− u(t0, x))
variance can be reduced
c© Copyright E. Platen NS of SDEs Chap. 16 653
Var(ZT ) = Var(
h(
Xt0,xT
))
+ α2 Var(
h(
Xt0,xT
))
− 2αCov(
h(
Xt0,xT
)
, h(
Xt0,xT
))
minimize the variance
αmin =Cov
(
h(
Xt0,xT
)
, h(
Xt0,xT
))
Var(
h(
Xt0,xT
))
c© Copyright E. Platen NS of SDEs Chap. 16 654
Example: Stochastic Volatility
• SDE
dSt = σt St dW1t
dσt = (κ− σt) dt+ ξ σt dW2t
(Ω,AT ,A, P )
• European call payoff
h(ST ) = (ST −K)+
c© Copyright E. Platen NS of SDEs Chap. 16 655
• adjusted price process
dSt = σtSt dW1t
dσt = (κ− σt) dt
• adjusted valuation function
u(t, St, σt) = E(
(ST −K)+∣
∣
∣At
)
• unbiased variate
ZT = (ST −K)+ − α((ST −K)+ − E(ST −K)+)
= (ST −K)+ − α((ST −K)+ − u(t0, s, σ))
unbiased variance reduced estimator
c© Copyright E. Platen NS of SDEs Chap. 16 656
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
• first exit time
τ = inft ≥ s : (t,Xs,xt
) 6∈ [s, T ) × Γ
c© Copyright E. Platen NS of SDEs Chap. 16 657
• operators
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
and
Lj f(t, x) =
d∑
i=1
bi,j(t, x)∂f(t, x)
∂xi
for (t, x) ∈ (0, T ) × Γ
c© Copyright E. Platen NS of SDEs Chap. 16 658
• valuation function
u(t, x) = E(
h(τ,Xt,xτ )
)
for (t, x) ∈ [0, T ] × Γ
assume
Mt = E(
h(
τ,X0,xτ
) ∣
∣At
)
square integrable(A, P )-martingale
c© Copyright E. Platen NS of SDEs Chap. 16 659
martingale representation theorem
=⇒
Mt = u(t,X0,xt∧τ )
= u(0, x) +
m∑
j=1
∫ t∧τ
0
ξjs dWjs
for t ∈ [0, T ]
c© Copyright E. Platen NS of SDEs Chap. 16 660
• given an approximation
u : [0, T ] × Γ → ℜ to u
u ∈ C1,2([0, T ] × Γ)
with
M jt =
∫ t∧τ
0
Lj u(s,X0,xs ) dW j
s
square integrable(A, P )-martingale
c© Copyright E. Platen NS of SDEs Chap. 16 661
u(τ,X0,xτ ) = u(τ,X0,x
τ ) = h(τ,X0,xτ )
=⇒
u(τ,X0,xτ ) = u(0, x) +
∫ τ
0
L0 u(t,X0,xt ) dt
+m∑
j=1
∫ τ
0
Lj u(t,X0,xt ) dW j
t
c© Copyright E. Platen NS of SDEs Chap. 16 662
=⇒
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(
1t<τL0 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
c© Copyright E. Platen NS of SDEs Chap. 16 663
Variance of the HP Estimator
Zτ = u(τ,X0,xτ ) −
m∑
j=1
∫ τ
0
Lj u(t,X0,xt ) dW j
t
= u(τ,X0,xτ ) −
m∑
j=1
∫ τ
0
Lj u(t,X0,xt ) dW j
t
= u(0, x) +
m∑
j=1
∫ τ
0
(
ξjt − Lj u(t,X0,xt )
)
dW jt
=⇒
c© Copyright E. Platen NS of SDEs Chap. 16 664
Var(Zτ ) = E
m∑
j=1
∫ τ
0
(
ξjt − Lj u(t,X0,xt )
)
dW jt
2
=
m∑
j=1
∫ T
0
E
(
1t<τ(
ξjt − Lj u(t,X0,xt )
)2)
dt
c© Copyright E. Platen NS of SDEs Chap. 16 665
• integrands
ξjt = Lj u(t,X0,xt )
=⇒
Var(Zτ ) =
m∑
j=1
∫ T
0
E
(
1t<τ(
(Lj u− Lj u) (t,X0,xt )
)2)
dt
if a good approximationu tou can be found,
so thatLj u is close toLj u
variance will be small
c© Copyright E. Platen NS of SDEs Chap. 16 666
• unbiased estimator
Zτ,α = u(τ,X0,xτ ) − α
m∑
j=1
∫ τ
0
Lj u(t,X0,xt ) dW j
t
= Zτ + (1 − α)
m∑
j=1
∫ τ
0
Lj u(t,X0,xt ) dW j
t
c© Copyright E. Platen NS of SDEs Chap. 16 667
An Example for the Heston Model
• SDEs
dSs,x1
t = uSs,x1
t dt+
√
vs,x2
t Ss,x1
t dW 1t
dvs,x2
t = κ(
θ − vs,x2
t
)
dt
+ ξ
√
vs,x2
t
(
dW 1t +
√
1 − 2 dW 2t
)
c© Copyright E. Platen NS of SDEs Chap. 16 668
• equivalent risk neutral martingale measure
dSs,x1
t = r Ss,x1
t dt+
√
vs,x2
t Ss,x1
t dW 1t
dvs,x2
t = κ(
θ − vs,x2
t
)
dt
+ ξ
√
vs,x2
t
(
dW 1t +
√
1 − 2 dW 2t
)
for t ∈ [s, T ] ands ∈ [0, T ], whereθ = θ − ξ (µ−r)κ
and
dW 1t =
µ− r√vt
dt+ dW 1t
dW 2t = dW 2
t
c© Copyright E. Platen NS of SDEs Chap. 16 669
• option price
c(0, x) = e−rT u(0, x)
where
u(0, x) = E(
(S0,x1
T −K)+)
• approximation
dSs,x1
t = r Ss,x1
t dt+
√
vs,x2
t Ss,x1
t dW 1t
dvs,x2
t = κ(
θ − vs,x2
t
)
dt
explicitly computed
vs,x2
t = θ + (x2 − θ) e−κ(t−s)
c© Copyright E. Platen NS of SDEs Chap. 16 670
Black-Scholes price
u(t, x) = E(
(St,x1
T −K)+)
= er(T−t)BS(x1,K, r, σt, T − t)
where
σt =
√
1
T − t
∫ T
t
vt,x2
z dz
=
√
θ − (x2 − θ)e−κ(T−t) − 1
κ(T − t)
c© Copyright E. Platen NS of SDEs Chap. 16 671
=⇒
(L0 − L0) f(t, x) = ξ x2
(
∂2f(t, x)
∂x1 ∂x2+
1
2ξ∂2f(t, x)
∂(x2)2
)
(L0 − L0) u(t, x) = ξ x2 er(T−t)
[
∂2BS(x1,K, r, σt, T − t)
∂x1 ∂σt
∂σt
∂x2
+1
2ξ
∂2BS(x1,K, r, σt, T − t)
∂σ2t
(
∂σt
∂x2
)2
+∂BS(x1,K, r, σt, T − t)
∂σt
∂2σt
∂(x2)2
]
∂BS
∂σt,
∂2BS
∂x1 ∂σt,
∂2BS
∂σ2t
,∂σt
∂x2and
∂2σt
∂(x2)2
can be computed
c© Copyright E. Platen NS of SDEs Chap. 16 672
Monte Carlo Simulation
0 = τ0 < τ1 < . . . < τN = T, ∆ =T
N
• predictor-corrector method of weak order 1.0
Yn+1 = Yn +(
α a(τn+1, Yn+1) + (1 − α) a(τn, Yn))
∆
+
m∑
j=1
(
η bj(τn+1, Yn+1) + (1 − η) bj(τn, Yn))
∆W jn
for n ∈ 0, 1, . . . , N−1 with predictor
Yn+1 = Yn + a(τn, Yn)∆ +
m∑
j=1
bj ∆W jn
c© Copyright E. Platen NS of SDEs Chap. 16 673
and modified drift coefficient values
a(τn, Yn) = a(τn, Yn) − ηd∑
i=1
m∑
j=1
bi,j(τn, Yn)∂bj(τn, Yn)
∂xi
α, η ∈ [0, 1] and∆W jn
Gaussian random two-point distributed random variables
Simulation Results
• raw Monte Carlo
intrinsic value(S0,x1
t −K)+
c© Copyright E. Platen NS of SDEs Chap. 16 674
0 0.1 0.2 0.3 0.4 0.5
10
20
30
40
50
Figure 16.1: Simulated outcomes for the intrinsic value(S0,x1
t −K)+, t ∈[0, T ].
c© Copyright E. Platen NS of SDEs Chap. 16 675
0 0.1 0.2 0.3 0.4 0.5
6.6
6.65
6.7
6.75
Figure 16.2: Simulated outcomes for the estimatorZt, t ∈ [0, T ].
c© Copyright E. Platen NS of SDEs Chap. 16 676
80
90
100
110
120
S0.005
0.01
0.015
0.02
0.025
v
-0.75-0.5
-0.25
0
0.25
80
90
100
110S
Figure 16.3: Diffusion operator values(L − L0) u as a function of asset
priceS and squared volatilityv.
c© Copyright E. Platen NS of SDEs Chap. 16 677
80
100
120K
0
0.1
0.2
0.3
0.4
t
-0.1-0.05
0
0.05
80
100
120K
Figure 16.4: Price differences between the Heston and corresponding Black-
Scholes model as a function of strikeK and timet.
c© Copyright E. Platen NS of SDEs Chap. 16 678
80 90 100 110 120 130
-0.15
-0.1
-0.05
0
0.05
Figure 16.5: Prices and corresponding error bounds as a function of strike
K.
c© Copyright E. Platen NS of SDEs Chap. 16 679
80
100
120K
0
0.1
0.2
0.3
0.4
t
0.2
0.21
0.22
80
100
120K
Figure 16.6: Implied volatility term structure for the Heston model.
c© Copyright E. Platen NS of SDEs Chap. 16 680
Exercises of Chapter 16
16.1 Construct an antithetic variance reduction method, where you have to simulate
the expectationE(1 + Z + 12(Z)2) for a standard Gaussian random variable
Z ∼ N(0, 1). For this purpose combine the raw Monte Carlo estimate
V +N =
1
N
N∑
k=1
(
1 + Z(ωk) +1
2(Z(ωk))
2
)
that uses outcomes ofZ with the antithetic one
V −N =
1
N
N∑
k=1
(
1 − Z(ωk) +1
2(−Z(ωk))
2
)
that uses in parallel the same outcomes with a negative sign.Demonstrate the
variance reduction that can be achieved for the estimator
VN =1
2(V +
N + V −N )
instead of using onlyV +N . Is VN an unbiased estimator?
c© Copyright E. Platen NS of SDEs Chap. 16 681
16.2 For calculating the expectationE((1 + Z + 12(Z)2)) use the control variate
V ∗N =
1
N
N∑
k=1
(1 + Z(ωk)) .
Analyze the variance reduction that can be achieved by the estimate
VN = V +N + α(γ − V ∗
N)
for α ∈ ℜ. For which choice ofγ ∈ ℜ one obtains an unbiased estimate? For
whichα ∈ ℜ one achieves the minimum variance ?
16.3 Check for the measure transformation variance reduction method andg(z) =
z2 thatg(X0,xT ) θT
θ0is non-random and equals
u(t, x) = E(
g(
X0,xT
) ∣
∣At
)
for t = 0, assuming the linear SDEs
dX0,xT = αX0,x
T dt+ βX0,xT dWt
c© Copyright E. Platen NS of SDEs Chap. 16 682
and
dX0,xT = α X0,x
T dt+ β X0,xT dWt
for t ∈ [0, T ] withX0,xT = X0,x
T = x0, where
dWt = dWt − d(t, X0,xt ) dt
and
dθt = θt d(t, X0,xt ) dWt
with
d(t, X0,xt ) = − β X0,x
t
u(t, X0,xt )
∂u(t, X0,xt )
∂x.
c© Copyright E. Platen NS of SDEs Chap. 16 683
16.4 Formulate a drift implicit simplified weak Euler schemefor the two-factor Hes-
ton model
dXt = (rd − rf)Xt + k√vtXt dW
1t
dvt = κ (vt − v) dt+ p√vt(
dW 1t +
√
1 − 2 dW 2t
)
,
for t ∈ [0, T ] with t ∈ [0, T ∗]X0 > 0 andv0 > 0, whereW 1 andW 2
are independent standard Wiener processes.
16.5 Consider the Heston model of Exercise 16.4 for = 0. Make also the diffusion
term in theX component of the simplified weak Euler scheme implicit and
correct appropriately the drift term in the resulting scheme.
c© Copyright E. Platen NS of SDEs Chap. 16 684
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