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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).
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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
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deterministic chain rule
=⇒d
dtf(Xt) = a(Xt)
∂
∂xf(Xt)
• operator
L = a∂
∂x
c© Copyright E. Platen NS of SDEs Chap. 4 108
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=⇒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
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=⇒
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
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• 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
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• 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
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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
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• 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
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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
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• 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
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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
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• 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
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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
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• 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
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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
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• 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
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+
∫ 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
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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
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+∂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
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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
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• 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
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• 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
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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
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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
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• 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
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∆ 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
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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
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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
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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
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• jth fraction
πjδ(t) = δjt
S(j)t
Sδt
,
d∑
j=0
πjδ(t) = 1
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• 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
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• 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
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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
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• 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
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• 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
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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
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(i) - natural negative sign for positive movements
(ii) - subadditivity
(iii) - positive homogeneity
(iv) - translation invariance
• there exist also convex risk measures
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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
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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
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• 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
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-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
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• 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
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• 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
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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
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• 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
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• 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
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• 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
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with variancec2 ∆
For Studentt distributed multivariate log-returns
ξ is Studentt distributed
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• 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
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• 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
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• 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
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• 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
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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
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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
)
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≈(
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
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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
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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
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• 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
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• 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
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• 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
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Multiple It o Integrals
Let andτ be two stopping times
0 ≤ ≤ τ ≤ T
multi-indexα= (j1, j2, . . ., jℓ) ∈ Mm
f ∈ Hα
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• 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
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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
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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 !
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• 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
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=⇒ 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
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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, . . .
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• 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 )
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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
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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
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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 ].
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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
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• 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′′
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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
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• Stratonovich coefficient function
fα
=
f for l = 0
Lj1f−αfor l ≥ 1
for α= (j1, . . ., jℓ)
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• 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)
,
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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
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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
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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)
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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 .
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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:
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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
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+
[
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
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Examples for Wagner Platen Expansions
• Vasicek interest rate model
drt = γ (r − rt) dt+ β dWt
for t ∈ [0, T ], r0 ≥ 0
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• 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
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• 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
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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
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• 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
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• 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
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• 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
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• 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
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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
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• 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
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Moments of Multiple It o Integrals
First Moments
α ∈ Mm \ v with ℓ(α) 6= n(α)
E(
Iα [f·],τ
∣
∣
∣A
)
= 0
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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
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• 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
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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.
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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).
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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.
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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
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• 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
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maximum step size
∆ = maxn∈0,1,...,N−1
∆n
• equidistant time discretization
τn = n∆
∆n ≡ ∆ = TN
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• 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
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Gaussian pseudo-random numbers generated
• abbreviation
f = f(τn, Yn)
Euler scheme
Yn+1 = Yn + a∆n + b∆Wn,
discrete time approximations
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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
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• linear interpolation
Yt = Ynt +t− τnt
τnt+1 − τnt
(Ynt+1 − Ynt)
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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
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• appreciation rate a
• volatility b 6= 0
• explicit solution
Xt = X0 exp
((
a− 1
2b2)
t+ bWt
)
• Wiener process
W = Wt, t ∈ [0, T ]
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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
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• 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 ?
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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
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Strong Approximation
• not specified acriterion for classification
• scenario simulation
approximates paths
testing ofcalibration methods
statisticalestimators
filtering
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• Monte-Carlo simulation
approximatesprobabilities
functionals
simulatesexpectations
moments
pricesof contingent claims
or risk measuresas Value at Risk
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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).
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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
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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
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• 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 ]
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• 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
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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
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• 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
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• 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
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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
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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∆.
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A Simulation Example
• Black-Scholes SDE
dXt = µXt dt+ σXt dWt
• exact solution
XT = X0 exp
(
µ− σ2
2
)
T + σWT
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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(∆)
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-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(∆).
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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
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• 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
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• 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)
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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
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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
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• 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
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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
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• 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
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=⇒
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
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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 )
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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
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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
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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
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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
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Convergence Theorem
E(|X0|2
)
< ∞
E
(
∣
∣
∣X0 − Y ∆0
∣
∣
∣
2) 1
2
≤ K1 ∆12
c© Copyright E. Platen NS of SDEs Chap. 5 247
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|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
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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
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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
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• linear regression
ln(ε(∆)) = −4.91021 + 0.95 ln(∆)
c© Copyright E. Platen NS of SDEs Chap. 5 251
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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
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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
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• 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
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• 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
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• 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
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• 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
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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
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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
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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
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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
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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
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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
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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
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• 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
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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
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• 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
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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
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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
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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√∆
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• 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√∆
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• 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
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• 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
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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.
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• 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
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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
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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
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• 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
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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
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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.
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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
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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
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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
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• 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
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• test equation
dxt
dt= λxt
with λ ∈ ℜ
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numerical scheme in recursive form
Yn+1 = G(λ∆)Yn
• A-stability region:
thoseλ∆ ∈ ℜ for which
|G(λ∆)| < 1
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• 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∆|
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-2 -1 0
-1
-0.5
0.5
1
Figure 5.7: Region ofA-stability for the deterministic Euler scheme.
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• 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
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StochasticA-Stability
• test equation
dXt = λXt dt+ dWt
λ ∈ ℜ
c© Copyright E. Platen NS of SDEs Chap. 5 290
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• 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
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• A-stability region
real numbersλ∆ for which
|GA(λ∆)| < 1
If A-stability region covers left half of the real axis
=⇒ thenA-stable
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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
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• 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
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• 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
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• 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∆)
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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
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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
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• 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
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• 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
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• 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
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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
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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√∆
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• 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
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• 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
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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
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• 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
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• 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
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• low order strong convergence
price to pay for numerical stability
• family of specific methods providing a balance between approximating diffusion
terms
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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| = +∞
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• 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
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• explicit solution
XT = exp
σWT − σ2
2T
X0
• absolute error
εT (∆) = E(|XT − YN |)
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Figure 5.8: Estimated absolute errorεT (∆) of the Euler method at timeT
for time step sizes∆ = 2−4 and2−5.
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Figure 5.9: Absolute error for the partially implicit method.
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Figure 5.10: Absolute error for the balanced implicit method.
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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)
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where
Cn = c0(τn, Yn)∆ +
m∑
j=1
cj(τn, Yn) |∆W jn|
c0, c1, . . . , cm
d× d-matrix-valued uniformly bounded functions
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• 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 < ∞
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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∆.
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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
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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
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• 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)
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• 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
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• for ξ1 = −1
default of the stock
• for ξk = δ − 1, k ∈ 1, 2, . . . with δ ∈ [0, 1]
δ recovery rate of a stock
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• 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
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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
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Random Jump Size
• Poisson jump measurepℓϕℓ
(·, ·)
• mark set E = ℜ\0
ϕℓ(E) < ∞
ℓ ∈ 1, 2, . . . , d
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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
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• 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
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• 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)
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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)
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• 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
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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
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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
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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
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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
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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
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• if noise is commutative =⇒ scheme simplifies
Platen (1982a)
γ = 1.0
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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.
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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.
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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
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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.
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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
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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
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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
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Y ∆T (ω1), Y
∆T (ω2), . . . , Y
∆T (ωN)
ωk ∈ Ω for k ∈ 1, 2, . . . , N
• discrete time weak approximationY ∆T
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• 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
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• 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
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• 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
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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
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• 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
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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
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• 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
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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
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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
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-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
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-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
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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
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• 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
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• 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
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-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
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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
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• 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
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∣
∣
∣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
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• 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
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• 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
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• 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
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• 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
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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
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• 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
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∆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
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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
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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
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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
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• multi-dimensional case
d,m ∈ 1, 2, . . .
additive noise
weak order3.0 Taylor scheme
c© Copyright E. Platen NS of SDEs Chap. 11 497
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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
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∆W j and∆Zj
independent pairs of
correlated Gaussian random variables
c© Copyright E. Platen NS of SDEs Chap. 11 499
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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
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• 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
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+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
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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
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-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
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• SDE with multiplicative noise
dXt = aXt dt+ bXt dWt
g(X) = x
c© Copyright E. Platen NS of SDEs Chap. 11 505
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-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
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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
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• 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
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• 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
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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
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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
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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
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• 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
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+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
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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
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• 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
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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
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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
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• 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
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+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
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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
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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
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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
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• 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
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• 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
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-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
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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
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-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
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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
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• 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
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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
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• 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
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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
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• 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
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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
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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
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with supporting values
Υ± = Yn + a∆ ± b√∆
∆W can be chosen as before
c© Copyright E. Platen NS of SDEs Chap. 11 537
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• implicit weak order 2.0 scheme
d,m ∈ 1, 2, . . .
c© Copyright E. Platen NS of SDEs Chap. 11 538
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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
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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
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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
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• 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
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and predictor
Yn+1 = Yn + a∆ +
m∑
j=1
bj ∆W j
c© Copyright E. Platen NS of SDEs Chap. 11 543
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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
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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
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• 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
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• 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
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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
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• multi-dimensional generalization
corrector
Yn+1 = Yn +1
2
(
a(
Yn+1
)
+ a)
∆ + φn
c© Copyright E. Platen NS of SDEs Chap. 11 549
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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
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supporting values
Rj± = Yn + a∆ ± bj
√∆ and U j
± = Yn ± bj√∆
c© Copyright E. Platen NS of SDEs Chap. 11 551
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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
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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
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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
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• simulation of functionals
u(s, y) = E(
g(Xs,yT )
∣
∣As
)
c© Copyright E. Platen NS of SDEs Chap. 11 555
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• 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
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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
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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
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• 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
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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
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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
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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
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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
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• 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
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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
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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
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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
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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
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• 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
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• 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
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• 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
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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
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• 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
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• 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
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• 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
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Figure 14.1: A-stability region for the Euler scheme.
c© Copyright E. Platen NS of SDEs Chap. 14 576
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• 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
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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
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• 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
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Figure 14.3: A-stability region for drift-implicit predictor-corrector Euler
method.
c© Copyright E. Platen NS of SDEs Chap. 14 580
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• 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
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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
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• 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
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Figure 14.5: A-stability region for the symmetric predictor-corrector Euler
method.
c© Copyright E. Platen NS of SDEs Chap. 14 584
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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
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Figure 14.6: A-stability region for fully implicit predictor-corrector Euler
method.
c© Copyright E. Platen NS of SDEs Chap. 14 586
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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
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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
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• Stability region those triplets(λ∆, α, p) for which Y is p-stable.
c© Copyright E. Platen NS of SDEs Chap. 14 589
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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• 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
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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
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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
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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
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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
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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
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• 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
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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
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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
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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
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• 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
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• 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
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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
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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
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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
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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
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• 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
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• 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
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• 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
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• 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
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• unbiased estimator
g(
X0,xT
) ΘT
Θ0
no particular choice ofdj made so far
c© Copyright E. Platen NS of SDEs Chap. 16 647
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• 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
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=⇒
g(
X0,xT
) ΘT
Θ0
is not random
c© Copyright E. Platen NS of SDEs Chap. 16 649
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• 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
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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
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• 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
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• 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
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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
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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
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• 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
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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
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• 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
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• 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
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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
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• 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
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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
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=⇒
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
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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
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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
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• 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
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• 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
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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
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• 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
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• 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
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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
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=⇒
(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
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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
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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
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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
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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
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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
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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
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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
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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
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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?
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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
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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
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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.
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Andersen, L. (2008). Simple and efficient simulation of the Heston stochastic volatility model.J. Comput. Finance11(3), 1–42.
Artzner, P., F. Delbaen, J. M. Eber, & D. . Heath (1997). Thinking coherently.Risk10, 68–71.
Barndorff-Nielsen, O. (1998). Processes of normal inverse Gaussian type.Finance Stoch.2(1), 41–68.
Barrett, R., M. Berry, T. F. Chan, J. Demmel, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, & H. van der Vorst (1994).Templates for the Solution ofLinear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia.
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Beskos, A., O. Papaspiliopoulos, & G. Roberts (2007). Monte-Carlo maximum likelihood estimation for discretely observed diffusion processes.University of Warwick and Lanchester University (working paper).
Bibby, B. M., M. Jacobsen, & M. Sørensen (2003). Estimating functions for discretely sampled diffusion-type models. In Y. Ait-Sahalia and L. P.Hansen (Eds.),Handbook of Financial Economics, Chapter 4, pp. 203–268. Amsterdam: North Holland.
Bjork, T., Y. Kabanov, & W. J. Runggaldier (1997). Bond market structure in the presence of marked point processes.Math. Finance7, 211–239.
Boyle, P. P. (1977). A Monte Carlo approach.J. Financial Economics4, 323–338.
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Boyle, P. P., M. Broadie, & P. Glasserman (1997). Monte Carlo methods for security pricing. Computational financial modelling.J. Econom. Dynam.Control 21(8-9), 1267–1321.
Brandt, M. & P. Santa-Clara (2002). Simulated likelihood estimation of diffusions with application to exchange rate dynamics in incomplete markets.J. Financial Economics63(2), 161–210.
Bratley, P., B. L. Fox, & L. Schrage (1987).A Guide to Simulation(2nd ed.). Springer.
Broadie, M. & J. Detemple (1997). The valuation of American options on multiple assets.Math. Finance7(3), 241–286.
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