Numerical Solution of Stochastic Differential Equations ... · Numerical Solution of Stochastic...
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Numerical Solution of Stochastic DifferentialEquations with Jumps in Finance
Eckhard PlatenSchool of Finance and Economics and School of Mathematical Sciences
University of Technology, Sydney
Kloeden, P.E.& Pl, E.: Numerical Solution of Stochastic Differential Equations
Springer, Applications of Mathematics23 (1992,1995,1999).
Pl, E.& Heath, D.: A Benchmark Approach to Quantitative Finance,Springer Finance (2010).
Pl, E.& Bruti-Niberati, N.: Numerical Solution of SDEs with Jumps in Finance,
Springer, Stochastic Modelling and Applied Probability64 (2010).
Jump-Diffusion Multi-Factor Models
Bjork, Kabanov & Runggaldier (1997)
Øksendal & Sulem (2005)
• Markovian
• explicit transition densities in special cases
• benchmark framework
• discrete time approximations
• suitable for simulation
• Markov chain approximations
c© Copyright E. Platen SDE Jump MC 1
Pathwise Approximations:
• scenario simulation of entire markets
• testing statistical techniques on simulated trajectories
• filtering hidden state variablesPl. & Runggaldier (2005, 2007)
• hedge simulation
• dynamic financial analysis
• extreme value simulation
• stress testing
=⇒ higher order strong schemespredictor-corrector methods
c© Copyright E. Platen SDE Jump MC 2
Strong Convergence
• Applications: scenario analysis, filtering and hedge simulation
• strong order γ if
εs(∆) =
√
E(
∣
∣XT − Y ∆N
∣
∣
2)
≤ K∆γ
c© Copyright E. Platen SDE Jump MC 3
Probability Approximations:
• derivative prices
• sensitivities
• expected utilities
• portfolio selection
• risk measures
• long term risk management
=⇒ Monte Carlo simulation, higher order weak schemes,
predictor-corrector,
variance reduction, Quasi Monte Carlo,
or Markov chain approximations, lattice methods
c© Copyright E. Platen SDE Jump MC 4
Weak Convergence
• Applications: derivative pricing, utilities, risk measures
• weak order β if
εw(∆) = |E(g(XT )) − E(g(Y ∆N ))| ≤ K∆β
c© Copyright E. Platen SDE Jump MC 5
Essential Requirements:
• parsimonious models
• long time horizons
• respect no-arbitrage in discrete time approximation
• numerically stable methods
• efficient methods for high-dimensional models
• higher order schemes
c© Copyright E. Platen SDE Jump MC 6
Continuous and Event Driven Risk
• Wiener processes W k, k ∈ {1, 2, . . .,m}
• counting processes pk
intensityhk
jump martingaleqk
dWm+kt = dqkt =
(
dpkt − hkt dt
) (
hkt
)− 12
k ∈ {1, 2, . . . , d−m}
W t = (W 1t , . . . ,W
mt , q
1t , . . . , q
d−mt )⊤
c© Copyright E. Platen SDE Jump MC 7
Primary Security Accounts
dSjt = S
jt−
(
ajt dt+
d∑
k=1
bj,kt dW k
t
)
Assumption 1
bj,kt ≥ −
√
hk−mt
k ∈ {m+ 1, . . . , d}.
Assumption 2
Generalized volatility matrixbt = [bj,kt ]dj,k=1 invertible.
c© Copyright E. Platen SDE Jump MC 8
• market price of risk
θt = (θ1t , . . . , θdt )
⊤ = b−1t [at − rt 1]
• primary security account
dSjt = S
jt−
(
rt dt+
d∑
k=1
bj,kt (θkt dt+ dW k
t )
)
• portfolio
dSδt =
d∑
j=0
δjt dS
jt
c© Copyright E. Platen SDE Jump MC 9
• fraction
πjδ,t = δ
jt
Sjt
Sδt
• portfolio
dSδt = Sδ
t−
{
rt dt+ π⊤δ,t− bt (θt dt+ dW t)
}
c© Copyright E. Platen SDE Jump MC 10
Assumption 3√
hk−mt > θkt
=⇒ numeraire portfolio (NP) exists
• generalized NP volatility
ckt =
θkt for k ∈ {1, 2, . . . ,m}θk
t
1−θk
t(h
k−m
t)−
12
for k ∈ {m+ 1, . . . , d}
• NP fractions
πδ∗,t = (π1δ∗,t, . . . , πd
δ∗,t)⊤ =
(
c⊤t b−1t
)⊤
c© Copyright E. Platen SDE Jump MC 11
• Numeraire Portfolio - Benchmark
dSδ∗
t = Sδ∗
t−
(
rt dt+ c⊤t (θt dt+ dW t))
• optimal growth rate
gδ∗
t = rt +1
2
m∑
k=1
(θkt )2
−d∑
k=m+1
hk−mt
ln
1 +θkt
√
hk−mt − θkt
+θkt
√
hk−mt
c© Copyright E. Platen SDE Jump MC 12
• benchmarked portfolio
Sδt =
Sδt
Sδ∗
t
Theorem 4 Any nonnegative benchmarked portfolioSδ is an
(A, P )-supermartingale, that is,
Sδt ≥ Et(S
δT )
0 ≤ t ≤ T < ∞=⇒ no strong arbitrage
Pl. & Heath (2010)
c© Copyright E. Platen SDE Jump MC 13
Multi-Factor Models
model under benchmark approach mainly:
• benchmarked primary security accounts
Sjt =
Sjt
Sδ∗
t
j ∈ {0, 1, . . . , d}
supermartingales, often SDE driftless,
(local martingales, sometimes martingales)
c© Copyright E. Platen SDE Jump MC 14
model savings account
S0t = exp
{∫ t
0
rs ds
}
=⇒ NP
Sδ∗
t =S0t
S0t
=⇒ stock
Sjt = S
jt S
δ∗
t
model additionally dividend rates
and foreign interest rates
c© Copyright E. Platen SDE Jump MC 15
Example
• benchmarked security:
dSt = St−
(
√
VtdWSt + dqt
)
• squared volatility:
• Bates model
dVt = ξ(η − Vt) dt+ q√
Vt dWVt
•32
model
d1
Vt
= ξ
(
η − 1
Vt
)
dt+ q
√
1
Vt
dWVt
c© Copyright E. Platen SDE Jump MC 16
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20
time
Figure 1: Simulated benchmarked primary security accounts.
c© Copyright E. Platen SDE Jump MC 17
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20
time
Figure 2: Simulated primary security accounts.
c© Copyright E. Platen SDE Jump MC 18
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 5 10 15 20
time
GOPEWI
Figure 3: Naive Diversification: NP and EWI ford = 50.
c© Copyright E. Platen SDE Jump MC 19
0
10
20
30
40
50
60
0 5 9 14 18 23 27 32
Figure 4: Benchmarked primary security accounts MMM.
c© Copyright E. Platen SDE Jump MC 20
0
50
100
150
200
250
300
350
400
450
0 5 9 14 18 23 27 32
Figure 5: Primary security accounts under the MMM.
c© Copyright E. Platen SDE Jump MC 21
0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
EWI
GOP
Figure 6: NP and EWI.
c© Copyright E. Platen SDE Jump MC 22
• fair security
benchmarked security martingale⇐⇒ fair
• minimal replicating portfolio
fair nonnegative portfolioSδ with Sδτ = Hτ
=⇒ minimal price
• real world pricing formula
VHτ(t) = Sδ∗
t Et
(
Hτ
Sδ∗
τ
)
No need for equivalent risk neutral probability measure!
c© Copyright E. Platen SDE Jump MC 23
Fair Hedging
• benchmarked fair portfolio
Sδt = Et
(
Hτ
Sδ∗
τ
)
=⇒ minimal price
• martingale representation
Hτ
Sδ∗
τ
= Et
(
Hτ
Sδ∗
τ
)
+
d∑
k=1
∫ τ
t
xkHτ
(s) dW ks +MHτ
(t)
MHτ- martingale (when pooled⇒ vanishing)
MHτandW k orthogonal
Follmer & Schweizer (1991), Du& Pl. (2011)
c© Copyright E. Platen SDE Jump MC 24
Numerical Solution of SDEs
Kloeden& Pl. (1999)
Milstein (1995)
Kloeden, Pl.& Schurz (2003)
Jackel (2002)
Glasserman (2004)
Pl. & Bruti-Liberati (2010)
• major problem:
propagation of errors during simulation
c© Copyright E. Platen SDE Jump MC 25
Simulation of SDEs with Jumps
• strong schemes(paths)exact simulationTaylor (Wagner-Pl. expansion)explicitderivative-freepredictor-correctorimplicit, balanced implicit
• weak schemes(probabilities)Taylor (Wagner-Pl. expansion)simplifiedexplicitderivative-freepredictor-corrector, implicit
c© Copyright E. Platen SDE Jump MC 26
Jump-Adapted Time Discretization
t0 t1 t2 t3 = T
regular
τ1
r
τ2
r jump times
t0 t1 t2 t3 t4 t5 = T
r r jump-adapted
c© Copyright E. Platen SDE Jump MC 27
• intensity of jump process
– regular schemes =⇒ high intensity
– jump-adapted schemes=⇒ low intensity
c© Copyright E. Platen SDE Jump MC 28
SDE with Jumps
dXt = a(t,Xt)dt+ b(t,Xt)dWt + c(t−, Xt−) dpt
X0 ∈ ℜd
• pt = Nt: Poisson process, intensityλ < ∞• pt =
∑Nt
i=1(ξi − 1): compound Poisson,ξi i.i.d r.v.
• Poisson random measure
c© Copyright E. Platen SDE Jump MC 29
• time discretization
tn = n∆
• discrete time approximation
Y ∆n+1 = Y ∆
n + a(Y ∆n )∆ + b(Y ∆
n )∆Wn + c(Y ∆n )∆pn
c© Copyright E. Platen SDE Jump MC 30
Literature on Strong Schemes with Jumps
• Pl (1982), Mikulevicius& Pl (1988)
=⇒ γ ∈ {0.5, 1, . . .} Taylor schemes and jump-adapted
• Maghsoodi (1996, 1998) =⇒ strong schemesγ ≤ 1.5
• Jacod & Protter (1998) =⇒ Euler scheme for semimartingales
• Gardon (2004) =⇒ γ ∈ {0.5, 1, . . .} strong schemes
• Higham & Kloeden (2005) =⇒ implicit Euler scheme
• Bruti-Liberati& Pl (2007) =⇒ γ ∈ {0.5, 1, . . .}explicit, implicit, derivative-free, predictor-corrector
c© Copyright E. Platen SDE Jump MC 31
Euler Scheme
• Euler scheme
Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn + c(Yn)∆pn
where
∆Wn ∼ N (0,∆) and ∆pn = Ntn+1−Ntn ∼ Poiss(λ∆)
• strong orderγ = 0.5
c© Copyright E. Platen SDE Jump MC 32
Strong Taylor Scheme
Wagner-Platen expansion (strong orderγ = 1.0) =⇒
Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn + c(Yn)∆pn + b(Yn)b′(Yn) I(1,1)
+ b(Yn) c′(Yn) I(1,−1) + {b(Yn + c(Yn)) − b(Yn)} I(−1,1)
+ {c (Yn + c(Yn)) − c(Yn)} I(−1,−1)
with
I(1,1) = 12{(∆Wn)
2 − ∆}, I(−1,−1) =1
2{(∆pn)
2 − ∆pn}
I(1,−1) =∑N(tn+1)
i=N(tn)+1 Wτi− ∆pn Wtn , I(−1,1) = ∆pn ∆Wn − I(1,−1)
• simulation jump timesτi : Wτi=⇒ I(1,−1) andI(−1,1)
• Computational effort heavily dependent on intensityλ
c© Copyright E. Platen SDE Jump MC 33
Derivative-Free Strong Schemes
avoid computation of derivatives
⇓
order1.0 derivative-free strong scheme
c© Copyright E. Platen SDE Jump MC 34
• implicit methods
Talay (1982)
Klauder & Petersen (1985)
Milstein (1988)
Hernandez & Spigler(1992, 1993)
Saito & Mitsui(1993a, 1993b)
Kloeden& Pl. (1992, 1999)
Milstein, Pl.& Schurz (1998)
Higham (2000)
Alcock & Burrage (2006)
• solve algebraic equation
c© Copyright E. Platen SDE Jump MC 35
• ad hoc attempts lead to explosions
• balanced implicit methods
Milstein, Pl.& Schurz (1998)
Alcock & Burrage (2006)
c© Copyright E. Platen SDE Jump MC 36
Implicit Strong Schemes
wide stability regions
⇓
implicit Euler scheme
order1.0 implicit strong Taylor scheme
c© Copyright E. Platen SDE Jump MC 37
Predictor-Corrector Euler Scheme
• corrector
Yn+1 = Yn +(
θ aη(Yn+1) + (1 − θ) aη(Yn))
∆n
+(
η b(Yn+1) + (1 − η) b(Yn))
∆Wn +
p(tn+1)∑
i=p(tn)+1
c (ξi)
aη = a− η b b′
• predictor
Yn+1 = Yn + a(Yn)∆n + b(Yn)∆Wn +
p(tn+1)∑
i=p(tn)+1
c (ξi)
θ, η ∈ [0, 1] degree of implicitness
c© Copyright E. Platen SDE Jump MC 38
Jump-Adapted Strong Approximations
jump-adapted time discretisation⇓
jump times included in time discretisation
• jump-adapted Euler scheme
Ytn+1− = Ytn + a(Ytn)∆tn + b(Ytn)∆Wtn
and
Ytn+1= Ytn+1− + c(Ytn+1−)∆pn
• strong orderγ = 0.5
c© Copyright E. Platen SDE Jump MC 39
Merton SDE :µ = 0.05, σ = 0.2,ψ = −0.2, λ = 10,X0 = 1, T = 1
0 0.2 0.4 0.6 0.8 1T
0
0.2
0.4
0.6
0.8
1X
Figure 7: Plot of a jump-diffusion path.
c© Copyright E. Platen SDE Jump MC 40
0 0.2 0.4 0.6 0.8 1T
-0.00125
-0.001
-0.00075
-0.0005
-0.00025
0
0.00025
0.0005Error
Figure 8: Plot of the strong error for Euler(red) and1.0 Taylor(blue) scheme.
c© Copyright E. Platen SDE Jump MC 41
Merton SDE :µ = −0.05, σ = 0.1, λ = 1,X0 = 1, T = 0.5
-10 -8 -6 -4 -2 0Log2dt
-25
-20
-15
-10
Log 2Error
15TaylorJA
1TaylorJA
1Taylor
EulerJA
Euler
Figure 9: Log-log plot of strong error versus time step size.
c© Copyright E. Platen SDE Jump MC 42
Literature on Weak Schemes with Jumps
• Mikulevicius& Pl (1991)
=⇒ jump-adapted orderβ ∈ {1, 2 . . .} weak schemes
• Liu & Li (2000) =⇒ orderβ ∈ {1, 2 . . .} weak Taylor, extrapo-
lation and simplified schemes
• Kubilius& Pl (2002) and Glasserman & Merener (2003)
=⇒ jump-adapted Euler with weaker assumptions on coefficients
• Bruti-Liberati&Pl (2006) =⇒ jump-adapted orderβ ∈ {1, 2 . . .}derivative-free, implicit and predictor-corrector schemes
c© Copyright E. Platen SDE Jump MC 43
Simplified Euler Scheme
• Euler scheme =⇒ weak orderβ = 1
• simplified Euler scheme
Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn + c(Yn) (ξn − 1)∆pn
• if ∆Wn and∆pn match the first 3 moments of∆Wn and∆pn up to
anO(∆2) error =⇒ weak orderβ = 1
•
P (∆Wn = ±√∆) =
1
2
c© Copyright E. Platen SDE Jump MC 44
Jump-Adapted Taylor Approximations
• jump-adapted Euler scheme =⇒ weak orderβ = 1
• jump-adapted order 2 weak Taylor scheme
Ytn+1− = Ytn + a∆tn + b∆Wtn +b b′
2
(
(∆Wtn)2 − ∆tn
)
+ a′ b∆Ztn
+1
2
(
a a′ +1
2a′ ′b2
)
∆2tn
+
(
a b′ +1
2b′′ b2
)
{∆Wtn ∆tn− ∆Ztn}
and
Ytn+1= Ytn+1− + c(Ytn+1−)∆pn
• weak orderβ = 2 (can be simplified and made derivative free)
c© Copyright E. Platen SDE Jump MC 45
Predictor-Corrector Schemes
• predictor-corrector =⇒ numerical stability and efficiency
• jump-adapted predictor-corrector Euler scheme
Ytn+1− = Ytn +1
2
{
a(Ytn+1−) + a}
∆tn + b∆Wtn
with predictor
Ytn+1− = Ytn + a∆tn + b∆Wtn
• weak orderβ = 1
c© Copyright E. Platen SDE Jump MC 46
-5 -4 -3 -2 -1Log2dt
-2
-1
0
1
2
3
Log2Error
PredCorrJA
ImplEulerJA
EulerJA
Figure 10: Log-log plot of weak error versus time step size.
c© Copyright E. Platen SDE Jump MC 47
Regular Approximations
• higher order schemes : time, Wiener and Poisson multiple integrals
• random jump size difficult to handle
• higher order schemes: computational effort dependent on intensity
c© Copyright E. Platen SDE Jump MC 48
Numerical Stability
• roundoff and truncation errors
• propagation of errors
• numerical stability priority over higher order
c© Copyright E. Platen SDE Jump MC 49
• 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 SDE Jump MC 50
• 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 SDE Jump MC 51
• 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 SDE Jump MC 52
Definition 5 Y = {Yt, t ≥ 0} is calledasymptotically stableif
P(
limt→∞
|Yt| = 0)
= 1.
impact of perturbations declines asymptotically over time
c© Copyright E. Platen SDE Jump MC 53
• stability region Γ
those pairs(λ∆, α) ∈ (−∞, 0)× [0, 1) for which approximationY
asymptotically stable
c© Copyright E. Platen SDE Jump MC 54
• 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 SDE Jump MC 55
• 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 SDE Jump MC 56
Figure 11: A-stability region for the Euler scheme
c© Copyright E. Platen SDE Jump MC 57
• 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 SDE Jump MC 58
Figure 12: -stability region for semi-drift-implicit predictor-corrector Euler
methodc© Copyright E. Platen SDE Jump MC 59
Figure 13: A-stability region for the predictor-correctorEuler method with
θ = 0 andη = 12
c© Copyright E. Platen SDE Jump MC 60
Figure 14: A-stability region for the symmetric predictor-corrector Euler
methodc© Copyright E. Platen SDE Jump MC 61
p-Stability
Pl. & Shi (2008)
Definition 6 For p > 0 a processY = {Yt, t > 0} is calledp-stable
if
limt→∞
E(|Yt|p) = 0.
Forα ∈ [0, 11+p/2
) andλ < 0 test SDE isp-stable.
c© Copyright E. Platen SDE Jump MC 62
• Stability region those triplets(λ∆, α, p) for which Y is p-stable.
c© Copyright E. Platen SDE Jump MC 63
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 SDE Jump MC 64
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 15: Stability region for the Euler scheme
c© Copyright E. Platen SDE Jump MC 65
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 16: Stability region for semi-drift-implicit predictor-corrector Euler
methodc© Copyright E. Platen SDE Jump MC 66
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 17: Stability region for the predictor-corrector Euler method with
θ = 0 andη = 12
c© Copyright E. Platen SDE Jump MC 67
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 SDE Jump MC 68
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 18: Stability region for semi-drift implicit Euler method
c© Copyright E. Platen SDE Jump MC 69
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 19: Stability region for full-drift implicit Euler method
c© Copyright E. Platen SDE Jump MC 70
• 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 SDE Jump MC 71
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 20: Stability region for a balanced implicit Euler method
c© Copyright E. Platen SDE Jump MC 72
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 21: Stability region for the simplified symmetric Euler method
c© Copyright E. Platen SDE Jump MC 73
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 22: Stability region for the simplified symmetric implicit Euler
Schemec© Copyright E. Platen SDE Jump MC 74
00.2
0.40.6
0.81
Α
-10-8-6-4-2
0
ΛD
0
0.5
1
1.5
2
p
Figure 23: Stability region for the simplified fully implicit Euler Scheme
c© Copyright E. Platen SDE Jump MC 75
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
c© Copyright E. Platen SDE Jump MC 76
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
c© Copyright E. Platen SDE Jump MC 77
u(0, x) = E(
h(τ,X0,xτ )
)
= E(
u(τ,X0,xτ )
)
= u(0, x) + E
(∫ τ
0
L0 u(t,X0,xt ) dt
)
= u(0, x) +
∫ T
0
E(
1{t<τ}L0 u(t,X0,x
t ))
dt
• unbiased estimator for u(0, x)
Zτ = u(0, x) +
∫ τ
0
L0 u(t,X0,xt ) dt
HP estimator
c© Copyright E. Platen SDE Jump MC 78
0 0.1 0.2 0.3 0.4 0.5
10
20
30
40
50
Figure 24: Simulated outcomes for the intrinsic value(S0,x1
t −K)+, t ∈[0, T ].
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0 0.1 0.2 0.3 0.4 0.5
6.6
6.65
6.7
6.75
Figure 25: Simulated outcomes for the estimatorZt, t ∈ [0, T ].
c© Copyright E. Platen SDE Jump MC 80
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