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Numerical Solution of Stochastic Differential Equations with Jumps in Finance

Eckhard Platen

School of Finance and Economics and School of Mathematical Sciences University of Technology, Sydney Kloeden, P.E. & Pl, E.: Numerical Solutions of Stochastic Differential Equations Springer, Applications of Mathematics 23 (1992,1995,1999). Pl, E. & Heath, D.: A Benchmark Approach to Quantitative Finance, Springer Finance (2006). Bruti-Liberati, N. & Pl, E.: Numerical Solutions of Stochastic Differential Equations with Jumps Springer, Applications of Mathematics (2008).

1 23

Springer Finance

A Benchmark Approach to Quantitative Finance

1

A Benchmark Approach to Quantitative Finance

S F

Platen · Heath

Eckhard Platen David Heath

Dieser Farbausdruck/pdf-file kann nur annähernd das endgültige Druckergebnis wiedergeben !

63575 15.5.06 designandproduction GmbH – Bender

Springer Finance

• E. Platen · D. Heath

The benchmark approach provides a general framework for financial market modeling, which extends beyond the standard risk neutral pricing theory. It allows for a unified treatment of portfolio optimization, derivative pricing, integrated risk management and insurance risk modeling. The existence of an equivalent risk neutral pricing measure is not required. Instead, it leads to pricing formulae with respect to the real world probability measure. This yields important modeling freedom which turns out to be necessary for the derivation

• f realistic, parsimonious market models.

The first part of the book describes the necessary tools from probability theory, statistics, stochastic calculus and the theory of stochastic differential equations with jumps. The second part is devoted to financial modeling under the bench- mark approach. Various quantitative methods for the fair pricing and hedging

• f derivatives are explained. The general framework is used to provide an under-

standing of the nature of stochastic volatility. The book is intended for a wide audience that includes quantitative analysts, postgraduate students and practitioners in finance, economics and insurance. It aims to be a self-contained, accessible but mathematically rigorous introduction to quantitative finance for readers that have a reasonable mathematical or quanti- tative background. Finally, the book should stimulate interest in the benchmark approach by describing some of its power and wide applicability.

• ISBN 3-540-26212-1

› springer.com

Jump-Diffusion Multi-Factor Models

Bj¨

• rk, Kabanov & Runggaldier (1997)
• continuous time
• Markovian
• explicit transition densities in special cases
• benchmark framework
• discrete time approximations
• suitable for simulation
• Markov chain approximations

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Pathwise Approximations:

• scenario simulation of entire markets
• testing statistical techniques on simulated trajectories
• filtering hidden state variables
• Pl. & Runggaldier (2005, 2007)
• hedge simulation
• dynamic financial analysis
• extreme value simulation
• stress testing

= ⇒ higher order strong schemes predictor-corrector methods

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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,

• r Markov chain approximations, lattice methods

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Essential Requirements:

• parsimonious models
• respect no-arbitrage in discrete time approximation
• numerically stable methods
• efficient methods for high-dimensional models
• higher order schemes, predictor-corrector

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Continuous and Event Driven Risk

• Wiener processes

W k, k ∈ {1, 2, . . ., m}

• counting processes

pk intensity hk jump martingale qk dW m+k

t

= dqk

t =

• dpk

t − hk t dt

hk

t

− 1

2

k ∈ {1, 2, . . . , d−m} W t = (W 1

t , . . . , W m t , q1 t , . . . , qd−m t

)⊤

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Primary Security Accounts

dSj

t = Sj t−

• aj

t dt + d

• k=1

bj,k

t

dW k

t

• Assumption 1

bj,k

t

≥ −

• hk−m

t

k ∈ {m + 1, . . . , d}. Assumption 2 Generalized volatility matrix bt = [bj,k

t

]d

j,k=1 invertible.

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• market price of risk

θt = (θ1

t , . . . , θd t )⊤ = b−1 t

[at − rt 1]

• primary security account

dSj

t = Sj t−

• rt dt +

d

• k=1

bj,k

t

(θk

t dt + dW k t )

• portfolio

dSδ

t = d

• j=0

δj

t dSj t

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• fraction

πj

δ,t = δj t

Sj

t

Sδ

t

• portfolio

dSδ

t = Sδ t−

• rt dt + π⊤

δ,t− bt (θt dt + dW t)

• Eckhard Platen

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Assumption 3

• hk−m

t

> θk

t

• generalized GOP volatility

ck

t =

     θk

t

for k ∈ {1, 2, . . . , m}

θk

t

1−θk

t (hk−m t

)− 1

2

for k ∈ {m + 1, . . . , d}

• GOP fractions

πδ∗,t = (π1

δ∗,t, . . . , πd δ∗,t)⊤ =

• c⊤

t b−1 t

⊤

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• Growth Optimal Portfolio

dSδ∗

t

= Sδ∗

t−

• rt dt + c⊤

t (θt dt + dW t)

• optimal growth rate

gδ∗

t

= rt + 1 2

m

• k=1

(θk

t )2

−

d

• k=m+1

hk−m

t

 ln  1 + θk

t

• hk−m

t

− θk

t

  + θk

t

• hk−m

t

 

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• benchmarked portfolio

ˆ Sδ

t = Sδ t

Sδ∗

t

Theorem 4 Any nonnegative benchmarked portfolio ˆ Sδ is an (A, P )-supermartingale. = ⇒ no strong arbitrage but there may exist: free lunch with vanishing risk (Delbaen & Schachermayer (2006)) free snacks or cheap thrills (Loewenstein & Willard (2000))

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Multi-Factor Model

model mainly:

• benchmarked primary security accounts

ˆ Sj

t = Sj t

Sδ∗

t

j ∈ {0, 1, . . . , d} supermartingales, often SDE driftless, local martingales, sometimes martingales

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savings account S0

t = exp

t rs ds

• =

⇒ GOP Sδ∗

t

= S0

t

ˆ S0

t

= ⇒ stock Sj

t = ˆ

Sj

t Sδ∗ t

additionally dividend rates foreign interest rates

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Example

Black-Scholes Type Market d ˆ Sj

t = − ˆ

Sj

t− d

• k=1

σj,k

t

dW k

t

hj

t, σj,k t

, rt

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Examples

• Merton jump-diffusion model

dXt = Xt− (µ dt + σ dWt + dpt) , ⇓ Xt = X0 e(µ− 1

2 σ2)t+σWt

Nt

• i=1

ξi

• Bates model

dSt = St−

• α dt +
• Vt dW S

t + dpt

• dVt = ξ(η − Vt) dt + θ
• Vt dW V

t

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0.5 1 1.5 2 2.5 3 5 10 15 20 time

Figure 1: Simulated benchmarked primary security accounts.

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1 2 3 4 5 6 7 8 9 10 5 10 15 20 time

Figure 2: Simulated primary security accounts.

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0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 15 20 time GOP EWI

Figure 3: Simulated GOP and EWI for d = 50.

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0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 15 20 time GOP index

Figure 4: Simulated accumulation index and GOP.

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Diversification

• diversified portfolios
• πj

δ,t

• ≤

K2 d

1 2 +K1 Eckhard Platen Bressanone07

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Theorem 5 In a regular market any diversified portfolio is an approximate GOP.

• Pl. (2005)
• robust characterization
• similar to Central Limit Theorem
• model independent

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10 20 30 40 50 60 5 9 14 18 23 27 32

Figure 5: Benchmarked primary security accounts.

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50 100 150 200 250 300 350 400 450 5 9 14 18 23 27 32

Figure 6: Primary security accounts under the MMM.

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10 20 30 40 50 60 70 80 90 100 5 9 14 18 23 27 32 EWI GOP

Figure 7: GOP and EWI.

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10 20 30 40 50 60 70 80 90 100 5 9 14 18 23 27 32

Market index GOP

Figure 8: GOP and market index.

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• fair security

benchmarked security (A, P )-martingale ⇐ ⇒ fair

• minimal replicating portfolio

fair nonnegative portfolio Sδ with Sδ

τ = Hτ

= ⇒ minimal nonnegative replicating portfolio

• fair pricing formula

VHτ (t) = Sδ∗

t

E Hτ Sδ∗

τ

• At
• No need for equivalent risk neutral probability measure!

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Fair Hedging

• fair portfolio

Sδ

t

• benchmarked fair portfolio

ˆ Sδ

t = E

Hτ Sδ∗

τ

• At
• martingale representation

Hτ Sδ∗

τ

= E Hτ Sδ∗

τ

• At
• +

d

• k=1

τ

t

xk

Hτ (s) dW k s + MHτ (t)

MHτ -(A, P )-martingale (pooled) E

• MHτ , W k

t

• = 0

F¨

• llmer & Schweizer (1991)

No need for equivalent risk neutral probability measure!

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Simulation of SDEs with Jumps

• strong schemes (paths)

Taylor explicit derivative-free implicit balanced implicit predictor-corrector

• weak schemes (probabilities)

Taylor simplified explicit derivative-free implicit, predictor-corrector

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• intensity of jump process

– regular schemes = ⇒ high intensity – jump-adapted schemes = ⇒ low intensity

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

X0 ∈ ℜd

• pt = Nt:

Poisson process, intensity λ < ∞

• pt = Nt

i=1(ξi − 1):

compound Poisson, ξi i.i.d r.v.

• Poisson random measure
• E

c(t−, Xt−, v) pφ(dv × dt)

• {(τi, ξi), i = 1, 2, . . . , NT }

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Numerical Schemes

• time discretization

tn = n∆

• discrete time approximation

Y ∆

n+1 = Y ∆ n + a(Y ∆ n )∆ + b(Y ∆ n )∆Wn + c(Y ∆ n )∆pn

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Strong Convergence

• Applications: scenario analysis, filtering and hedge simulation
• strong order γ if

εs(∆) =

• E
• XT − Y ∆

N

• 2

≤ K ∆γ

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Weak Convergence

• Applications: derivative pricing, utilities, risk measures
• weak order β if

εw(∆) = |E(g(XT )) − E(g(Y ∆

N ))| ≤ K∆β

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Literature on Strong Schemes with Jumps

• Pl (1982), Mikulevicius & Pl (1988)

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

• Maghsoodi (1996, 1998)

= ⇒ strong schemes γ ≤ 1.5

• Jacod & Protter (1998)

= ⇒ Euler scheme for semimartingales

• Gardo`

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

• Higham & Kloeden (2005)

= ⇒ implicit Euler scheme

• Bruti-Liberati & Pl (2005)

= ⇒ γ ∈ {0.5, 1, . . .} explicit, implicit, derivative-free, predictor-corrector

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Euler Scheme

• Euler scheme

Yn+1 = Yn + a(Yn)∆ + b(Yn)∆Wn + c(Yn)∆pn where ∆Wn ∼ N (0, ∆) and ∆pn = Ntn+1−Ntn ∼ P oiss(λ ∆)

• γ = 0.5

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Strong Taylor Scheme

Wagner-Platen expansion = ⇒

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) = 1

2{(∆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 times τi :

Wτi = ⇒ I(1,−1) and I(−1,1)

• Computational effort heavily dependent on intensity λ

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Derivative-Free Strong Schemes avoid computation of derivatives

⇓

• rder 1.0 derivative-free strong scheme

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Implicit Strong Schemes wide stability regions

⇓

implicit Euler scheme

• rder 1.0 implicit strong Taylor scheme

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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 +

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

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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 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

• γ = 0.5

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

0.2 0.4 0.6 0.8 1

T

0.2 0.4 0.6 0.8 1

X

Figure 9: Plot of a jump-diffusion path.

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0.2 0.4 0.6 0.8 1

T

• 0.00125
• 0.001
• 0.00075
• 0.0005
• 0.00025

0.00025 0.0005

Error

Figure 10: Plot of the strong error for Euler(red) and 1.0 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

Log2dt

• 25
• 20
• 15
• 10

Log2Error

15TaylorJA 1TaylorJA 1Taylor EulerJA Euler

Figure 11: Log-log plot of strong error versus time step size.

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Literature on Weak Schemes with Jumps

• Mikulevicius & Pl (1991)

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

• Liu & Li (2000)

= ⇒

• rder β ∈ {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 (?)

= ⇒ jump-adapted order β ∈ {1, 2 . . .} derivative-free, implicit and predictor-corrector schemes

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Simplified Euler Scheme

• Euler scheme

= ⇒ β = 1

• simplified Euler scheme

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

• if ∆ ˆ

Wn and ∆ˆ pn match the first 3 moments of ∆Wn and ∆pn up to an O(∆2) error = ⇒ β = 1

• P (∆ ˜

Wn = ± √ ∆) = 1 2

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Jump-Adapted Taylor Approximations

• jump-adapted Euler scheme

= ⇒ β = 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

• ∆2

tn +

• a b′ + 1

2b′′ b2

• {∆Wtn ∆tn − ∆Ztn}

and Ytn+1 = Ytn+1− + c(Ytn+1−) ∆pn

• β = 2

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Predictor-Corrector Schemes

• predictor-corrector

= ⇒ 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

• β = 1

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• 5
• 4
• 3
• 2
• 1

Log2dt

• 2
• 1

1 2 3

Log2Error

PredCorrJA ImplEulerJA EulerJA

Figure 12: Log-log plot of weak error versus time step size.

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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

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Conclusions

• low intensity

= ⇒ jump-adapted higher order predictor-corrector

• high intensity

= ⇒ regular schemes

• distinction between strong and weak predictor-corrector schemes

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References

Bj¨

• rk, T., Y. Kabanov, & W. Runggaldier (1997). Bond market structure in the presence of marked point
• processes. Math. Finance 7, 211–239.

Bruti-Liberati, N. & E. Platen (2005). On the strong approximation of jump-diffusion processes. Technical report, University of Technology, Sydney. QFRC Research Paper 157. Delbaen, F. & W. Schachermayer (2006). The Mathematics of Arbitrage. Springer Finance. Springer. F¨

• llmer, H. & M. Schweizer (1991). Hedging of contingent claims under incomplete information. In
• M. H. A. Davis and R. J. Elliott (Eds.), Applied Stochastic Analysis, Volume 5 of Stochastics

Monogr., pp. 389–414. Gordon and Breach, London/New York. Gardo` n, A. (2004). The order of approximations for solutions of Itˆ

• -type stochastic differential equations

with jumps. Stochastic Analysis and Applications 22(3), 679–699. Glasserman, P. & N. Merener (2003). Numerical solution of jump-diffusion LIBOR market models. Fi- nance Stoch. 7(1), 1–27. Higham, D. & P. Kloeden (2005). Numerical methods for nonlinear stochastic differential equations with

• jumps. Numer. Math. 110(1), 101–119.

Jacod, J. & P. Protter (1998). Asymptotic error distribution for the Euler method for stochastic differential

• equations. Ann. Probab. 26(1), 267–307.

Kubilius, K. & E. Platen (2002). Rate of weak convergence of the Euler approximation for diffusion processes with jumps. Monte Carlo Methods Appl. 8(1), 83–96. Liu, X. Q. & C. W. Li (2000). Weak approximation and extrapolations of stochastic differential equations

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with jumps. SIAM J. Numer. Anal. 37(6), 1747–1767. Loewenstein, M. & G. A. Willard (2000). Local martingales, arbitrage, and viability: Free snacks and cheap thrills. Econometric Theory 16(1), 135–161. Maghsoodi, Y. (1996). Mean-square efficient numerical solution of jump-diffusion stochastic differential

• equations. SANKHYA A 58(1), 25–47.

Maghsoodi, Y. (1998). Exact solutions and doubly efficient approximations of jump-diffusion Itˆ

• equa-
• tions. Stochastic Anal. Appl. 16(6), 1049–1072.

Mikulevicius, R. & E. Platen (1988). Time discrete Taylor approximations for Ito processes with jump

• component. Math. Nachr. 138, 93–104.

Mikulevicius, R. & E. Platen (1991). Rate of convergence of the Euler approximation for diffusion pro-

• cesses. Math. Nachr. 151, 233–239.

Platen, E. (1982). An approximation method for a class of Itˆ

• processes with jump component. Liet. Mat.
• Rink. 22(2), 124–136.

Platen, E. (2005). Diversified portfolios with jumps in a benchmark framework. Asia-Pacific Financial Markets 11(1), 1–22. Platen, E. & W. J. Runggaldier (2005). A benchmark approach to filtering in finance. Asia-Pacific Finan- cial Markets 11(1), 79–105. Platen, E. & W. J. Runggaldier (2007). A benchmark approach to portfolio optimization under partial

• information. Technical report, University of Technology, Sydney. QFRC Research Paper 191.

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