Numerical Solution of Stochastic Differential ... - Semantic Scholar
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).
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A Benchmark Approach to Quantitative Finance
15.5.06 designandproduction GmbH – Bender
ISBN 3-540-26212-1
1 A Benchmark Approach to Quantitative Finance
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 benchmark approach. Various quantitative methods for the fair pricing and hedging of derivatives are explained. The general framework is used to provide an understanding 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 quantitative background. Finally, the book should stimulate interest in the benchmark approach by describing some of its power and wide applicability.
Eckhard Platen David Heath
63575
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 of realistic, parsimonious market models.
Platen · Heath
E. Platen · D. Heath
1 23
Jump-Diffusion Multi-Factor Models Bj¨ork, Kabanov & Runggaldier (1997) • continuous time • Markovian • explicit transition densities in special cases • benchmark framework • discrete time approximations • suitable for simulation • Markov chain approximations
Eckhard Platen
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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
Eckhard Platen
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2
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
Eckhard Platen
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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
Eckhard Platen
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4
Continuous and Event Driven Risk • Wiener processes • counting processes
W k , k ∈ {1, 2, . . ., m} pk
intensity hk jump martingale q k dWtm+k
=
dqtk
=
dpkt
k ∈ {1, 2, . . . , d−m}
−
hkt
k − 12 dt ht
W t = (Wt1 , . . . , Wtm , qt1 , . . . , qtd−m )⊤
Eckhard Platen
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Primary Security Accounts
j dStj = St−
ajt dt +
d X
k bj,k t dWt
k=1
!
Assumption 1 bj,k ≥− t k ∈ {m + 1, . . . , d}.
q
hk−m t
Assumption 2 d Generalized volatility matrix bt = [bj,k ] t j,k=1 invertible.
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• market price of risk [at − rt 1] θ t = (θt1 , . . . , θtd )⊤ = b−1 t
• primary security account j dStj = St−
rt dt +
d X
k k bj,k t (θt dt + dWt )
k=1
!
• portfolio dStδ =
d X
δtj dStj
j=0
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• fraction j πδ,t = δtj
Stj Stδ
• portfolio o n δ dStδ = St− rt dt + π ⊤ δ,t− bt (θ t dt + dW t )
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Assumption 3 q
hk−m > θtk t
• generalized GOP volatility
ckt =
1−θtk
θtk
for k ∈ {1, 2, . . . , m}
θtk
for k ∈ {m + 1, . . . , d}
k−m − 1 (ht ) 2
• GOP fractions π δ∗ ,t =
Eckhard Platen
(πδ1∗ ,t , . . . , πδd∗ ,t )⊤
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=
c⊤ t
b−1 t
⊤ 9
• Growth Optimal Portfolio dStδ∗
=
δ∗ St−
rt dt +
c⊤ t
(θ t dt + dW t )
• optimal growth rate gtδ∗ = rt +
−
Eckhard Platen
m 1 X
2
(θtk )2
k=1
d X
k=m+1
hk−m t
ln 1 + q
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θtk hk−m − θtk t
+ q
θtk hk−m t
10
• benchmarked portfolio ˆδ S t
Theorem 4
=
Stδ Stδ∗
ˆδ is an Any nonnegative benchmarked portfolio S
(A, P )-supermartingale. =⇒
no strong arbitrage
but there may exist: free lunch with vanishing risk free snacks or cheap thrills Eckhard Platen
(Delbaen & Schachermayer (2006)) (Loewenstein & Willard (2000)) Bressanone07
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Multi-Factor Model model mainly: • benchmarked primary security accounts ˆtj = S
Stj Stδ∗
j ∈ {0, 1, . . . , d} supermartingales, often SDE driftless, local martingales, sometimes martingales
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savings account St0 = exp =⇒
GOP Stδ∗
=⇒
Z
t
rs ds
0
St0 = 0 ˆt S
stock ˆtj S δ∗ Stj = S t
additionally dividend rates foreign interest rates
Eckhard Platen
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Example Black-Scholes Type Market
ˆtj = −S ˆj dS t−
d X
σtj,k dWtk
k=1
hjt , σtj,k , rt
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Examples • Merton jump-diffusion model dXt = Xt− (µ dt + σ dWt + dpt ) , ⇓ 2 (µ− 1 2 σ )t+σWt
Xt = X0 e
Nt Y
ξi
i=1
• Bates model
p
Vt dWtS + dpt p dVt = ξ(η − Vt ) dt + θ Vt dWtV dSt = St− α dt +
Eckhard Platen
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3
2.5
2
1.5
1
0.5
0 0
5
10 time
15
20
Figure 1: Simulated benchmarked primary security accounts. Eckhard Platen
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10 9 8 7 6 5 4 3 2 1 0 0
5
10 time
15
20
Figure 2: Simulated primary security accounts. Eckhard Platen
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4.5 GOP EWI 4
3.5
3
2.5
2
1.5
1
0.5 0
5
10 time
15
20
Figure 3: Simulated GOP and EWI for d = 50. Eckhard Platen
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4.5 GOP index 4
3.5
3
2.5
2
1.5
1
0.5 0
5
10 time
15
20
Figure 4: Simulated accumulation index and GOP. Eckhard Platen
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Diversification • diversified portfolios j πδ,t ≤
Eckhard Platen
K2 1
d 2 +K1
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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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60
50
40
30
20
10
0 0
5
9
14
18
23
27
32
Figure 5: Benchmarked primary security accounts.
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450
400
350
300
250
200
150
100
50
0 0
5
9
14
18
23
27
32
Figure 6: Primary security accounts under the MMM.
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100 EWI GOP
90
80
70
60
50
40
30
20
10
0 0
5
9
14
18
23
27
32
Figure 7: GOP and EWI.
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100
Market index GOP
90
80
70
60
50
40
30
20
10
0 0
5
9
14
18
23
27
32
Figure 8: GOP and market index.
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• fair security benchmarked security (A, P )-martingale
⇐⇒
fair
• minimal replicating portfolio fair nonnegative portfolio S δ with Sτδ = Hτ =⇒
minimal nonnegative replicating portfolio
• fair pricing formula VHτ (t) = Stδ∗ E
Hτ At δ ∗ Sτ
No need for equivalent risk neutral probability measure! Eckhard Platen
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Fair Hedging • fair portfolio
Stδ
• benchmarked fair portfolio ˆδ = E S t
Hτ At δ ∗ S τ
• martingale representation d Z τ X Hτ Hτ k k + (s) dW = E x A t Hτ s + MHτ (t) δ δ ∗ ∗ Sτ Sτ k=1 t MHτ -(A, P )-martingale E
(pooled)
M Hτ , W
k
t
=0
F¨ollmer & Schweizer (1991) No need for equivalent risk neutral probability measure! Eckhard Platen
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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 Eckhard Platen
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• intensity of jump process – regular schemes
=⇒
– jump-adapted schemes
Eckhard Platen
high intensity =⇒
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 < ∞ PNt • pt = i=1 (ξi − 1): compound Poisson, ξi i.i.d r.v. • Poisson random measure Z c(t−, Xt− , v) pφ (dv × dt) E
• {(τi , ξi ), i = 1, 2, . . . , NT } Eckhard Platen
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Numerical Schemes • time discretization tn = n∆ • discrete time approximation
∆ Yn+1 = Yn∆ + a(Yn∆ )∆ + b(Yn∆ )∆Wn + c(Yn∆ )∆pn
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Strong Convergence • Applications: scenario analysis, filtering and hedge simulation • strong order γ if εs (∆) =
Eckhard Platen
r
2 E XT − YN∆ ≤ K ∆γ
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Weak Convergence • Applications: derivative pricing, utilities, risk measures • weak order β if εw (∆) = |E(g(XT )) − E(g(YN∆ ))| ≤ 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
• Jacod & Protter (1998) • Gardo`n (2004)
=⇒
strong schemes γ ≤ 1.5
=⇒
• Maghsoodi (1996, 1998)
=⇒
Euler scheme for semimartingales
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) = I(1,−1) =
2 1 {(∆W ) n 2
PN (t n+1 )
i=N (t n )+1
− ∆},
Wτ i − ∆pn Wt n ,
• simulation jump times τi :
Wτi =⇒
I(−1,−1) =
1 {(∆pn )2 − ∆pn } 2
I(−1,1) = ∆pn ∆Wn − I(1,−1)
I(1,−1) and I(−1,1)
• Computational effort heavily dependent on intensity λ Eckhard Platen
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Derivative-Free Strong Schemes avoid computation of derivatives ⇓
order 1.0 derivative-free strong scheme
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Implicit Strong Schemes wide stability regions ⇓
implicit Euler scheme order 1.0 implicit strong Taylor scheme
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Predictor-Corrector Euler Scheme • corrector Yn+1
= Yn + θ a ¯η (Y¯n+1 ) + (1 − θ) a ¯η (Yn ) ∆n
+ η b(Y¯n+1 ) + (1 − η) b(Yn ) ∆Wn +
p(tn+1 )
X
c (ξi )
i=p(tn )+1
a ¯η = a − η b b′ • predictor p(tn+1 )
Y¯n+1 = Yn + a(Yn ) ∆n + b(Yn ) ∆Wn +
X
c (ξi )
i=p(tn )+1
θ, η ∈ [0, 1] degree of implicitness Eckhard Platen
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Jump-Adapted Time Discretization
regular t0
t0
Eckhard Platen
t1
t1
t2 r τ1
r τ2
r t2
r t3
t3 = T
jump times
jump-adapted t4
t5 = T
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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 Eckhard Platen
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Merton SDE : µ = 0.05, σ = 0.2, ψ = −0.2, λ = 10, X0 = 1, T = 1 1
0.8
X
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
T
Figure 9: Plot of a jump-diffusion path. Eckhard Platen
Bressanone07
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0.0005
0.00025
Error
0
-0.00025
-0.0005
-0.00075
-0.001
-0.00125 0
0.2
0.4
0.6
0.8
1
T
Figure 10: Plot of the strong error for Euler(red) and 1.0 Taylor(blue) scheme. Eckhard Platen
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Merton SDE : µ = −0.05, σ = 0.1, λ = 1, X0 = 1, T = 0.5
Log2 Error
-10
-15
-20 Euler EulerJA 1Taylor -25
1TaylorJA 15TaylorJA
-10
-8
-6
-4
-2
0
Log2 dt
Figure 11: Log-log plot of strong error versus time step size.
Eckhard Platen
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Literature on Weak Schemes with Jumps • Mikulevicius & Pl (1991) =⇒ jump-adapted order β ∈ {1, 2 . . .} weak schemes • Liu & Li (2000) =⇒ order β ∈ {1, 2 . . .} weak Taylor, extrapolation 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
Eckhard Platen
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Simplified Euler Scheme • Euler scheme
=⇒
β=1
• simplified Euler scheme ˆ n + c(Yn ) (ξˆn − 1)∆pˆn Yn+1 = Yn + a(Yn )∆ + b(Yn )∆W
ˆ n and ∆pˆn match the first 3 moments of ∆Wn and ∆pn up to • if ∆W an O(∆2 ) error =⇒ β = 1
•
√
˜ n = ± ∆) = P (∆W Eckhard Platen
Bressanone07
1 2 46
Jump-Adapted Taylor Approximations • jump-adapted Euler scheme
=⇒
β=1
• jump-adapted order 2 weak Taylor scheme b b′
2
Ytn+1 − = Ytn + a∆tn + b∆Wtn + (∆Wtn ) − ∆tn + a′ b ∆Ztn 2 1 1 ′′ 2 1 ′′ 2 ′ 2 ′ a a + a b ∆tn + a b + b b {∆Wtn ∆tn − ∆Ztn } + 2 2 2 and Ytn+1 = Ytn+1 − + c(Ytn+1 − ) ∆pn
• β=2 Eckhard Platen
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47
Predictor-Corrector Schemes • predictor-corrector
=⇒
stability and efficiency
• jump-adapted predictor-corrector Euler scheme o 1n a(Y¯tn+1 − ) + a ∆tn + b∆Wtn Ytn+1 − = Ytn + 2 with predictor Y¯tn+1 − = Ytn + a∆tn + b∆Wtn
• β=1
Eckhard Platen
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48
3 EulerJA
ImplEulerJA
2
Log2 Error
PredCorrJA 1
0
-1
-2
-5
-4
-3
-2
-1
Log2 dt
Figure 12: Log-log plot of weak error versus time step size. Eckhard Platen
Bressanone07
49
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
Eckhard Platen
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Conclusions • low intensity • high intensity
=⇒ =⇒
jump-adapted higher order predictor-corrector
regular schemes
• distinction between strong and weak predictor-corrector schemes
Eckhard Platen
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References Bj¨ork, 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¨ollmer, 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 Itˆo-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. Finance 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 Itˆo equations. 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 processes. Math. Nachr. 151, 233–239. Platen, E. (1982). An approximation method for a class of Itˆo 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 Financial 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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Springer Finance
SF Springer Finance
›
springer.com
Dieser Farbausdruck/pdf-file kann nur annähernd das endgültige Druckergebnis wiedergeben !
� ������ ������
A Benchmark Approach to Quantitative Finance
15.5.06 designandproduction GmbH – Bender
ISBN 3-540-26212-1
1 A Benchmark Approach to Quantitative Finance
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 benchmark approach. Various quantitative methods for the fair pricing and hedging of derivatives are explained. The general framework is used to provide an understanding 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 quantitative background. Finally, the book should stimulate interest in the benchmark approach by describing some of its power and wide applicability.
Eckhard Platen David Heath
63575
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 of realistic, parsimonious market models.
Platen · Heath
E. Platen · D. Heath
1 23
Jump-Diffusion Multi-Factor Models Bj¨ork, Kabanov & Runggaldier (1997) • continuous time • Markovian • explicit transition densities in special cases • benchmark framework • discrete time approximations • suitable for simulation • Markov chain approximations
Eckhard Platen
Bressanone07
1
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
Eckhard Platen
Bressanone07
2
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
Eckhard Platen
Bressanone07
3
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
Eckhard Platen
Bressanone07
4
Continuous and Event Driven Risk • Wiener processes • counting processes
W k , k ∈ {1, 2, . . ., m} pk
intensity hk jump martingale q k dWtm+k
=
dqtk
=
dpkt
k ∈ {1, 2, . . . , d−m}
−
hkt
k − 12 dt ht
W t = (Wt1 , . . . , Wtm , qt1 , . . . , qtd−m )⊤
Eckhard Platen
Bressanone07
5
Primary Security Accounts
j dStj = St−
ajt dt +
d X
k bj,k t dWt
k=1
!
Assumption 1 bj,k ≥− t k ∈ {m + 1, . . . , d}.
q
hk−m t
Assumption 2 d Generalized volatility matrix bt = [bj,k ] t j,k=1 invertible.
Eckhard Platen
Bressanone07
6
• market price of risk [at − rt 1] θ t = (θt1 , . . . , θtd )⊤ = b−1 t
• primary security account j dStj = St−
rt dt +
d X
k k bj,k t (θt dt + dWt )
k=1
!
• portfolio dStδ =
d X
δtj dStj
j=0
Eckhard Platen
Bressanone07
7
• fraction j πδ,t = δtj
Stj Stδ
• portfolio o n δ dStδ = St− rt dt + π ⊤ δ,t− bt (θ t dt + dW t )
Eckhard Platen
Bressanone07
8
Assumption 3 q
hk−m > θtk t
• generalized GOP volatility
ckt =
1−θtk
θtk
for k ∈ {1, 2, . . . , m}
θtk
for k ∈ {m + 1, . . . , d}
k−m − 1 (ht ) 2
• GOP fractions π δ∗ ,t =
Eckhard Platen
(πδ1∗ ,t , . . . , πδd∗ ,t )⊤
Bressanone07
=
c⊤ t
b−1 t
⊤ 9
• Growth Optimal Portfolio dStδ∗
=
δ∗ St−
rt dt +
c⊤ t
(θ t dt + dW t )
• optimal growth rate gtδ∗ = rt +
−
Eckhard Platen
m 1 X
2
(θtk )2
k=1
d X
k=m+1
hk−m t
ln 1 + q
Bressanone07
θtk hk−m − θtk t
+ q
θtk hk−m t
10
• benchmarked portfolio ˆδ S t
Theorem 4
=
Stδ Stδ∗
ˆδ is an Any nonnegative benchmarked portfolio S
(A, P )-supermartingale. =⇒
no strong arbitrage
but there may exist: free lunch with vanishing risk free snacks or cheap thrills Eckhard Platen
(Delbaen & Schachermayer (2006)) (Loewenstein & Willard (2000)) Bressanone07
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Multi-Factor Model model mainly: • benchmarked primary security accounts ˆtj = S
Stj Stδ∗
j ∈ {0, 1, . . . , d} supermartingales, often SDE driftless, local martingales, sometimes martingales
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savings account St0 = exp =⇒
GOP Stδ∗
=⇒
Z
t
rs ds
0
St0 = 0 ˆt S
stock ˆtj S δ∗ Stj = S t
additionally dividend rates foreign interest rates
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Example Black-Scholes Type Market
ˆtj = −S ˆj dS t−
d X
σtj,k dWtk
k=1
hjt , σtj,k , rt
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Examples • Merton jump-diffusion model dXt = Xt− (µ dt + σ dWt + dpt ) , ⇓ 2 (µ− 1 2 σ )t+σWt
Xt = X0 e
Nt Y
ξi
i=1
• Bates model
p
Vt dWtS + dpt p dVt = ξ(η − Vt ) dt + θ Vt dWtV dSt = St− α dt +
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3
2.5
2
1.5
1
0.5
0 0
5
10 time
15
20
Figure 1: Simulated benchmarked primary security accounts. Eckhard Platen
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10 9 8 7 6 5 4 3 2 1 0 0
5
10 time
15
20
Figure 2: Simulated primary security accounts. Eckhard Platen
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4.5 GOP EWI 4
3.5
3
2.5
2
1.5
1
0.5 0
5
10 time
15
20
Figure 3: Simulated GOP and EWI for d = 50. Eckhard Platen
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4.5 GOP index 4
3.5
3
2.5
2
1.5
1
0.5 0
5
10 time
15
20
Figure 4: Simulated accumulation index and GOP. Eckhard Platen
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Diversification • diversified portfolios j πδ,t ≤
Eckhard Platen
K2 1
d 2 +K1
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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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60
50
40
30
20
10
0 0
5
9
14
18
23
27
32
Figure 5: Benchmarked primary security accounts.
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450
400
350
300
250
200
150
100
50
0 0
5
9
14
18
23
27
32
Figure 6: Primary security accounts under the MMM.
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100 EWI GOP
90
80
70
60
50
40
30
20
10
0 0
5
9
14
18
23
27
32
Figure 7: GOP and EWI.
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100
Market index GOP
90
80
70
60
50
40
30
20
10
0 0
5
9
14
18
23
27
32
Figure 8: GOP and market index.
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• fair security benchmarked security (A, P )-martingale
⇐⇒
fair
• minimal replicating portfolio fair nonnegative portfolio S δ with Sτδ = Hτ =⇒
minimal nonnegative replicating portfolio
• fair pricing formula VHτ (t) = Stδ∗ E
Hτ At δ ∗ Sτ
No need for equivalent risk neutral probability measure! Eckhard Platen
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Fair Hedging • fair portfolio
Stδ
• benchmarked fair portfolio ˆδ = E S t
Hτ At δ ∗ S τ
• martingale representation d Z τ X Hτ Hτ k k + (s) dW = E x A t Hτ s + MHτ (t) δ δ ∗ ∗ Sτ Sτ k=1 t MHτ -(A, P )-martingale E
(pooled)
M Hτ , W
k
t
=0
F¨ollmer & Schweizer (1991) No need for equivalent risk neutral probability measure! Eckhard Platen
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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 Eckhard Platen
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• intensity of jump process – regular schemes
=⇒
– jump-adapted schemes
Eckhard Platen
high intensity =⇒
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 < ∞ PNt • pt = i=1 (ξi − 1): compound Poisson, ξi i.i.d r.v. • Poisson random measure Z c(t−, Xt− , v) pφ (dv × dt) E
• {(τi , ξi ), i = 1, 2, . . . , NT } Eckhard Platen
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Numerical Schemes • time discretization tn = n∆ • discrete time approximation
∆ Yn+1 = Yn∆ + a(Yn∆ )∆ + b(Yn∆ )∆Wn + c(Yn∆ )∆pn
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Strong Convergence • Applications: scenario analysis, filtering and hedge simulation • strong order γ if εs (∆) =
Eckhard Platen
r
2 E XT − YN∆ ≤ K ∆γ
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Weak Convergence • Applications: derivative pricing, utilities, risk measures • weak order β if εw (∆) = |E(g(XT )) − E(g(YN∆ ))| ≤ 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
• Jacod & Protter (1998) • Gardo`n (2004)
=⇒
strong schemes γ ≤ 1.5
=⇒
• Maghsoodi (1996, 1998)
=⇒
Euler scheme for semimartingales
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) = I(1,−1) =
2 1 {(∆W ) n 2
PN (t n+1 )
i=N (t n )+1
− ∆},
Wτ i − ∆pn Wt n ,
• simulation jump times τi :
Wτi =⇒
I(−1,−1) =
1 {(∆pn )2 − ∆pn } 2
I(−1,1) = ∆pn ∆Wn − I(1,−1)
I(1,−1) and I(−1,1)
• Computational effort heavily dependent on intensity λ Eckhard Platen
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Derivative-Free Strong Schemes avoid computation of derivatives ⇓
order 1.0 derivative-free strong scheme
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Implicit Strong Schemes wide stability regions ⇓
implicit Euler scheme order 1.0 implicit strong Taylor scheme
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Predictor-Corrector Euler Scheme • corrector Yn+1
= Yn + θ a ¯η (Y¯n+1 ) + (1 − θ) a ¯η (Yn ) ∆n
+ η b(Y¯n+1 ) + (1 − η) b(Yn ) ∆Wn +
p(tn+1 )
X
c (ξi )
i=p(tn )+1
a ¯η = a − η b b′ • predictor p(tn+1 )
Y¯n+1 = Yn + a(Yn ) ∆n + b(Yn ) ∆Wn +
X
c (ξi )
i=p(tn )+1
θ, η ∈ [0, 1] degree of implicitness Eckhard Platen
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Jump-Adapted Time Discretization
regular t0
t0
Eckhard Platen
t1
t1
t2 r τ1
r τ2
r t2
r t3
t3 = T
jump times
jump-adapted t4
t5 = T
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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 Eckhard Platen
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Merton SDE : µ = 0.05, σ = 0.2, ψ = −0.2, λ = 10, X0 = 1, T = 1 1
0.8
X
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
T
Figure 9: Plot of a jump-diffusion path. Eckhard Platen
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0.0005
0.00025
Error
0
-0.00025
-0.0005
-0.00075
-0.001
-0.00125 0
0.2
0.4
0.6
0.8
1
T
Figure 10: Plot of the strong error for Euler(red) and 1.0 Taylor(blue) scheme. Eckhard Platen
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Merton SDE : µ = −0.05, σ = 0.1, λ = 1, X0 = 1, T = 0.5
Log2 Error
-10
-15
-20 Euler EulerJA 1Taylor -25
1TaylorJA 15TaylorJA
-10
-8
-6
-4
-2
0
Log2 dt
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) =⇒ order β ∈ {1, 2 . . .} weak Taylor, extrapolation 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 ˆ n + c(Yn ) (ξˆn − 1)∆pˆn Yn+1 = Yn + a(Yn )∆ + b(Yn )∆W
ˆ n and ∆pˆn match the first 3 moments of ∆Wn and ∆pn up to • if ∆W an O(∆2 ) error =⇒ β = 1
•
√
˜ n = ± ∆) = P (∆W Eckhard Platen
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Jump-Adapted Taylor Approximations • jump-adapted Euler scheme
=⇒
β=1
• jump-adapted order 2 weak Taylor scheme b b′
2
Ytn+1 − = Ytn + a∆tn + b∆Wtn + (∆Wtn ) − ∆tn + a′ b ∆Ztn 2 1 1 ′′ 2 1 ′′ 2 ′ 2 ′ a a + a b ∆tn + a b + b b {∆Wtn ∆tn − ∆Ztn } + 2 2 2 and Ytn+1 = Ytn+1 − + c(Ytn+1 − ) ∆pn
• β=2 Eckhard Platen
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Predictor-Corrector Schemes • predictor-corrector
=⇒
stability and efficiency
• jump-adapted predictor-corrector Euler scheme o 1n a(Y¯tn+1 − ) + a ∆tn + b∆Wtn Ytn+1 − = Ytn + 2 with predictor Y¯tn+1 − = Ytn + a∆tn + b∆Wtn
• β=1
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3 EulerJA
ImplEulerJA
2
Log2 Error
PredCorrJA 1
0
-1
-2
-5
-4
-3
-2
-1
Log2 dt
Figure 12: Log-log plot of weak error versus time step size. Eckhard Platen
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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 • high intensity
=⇒ =⇒
jump-adapted higher order predictor-corrector
regular schemes
• distinction between strong and weak predictor-corrector schemes
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References Bj¨ork, 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¨ollmer, 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 Itˆo-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. Finance 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 Itˆo equations. 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 processes. Math. Nachr. 151, 233–239. Platen, E. (1982). An approximation method for a class of Itˆo 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 Financial 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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