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Simulation of Diversified Portfolios in a Continuous
Financial Market
Eckhard PlatenSchool of Finance and Economics and School of Mathematical Sciences
University of Technology, Sydney
Platen, E.& Heath, D.: A Benchmark Approach to Quantitative FinanceSpringer Finance, 700 pp., 199 illus., Hardcover, ISBN-10 3-540-26212-1 (2006).
Le, T. & Platen. E.: Approximating the growth optimal portfolio with a diversifiedworld stock index.J. Risk Finance7(5), 559–574 (2006).
Platen, E.& Rendek, R.: Simulation of diversified portfolios in a continuous financial market.(working paper) (2010).
Benchmark Approach
Pl. & Heath (2006)
• Diversification Theorem
well diversified portfolio approximates numeraire portfolio
• growth optimal portfolio equals numeraire portfolio, Long(1990)
• benchmark in portfolio optimization
• numeraire in derivative pricing
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 1
0 1000 2000 3000 4000 5000 6000 7000 8000 90000
20,000
40,000
60,000
80,000
100,000
120,000
MCI
EWI114
Figure 1: EWI114 and MCI.
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 2
Supermartingale Property
Assume numeraire portfolioSδ∗
tnas benchmark s.t.
for all nonnegative portfoliosSδtn
Sδtn
Sδ∗
tn
= Sδtn
≥ Etn
(
Sδtn+1
)
Sδ∗ - best performing portfolio, numeraire portfolio, growth optimal
Kelly (1956), Long (1990), Becherer (2001), Pl. (2002),
Buhlmann& Pl. (2003), Pl.& Heath (2006), Karatzas & Kardaras (2007)
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 3
• Benchmarked numeraire portfolio
Sδ∗
tn= 1 =⇒
Sδ∗
tn+1− Sδ∗
tn
Sδ∗
tn
= 0 a.s.
• approximateSδ∗
strictly positive
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 4
• sequence of strictly positive, benchmarked portfolios
(
Sδℓ
)
ℓ∈{1,2,...}
with Sδℓ
0 = 1
is a sequence ofapproximate numeraire portfolios
if for ε > 0
limℓ→∞
Ptn
(
1√tn+1 − tn
∣
∣
∣
∣
∣
Sδℓ
tn+1− Sδℓ
tn
Sδℓ
tn
∣
∣
∣
∣
∣
≥ ε
)
= 0
for all n ∈ {0, 1, . . .}
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 5
• financial market issemi-regular if for all ε > 0
limℓ→∞
P
∣
∣
∣
∣
∣
∣
1√ℓ
ℓ∑
j=1
σj,ktn
∣
∣
∣
∣
∣
∣
≥ ε
= 0
for all k ∈ {1, 2, . . .} andn ∈ {0, 1, . . .}
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 6
Diversification Theorem:
In a semi-regular market each sequence of EWIs
is a sequence of approximate numeraire portfolios.
Pl. (2005), Pl.& Rendek (2009)
Numeraire portfolio has only non-diversifiable risk!
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 7
Equally Weighted Index
EWI
πjδEWI,t
=1
d
j ∈ {1, 2, . . . , d}
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 8
Inverse Transform Method
• random variableY - distribution functionFY
• uniformly distributed random variable0 < U < 1
FY distributed random variabley(U)
U = FY (y(U))
y(U) = F−1
Y (U)
More generally
y(U) = inf{y : U ≤ FY (y)}
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 9
Transition Density of a Matrix SR-process
d × m matrix
p(s,X; t, Y) =pδ
(
ϕ(s), Xss
;ϕ(t), Yst
)
st
• ϕ-time
ϕ(t) = ϕ(0) +b2
4c s0(1 − exp{−c t})
st = s0 exp{c t} for t ∈ [0,∞), s0 > 0, c < 0 andb 6= 0
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 10
0 5 10
0
5
10
15
0 5 10
0
5
10
15
0 5 10
0
5
10
15
0 5 10
0
5
10
15
Figure 2: Matrix valued square root process
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 11
0 50 100 1500
2
4
6
8
10
12
Figure 3: Simulated benchmarked primary security accountsunder the
Black-Scholes model
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 12
0 50 100 1500
100
200
300
400
500
600
GOP
EWI
MCI
Figure 4: Simulated GOP, EWI and MCI under the Black-Scholes model
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 13
0 50 100 1500.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
benchmarked MCI
benchmarked EWI
Figure 5: Simulated benchmarked GOP, EWI and MCI under the Black-
Scholes model
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 14
Multi-asset Heston Model
Heston (1993)
• matrix SDEs
dSt = diag(
√
V t
)
diag(
St
) (
AdW1
t + BdW2
t
)
dV t = (a − EV t) dt + Fdiag(√
V t
)
dW1
t
St = (S0t , S
1t , . . . , S
dt )
⊤
• independentvectors of correlated Wiener processes
Wk
t = CkW kt
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 15
• exact simulation in Broadie & Kaya (2006) (complicated, slow)
• simplified almost exact simulation Pl.& Rendek (2010)
• squared volatility Vjti+1
sampling directly from the noncentral chi-square distribution
exact
• almost exact simulation
Xt = ln(St)
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 16
• log-asset price
Xjti+1
= Xjti+
j
γj
(
Vjti+1
− Vjti− aj∆
)
+
(
jκj
γj
− 1
2
)∫ ti+1
ti
V ju du
+√
1 − 2j
∫ ti+1
ti
√
VjudW 2,j
u
with∫ ti+1
ti
√
VjudW 2,j
u
conditionally Gaussian:
mean zero, variance∫ ti+1
tiV ju du
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 17
• approximate∫ ti+1
ti
V judu
• trapezoidal rule∫ ti+1
ti
V judu ≈ ∆
2
(
Vjti+ V
jti+1
)
=⇒∫ ti+1
ti
√
VjudW 2,j
u ≈ N(
0,∆
2
(
Vjti+ V
jti+1
)
)
achieved with high accuracy
efficient almost exact simulation technique
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 18
0 50 100 1500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Figure 6: Simulated squared volatility under the Heston model
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 19
0 50 100 1500.4
0.5
0.6
0.7
0.8
0.9
1
1.1
benchmarked MCI
benchmarked EWI
Figure 7: Simulated benchmarked GOP, EWI and MCI under the Heston
model
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 20
Multi-asset ARCH-diffusion Model
Nelson (1990), Frey (1997)
dSt = diag(
√
V t
)
diag(
St
) (
AdW1
t + BdW2
t
)
dV t = (a − EV t) dt + Fdiag(V t) dW1
t
• squared volatility
Vjti+1
= exp
{(
−κj − 1
2γ2j
)
ti+1 + γjW1,jti+1
}
= ×(
Vjt0
+ aj
i∑
k=0
∫ tk+1
tk
exp
{(
κj +1
2γ2j
)
s − γjW1,js
}
ds
)
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 21
0 50 100 1500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Figure 8: Simulated squared volatility under the ARCH-diffusion model
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 22
0 50 100 150
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
benchmarked EWI
benchmarked MCI
Figure 9: Simulated benchmarked GOP, EWI and MCI under the ARCH-
diffusion model
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 23
Geometric Ornstein-Uhlenbeck Volatility Model
dSt = diag(exp{V t}) diag(
St
) (
AdW1
t + BdW2
t
)
dV t = (a − EV t) dt + FdW1
t
simulation forexp{V j} exact
log-asset price as before
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 24
0 50 100 1500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Figure 10: Simulated squared volatility under the geometric Ornstein-
Uhlenbeck volatility model
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 25
0 50 100 1500.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
benchmarked EWI
benchmarked MCI
Figure 11: Simulated benchmarked GOP, EWI and MCI under the geometric
Ornstein-Uhlenbeck volatility model
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 26
Minimal Market Model
Pl. (2001), Pl.& Heath (2006)
• ϕ-time
ϕj(t) =α
j0
4ηjexp{ηjt}
• benchmarked primary security account
dSj(ϕj(t)) = −2(
Sj(ϕj(t)))
32
dW j(ϕj(t))
strict supermartingale
Sj(
ϕj(ti+1))
=1
∑4
k=1
(
wk + Wk,jti+1
)2
exact simulation
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 27
0 50 100 1500
2
4
6
8
10
12
Figure 12: Simulated benchmarked primary security accounts under the
MMM
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 28
0 50 100 1500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 13: Simulated squared volatility under the MMM
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 29
0 50 100 1500
0.2
0.4
0.6
0.8
1
1.2
1.4
benchmarked MCI
benchmarked EWI
Figure 14: Simulated benchmarked GOP, EWI and MCI under the MMM
model
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 30
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Stoch.5, 327–341.
Broadie, M. & O. Kaya (2006). Exact simulation of stochasticvolatility and other affinejump diffusion processes.Oper. Res.54, 217–231.
Buhlmann, H. & E. Platen (2003). A discrete time benchmark approach for insurance andfinance.ASTIN Bulletin33(2), 153–172.
Frey, R. (1997). Derivative asset analysis in models with level-dependent and stochasticvolatility. Mathematics of Finance, Part II.CWI Quarterly10(1), 1–34.
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with ap-plications to bond and currency options.Rev. Financial Studies6(2), 327–343.
Karatzas, I. & C. Kardaras (2007). The numeraire portfolio in semimartingale financial mod-els.Finance Stoch.11(4), 447–493.
Kelly, J. R. (1956). A new interpretation of information rate.Bell Syst. Techn. J.35, 917–926.
Long, J. B. (1990). The numeraire portfolio.J. Financial Economics26, 29–69.
Nelson, D. B. (1990). ARCH models as diffusion approximations.J. Econometrics45, 7–38.
Platen, E. (2001). A minimal financial market model. InTrends in Mathematics, pp. 293–
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 31
301. Birkhauser.
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540–558.
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Financial Markets11(1), 1–22.
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Finance. Springer.
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Technical report, University of Technology, Sydney. QFRC Research Paper, draft.
Platen, E. & R. Rendek (2010). Almost exact simulation of multi-dimensional stochastic
volatility models. (working paper).
c© Copyright E. Platen BFS Simulation of Diversified Portfolios 32