simulation of diversiﬁed portfolios in a continuous ... · simulation of diversiﬁed portfolios...

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.


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)


• Benchmarked numeraire portfolio

Sδ∗

tn= 1 =⇒

Sδ∗

tn+1− Sδ∗

tn

Sδ∗

tn

= 0 a.s.

• approximateSδ∗

strictly positive


• 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, . . .}


• 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, . . .}


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!


Equally Weighted Index

EWI

πjδEWI,t

=1

d

j ∈ {1, 2, . . . , d}


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)}


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


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


0 50 100 1500

2

4

6

8

10

12

Figure 3: Simulated benchmarked primary security accountsunder the

Black-Scholes model


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


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


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


• 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)


• 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


• 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


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


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


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

)


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


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


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


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


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


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


0 50 100 1500

2

4

6

8

10

12

Figure 12: Simulated benchmarked primary security accounts under the

MMM


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


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


ReferencesBecherer, D. (2001). The numeraire portfolio for unboundedsemimartingales.Finance

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–


301. Birkhauser.

Platen, E. (2002). Arbitrage in continuous complete markets. Adv. in Appl. Probab.34(3),

540–558.

Platen, E. (2005). Diversified portfolios with jumps in a benchmark framework.Asia-Pacific

Financial Markets11(1), 1–22.

Platen, E. & D. Heath (2006).A Benchmark Approach to Quantitative Finance. Springer

Finance. Springer.

Platen, E. & R. Rendek (2009). Approximating the numeraire portfolio by diversification.

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).


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