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Beyond the Classical Paradigm
Eckhard PlatenUniversity of Technology Sydney
Pl. & Heath (2006, 2010), A Benchmark Approach to Quantitative Finance. Springer
Pl. & Bruti-Liberati (2010), Numerical Solution of Stochastic DifferentialEquations with Jumps in Finance. Springer
Baldeaux & Pl. (2013). Functionals of MultidimensionalDiffusions with Applications to Finance. Springer
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index
K*savings bond
"risk neutral" put
fair put
Risk neutral and fair put on index
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index
K*savings bond
"risk neutral" put
fair put
Benchmarked “risk neutral” and fair put on index
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Complications with Classical Approach
• inverse of 3-dimensional Bessel process; Delbaen & Schachermayer(1995)
• 32
volatility model; Pl. (1997), Heston (1997), Lewis (2000), Pl. (2001),...
• some other stochastic volatility models; Sin (1998), ...
• allowing some classical arbitrage; Loewenstein & Willard (2000),Pl. (2002), Fernholz & Karatzas (2005), ...
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Numeraire Portfolio S∗t as Benchmark
• tradeable for modeling, investing and pricing
• supermartingale property:
Sδt =
Sδt
S∗t
≥ E(Sδs |Ft)
0 ≤ t ≤ s ≤ ∞Long (1990), Becherer (2001), Pl. (2002), Goll & Kallsen (2003), Pl. &Heath (2006), Christensen & Pl. (2005), Karatzas & Kardaras (2007),...
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• growth optimal portfolio is NP S∗ti
Kelly (1956), ..., MacLean et al. (2011)
• long term growth rate
g = limt→∞
sup1
tln(
S∗t
S∗0
) -maximal P-a.s.
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Diversification Approximates NP
• diversification theoremPl. (2005), Le & Pl. (2006)
• naive diversificationPl. & Rendek (2012)model independent
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1973 1980 1988 1995 2003 20110
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9
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15x 10
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EWI114MCI
EWI114 and MCI
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1973 1980 1988 1995 2003 20111.5
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2.5
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3.5
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4.5
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5.5
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log(EWI114)log(MCI)
Logarithms of EWI114 and MCI
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Maximum Drawdown Constrained Portfolios
Kardaras, Obloj & Pl. (2012)Cheredito, Nikeghbali & Pl. (2012)
X - set of nonnegative continuous discounted portfolios
• running maximumfor X = {Xt, t ≥ 0} ∈ X
X∗t = sup
u∈[0,t]
Xu
• relative drawdownXt
X∗t
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• maximum relative drawdown
express attitude towards risk by restricting to X ∈ αX , where
Xt
X∗t
≥ α ,α ∈ [0, 1)
pathwise criterion
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• maximum drawdown constrained portfolioα ∈ [0, 1), X ∈ X
αXt = α(X∗t )
1−α + (1 − α)Xt(X∗t )
−α
= 1 +
∫ t
0
(1 − α)(X∗s )
−αdXs ≥ ααX∗t
Grossman & Zhou (1993), Cvitanic & Karatzas (1994)
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=⇒ SDEdαXt
αXt
= απt
dXt
Xt
fraction
απt = 1 −(1 − α)Xt
X∗t
α + (1 − α)Xt
X∗t
model independentdepends only on Xt
X∗t
and ααS∗
t - invests in S∗ and savings account
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1871 1899 1927 1955 1983 2011−1
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2
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ln(X)
ln(X*)
Logarithm of discounted S&P500 and its running maximum
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1871 1899 1927 1955 1983 20110
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0.6
0.8
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1.2
relative drawdown
Relative drawdown of discounted S&P500
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• asymptotic maximum long term growth rateKardaras, Obloj & Pl. (2012)
limt→∞
1
tlog(αS∗
t ) = (1 − α) limt→∞
1
tlog(S∗
t )
αg = (1 − α)g
restricted drawdown =⇒ reduced maximum growth ratelong term view with short term attitude towards riskrealistic alternative to Markowitz mean-variance approachand utility maximization
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1871 1900 1928 1956 1984 2012−1
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Logarithm of drawdown constrained portfolios
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Shortest Expected Market Time to Reach a Level
Kardaras & Pl. (2010)
Works together with maximum drawdown constraint⇒ new type of fund management
• long term view
• short term attitude
• pathwise properties
• model independent
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Relative drawdown of MCI and EWI114
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Utility Maximization
Kardaras & Pl. (2013), Pl. & Heath (2006)
Sδt = E(U ′−1(S0
T )S0T |Ft)
• two fund separation ⇒ some efficient frontier
• substituting market portfolio by NP ⇒ modified CAPM
• does not maximize Sharpe ratio for jump caseChristensen & Pl. (2007)
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Last Passage Times
Nikeghbali & Pl. (2008, 2013); Profeta, Roynette & Yor (2010)
Nt-local martingale, no positive jumps, limt→∞ Nt = 0
⇒N0
limt→∞ supt≥0 Nt
∼ U(0, 1)
• model independent
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Trajectories of Nt under MMM
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Running maxima
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Inverse of maxima
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Valuation and Pricing
• supermartingale property for all price processes ⇒ NUPBR
SδHTt ≥ E(
HT
S∗T
|Ft)
• martingale is the minimal supermartingale, Du & Pl. (2013)⇒ minimal price:real world pricing formula
SδHTt = E(
HT
S∗T
|Ft)
⇒ SδHTt = S∗
t E(HT
S∗T
|Ft)
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If S∗T and HT independent ⇒ Actuarial pricing formula
SδHTt = P (t, T )E(HT |Ft)
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If Radon-Nikodym derivative Λt =S0
t
S00
martingale ⇒
Risk neutral pricing formula
SδHTt = E(
ΛT
Λt
S0t HT
S0T
|Ft) = EQ(S0t HT
S0T
|Ft)
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Benchmarked Risk Minimization
Follmer & Sondermann (1986), Follmer & Schweizer (1991)Du & Pl. (2013), Biagini, Cretarola & Pl. (2014)
• jth benchmarked primary security account
Sjt
local martingale
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Dynamic Trading Strategy
v = {vt = (ηt, ϑ1t , . . . , ϑ
dt )
>, t ∈ [0,∞)} forms benchmarked priceprocess
V vt = ϑ>
t St + ηt
ϑ predictable,η adapted, η0 = 0, monitors non-self-financing part of supermartingale
V vt = V v
0 +
∫ t
0
ϑ>s dSs + ηt
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• benchmarked contingent claim HT
v delivers HT if
V vT = HT
=⇒ replicable if self-financing
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Benchmarked P&L
Ct = V vt −
d∑j=1
∫ t
0
ϑjudS
ju − V v
0
⇒ Ct = ηt for t ∈ [0,∞)
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• benchmarked P&L usually fluctuating
• intrinsic risk
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What criterion would be most natural?
• symmetric view with respect to all primary security accounts,in particular the domestic savings account
• pooling in large trading book=⇒ vanishing total hedge error, P&L
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Pooling
HT,l; V vl with Cvl independent square integrable martingale with
E
((C
vlt
Vvl0
)2)
≤ Kt < ∞ for l ∈ {1, 2, . . . },
well diversified trading book holds equal fractions at initial time:
total benchmarked wealth Ut =U0
m
∑ml=1
Vvlt
Vvl0
total benchmarked P&L
Cm(t) =U0
m
m∑l=1
Cvlt
V vl0
,
⇒
limm→∞
Cm(t) = 0
P -a.s.
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• VHTset of strategies v delivering HT with orthogonal benchmarked
P&L
that is, η and ηS are local martingales
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• market participants prefer more for less
⇒ Benchmarked Risk Minimization
For HT strategy v ∈ VHTbenchmarked risk minimizing (BRM) if for all
v ∈ VHTprice V v
t is minimal
V vt ≤ V v
t
P -a.s. for all t ∈ [0, T ].
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Regular Benchmarked Contingent Claims
HT is called regular if
HT = Et(HT ) +
d∑j=1
∫ T
t
ϑj
HT(s)dSj
s + ηHT(T ) − ηHT
(t)
ϑHT- predictable
ηHT- local martingale, adapted, orthogonal to S
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⇒ regular HT has BRM strategy v with
V vt = E(HT |Ft) ,
and orthogonal benchmarked P&L: Ct = ηHT(t)
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• general semimartingale market
• no second moments required
• no risk neutral measure required
• takes evolving information about nonhedgeable part of claim into ac-count
• HT nonhedgeable ⇒ δ0t = E(HT |Ft)
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• hedge designed for pooling
• under minimal martingale measure, see Schweizer (1995),prices as under local risk-minimization but hedge different
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Conjectured Model for Well-diversified Equity Indices
Filipovic & Pl. (2009), Pl. & Rendek (2012)
• well diversified portfolio ≈ NP
• discounted NP time transformed squared Bessel process
• normalized NP
dYτ = (1 − Yτ )dτ +√YτdW (τ )
• nature of feedback in diversified wealth
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• market activity timedτt
dt=
1
Zt
• inverse market activity Zt
trading behaviour exaggerates volatility, Kahneman & Tversky (1979)fast moving
dZt = (γ − εZt)dt +√γZtdWt
dWt =√ZtdW (τt)
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⇒ discounted index model
St = AτtYτt
Aτt = A0 exp{aτt}
a ≥ 1 ⇒ S0supermartingale
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⇒
• volatility
σt =
√1
YτtZt
• expected rate of return
µt =a − 1
Zt
+ σ2t
• long term average market time
τt ≈2ε
γt
• almost exact simulation
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Simulated market activity
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Estimated market activity of simulated index
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• conjectured parsimonious model:
3 initial parameters: A0, Y0, Z0
3 structural parameters: a, ε, γ1 driving Brownian motion (nondiversifiable equity risk)
highly tractable, Heath & Pl. (2014), Baldeaux & Pl. (2013)
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• for long term tasks average market activity M = const.
if S0 local martingale ⇒
3 initial parameters: A0, Y0, Z0 = 2M
1 driving Brownian motion
analytically tractableexplicit call, put, binary, bond, ...
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• Popper (1935, 2002): one can only falsify models
• conjectured model difficult to falsify: MCI, EWI114, S&P500, ...
• most known index models can be falsifiedby some of 10 listed stylized empirical facts, Pl. & Rendek (2014)
• ideal for pricing long-dated pension and insurance contractsBaldeaux & Pl. (2013), Heath & Pl. (2014), Baldeaux, Grasselli & Pl.(2014) (FX)
• model leads beyond classical paradigm
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• BA generalizes classical theory
• several ”puzzles” can be naturally explained
• no representative agent employed
• long term model as simple as BS model
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• significant performance improvement in long term fund management
• major savings in pension and insurance products
• systematic, efficient approach to risk measurement and regulation, Pl. &Stahl (2005)
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