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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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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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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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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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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• 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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