Polynomial processes in stochastic portfolio theory · Introduction Overview Introduction and overview Stochastic portfolio theory (SPT) I theory for analyzing stock market and portfolio

Polynomial processes in stochastic portfolio theory

Christa Cuchiero

University of Vienna

School and workshop on dynamical models in financeLausanne, May 24, 2017

Christa Cuchiero (University of Vienna) Polynomial processes in SPT Lausanne, May 2016 1 / 40

Introduction Overview

Introduction and overview

Stochastic portfolio theory (SPT)

I theory for analyzing stock market and portfolio behavior;

I introduced in several papers (1982, 1999, 2001) and a monograph(2002) by Robert Fernholz.

Major goals within SPT are

I the analysis of the relative performance of a portfolio with respect tothe market portfolio ⇒ Specificity of SPT: market portfolio is chosenas numeraire

I the development and analysis of tractable models which work in highdimensions and match empirically observed facts, e.g, the dynamicsand shape of the capital distribution curves;

I to understand various aspects of relative arbitrages, in particular theproperties of portfolios generating them (e.g., so-called functionallygenerated and/or long only portfolios, their implementation on differenttime horizons, etc.)
































Introduction Literature overview

A (very incomplete) literature overview of SPT

The first instance of the ideas of SPT is the article “Stochastic PortfolioTheory and Stock Market Equilibrium” (Journal of Finance, 1982) byRobert Fernholz and Brian Shay.

Robert Fernholz further developed it in the papers

I “On the diversity of equity markets”(1999),

I “Equity portfolios generated by functions of ranked market weights”(2001) and in

I the monograph “Stochastic Portfolio Theory” (2002).

Since then a lot of research has been conducted in this area, in particular byAdrian Banner, Daniel Fernholz, Robert Fernholz, Irina Goia, TomoyukiIchiba, Ioannis Karatzas, Kostas Kardaras, Soumik Pal, Radka Pickova,Johannes Ruf, Mykhaylo Shkolnikov, Alexander Vervuurt etc., which ispartly summarized in the...

...Overview article by Robert Fernholz and Ioannis Karatzas, StochasticPortfolio Theory: an Overview (2009) and Alexander Vervuurt, Topics instochastic portfolio theory (2015).


Introduction Motivation

Motivation

Model classes used in SPT can be broadly categorized into

I Rank based models (Atlas and hybrid Atlas model) (Ichiba,Papathanakos, Banner, Karatzas, Fernholz (2011,2013))

I Diverse models (Fernholz (2002)) (No single company is allowed todominate the entire market in terms of relative capitalization)

I Sufficiently volatile models with volatility stabilized models (Fernholz&Karatzas (2005)) and generalized volatility stabilized models (Pickova(2014)) as subclasses.

In terms of practical applicability there are however two shortcomings,namely

I unrealistic modeling assumptions, in particular no correlation betweenthe assets;

I limited tractability except for volatility stabilized models, whichhowever share the disadvantage of no correlation.



Motivation

Model classes used in SPT can be broadly categorized into

I Rank based models (Atlas and hybrid Atlas model) (Ichiba,Papathanakos, Banner, Karatzas, Fernholz (2011,2013))

I Diverse models (Fernholz (2002)) (No single company is allowed todominate the entire market in terms of relative capitalization)

I Sufficiently volatile models with volatility stabilized models (Fernholz&Karatzas (2005)) and generalized volatility stabilized models (Pickova(2014)) as subclasses.

In terms of practical applicability there are however two shortcomings,namely

I unrealistic modeling assumptions, in particular no correlation betweenthe assets;

I limited tractability except for volatility stabilized models, whichhowever share the disadvantage of no correlation.



Goal of this talk

Problem

High dimensional realistic modeling of say 500 stocks with the aim topreserve tractability in view of calibration and relative arbitrages;incorporate correlations;match the dynamics of the ranked marked weights.



Goal of this talk

Introduce market weight and asset price models based on polynomialprocesses

I which constitutes the most tractable class when it comes to modelingthe the market weights directly,

I of which volatility stabilized models are a specific example,

I whose structural properties fit the transformation between marketcapitalizations and market weights particularly well.

I to treat high-dimensionality (extension by introducing measure valuedpolynomial processes.)


Setting and notions in stochastic portfolio theory The financial market

The financial market model - Setting

Filtered probability space: (Ω,F , (Ft)t∈[0,T ],P) for some finite time horizonT and some right continuous filtration (Ft)t∈[0,T ].

(Undiscounted) asset capitalizations are modeled by an Rd+-valued

semimartingale S .

Σ denotes the total capitalization of the considered equity market, i.e.

Σ =d∑

i=1

S i ,

and require that Σ is a.s. strictly positive, i.e. P[Σt > 0,∀t ≥ 0] = 1.

We consider trading strategies investing only in these d assets and do notintroduce a bank account.


Setting and notions in stochastic portfolio theory The financial market

Trading strategiesWe call a predictable, Rd -valued, S-integrable process ϑ a tradingstrategy with wealth process V v ,ϑ, where

V v ,ϑt =

d∑i=1

ϑitSit , 0 ≤ t ≤ T , with V v ,ϑ

0 = v ,

is the time t-value of the investment according to the strategy ϑ.

A trading strategy ϑ is called self-financing if V v ,ϑ satisfies

V v ,ϑt = v +

∫ t

0ϑsdSs , 0 ≤ t ≤ T .

A trading strategy is called v -admissible if V v ,ϑ ≥ 0 and V v ,ϑ− ≥ 0.

A trading strategy is called long-only, if it never sells any stock short,i.e. ϑ takes values in Rd

+.


Setting and notions in stochastic portfolio theory The market portfolio

The market portfolio and the relative wealth processSince one main interest is the performance of a portfolio with respect to themarket, the natural quantity to consider are the market weights, denoted byµ = (µ1, . . . , µd).

This amounts to choosing the total capitalization Σ as numeraire

µi =S i∑di=1 S

i=

S i

Σ, i ∈ 1, . . . , d.

µ takes values in ∆d , which denotes the unit simplex, i.e.,

∆d =

z ∈ [0, 1]d |

d∑i=1

z i = 1

.

For q ∈ R++ and a portfolio π the relative wealth process with respect tothe market portfolio is given by

Y q,ϑt =

V qΣ0,ϑt

Σt, 0 ≤ t ≤ T , with Y q,ϑ

0 =qΣ0

Σ0= q,


Setting and notions in stochastic portfolio theory Relative arbitrage with respect to the market portfolio

Relative arbitrage with respect to the market portfolio

Definition

A 1-admissible selffinancing trading strategy constitutes a relativearbitrage opportunity with respect to the market portfolio over the timehorizon [0,T ] if

P[Y 1,Σ0ϑT ≥ 1

]= 1 and P

[Y 1,Σ0ϑT > 1

]> 0

and a strong relative arbitrage opportunity if

P[Y 1,Σ0ϑT > 1

]= 1

holds true.

Note that the existence of relative arbitrages achieved with portfoliosdepends only on the market weight process µ.


Part I

Polynomial models in SPT - Theoretical part


Polynomial processes Definition

Polynomial processesPm: finite dimensional vector space of polynomials up to degreem ≥ 0 on D where D is a closed subset of Rn,

Pm :=

D 3 x 7→m∑|k|=0

αkxk,∣∣∣αk ∈ R

,

where we use multi-index notation k = (k1, . . . , kn) ∈ Nn0,

|k| = k1 + · · ·+ kn and xk = xk11 · · · xknn .

P denotes the vector space of all polynomials on D.

Definition

We call a D-valued time-homogeneous Markov process polynomial if for allk ∈ N, all f ∈ Pk , initial values x ∈ D and t ∈ [0,T ],

x 7→ Ex [f (Xt)] ∈ Pk .


Polynomial processes Definition

Polynomial processesPm: finite dimensional vector space of polynomials up to degreem ≥ 0 on D where D is a closed subset of Rn,

Pm :=

D 3 x 7→m∑|k|=0

αkxk,∣∣∣αk ∈ R

,

where we use multi-index notation k = (k1, . . . , kn) ∈ Nn0,

|k| = k1 + · · ·+ kn and xk = xk11 · · · xknn .

P denotes the vector space of all polynomials on D.

Definition

We call a D-valued time-homogeneous Markov process polynomial if for allk ∈ N, all f ∈ Pk , initial values x ∈ D and t ∈ [0,T ],

x 7→ Ex [f (Xt)] ∈ Pk .


Polynomial processes Characterization

Polynomial processes - CharacterizationTheorem (C., Keller-Ressel, Teichmann (2012))

Let m ≥ 2. Then for a Markovian Ito-semimartingale X with state spaceD and Ex [

∫‖ξ‖mK (Xt , dξ)] <∞ for all x ∈ D, t ∈ [0,T ] the following

are equivalent:1 X is a polynomial process.2 The differential characteristics of X denoted by

(b(Xt), c(Xt),K (Xt , dξ))t∈[0,T ] (with respect to the “truncation”function χ(ξ) = ξ) satisfy

bi (x) ∈ P1 i ∈ 1, . . . , n,

cij(x) +

∫ξiξjK (x , dξ) ∈ P2 i , j ∈ 1, . . . , n,∫ξkK (x , dξ) ∈ P|k| |k| = 3, . . . .



Polynomial processes - Tractability

Expectations of polynomials of Xt can be computed via matrixexponentials, more precisely for every k ∈ N, there exists a linear mapA on Pk , such that for all t ≥ 0, the semigroup (Pt) restricted to Pkcan be written as

Pt |Pk= etA.

⇒ Easy and efficient computation of moments without knowing thethe probability distribution or characteristic function.

Pathwise estimation techniques of the integrated covariance areparticular well suited to estimate parameters in high dimensionalsituations.



Relation to polynomial jump diffusions

Remark

Consider the following linear operator G : P → P defined by

Gf (x) =n∑

i=1

Di f (x)bi (x) +1

2

n∑i ,j=1

Dij f (x)cij(x)

+

∫Rn

(f (x + ξ)− f (x)−

n∑i=1

Di f (x)ξi

)K (x , dξ),

with (b, c ,K ) as above, which is usually called extended infinitesimalgenerator. A unique solution to the martingale problem for G thencorresponds to a polynomial process in the above sense.


Volatility stabilized market models Properties

Volatility stabilized market models - IntroductionVolatility stabilized market models (R. Fernholz and I. Karatzas (2005)) areexamples of polynomial processes with the following properties:

The wealth process of the market portfolio corresponds to a specific BlackScholes model;

The individual stocks reflect the fact that log prices of smaller stocks tendto have greater volatility than the log prices of larger ones,

They exhibit a constant positive excess growth rate

γµ∗ =1

2

d∑i=1

cµiiµi

=d − 1

2,

which is actually the defining property of sufficiently volatile models.

Such a positive excess growth rate allows for strong relative arbitrage withlong only portfolios on sufficiently long time horizons via functionallygenerated portfolios.

Strong relative arbitrage with long only portfolios on arbitrary time horizonsis also possible.


Volatility stabilized market models The model

The model

The dynamics of the asset prices in volatility stabilized market modelsare defined through

dS it = S i

t

(1 + α

2µitdt +

1√µit

dW it

), S i

0 = s i , i ∈ 1, . . . , d,

where α ≥ 0 and (W 1, . . . ,W d) is a standard Brownian motion.

Recalling Σ =∑d

i=1 Si and µi = S i

Σ , we can rewrite

dS it =

1 + α

2Σtdt +

√S itΣtdW

i ,

from which the polynomial property is easily seen.

We here consider weak solutions to such SDEs or equivalentlysolutions to the corresponding martingale problem.


Volatility stabilized market models Polynomial property

Polynomial property of the modelProposition

The volatility stabilized model (S1, . . . ,Sd) satisfies the following properties:

1 (S1, . . . ,Sd) is a polynomial diffusion process on Rd++ whose differential

characteristics are of the form

bSi,t =1 + α

2

d∑j=1

S jt , cSii,t = S i

t

d∑j=1

S jt

, cSij,t = 0.

2 The dynamics of the wealth process Σ of the market portfolio are describedby a Black Scholes model of the form

dΣt = Σt

(d(1 + α)

2dt + dBt

),

for some Brownian motion B, which is named stabilization by volatility(and drift in the case α > 0).



The market weights process

Proposition (compare I.Goia, S.Pal (2009))

The dynamics of the market weights (µ1, . . . , µd) can be described by amultivariate Jacobi process of the form

dµit =

(1 + α

2− d(1 + α)

2µit

)dt +

√µit(1− µi

t)dZit −

∑i 6=j

µit

√µjtdZ

jt ,

where Z denotes a d-dimensional standard Brownian motion. In particular(µ1, . . . , µd) is a polynomial diffusion process with respect to its natural filtration(made right continuous), with state space ∆d and differential characteristics ofthe form

bµi,t =1 + α

2− µi

t

d(1 + α)

2, cµii,t = µi

t(1− µit), cµij,t = −µi

tµjt .



The joint process

Proposition

The joint process (µ,S) is a polynomial process taking values in∆d × Rd

++. In particular, the instantaneous covariance between µi and S j

as well as µj and S i for i 6= j is given by

cµ,Sij ,t = cµ,Sji ,t = −µitSjt = −µjtS i

t , i 6= j

and between S it and µit by

cµ,Sii ,t = S it − µitS i

t .

Moreover, the instantaneous covariance between Σ and µi vanishes, i.e.cµ,Σi ,t = 0 for all i ∈ 1, . . . , d. Hence µ and Σ are independent.


Polynomial market weight and asset price models Definition and characterization

Polynomial market weight and asset price models

Definition

Let (µ,S) be a stochastic process taking values in ∆d × Rd+ such that

Σ > 0 and µi = S i

Σ for all i ∈ 1, . . . , d. We call (µ,S) a polynomialmarket weight and asset price model if

1 the weights process µ is Markovian with respect to its naturalfiltration (made right continuous) and

2 the joint process (µ,S) is a polynomial process.

Theorem (C. 2016)

The following assertions are equivalent:1 The process (µ,S) is a polynomial market weight and asset price

model with continuous trajectories.2 The processes µ and Σ are independent polynomial processes on ∆d

and R++ respectively.



Polynomial market weight and asset price models

Definition

Let (µ,S) be a stochastic process taking values in ∆d × Rd+ such that

Σ > 0 and µi = S i

Σ for all i ∈ 1, . . . , d. We call (µ,S) a polynomialmarket weight and asset price model if

1 the weights process µ is Markovian with respect to its naturalfiltration (made right continuous) and

2 the joint process (µ,S) is a polynomial process.

Theorem (C. 2016)

The following assertions are equivalent:1 The process (µ,S) is a polynomial market weight and asset price

model with continuous trajectories.2 The processes µ and Σ are independent polynomial processes on ∆d

and R++ respectively.



Admissible parametersDefinition

We call a triplet (βµ,Bµ, γµ) with βµ ∈ Rd , Bµ ∈ Rd×d and γµ ∈ Sdadmissible simplex parameter set if

γµ has nonnegative off-diagonal elements and all its diagonalelements are equal to 0,(Bµ)>1 + (β>1)1 = 0 and βµi + Bµij ≥ 0 for all i , j 6= i ∈ 1, . . . , d.

Remark

Note that the drift part could also be written as

β + Bµµ = Bµµ,

where Bµij = βi + Bµij for all i , j ∈ 1, . . . , d satisfies (Bµ)>1 = 0 andBij ≥ 0 for all i , j ∈ I , i 6= j . In order to be consistent with the literature,we however keep the notation with a constant term.



Polynomial market weight and asset price modelsTheorem (cont.)

Each of the conditions above implies the following assertion:There exists an admissible simplex parameter set (βµ,Bµ, γµ) and parametersκ, ϕ ∈ R+ satisfying 2κ− ϕ ≥ 0, λ, σ ∈ R, such that the differentialcharacteristics of (µ,Σ) are given by

bµi,t = βµi +d∑

j=1

Bµij µjt , cµii,t =

∑i 6=j

γµij µitµ

jt , cµij,t = −γµij µ

itµ

jt , i 6= j ,

bΣt = κ+ λΣt , cΣ

t = ϕΣt + σ2Σ2t , cΣ,µ

i,t = 0.

Conversely, for an admissible simplex parameter set (βµ,Bµ, γµ) and parametersκ, ϕ ∈ R+, λ, σ ∈ R, there exists a polynomial market weight and asset pricemodel whose differential characteristics are of the above form.

This builds on the characterization of polynomial diffusions on ∆d obtained byFilipovic and Larsson (2016).



Polynomial market weight and asset price model

Corollary

The following assertions are equivalent:

1 The process (µ,S) is a polynomial market weight and asset price model withcontinuous trajectories such that S is polynomial with respect to its ownfiltration.

2 The processes µ and Σ are independent polynomial processes on ∆d andR++ respectively with the additional property that Σ is a Black -Scholesmodel of the form

dΣt = λΣtdt + σΣtdBt

for some parameters λ, σ ∈ R and B a Brownian motion.

RemarkIf µ and S are polynomial processes each of which with respect to its naturalfiltration but not jointly, then more flexibility for Σ is possible.



Polynomial market weight and asset price model

Corollary

The following assertions are equivalent:

1 The process (µ,S) is a polynomial market weight and asset price model withcontinuous trajectories such that S is polynomial with respect to its ownfiltration.

2 The processes µ and Σ are independent polynomial processes on ∆d andR++ respectively with the additional property that Σ is a Black -Scholesmodel of the form

dΣt = λΣtdt + σΣtdBt

for some parameters λ, σ ∈ R and B a Brownian motion.

RemarkIf µ and S are polynomial processes each of which with respect to its naturalfiltration but not jointly, then more flexibility for Σ is possible.


Polynomial market weight and asset price models Properties

Properties

In polynomial market weight and asset prices models the wealth process ofthe market portfolio can be any polynomial diffusion on R+, in particular itcan be an affine process.

It is necessarily a Black & Scholes model if S is a polynomial process on Rd+

(in its natural filtration).

The individual stocks can still reflect the fact that log prices of smallerstocks tend to have greater volatility than the log prices of larger ones, butallow additionally for correlation, in particular we have

c log Sii =

1

µi

(σ2S i +

∑i 6=j γ

µij S

j + ϕµi

S

), c log S

ij = −γµij + σ2 +ϕ

S.

Positive excess growth rate: γµ∗ = 12

∑di=1

cµiiµi ≥ mini,j γ

µij

d−12 whenever

µ ∈ ∆d and γµ is non-degenerate, which implies the existence of certainfunctionally generated relative arbitrage opportunities over sufficiently longtime horizons



Properties





c log Sii =

1

µi

(σ2S i +

∑i 6=j γ

µij S

j + ϕµi

S

), c log S


S.


∑di=1


µij

d−12 whenever




Properties





c log Sii =

1

µi

(σ2S i +

∑i 6=j γ

µij S

j + ϕµi

S

), c log S


S.


∑di=1


µij

d−12 whenever




Properties





c log Sii =

1

µi

(σ2S i +

∑i 6=j γ

µij S

j + ϕµi

S

), c log S


S.


∑di=1


µij

d−12 whenever



Polynomial market weight and asset price models Extension with jumps

Extension with jumpsProposition (C. 2016)

Let µ and Σ be independent polynomial processes (both componentspossibly with jumps) on ∆d and R++ respectively. Assume that therespective jump measures satisfy

µ 7→∫

(ξµ)kK (µ, dξµ) ∈ P2(∆d) for |k| = 2

Σ 7→∫

(ξΣ)2K (Σ, dξΣ) ∈ P2(R++).

Define S i = µiΣ for i ∈ 1, . . . , d. Then (µ,S) is a polynomial marketweight and asset price model.

Remark

The assumptions on Σ are satisfied if Σ is for instance an affine CIRprocess with jumps.


Polynomial market weight and asset price models Extension with jumps

Extension with jumpsProposition (C. 2016)

Let µ and Σ be independent polynomial processes (both componentspossibly with jumps) on ∆d and R++ respectively. Assume that therespective jump measures satisfy

µ 7→∫

(ξµ)kK (µ, dξµ) ∈ P2(∆d) for |k| = 2

Σ 7→∫

(ξΣ)2K (Σ, dξΣ) ∈ P2(R++).

Define S i = µiΣ for i ∈ 1, . . . , d. Then (µ,S) is a polynomial marketweight and asset price model.

Remark

The assumptions on Σ are satisfied if Σ is for instance an affine CIRprocess with jumps.


Polynomial market weight and asset price models Relative arbitrage

Polynomial market weight models allowing for relativearbitrageTheorem (C. 2016)

Let T ∈ (0,∞) denote some finite time horizon and let µ be a polynomialdiffusion process for the market weights on ∆d with µ0 ∈ ∆d , being described byan admissible simplex parameter set (βµ,Bµ, γµ) with γµij > 0 for alli 6= j ∈ 1, . . . , d. Then the following assertions are equivalent:

1 The model satisfies NUPBR, i.e

limn→∞

supϑ∈J (µ)

P[Y 1,ϑT ≥ n] = 0

and there exist strong relative arbitrage opportunities.

2 There exists some i ∈ 1, . . . , d such that bµi > 0 for some elements inµi = 0 and for all such indices i , we have

2βµi + mini 6=j

(2Bµij − γµij ) ≥ 0.



Connection with boundary non-attainment

The above theorem is related to the boundary non-attainment of theprocess µ as follows:

Proposition

Let µ be a polynomial diffusion process for the market weights on ∆d

described by an admissible simplex parameter set (βµ,Bµ, γµ) with µi0 > 0P-a.s. Then the following assertions are equivalent:

1 For all t > 0, µit > 0 P-a.s.2 2βµi + mini 6=j(2Bµij − γ

µij ) ≥ 0.



Unique martingale deflator and completeness

Proposition

Let µ be a polynomial diffusion process for the market weights on ∆d asof the Theorem above satisfying one of the equivalent conditions (i) or(ii). Then the following assertions hold true:

1 There exists a unique strictly positive local martingale deflator Z .2 The model is complete in the sense that every boundedFT -measurable claim Y can be replicated via some strategy ψ. Moreprecisely,

Y = E [YZT ] +

∫ T

0ψ>s dµs , P-a.s.

holds true.


Implementation of the optimal arbitrage strategy

Optimal arbitrage - DefinitionDefinition

We denote by UT the superhedging price of 1 at time T > 0, that is,

UT := infq ≥ 0 | ∃ϑ ∈ J (µ) with Y q,ϑT ≥ 1 P-a.s.

and we call 1UT

optimal arbitrage.

Remark

By the superhedging duality and completeness, we have UT = E [ZT ] and the“price at time t of the optimal arbitrage” is given by

g(t, µt) =E [ZT |Ft ]

Zt=

E [1E (µT )ZT |Ft ]

Zt

P-a.s.= EQ [1E (µT )|Ft ], (1)

where Q denotes the so called Follmer measure for which P Q holds and underwhich µ is a martingale until it leaves the set E defined viaE := µ ∈ ∆d |µj > 0 for all j /∈ J, where J = j1, . . . , jk, 0 ≤ k ≤ d , denotethe set of indices for which bµji = 0 on µji = 0.



Optimal arbitrage - DefinitionDefinition

We denote by UT the superhedging price of 1 at time T > 0, that is,

UT := infq ≥ 0 | ∃ϑ ∈ J (µ) with Y q,ϑT ≥ 1 P-a.s.

and we call 1UT

optimal arbitrage.

Remark

By the superhedging duality and completeness, we have UT = E [ZT ] and the“price at time t of the optimal arbitrage” is given by

g(t, µt) =E [ZT |Ft ]

Zt=

E [1E (µT )ZT |Ft ]

Zt

P-a.s.= EQ [1E (µT )|Ft ], (1)

where Q denotes the so called Follmer measure for which P Q holds and underwhich µ is a martingale until it leaves the set E defined viaE := µ ∈ ∆d |µj > 0 for all j /∈ J, where J = j1, . . . , jk, 0 ≤ k ≤ d , denotethe set of indices for which bµji = 0 on µji = 0.



Computational aspects

Proposition (C. (2016))

Consider a polynomial diffusion market weight model satisfying eitherCondition (i) or equivalently (ii) of the above Theorem. Then for everyε > 0 there exists a time-dependent polynomial µ 7→ pε(t, µ) and astrategy defined via

ϑi ,εt = Dipε(t, µt) + pε(t, µt)−

d∑j=1

µjtDjpε(t, µt)

such that P[Y 1,ϑε

T > 1] ≥ 1− ε. Moreover, as ε→ 0, Y 1,ϑε

T convergesP-a.s. to the optimal arbitrage.


Part II

Polynomial models in SPT - Calibration

results

(based on ongoing joint work with K.Gellert, M. Giuricich, A. Platts,S.Sookdeo and J.Teichmann)


Calibration of polynomial market weight and asset pricemodel

For the implementation of optimal arbitrages only the covariance structure isof importance⇒ No drift estimation, only covariance estimation

Model:

dµt = · · · dt +√cµt dWt , dΣt = · · · dt + σΣtdBt ,

where cµii =∑

i 6=j γµij µ

iµj and cµij = −γµij µiµj with γµij = γµji ≥ 0 for all i 6= j .

Pathwise estimation of the integrated covariance 〈µi , µj〉T and 〈Σ,Σ〉 toobtain γij and σ

The structure of cµ allows to obtain γµ from any estimator of the integratedcovariance of logµ since

1

T

∫ T

0

clog(µ)ij,t dt = −γij , i 6= j .

This simple relationship enables to estimate the parameters of theinstantaneous covariance in any dimension.


Data and modeling assumptions

300 stocks representing the MSCI World Index, from August 2006 toOctober 2007, daily data.

Challenge: How to estimate γµ (44700 parameters) on the basis ofthis scarce data set?

Assumption: Significant correlations exist only for similar magnitudemarket capitalizations


Comparison market and simulated trajectories

Market trajectories

0 50 100 150 200 250 300

time (days)

0

5

10

15

Ci t

#104

Simulated trajectories

0 50 100 150 200 250 300

time (days)

0

2

4

6

8

10

12

14

16

18

Ci t

#104

Stochastic volatility (coming from the influences of other stocks) andcorrelation structure of market and simulated data is similar despite aslightly higher volatility in the simulated data over time.


Capital distribution curvesPreservation of the shape as well as reasonable dynamic behavior

0 1 2 3 4 5 6

log(rank)

-10

-9

-8

-7

-6

-5

-4

-3

log(7

i )

initialfinal


Outlook Extension to measure-valued polynomial processes

Outlook - Extension to measure-valued polynomialprocesses

Problem

Despite the tractability of the model class, the high dimensionality of themarket weights process is crucial ⇒ Develop a large financial markets per-spective

Possible Remedy

Take one underlying measure-valued process as model for a highdimensional market which can contain possibly a continuum of assets.Build prices of single assets via integrating some deterministicfunctions against the random measures.



Outlook - Extension to measure-valued polynomialprocesses

Problem

Despite the tractability of the model class, the high dimensionality of themarket weights process is crucial ⇒ Develop a large financial markets per-spective

Possible Remedy

Take one underlying measure-valued process as model for a highdimensional market which can contain possibly a continuum of assets.Build prices of single assets via integrating some deterministicfunctions against the random measures.



Outlook - Mathematical and financial setupMathematical setup:

E : (locally) compact Polish space

M1(E ): space of probability measures on E

(Xt)t∈[0,T ]: M1(E )-valued Markov process

Financial setup:

Market weights: Let C be some (finite or infinite) subset of C (E ) and let Xbe M1(E )-valued. Then the market weights can be defined by

µϕ := 〈ϕ,Xt〉 :=

∫E

ϕ(x)Xt(dx),

appropriate ϕ summing up to one.

Specifying X as a measure valued polynomial process leads to similartractability but to a much richer class of models enabling to treat an(uncountably) infinite number of assets by one process.



Outlook - Mathematical and financial setupMathematical setup:

E : (locally) compact Polish space

M1(E ): space of probability measures on E

(Xt)t∈[0,T ]: M1(E )-valued Markov process

Financial setup:

Market weights: Let C be some (finite or infinite) subset of C (E ) and let Xbe M1(E )-valued. Then the market weights can be defined by

µϕ := 〈ϕ,Xt〉 :=

∫E

ϕ(x)Xt(dx),

appropriate ϕ summing up to one.

Specifying X as a measure valued polynomial process leads to similartractability but to a much richer class of models enabling to treat an(uncountably) infinite number of assets by one process.



Open questions

Connection between asset prices and market weights within ameasure-valued polynomial framework?

Existence of arbitrages?

Existence of supermartingale deflators?

Functionally generated portfolios, in particular infinite dimension?

Ito type formulas and stochastic integration in the sense of Follmerfor measure valued processes?

Implications for capital distribution curve modeling? Capitaldistribution curves are one to one with the empirical measure1d

∑di=1 δS i

t, which for quite general specifications of S turns out to be

a probability measure valued polynomial process.


Conclusion

Conclusion and Outlook

Conclusion:

I Introduction of polynomial market weight and asset price models asgeneralization of volatility stabilized market models

I Characterization of the existence of relative arbitrage in these models

I Tractability properties to implement for instance (optimal) relativearbitrages

I Successful calibration of a polynomial market weight model for 300stocks based on relatively sparse data using a very direct method

I Promising empirical features, e.g. capital distribution curve

Outlook:

I Extension to measure-valued polynomial processes to treat highdimensionality and study limit behaviors

Thank you for your attention!


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Polynomial processes in stochastic portfolio theory · Introduction Overview Introduction and overview Stochastic portfolio theory (SPT) I theory for analyzing stock market and portfolio