on the wealth dynamics of self-financing portfolios under endogenous prices

8/9/2019 On the Wealth Dynamics of Self-Financing Portfolios Under Endogenous Prices

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On the Wealth Dynamics of Self-financing

Portfolios under Endogeneous Prices

Jan Palczewski

Faculty of MathematicsUniversity of Warsaw

and

School of MathematicsUniversity of Leeds

Vienna, September 2007

Joint work with Jesper Pedersen and Klaus Schenk-Hopp.

Jan Palczewski On the Wealth Dynamics under Endogeneous Prices AMaMeF 2007, Vienna 1 / 16


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Motivation

Mathematical Finance

Classical continuoustime theory

Price process given

Option pricing

Optimal investment

Economics

Supply and demandPrices by market

clearing

Interaction of investors

Evolution of investors wealth

Price formation

Optimal strategies



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Classical continuous-time finance

Investors are price-takers

Trades have no impact on the marketDynamics of asset prices are given by a stochastic

process, e.g.

St = S0 exp(t + Bt).

There is infinite supply of financial assets

There is infinite divisibility of financial assets

Standing assumption

Small investors!!!

Infinite divisibility of financial assets = big investors



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Large trader and large trades

1 Option hedging has significant impact on stock prices

Empirical proofsLarge trader models: Frey (1998), Platen and Schweizer(1998), Bank and Baum (2004)

2 Large trades cannot be performed without being noticed

splitting large trades into smaller to lower market impact algorithmic tradingusing strategies based on econometric and mathematicalreasoning: Keym and Madhavan (1996), He and Mamaysky(2005)strategies based on analysis of limit order books

Limitations

only one large trader

traders impact on the market is ad-hoc specified



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Equilibrium with heterogeneous agents

many investors, heterogeneous beliefs

dividends

investors are utility maximizers

prices determined to clear the market

one-period models and overlapping generations (De Long,

Shleifer, Summers, Waldmann)

dynamic models are very complicated and often

unsolvable (Hommes)



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The Market

Asset k k= 1, 2

Price Sk(t)

Cumulative dividends Dk(t)

Dk(t) = t

0

k(s)ds

Assets in net supply of 1.



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The Market

Asset k k= 1, 2

Price Sk(t)

Cumulative dividends Dk(t)

Dk(t) = t

0

k(s)ds

Assets in net supply of 1.

Investor i i= 1, 2

Wealth Vi(t)

Consumption rate cVi(t)

Constant proportions

trading strategy (i1, i2)

Portfolio

number of shares of asset k:ikV

i(t)

Sk(t)



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Wealth dynamics

dVi(t) = capital gains + dividends consumption



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Wealth dynamics


Capital gains

2

k=1

ikVi(t)

Sk(t)dS

k(t)



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Wealth dynamics


Capital gains

2

k=1

ikVi(t)

Sk(t)dS

k(t)

Dividends

2

k=1

ikVi(t)

Sk(t)dDk(t)



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Wealth dynamics


Capital gains

2

k=1

ikVi(t)

Sk(t)dS

k(t)

Dividends

2

k=1

ikVi(t)

Sk(t)dDk(t)

Consumption

cVi

(t)dt



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Wealth dynamics


dVi(t) =2

k=

1

ikVi(t)

Sk(t)

dSk(t) + dDk(t)

cVi(t)dt



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Market clearing

Market clearing condition

1kVi(t)

Sk(t)+

2kVi(t)

Sk(t)= 1, k = 1,2.

Equivalent to the net clearing condition:

d1

k(t) + d2

k(t) = 0, k = 1,2.



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Price formation

Dividend intensities k(t)+

Investment strategies (i1, i2)

+

Investors wealth dynamics+

Market clearing condition

Asset prices Sk(t), k = 1,2


P i f i


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Price formation

Theorem

1 For any feasible

V1(0),V2(0)

there exists a unique

V1(t),V2(t) satisfyingwealth dynamicsandmarketclearing condition.

2 Asset price dynamics are given by

Sk(t) = 1kV

1(t) + 2kV2(t), k = 1,2.


M k i di id d i t iti


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Markovian dividend intensities

Relative dividend intensity (t) =1(t)

1(t) + 2(t)

[0,1]

Assumptions

1 (t) is a positively recurrent Markov process

2 its state space is countable3 its initial distribution is stationary (stationary economy)

Theorem

Relative dividend intensity process is ergodic:

limt

1

t

t0

(s)ds = E(0).


S l ti d i


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Selection dynamics

Theorem

If investor 1 follows strategy

= (11, 12) = (E(0),1 E(0))

and investor 2follows a strategy(21, 22) =

then

limt

1

t

t0

V1(s)

V1(s) + V2(s)ds = 1.

Remarks

1 is based on fundamental valuation.2 Relative wealth of investor 2 converges to zero.

3 At odds with findings in discrete-time evolutionary models

(Evstigneev, Hens, Schenk-Hopp).


P i d i


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Price dynamics

If one of the investors follows trading strategy then assetprices converge:

limt

1

t

t

0

S1(s)

S1(s) + S2(s)ds = E(0).


Price dynamics


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Price dynamics

If one of the investors follows trading strategy then assetprices converge:

limt

1

t

t

0

S1(s)

S1(s) + S2(s)ds = E(0).

Fundamental valuation

E1(0)

E1(0) + E2(0)

Our valuation

E

1(0)

1(0) + 2(0)

Remarks

1 Fundamental valuation comes as a result of computing

average historical payoffs.

2 Our valuation is a fundamentally different benchmark.


Almost sure convergence


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Almost sure convergence

Assumption

For every state x

Ex(x)

2


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Proof

What we hoped to do

Linearization and Lagrange multipliers

Multiplicative Ergodic Theorem

Why? It works fine in discrete-time.

Continous-time setting supprised us. Lagrange multiplier at

the steady state is zero!

What we have done

Domination by a Ricatti-type equation with random

coefficients.One coefficient depending on the solution of the original

problem.

Arcsine law.


Summary


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Summary

Heterogeneous investors in continuous time model

Wealth dynamics

Optimal investment strategies

Asset pricing - new valuation benchmark

Open problems

Time varying investment strategies

More agents


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