Information Asymmetry and Optimal Transport

IntroductionMulti-asset Kyle’s model

Risk averse insider

Information Asymmetry and Optimal Transport

Ibrahim EKREN

Florida State UniversityJoint works with F. Cocquemas, A. Lioui and S. Bose

October 2020University of Michigan

Ibrahim EKREN Information Asymmetry and Optimal Transport


Risk averse insider

Table of contents

1 IntroductionObjectivesThe model

2 Multi-asset Kyle’s modelLiteratureOptimal transportThe main resultNumerical results

3 Risk averse insiderThe modelExistence equilibriumProperties of the equilibrium



Risk averse insider

ObjectivesThe model

Table of Contents






Risk averse insider

ObjectivesThe model

Objectives

Understand the consequences of long-lived asymmetricinformation on price formation.

Obtain the ”(permanent) price impact” from the interactionof market participants with superior/inferior information.

Develop novel tools to handle information asymmetry.

Market maker(s) who has inferior information quotes fairprices and protect himself from the insider that has superiorinformation.

The pricing rule of the market maker provides a relationbetween the prices and the volumes.



Risk averse insider

ObjectivesThe model

Objectives

Understand the consequences of long-lived asymmetricinformation on price formation.

Obtain the ”(permanent) price impact” from the interactionof market participants with superior/inferior information.

Develop novel tools to handle information asymmetry.

Market maker(s) who has inferior information quotes fairprices and protect himself from the insider that has superiorinformation.

The pricing rule of the market maker provides a relationbetween the prices and the volumes.



Risk averse insider

ObjectivesThe model

Asymmetric information

Informed trader (aka Insider): knows thatTSLA will hit $800 in one yearObjective : Make the most profit from privateinformation

Problem 1 : Timing the ”injection of privateinformation”Problem 2 : Trading in stock or options

Problem 3 : Dependence of the equilibrium in the risk aversion(or neutrality) of the insider.



Risk averse insider

ObjectivesThe model


Informed trader (aka Insider): knows thatTSLA will hit $800 in one yearObjective : Make the most profit from privateinformationProblem 1 : Timing the ”injection of privateinformation”

Problem 2 : Trading in stock or options




Risk averse insider

ObjectivesThe model


Informed trader (aka Insider): knows thatTSLA will hit $800 in one yearObjective : Make the most profit from privateinformationProblem 1 : Timing the ”injection of privateinformation”Problem 2 : Trading in stock or options




Risk averse insider

ObjectivesThe model






Risk averse insider

ObjectivesThe model






Risk averse insider

ObjectivesThe model

Kyle’s model with one stock

Introduced in Kyle (1985) (see also Back (1992))



Risk averse insider

LiteratureOptimal transportThe main resultNumerical results

Table of Contents






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Kyle’s model with one stock and one option

Introduced in Back (1993)



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The data of the fundamental model with a call option

Consider a market with a stock and a call option on the stock(strike K). We fix the maturity T for the option and theinformation. Interest rate is 0.

Three market participants:

an insider who knows the value of the stock vS at time T . Hiscumulative demand at time t is Xt ∈ R2.uninformed noise traders who trade due to exogenous needs.Their cumulative demand is denoted Zt = ΣBt ∈ R2 where Bis a 2D Brownian motion independent from vS .a market maker who receives the total orders Yt = Xt + Zt

(without being able to disentangle them) and quotes a price Pt

based on (Ys)s≤t and νS = L(vS).

We denote v = (vS , (vS −K)+) ∈ R2 and ν = L(v), asingular measure on R2.



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Definition of equilibrium (risk neutral insider)

An equilibrium between the insider and the market maker

consists in a trading strategy for the insider X∗t ∈ F

v,Zt and

pricing rule Pt = H∗(t, Y·) such that:

If the market maker uses the pricing rule Pt = H∗(t, Y·), thenX∗ maximizes the expected wealth of the insider

supX

E

[∫ T

0

(v − Pt)>dXt −

2∑i=1

〈Xi, P i〉T |F v,Z0

]

If the insider uses the strategy X∗, the price of the assetPt = H∗(t,X∗· + Z·) is fair, i.e.

Pt = E[v|FYt ].



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Literature

Kyle (1985): Only one stock, ν is Gaussian, Linear-quadraticstructure.Back (1992): Only one stock, ν is absolutely continuous, viaan HJB equation whose final condition is a priori not known.Back (1993): One stock and one option with a very particularunrealistic assumption on the covariance of (Zt).Cetin and Danilova (2016): Market makers are risk averse.Collin-Dufresne and Fos (2016): (Zt) has stochastic quadraticvariation.Baudoin, Bouchaud, Baruch, Cho, Corcuera, Donnelly, Foster,Holden, Ma, Oksendal, Pedersen, Subrahmanyam, Vayanos,Viswanathan.Solution methods require an ansatz on the expected utility ofthe insider and existence of equilibrium is usually obtained forexplicitly solvable models.



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The main challenge to establish an equilibrium

The insider does not only control the state Yt, but alsocontrols the believe of the market maker on v.

In a Gaussian framework, the Kalman filter allows adescription of the evolution of this believe.

If ν is not Gaussian (which is the case with options), then onewould need to study the filtering SPDE which is not tractable.

Our main insight to find an equilibrium: Keep everythingGaussian and linear in the space of Y (volumes) and mapthese Gaussian distributions to the distribution ν of the pricev via an optimal transport map.

We filter in the Gaussian Y -space and price via an optimaltransport map.



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Brenier’s theorem

Theorem (Brenier (1991) and McCann (1995))

Let µ be an absolutely continuous probability measure on R2.Then, there exists a unique (up to an additive constant) convexfunction Γ so that ∇Γ pushes forward µ to ν. If ν is also absolutelycontinuous then, (∇Γ)−1 = ∇(Γc) pushes forward ν to µ.

Γ is also the minimizer for the problem

inf

{∫|x−Ψ(x)|2µ(dx) : Ψ]µ = ν

}.

If both measures admit smooth densities, then, Γ is convexand solves the Monge-Ampere equation

fµ(x) = Det(∇2Γ(x))fν(∇Γ(x)).



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Existence of equilibrium

Take µ = L(ZT ). An equilibrium pricing rule is

H∗(t, y) = E[∇Γ(y + ZT − Zt)]The market maker prices by transporting the volumes to thedistribution of the price.An equilibrium strategy for the insider is to construct aBrownian bridge to ”(∇Γ)−1(v)” via the trading rate

dX∗t =

”(∇Γ)−1(v)”− YtT − t

dt.

Theorem (Cocquemas, E., Lioui (2020))

The couple (H∗, X∗) forms an equilibrium.

The computation of the Brenier map is the main challenge in highdimension.



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Properties of the equilibrium

We only do filtering in a Gaussian space.

All non-linearity is hidden in the Brenier map.

Price impact matrix is a martingale. Market depth is asubmartingale.

All statements hold for multiple assets and options.

H∗ and its derivatives provide the dependence of the prices (ofall assets) and volatilities on the volumes (cross price impact).

We can extract an information based dynamics for the impliedvolatility smile of the stock.

We can obtain the dynamics of the belief of the market makeron v.



Risk averse insider



We only do filtering in a Gaussian space.

All non-linearity is hidden in the Brenier map.

Price impact matrix is a martingale. Market depth is asubmartingale.

All statements hold for multiple assets and options.

H∗ and its derivatives provide the dependence of the prices (ofall assets) and volatilities on the volumes (cross price impact).

We can extract an information based dynamics for the impliedvolatility smile of the stock.

We can obtain the dynamics of the belief of the market makeron v.



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Pricing rule

We compute the pricing rule in a market with one stock andone call option.

The market maker has a Lognormal belief.



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Evolution of prices

We simulate the order flow{(ZSt , ZCt ) : t ∈ [0, 1]}.Using the pricing function,we price both the stock andthe option.

Depending on the orderflows, the price of the calloption deviates from theBlack-Scholes prices.



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Implied volatility smile dynamics

We simulate 20000 orderflows in three differentmarkets with one stock and3 put options.

Dependence of the impliedvolatility smile on thecovariance structure of(ZS , ZP70, ZP100, ZP130).

We plot the smile

IV (70) + IV (130) − 2IV (100)

130 − 70.



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Sanity check

We regress the implied volatility of each option on the volumes

IV (P70) IV (P100) IV (P130) IV Curvature

Cst 0.192∗∗∗ 0.194∗∗∗ 0.189∗∗∗ −10.255∗∗

(0.003) (0.002) (0.004) (5.177)

ZS 0.017∗∗∗ 0.027∗∗∗ 0.042∗∗∗ 10.537∗∗∗

(0.002) (0.002) (0.003) (2.671)

ZP70 0.045∗∗∗ 0.002∗∗ 0.002 70.874∗∗∗

(0.001) (0.001) (0.001) (3.748)

ZP100,ATM 0.020∗∗∗ 0.087∗∗∗ 0.038∗∗∗ −192.426∗∗∗

(0.002) (0.004) (0.002) (12.127)

ZP130 −0.010∗∗∗ −0.008∗∗∗ 0.059∗∗∗ 107.529∗∗∗

(0.002) (0.002) (0.004) (6.705)

Time t 0.044∗∗∗ 0.048∗∗∗ 0.066∗∗∗ 23.516

(0.008) (0.006) (0.012) (16.886)

Obs. 20,000 20,000 20,000 20,000

Adj. R2 0.790 0.705 0.673 0.763

Res. S.E. 0.033 0.025 0.042 0.001

∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01



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Computational Challange

The Brenier map is essentially only explicit for ν Gaussian orin 1D.

LP, Sinkhorn, Greenkhorn etc... : curse of dimensionality.

Ongoing project : Neural network based approach.

Computing the Brenier map ∇Γ via an optimization problem.

Objective: Solve the heat equation and compute the Breniermap together with neural networks by computing

infθCostparabolic(H

θ(·, ·)) + CostBrenier(Hθ(1, ·))

and obtain an approximate pricing rule as Hθ∗(t, y).



Risk averse insider


Computational Challange

The Brenier map is essentially only explicit for ν Gaussian orin 1D.

LP, Sinkhorn, Greenkhorn etc... : curse of dimensionality.

Ongoing project : Neural network based approach.

Computing the Brenier map ∇Γ via an optimization problem.

Objective: Solve the heat equation and compute the Breniermap together with neural networks by computing

infθCostparabolic(H

θ(·, ·)) + CostBrenier(Hθ(1, ·))

and obtain an approximate pricing rule as Hθ∗(t, y).



Risk averse insider

The modelExistence equilibriumProperties of the equilibrium

Table of Contents






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Risk averse insider

We return to the one asset case. v ∈ R and ν = L(v).

We assume that the insider has a CARA utility function

supX

E[−γ exp (−γWT (X,H∗)) |F v,Z0

].

A forward-backward interaction between the market makerand the insider is expected.

The relevant state variable is not known.

Cho (2004) takes a restrictive definition of equilibrium andshows that an equilibrium exists if and only if ν is Gaussian.

In particular in such a case the price impact is deterministic.

The main difficulty is to find an appropriate final condition forthe interaction between the insider and the market maker.

We identify this final condition as a Brenier map.



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supX


].









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supX


].









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The fixed point condition

If φ is known, the optimality condition yields to the followingbackward quasilinear PDE for the pricing rule H

Ht(t, ξ) +σ2Hξξ(t, ξ)

2(1− γσ2Hξ(t, ξ)(T − t))2= 0

H(T, ξ)=φξ(ξ)

To generate the relevant Markov bridges, the forwardcomponent has to satisfy

dξ0t =σdBt

1− γσ2Hξ(t, ξ0t )(T − t)

with the transport type constraint which is

φ is the Brenier map that pushes L(ξ0T ) to ν.



Risk averse insider


The fixed point condition

If φ is known, the optimality condition yields to the followingbackward quasilinear PDE for the pricing rule H

Ht(t, ξ) +σ2Hξξ(t, ξ)

2(1− γσ2Hξ(t, ξ)(T − t))2= 0

H(T, ξ)=φξ(ξ)

To generate the relevant Markov bridges, the forwardcomponent has to satisfy

dξ0t =σdBt

1− γσ2Hξ(t, ξ0t )(T − t)

with the transport type constraint which is

φ is the Brenier map that pushes L(ξ0T ) to ν.



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A system of forward (Fokker-Planck) and backward(quasilinear parabolic) equations with a transport type fixedpoint condition at maturity.

Mean-field-like system where the final condition isendogenously determined via the fixed point condition.

Assumption

dν(x) = e−V (x)dx for some strongly convex function V .

Cafarelli’s contraction theorem: The Brenier map pushing µforward to ν is Lipschitz.

Theorem (Bose and E. (2020))

For γ small enough such a fixed point exists.



Risk averse insider



A system of forward (Fokker-Planck) and backward(quasilinear parabolic) equations with a transport type fixedpoint condition at maturity.

Mean-field-like system where the final condition isendogenously determined via the fixed point condition.

Assumption

dν(x) = e−V (x)dx for some strongly convex function V .

Cafarelli’s contraction theorem: The Brenier map pushing µforward to ν is Lipschitz.

Theorem (Bose and E. (2020))

For γ small enough such a fixed point exists.



Risk averse insider


Equilibrium strategies

One can show that there exists a smooth deterministicincreasing function χ so that

χ(t, ξt) = χ(0, 0) + Yt +

∫ t

0α(s, χ(t, ξs))ds.

The market maker tracks ξt and uses the pricing rule

Pt = H(t, ξt).

Via Doob’s h-transform, the insider generates a Markov bridgefor (ξt) using the strategy

dXt ∝(φξ)

−1(v)− ξtT − t

dt.



Risk averse insider


Proof

To find a fixed point, we use Schauder’s fixed point theorem.

Caffarelli’s contraction theorem provides a compact spacewhere we can find a fixed point.

We need continuous dependence estimates for the quasilinearequation (up to the second derivatives).

We establish a novel stochastic representation for this type ofequations.

Using Levy’s parametrix method, the Fokker-Planck equationadmits a representation via the solution to the backwardequation.



Risk averse insider


Proof

To find a fixed point, we use Schauder’s fixed point theorem.

Caffarelli’s contraction theorem provides a compact spacewhere we can find a fixed point.

We need continuous dependence estimates for the quasilinearequation (up to the second derivatives).

We establish a novel stochastic representation for this type ofequations.

Using Levy’s parametrix method, the Fokker-Planck equationadmits a representation via the solution to the backwardequation.



Risk averse insider


Properties of the state variable

ξ is endogenously determined via the fixed point condition.

The strategy of the insider renders (ξt) a Markov bridge.

The terminal value of ξ is given by (∇φ)−1(v).

The order flow of the insider is not predictable by the marketmaker, (ξt) and (Pt) are martingale in their own filtration.

Price converges to v.



Risk averse insider



The dynamics of the price is given by

dPt =Hξ(t, ξt)

1− γσ2(T − t)Hξ(t, ξt)dYt = λtdYt

Price impact λt depends on ξt and is a supermartingale forthe market maker.

The market depth 1λt

is a submartingale.

γ = 0 corresponds to Back (1992). In this case, λt is constant(martingale).

Due to risk aversion, close to maturity, the insider will trademore moderately and the market maker accepts to providemore liquidity close to maturity.



Risk averse insider



The dynamics of the price is given by

dPt =Hξ(t, ξt)

1− γσ2(T − t)Hξ(t, ξt)dYt = λtdYt

Price impact λt depends on ξt and is a supermartingale forthe market maker.

The market depth 1λt

is a submartingale.

γ = 0 corresponds to Back (1992). In this case, λt is constant(martingale).

Due to risk aversion, close to maturity, the insider will trademore moderately and the market maker accepts to providemore liquidity close to maturity.



Risk averse insider


Comparison to literature

Similar to Cho (2004), if ν is Gaussian, the market depth isdeterministic and

d 1λt

dt= γσ2.

The drift of the market depth is minimal for Gaussian beliefs.

With ν non-Gaussian, the insider needs to face an additionalrisk which is the uncertainty on the price impact.

In average, the market depth increases more withnon-Gaussian beliefs.

We can compute the expected utility of the insider via theforward-backward system.



Risk averse insider


Numerical results

Fokker-Planck admits a representation using the backwardequation.

The backward equation admits a stochastic representationcomputable via a gradient descent.

In 1D the computation of the Brenier map is explicit via theCDFs of the distribution (in multi-dimension we need to studya Monge-Ampere equation).

We can numerically find a fixed point.



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Comparing to Cho (2004)

−4 −2 0 2 4xi_T

−7.5

−5.0

−2.5

0.0

2.5

5.0

7.5

10.0

P(T,xi_T)

Pricing rule at maturit with Gaussian beliefs

Figure: Dashed: Cho (2004). Blue: Numerically computed fixed pointIbrahim EKREN Information Asymmetry and Optimal Transport


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Transport map at maturity

−10.0 −7.5 −5.0 −2.5 0.0 2.5 5.0 7.5 10.0xi_T

10

12

14

16

18

20

P(T,xi_T)

Pricing rule at maturity with uniform belief

Figure: Uniform beliefIbrahim EKREN Information Asymmetry and Optimal Transport


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Transport map at maturity

−4 −2 0 2 4xi_T

0

10

20

30

40

50

60

P(T,xi_T)

Pricing rule at maturity with lognormal belief

Figure: Lognormal beliefIbrahim EKREN Information Asymmetry and Optimal Transport


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Evolution of Belief

10 12 14 16 18 20Price level

0.0

0.2

0.4

0.6

0.8

1.0Cd

f of v

Conditionl cdf of the finl price vs the initil cdf

Figure: CDF of v conditional to (t, ξt) ∈ {(0, 0), (0.5, 0), (0.95, 0)} versusthe initial CDF of v ∼Unif(10, 20).



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Evolution of Belief

10 12 14 16 18 20Price level

0.0

0.2

0.4

0.6

0.8

1.0Cd

f of v


Figure: (t, ξt) ∈ {(0.5, 0.5), (0.95, 0.5)}.



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Evolution of Belief

10 12 14 16 18 20Price level

0.0

0.2

0.4

0.6

0.8

1.0Cd

f of v


Figure: (t, ξt) ∈ {(0.5,−0.5), (0.95,−0.5)}.



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Summary

We provide a novel method to find equilibrium in financialmarkets with long-lived asymmetric information based onoptimal transport.

In order to be able to construct Markov bridges, we postulatethat the pricing rule at maturity is a Brenier map pushing thedistribution of the state at maturity µ to the belief of themarket maker, ν.

Brenier map completely elucidates the coupling between theproblems of the insider and the market maker.

The relevant Brenier map is found via a fixed point conditionfor a system of forward (Focker-Planck) and backward(quasilinear) equation.

We discussed properties of the equilibria we obtain.



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Thank you

THANK YOU!



Risk averse insider


Back, Kerry. ”Insider trading in continuous time.” The Reviewof Financial Studies 5.3 (1992): 387-409.

Back, Kerry. ”Asymmetric information and options.” TheReview of Financial Studies 6.3 (1993): 435-472.

Bose, Shreya, and Ibrahim Ekren. ”Kyle-Back Models with riskaversion and non-Gaussian Beliefs.” arXiv preprintarXiv:2008.06377 (2020).

Brenier, Yann. ”Polar factorization and monotonerearrangement of vector-valued functions.” Communicationson pure and applied mathematics 44.4 (1991): 375-417.

Caffarelli, Luis A. ”Monotonicity Properties of OptimalTransportation and the FKG and Related Inequalities.”Communications in Mathematical Physics 214.3 (2000):547-563.



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Cetin, Umut, and Albina Danilova. ”Markovian Nashequilibrium in financial markets with asymmetric informationand related forward-backward systems.” The Annals of AppliedProbability 26.4 (2016): 1996-2029.

Cetin, Umut, and Albina Danilova. ”On pricing rules andoptimal strategies in general Kyle-Back models.” arXivpreprint arXiv:1812.07529 (2018).

Cho, Kyung-Ha. ”Continuous auctions and insider trading:uniqueness and risk aversion.” Finance and Stochastics 7.1(2003): 47-71.

Cocquemas, Francois, Ekren, Ibrahim, and Lioui Abraham. ”AGeneral Solution Technique For Insider Problems.” arXivpreprint arXiv:2006.09518.

Collin-Dufresne, Pierre, and Vyacheslav Fos. ”Insider trading,stochastic liquidity, and equilibrium prices.” Econometrica 84.4(2016): 1441-1475.



Risk averse insider


Corcuera, Jose Manuel ,and Di Nunno, Giulia.”Path-dependent Kyle equilibrium model.” arXiv preprintarXiv:2006.06395 (2020)

Garcia del Molino, Luis Carlos, et al. ”The Multivariate KyleModel: More is Different.” SIAM Journal on FinancialMathematics 11.2 (2020): 327-357.

Holden, Craig W., and Avanidhar Subrahmanyam. ”Riskaversion, imperfect competition, and long-lived information.”Economics Letters 44.1-2 (1994): 181-190.

Kyle, Albert S. ”Continuous auctions and insider trading.”Econometrica: Journal of the Econometric Society (1985):1315-1335.

Ladyzenskaja, Olga A., Vsevolod Alekseevich Solonnikov, andNina N. Uralceva. Linear and quasi-linear equations ofparabolic type. Vol. 23. American Mathematical Soc., 1988.


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Information Asymmetry and Optimal Transport