Post on 09-May-2022
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
IntroductionMulti-asset Kyle’s model
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
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
ObjectivesThe model
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
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
ObjectivesThe model
Asymmetric information
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
Problem 3 : Dependence of the equilibrium in the risk aversion(or neutrality) of the insider.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
ObjectivesThe model
Asymmetric information
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
Problem 3 : Dependence of the equilibrium in the risk aversion(or neutrality) of the insider.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
ObjectivesThe model
Asymmetric information
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
Problem 3 : Dependence of the equilibrium in the risk aversion(or neutrality) of the insider.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
ObjectivesThe model
Asymmetric information
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
Problem 3 : Dependence of the equilibrium in the risk aversion(or neutrality) of the insider.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
ObjectivesThe model
Kyle’s model with one stock
Introduced in Kyle (1985) (see also Back (1992))
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
Kyle’s model with one stock and one option
Introduced in Back (1993)
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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 ].
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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)).
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
Pricing rule
We compute the pricing rule in a market with one stock andone call option.
The market maker has a Lognormal belief.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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).
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
LiteratureOptimal transportThe main resultNumerical results
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).
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
Existence of equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
Existence of equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
Properties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
Properties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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).
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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: (t, ξt) ∈ {(0.5, 0.5), (0.95, 0.5)}.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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: (t, ξt) ∈ {(0.5,−0.5), (0.95,−0.5)}.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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.
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
Thank you
THANK YOU!
Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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Ibrahim EKREN Information Asymmetry and Optimal Transport
IntroductionMulti-asset Kyle’s model
Risk averse insider
The modelExistence equilibriumProperties of the equilibrium
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Ibrahim EKREN Information Asymmetry and Optimal Transport