Equilibrium option price with competing market makers

IntroductionThe model

Option market equilibriumFuture Work

Equilibrium option price with competing marketmakers

Ibrahim EKREN

Florida State UniversityJoint work in progress with S. Nadtochiy and Y. Stoev

October 2018, University of Michigan

Ibrahim EKREN Equilibrium option price with competing market makers



Table of contents

1 IntroductionObjectiveLiterature

2 The modelAuxiliary problem with N = 1Interaction of N market makers in the stock marketNash equilibrium in stock market

3 Option market equilibriumOption marketCandidate Nash equilibriumThe main result

4 Future Work




ObjectiveLiterature

Objective

Compute the price of a less liquid derivative using theequilibrium between market makers and the stock market.

Obtain the ”impact curve” in the less liquid derivativesmarket given the ”impact curve” in the stock market.

Market makers can adjust their quotes of the option price toattract clients.

Understand the reaction of the market makers against anorder flow of options.

Extensions.




ObjectiveLiterature

Classical mathematical finance

Markets are frictionless.

In the classical theory of mathematical finance, there is aunique price process S and agents can do transaction withoutaffecting S.

The optimal investment strategies or hedging strategies arediffusive.




ObjectiveLiterature

Price Impact

In reality, there is an adverse effect of your transactions on theprice.

If you buy a lot of shares very fast then you will drive the priceup (⇒ price impact)

There is growing literature which models this effect.

Phenomenological approach: reduced-form models for theimpact observed in the market.

Study the effect of this impact structure on hedging problems.

Almgren, Bichuch, Bouchaud, Bayraktar, Bertsimas, Cetin,Chris, E, Garleanu, Gatheral, Guasoni, Jarrow, Muhle-Karbe,Obizawa, Pederson, Protter, Touzi, Shreve, Soner, Wang.




Auxiliary problem with N = 1Interaction of N market makers in the stock marketNash equilibrium in stock market

Market participants

In our model there are three types of market components:

Client(s) that wants to trade an option with payoff H(s).N Market makers with N > 1.Stock market.

Clients want to trade the option in the OTC market.

We assume that the market makers are symmetric and havethe same exposure at the beginning of times.

Each market maker is free to quote any price to the client.

The best price gets the order from the client.

Equally distributed among market makers if the same bestprice is quoted by many market makers.





Market participants

In our model there are three types of market components:

Client(s) that wants to trade an option with payoff H(s).N Market makers with N > 1.Stock market.

Clients want to trade the option in the OTC market.

We assume that the market makers are symmetric and havethe same exposure at the beginning of times.

Each market maker is free to quote any price to the client.

The best price gets the order from the client.

Equally distributed among market makers if the same bestprice is quoted by many market makers.





Market participants





Market participants





Comparison to the literature

Price formation of less liquid derivatives from inventory risk ofmarket makers.

Golsten/Milgrom (1985) : information asymmetry and themarket makers cannot offset their risk in the stock market.

Bank/Kramkov (2015) : The market makers trade amongeach other to attain an equilibrium.

Our equilibrium price is a type of utility indifference pricesimilar to Davis/Yoshikawa(2010).

Bank/E/Muhle-Karbe (2018): market makers choosequantities to trade.

Utility maximization problem is studied under assumption byO. Gueant.





Utility maximization





Evolution of the state

We first study the utility maximization problem of a marketmaker facing linear price impact.

Bachelier modelSt = s+ σWt.

The market maker holds Q options ⇒ QH(ST ) to be hedgedin the stock market.

Her position in stock at time t ≥ 0 is

πt = π +

∫ t

0νrdr where ν is her control.

Wealth accumulated

under linear price impact

∫ t

0πrdSr

− η∫ t

0ν2rdr






We first study the utility maximization problem of a marketmaker facing linear price impact.

Bachelier modelSt = s+ σWt.

The market maker holds Q options ⇒ QH(ST ) to be hedgedin the stock market.

Her position in stock at time t ≥ 0 is

πt = π +

∫ t

0νrdr where ν is her control.

Wealth accumulated under linear price impact∫ t

0πrdSr − η

∫ t

0ν2rdr





Utility of the market maker

Denote γ > 0 risk aversion parameter, define V (0, s, x, π,Q)

supν

E[− exp

(−γ(XT + πTST

−l(πT )2

+QH(ST )) )]

.

where XT = x−∫ T0 νr(Sr + ηνr)dr.

Without price impact η = 0. There exist π0(t, s) such that

E[H(ST )] +

∫ T

0π0(r, Sr)dSr = H(ST ), P− a.s.

The optimal strategy is πr = −Qπ0(r, Sr)1r<T and

V (0, s, x, π,Q) = −e−γ(x+πs+QE[H(ST )]).





Utility of the market maker

Denote γ > 0 risk aversion parameter, define V (0, s, x, π,Q)

supν

E[− exp

(−γ(XT + πTST−l(πT )2 +QH(ST )

) )].

where XT = x−∫ T0 νr(Sr + ηνr)dr.

Without price impact η = 0. There exist π0(t, s) such that

E[H(ST )] +

∫ T

0π0(r, Sr)dSr = H(ST ), P− a.s.

The optimal strategy is πr = −Qπ0(r, Sr)1r<T and

V (t, s, x, π,Q) = −e−γ(x+πs+QEt,s[H(ST )]).





A second order parabolic PDE

For general η > 0 we introduce the function u

u(t, s, π,Q) := ln (−V (t, s, x, π,Q))+γ(x+πs+QEt,s[H(ST )])

that has quadratic growth in π and solves the PDE

0 = ∂tu+σ2

2∂ssu−

1

4ηγ(∂πu)2+

σ2

2

(∂su− γ(π +Q∂sπ

0(t, s)))2

Degenerate second order, quadratic concave+convexHamiltonian.

If H is Lipschitz continuous. u is Lipschitz continuous in s.

Comparison of viscosity solutions holds among functionsquadratic in π and Lipschitz in s ⇒ u is continuous.





Boundedness of controls

Theorem (Main result for the utility maximization problem)

If H is Lipschitz continuous, then V is C1+α in (s, x, π,Q). Theunique optimal control is of feedback form and it satisfies

ν∗(t, s, π,Q) = −κ(t)πt + EQν∗[∫ T

tK(t, r)(∂sur −Qπ0r )dr

]πr = π +

∫ r

tν∗rdr

This is in fact a FBSDE’s whose wellposedness was unknown.

Without ∂su this would be the formula obtained inBank/Soner/Voss (2015) in a linear quadratic framework.





Methodology(outline)

Improve the regularity of the viscosity solution fromcontinuous to C1+α.

Penalise |ν| ≤ ε−1 ⇒ strong convexity.

Existence of unique optimiser (νε,t,s,π,Qr )r∈[t,T ] that dependscontinuously in (s, π,Q)

Exploit the first order condition to show that (νε,t,s,π,Qr )r∈[t,T ]is a solution to an FBSDE.

FBSDE implies that (νε,t,s,π,Qr )r∈[t,T ] is bounded uniformly inε > 0.

⇒ Bounded controls for both the penalised and the limitingproblem.





Methodology

The feedback optimal control is a solution of

∂tν∗ +

σ2

2∂ssν

∗ + ∂πν∗ν∗ + Z(t, s, π)

(σ

2η+ ∂sν

∗)

= 0

where Z(t, s, π) :=(∂su− γ(π +Q∂sπ

0(t, s))).

On the other hand, the envelope theorem shows that thevalue and its derivatives are continuous in (s, π,Q) and

(∂πu)(t, s, π,Q) = ∂π(J )(t, s, π,Q, ν∗,t,s,π,Q)

Theorem

ν∗ = −∂πu2ηγ is Holder continuous in π, s,Q and decreasing in π.

Byproduct: exponential utility maximization with linear priceimpact and smooth target position⇒ Bounded controls.





Interacting market makers





Methodology

To find a Nash equilibrium, for any (Qr) we need to havesolutions to the equation

π∗t = π +

∫ t

0ν∗(r, Sr, π

∗r , Qr)dr

This is an ODE with random coefficients and ν∗ is onlyHolder continuous in π

However, ν∗ is decreasing in π. Maximum of two solutions isa solution ⇒ unique solution.





Trading in the stock market

We now assume that N market makers start from the samestate (π, x,Q).

They interact through the price impact they generate

πit = π +

∫ t

0νirdr,

Xit = x−

∫ t

0νir

Sr + η

νir +

N∑j 6=i

νjr

dr.

Almgren and Chriss model with interaction.

The market makers need to make the trade off between theirwish to track π0(t, St), the impact they generate, and theimpact of other market makers.





Nash equilibrium

Assume that market makers have the same initial state (π,Q).

Definition

A Nash equilibrium is a strategy (νr) such that if all marketmakers except the ith use the strategy (νr) then it is optimal forthe ith market makers to use the strategy (νr); i.e.,

J i(ν, . . . , ν, . . . ν) ≥ J i(ν, . . . , νi, . . . ν), for all νi

Denote η̄ := η(N+1)2 and V̄ the value functions where the

market impact parameter η is replaced by η̄.





Description of Nash equilibrium

In order to exhibit our candidate equilibrium strategy define

π∗t = π +

∫ t

0ν̄∗(r, Sr, π

∗r , Q)dr

and consider ν̄∗(r, Sr, π∗r , Q) as a mapping of (r, ω) and NOT

of feedback type for any type of market maker

(r, ω) 7→ ν∗r (ω) := ν̄∗(r, ω, Sr(ω), π∗r (ω), Q).

We assume that all market makers j 6= i pre-commits to thisopen-loop strategy (ν∗r (ω))

Regardless of the actions of the ith agentRegardless of their current positions





Description of Nash equilibrium


π∗t = π +

∫ t

0ν̄∗(r, Sr, π

∗r , Q)dr

and consider ν̄∗(r, Sr, π∗r , Q) as a mapping of (r, ω) and NOT

of feedback type for any type of market maker

(r, ω) 7→ ν∗r (ω) := ν̄∗(r, ω, Sr(ω), π∗r (ω), Q).

We assume that all market makers j 6= i pre-commits to thisopen-loop strategy (ν∗r (ω))

Regardless of the actions of the ith agentRegardless of their current positions





Methodology of the proof

ν∗ not of feed back type⇒ Convex problem for agent i:

νi 7→ J i(ν∗, . . . , νi, . . . ν∗).

Is ν∗ a maximiser?

Two methods:Dynamic programming

The value function is martingale if i uses ν∗

supermartingale otherwise.

Maximum principle

Martingality type condition for the value functionand its derivativesONLY along optimal trajectories.





Methodology of the proof

ν∗ not of feed back type⇒ Convex problem for agent i:

νi 7→ J i(ν∗, . . . , νi, . . . ν∗).

Is ν∗ a maximiser?

Two methods:Dynamic programming

The value function is martingale if i uses ν∗

supermartingale otherwise.

Maximum principle

Martingality type condition for the value functionand its derivativesONLY along optimal trajectories.





Main result for the stock market interaction

Theorem

ν∗ is a Nash equilibrium and starting from the position (s, π,Q),any market makers’ position is π∗ and

V̄ (0, s, x, π,Q) = supνiJ i(ν∗, . . . , νi, . . . ν∗).

If all other market makers pre-commit to ν∗, the ith marketmaker has to pre-commit to ν∗.

A posteriori, π∗ is in fact the position of each market maker.





Relations between envelop theorem and Maximum principle

V̄ is defined for one market maker as

V̄ (t, s, x, π,Q) = supνiJ (t, s, x, π,Q, ν)

= J (t, s, x, π,Q, νt,s,x,π,Q)

Envelop theorem yields

∂V̄ (t, s, x, π,Q) = (∂J )(t, s, x, π,Q, νt,s,x,π,Q)

We have a good description of the properties of the valuefunction and its derivatives along the optimal trajectory.

Convexity and these properties are what we need for theMaximum principle.





Methodology

The choice η̄ = N+12 η implies that

ν̄∗t = arg maxν

(−∂xV̄tν(St + η(ν + (N − 1)ν̄∗t )) + ∂πV̄tν

).

Each market maker reevaluates her market impact toη̄ = N+1

2 η.

Then the market maker applies the optimal policycorresponding to the value function of the optimal investmentproblem with one agent and market impact η̄.

The only parameters needed are (t, s) and initial endowments(π,Q).





Methodology

The choice η̄ = N+12 η implies that

ν̄∗t = arg maxν

(−∂xV̄tν(St + η(ν + (N − 1)ν̄∗t )) + ∂πV̄tν

).

Each market maker reevaluates her market impact toη̄ = N+1

2 η.

Then the market maker applies the optimal policycorresponding to the value function of the optimal investmentproblem with one agent and market impact η̄.

The only parameters needed are (t, s) and initial endowments(π,Q).




Option marketCandidate Nash equilibriumThe main result

Market participants





Order flow of option

Each market makers starts with the position

πi0 = π stock,Qi

0 = Q options.

We now assume that there is an order flow of option.

qtdt during the period dt.

Market makers are free to choose the option price P it .

The order goes to the best price or equally distributed if sameprice is given by many market makers.

The state of each market maker is (t, St, πit, Q

it).

The control of the agent is (P it , νit).






Assume that every market maker except ith chooses thestrategies (Pt, νt) and the ith market maker chooses thestrategies (P it , ν

it)

Then the evolution of the state is

dπit = νitdt

dQit =

((1

N1P i

t=Pt+ 1P i

t>Pt

)(qt)

− −(

1

N1P i

t=Pt+ 1P i

t<Pt

)(qt)

+

)dt,

dXit = −P it dQit − νit(St + η(νit + (n− 1)νt))dt

The utility of the market maker is J i(π, s, x,Q; νi, pi, ν, p, q)

E[− exp

(−γ(XiT + πiTST−l(πiT )2 +QiTH(ST )

))].





Utility of a market maker

The market makers optimize criterion of type

supνi,piJ i(π, s, x,Q; νi, pi, ν, p, q).

Definition

A Nash equilibrium is a strategy (p, ν) such that if all marketmakers except the ith use the strategy (p, ν), then it is optimal forthe ith market makers to use the strategy (p, ν), i.e.

infqtJ i(π, s, x,Q; ν, p, ν, p, q) ≥ inf

qtJ i(π, s, x,Q; νi, pi, ν, p, q),

for all pi, νi.





Utility of a market maker

The market makers optimize criterion of type

supνi,pi

infqtJ i(π, s, x,Q; νi, pi, ν, p, q).

Definition

A Nash equilibrium is a strategy (p, ν) such that if all marketmakers except the ith use the strategy (p, ν), then it is optimal forthe ith market makers to use the strategy (p, ν), i.e.

infqtJ i(π, s, x,Q; ν, p, ν, p, q) ≥ inf

qtJ i(π, s, x,Q; νi, pi, ν, p, q),

for all pi, νi.





Candidate strategy in the stock market


Qt := Q+

∫ t

0

qrNdr

π∗t = π +

∫ t

0ν̄∗(r, Sr, π

∗r , Qr)dr

and consider ν̄∗(r, Sr, π∗r , Qr) as a mapping of (r, ω) and

NOT of feedback type

(r, ω) 7→ ν∗r (ω) := ν̄∗(r, ω, Sr(ω), π∗r (ω), Qr(ω)).

We assume that all market makers j 6= i pre-commits to thisopen-loop strategy (ν∗r )

Regardless of the actions of the i agentsRegardless of their current positions





Candidate pricing rule: Utility indifference price

Define the process

p∗t =∂QV̄ (t, St, 0, π

∗t , Qt)

∂xV̄ (t, St, 0, π∗t , Qt).

Similarly, consider it as a mapping

(t, ω) 7→ p∗(t, ω) :=∂QV̄ (t, St(ω), 0, π∗t (ω), Qt(ω))

∂xV̄ (t, St(ω), 0, π∗t (ω), Qt(ω))

N − 1 agents offers this (open-loop) price for the optionregardless of their agent’s state or the ith.

Is it optimal for the ith market maker to follow the strategy(p∗, ν∗)?





Main result

Theorem

(p∗, ν∗) is a Nash equilibrium and starting from the position(s, π,Q), the market makers stay symmetric and

V̄ (0, s, x, π,Q) = sup(pi,νi)

infqtJ i(π, s, x,Q : νi, pi, ν, p, q).

Additionally, (p∗, ν∗, q ≡ 0) is a saddle point.

The proof is based on the optimality of ν∗ for the case q ≡ 0.





Methodology

For simplicity assume V̄ is smooth. Application of Ito’sformula yields

dV̄ (t, St, Xit , π

∗t , Qt) =

(∂πV̄ ν

it − ∂xV̄ ν

it(St + η(νit + (N − 1)νt))

)dt

+

(∂tV̄ +

σ2

2∂ssV̄

)dt+

(∂QV̄ − pit∂xV̄

)dQit

+ dMt

The optimality of (ν∗, p∗) cancels the drift.

Marginal utility indifference price.

This is also a locally order flow indifference price.





Methodology

For simplicity assume V̄ is smooth. Application of Ito’sformula yields

dV̄ (t, St, Xit , π

∗t , Qt) =

(∂πV̄ ν

it − ∂xV̄ ν

it(St + η(νit + (N − 1)νt))

)dt

+

(∂tV̄ +

σ2

2∂ssV̄

)dt+

(∂QV̄ − pit∂xV̄

)dQit

+ dMt

The optimality of (ν∗, p∗) cancels the drift.

Marginal utility indifference price.

This is also a locally order flow indifference price.





Properties of the price

The option price only depends on (t, St, π∗t , Qt).

A posteriori (π∗t , Qt) is the position of each market maker atequilibrium.

Unlike in Bank/E/Muhle-Karbe (2018), the market makers donot need to predict the future order flow and an equilibriumcan be reached with only the knowledge of the current state.

Recall

P ∗t =∂QV̄ (t, St, 0, π

∗t , Qt)

∂xV̄ (t, St, 0, π∗t , Qt)

= EQ(ν∗,t,St,π̂t,Qt )t [H(ST )].

where Q(ν∗,t,St,π̂t,Qt) is the measure under which

(dWr − Z(r, Sr, π∗,t,St,π̂t,Qtr )dr)r≥t

is a martingale.





Properties of the price

The option price only depends on (t, St, π∗t , Qt).

A posteriori (π∗t , Qt) is the position of each market maker atequilibrium.

Unlike in Bank/E/Muhle-Karbe (2018), the market makers donot need to predict the future order flow and an equilibriumcan be reached with only the knowledge of the current state.

Recall

P ∗t =∂QV̄ (t, St, 0, π

∗t , Qt)

∂xV̄ (t, St, 0, π∗t , Qt)= EQ(ν∗,t,St,π̂t,Qt )

t [H(ST )].

where Q(ν∗,t,St,π̂t,Qt) is the measure under which

(dWr − Z(r, Sr, π∗,t,St,π̂t,Qtr )dr)r≥t

is a martingale.





Summary

We study the utility maximization problem with linear priceimpact and exponential utility.

Regularity of the value function and existence of feedbackoptimal control.

Interaction of N market makers that hold a constant Qoptions.

Exhibit an equilibrium price for the option for an order flow(Qt).

The price does not require predictions of the future orderflows.




Future work

Is the robustness in q necessary?

The same results hold if the market makers are averse touncertainty in fraction of proportion of order flow captured.

The utility of the agent i is

supνi,pi

infξit

J i(π, s, x,Q; νi, pi, ν, p, ξi)

where ξi is the (random) proportion of the order flowcaptured by the i market maker i.e.,

dQit =

((ξit1Pi

t=Pt+ 1

Pit>Pt

)(qt)

− −(ξit1Pi

t=Pt+ 1

Pit<Pt

)(qt)

+)dt,

P ∗ is still a Nash equilibrium for any order flow q.




Future work

Is the robustness in q necessary?

The same results hold if the market makers are averse touncertainty in fraction of proportion of order flow captured.

The utility of the agent i is

supνi,pi

infξit

J i(π, s, x,Q; νi, pi, ν, p, ξi)

where ξi is the (random) proportion of the order flowcaptured by the i market maker i.e.,

dQit =

((ξit1Pi

t=Pt+ 1

Pit>Pt

)(qt)

− −(ξit1Pi

t=Pt+ 1

Pit<Pt

)(qt)

+)dt,

P ∗ is still a Nash equilibrium for any order flow q.




Future work

The conjecture is that for N →∞, we do not need the robustutility we can take the utility of form

sup(pi,νi)

J i(π, s, x,Q; νi, pi, ν, p).

Properties of V N for large N?

Optimal liquidation problem facing such an option price?

Functional properties

H 7→ EQ(ν∗,0,s,π,Q)[H(ST )] as η → 0?

Defining implied market impact.




THANK YOU!


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Equilibrium option price with competing market makers