Optimal make-take fees for market making regulation O. El Euch, T. Mastrolia, M. Rosenbaum and N. Touzi Ecole Polytechnique 11 September 2018 El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 1
Table of contents Introduction 1 The model 2 Solving the market maker problem 3 Solving the exchange problem 4 El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 2
Table of contents Introduction 1 The model 2 Solving the market maker problem 3 Solving the exchange problem 4 El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 3
Introduction Exchanges in competition With the fragmentation of financial markets, exchanges are nowadays in competition. Traditional international exchanges are now challenged by alternative trading venues. Consequently, they have to find innovative ways to attract liquidity on their platforms. A possible solution : using a make-taker fees system, that is charging in an asymmetric way liquidity provision and liquidity consumption. El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 4
Introduction A controversial topic Make-take fees policies are seen as a major facilitating factor to the emergence of a new type of market makers aiming at collecting fee rebates : the high frequency traders. As stated by the Securities and Exchanges commission : “Highly automated exchange systems and liquidity rebates have helped establish a business model for a new type of professional liquidity provider that is distinct from the more traditional exchange specialist and over-the-counter market maker.” El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 5
Introduction HFT market makers The concern with high frequency traders becoming the new liquidity providers is two-fold. Their presence implies that slower traders no longer have access to the limit order book, or only in unfavorable situations when high frequency traders do not wish to support liquidity. They tend to leave the market in time of stress. El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 6
Introduction Our aim Providing a quantitative and operational answer to the question of relevant make-take fees. We take the position of an exchange (or of the regulator) wishing to attract liquidity. The exchange is looking for the best make-take fees policy to offer to market makers in order to maximize its utility. In other words, it aims at designing an optimal contract with the (unique) market marker to create an incentive to increase liquidity. Principal/agent type approach : the wealth of the principal (exchange) depends on the agent’s (market maker) effort (essentially his spread), but the principal cannot directly control the effort. El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 7
Table of contents Introduction 1 The model 2 Solving the market maker problem 3 Solving the exchange problem 4 El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 8
The market maker Market maker’s controls The market maker has a view on the efficient price (midprice) of the asset S t = S 0 + σ W t , where σ is the price volatility. He fixes the ask and bid prices P a t = S t + δ a P b t = S t − δ b t , t . El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 9
The order flow Arrival of market orders We model the arrival of buy (resp. sell) market orders by a point process ( N a t ) t ≥ 0 (resp. ( N b t ) t ≥ 0 ) with intensity ( λ a t ) t ≥ 0 (resp. ( λ b t ) t ≥ 0 ). The inventory of the market maker Q t = N b t − N a t . We consider a threshold inventory ¯ q above which the market maker stops quoting on the ask or bid side. From financial economics arguments : λ a t = λ ( δ a λ b t = λ ( δ b t )1 { Q t > − ¯ q } , t )1 { Q t < ¯ q } . where λ ( x ) = Ae − k ( x + c ) /σ . El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 10
Martingale processes Equivalent probabilities The market maker controls the spread δ = ( δ a , δ b ). We define the associated probability P δ such that � t N a ,δ � = N a λ ( δ a t − s )1 { Q s > − ¯ q } ds t 0 and � t N b ,δ � = N b λ ( δ b t − s )1 { Q s < ¯ q } ds t 0 are martingales. El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 11
The market maker viewpoint The profit and loss of the market maker We consider a final time horizon T > 0. The cash flow of the market maker � t � t X δ P a u dN a P b u dN b t = u − u . 0 0 The inventory risk of the market maker is Q t S t . For a given contract ξ given by the exchange, seen as an F T measurable random variable, the market maker chooses his spread δ by maximizing his utility. El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 12
The market maker optimization problem The market maker problem Under the exchange incentive policy ξ , the market maker solves now E δ � � �� − γ ( X δ V MM ( ξ ) = sup − exp T + Q T S T + ξ ) . δ We obtain an optimal response given by ˆ δ t ( ξ ) = (ˆ t ( ξ ) , ˆ δ a δ b t ( ξ )) . We will only consider contracts such that V MM ( ξ ) is above a threshold utility value R : C = { ξ F T -measurable such that V MM ( ξ ) > R } + integrability conditions. For ξ = 0, well studied problem since Avellaneda and Stoikov. El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 13
The exchange viewpoint We assume that the exchange Earns c > 0 for each market order occurring in its platform. Pays the incentive policy ξ to the market maker. The profit and loss of the exchange is c ( N a T + N b T ) − ξ. The exchange problem The exchange designs the contract ξ by solving δ ( ξ ) � � �� ˆ − η ( c ( N a T + N b V E = sup − exp T ) − ξ ) , E ξ ∈C where η is the risk aversion of the exchange. El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 14
Table of contents Introduction 1 The model 2 Solving the market maker problem 3 Solving the exchange problem 4 El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 15
Solving the market maker problem for a given contract Dynamic programming principle We fix ξ and compute the best response of the market maker. Let τ be a stopping time with values in [ t , T ] and µ ∈ A τ , where A τ denotes the restriction of the set of admissible controls A to controls on [ τ, T ]. � u + Q u dS u ) e − γξ � � T Let J T ( τ, µ ) = E µ τ ( µ a u dN a u + µ b u dN b − e − γ and τ V τ = ess sup J T ( τ, µ ) . µ ∈A τ Dynamic programming principle : � � � τ t ( δ a u dN a u + δ b u dN b E δ − e − γ u + Q u dS u ) V τ V t = ess sup . t δ ∈A El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 16
Solving the market maker problem for a given contract A convenient super-martingale Let � t 0 δ a u dN a u + δ b u dN b U δ t = V t e − γ u + Q u dS u . U δ 0 = V 0 and � � T � 0 δ a u dN a u + δ b u dN b T = − e − γ u + Q u dS u + ξ U δ . From the DPP, we get that U δ t is a P δ − super-martingale. We want to find the optimal controls ( δ a , δ b ) turning it into a martingale. To do so we find a suitable representation of U δ t . El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 17
Solving the market maker problem for a given contract Doob-Meyer and martingale representation t , where M δ is a P δ − martingale and A δ is Doob-Meyer : U δ t = M δ t − A δ an integrable non-decreasing predictable process starting at zero. Martingale representation theorem : There exists a predictable process Z δ = ( � � Z δ, S , � Z δ, a , � Z δ, b ) such that M δ t can be represented as � t � t � t Z δ � Z δ, a � Z δ, b � λ ( δ a λ ( δ b V 0 + r . d χ r − r )1 { Q r > − ¯ q } dr − r )1 { Q r < ¯ q } dr , r r 0 0 0 with χ = ( S , N a , N b ). El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 18
Solving the market maker problem for a given contract Reducing the class of contracts Let Y be the process defined by V t = − e − γ Y t . Y T = ξ and using Ito’s formula together with the previous result and the martingale property of U t for the optimal controls we get dY t = Z a t dN a t + Z b t dN b t + Z S t dS t − H ( Z t , Q t ) dt , for an explicit function H and where the Z i do not depend on δ . Any contract ξ can be (uniquely) represented under the preceding form ! We can restrict ourselves to such contracts. Natural financial interpretation of the contracts : The exchange rewards the market maker by Z a (resp. Z b ) for each buy (resp.sell) market order. The exchange participates to the market/inventory risk of the market maker by taking − Z S of his share. The market maker pays a continuous coupon H ( Z t , Q t ) dt . El Euch, Mastrolia, Rosenbaum, Touzi Optimal make-take fees for market making regulation 19
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