Introduction Optimization Estimation Market maker simulations Conclusion High-Frequency Trading in a Limit Order Book Sasha Stoikov (with M. Avellaneda) Cornell University February 9, 2009
Introduction Optimization Estimation Market maker simulations Conclusion The limit order book
Introduction Optimization Estimation Market maker simulations Conclusion Motivation • Two main categories of traders 1 Liquidity taker: buys at the ask, sell at the bid 2 Liquidity provider: waits to buy at the bid, sell at the ask
Introduction Optimization Estimation Market maker simulations Conclusion Motivation • Two main categories of traders 1 Liquidity taker: buys at the ask, sell at the bid 2 Liquidity provider: waits to buy at the bid, sell at the ask • How do liquidity providers (market makers) make money? 1 Making the bid/ask spread 2 Managing their risk by adjusting the quantities/prices
Introduction Optimization Estimation Market maker simulations Conclusion Motivation • Two main categories of traders 1 Liquidity taker: buys at the ask, sell at the bid 2 Liquidity provider: waits to buy at the bid, sell at the ask • How do liquidity providers (market makers) make money? 1 Making the bid/ask spread 2 Managing their risk by adjusting the quantities/prices • Factors affecting the optimal bid/ask prices: 1 Inventory risk • The stock mid price: S • The stock volatility: σ • The risk aversion: γ • The liquidity: λ ( · )
Introduction Optimization Estimation Market maker simulations Conclusion Motivation • Two main categories of traders 1 Liquidity taker: buys at the ask, sell at the bid 2 Liquidity provider: waits to buy at the bid, sell at the ask • How do liquidity providers (market makers) make money? 1 Making the bid/ask spread 2 Managing their risk by adjusting the quantities/prices • Factors affecting the optimal bid/ask prices: 1 Inventory risk • The stock mid price: S • The stock volatility: σ • The risk aversion: γ • The liquidity: λ ( · ) 2 Adverse selection risk
Introduction Optimization Estimation Market maker simulations Conclusion Outline 1 Optimization • The maximal utility problem • Optimal bid and ask prices • Some approximations • P&L profiles of the optimal strategy
Introduction Optimization Estimation Market maker simulations Conclusion Outline 1 Optimization • The maximal utility problem • Optimal bid and ask prices • Some approximations • P&L profiles of the optimal strategy 2 Estimation • Modeling the order book • Estimating model parameters • Steady state quantities
Introduction Optimization Estimation Market maker simulations Conclusion Outline 1 Optimization • The maximal utility problem • Optimal bid and ask prices • Some approximations • P&L profiles of the optimal strategy 2 Estimation • Modeling the order book • Estimating model parameters • Steady state quantities 3 Simulation • A market making algorithm • Autocorrelation in the order flow
Introduction Optimization Estimation Market maker simulations Conclusion Outline 1 Optimization • The maximal utility problem • Optimal bid and ask prices • Some approximations • P&L profiles of the optimal strategy 2 Estimation • Modeling the order book • Estimating model parameters • Steady state quantities 3 Simulation • A market making algorithm • Autocorrelation in the order flow 4 Conclusion
Introduction Optimization Estimation Market maker simulations Conclusion The mid price of the stock • Brownian motion dS t = σ dW t
Introduction Optimization Estimation Market maker simulations Conclusion The mid price of the stock • Brownian motion dS t = σ dW t • Geometric Brownian motion dS t = σ dW t S t
Introduction Optimization Estimation Market maker simulations Conclusion The mid price of the stock • Brownian motion dS t = σ dW t • Geometric Brownian motion dS t = σ dW t S t • Trading at the mid-price is not allowed. However, we may quote limit orders p b and p a around the mid-price.
Introduction Optimization Estimation Market maker simulations Conclusion The arrival of buy and sell orders • Controls: p a t and p b t
Introduction Optimization Estimation Market maker simulations Conclusion The arrival of buy and sell orders • Controls: p a t and p b t • Number of stocks bought N b t is Poisson with intensity λ b ( p b − s ), an increasing function of p b • Number of stocks sold N a t is Poisson with intensity λ a ( p a − s ), a decreasing function of p a
Introduction Optimization Estimation Market maker simulations Conclusion The arrival of buy and sell orders • Controls: p a t and p b t • Number of stocks bought N b t is Poisson with intensity λ b ( p b − s ), an increasing function of p b • Number of stocks sold N a t is Poisson with intensity λ a ( p a − s ), a decreasing function of p a • The wealth in cash dX t = p a dN a t − p b dN b t The inventory q t = N b t − N a t
Introduction Optimization Estimation Market maker simulations Conclusion The market maker’s objective • Maximize exponential utility � − e − γ ( X T + q T S T ) � u ( s , x , q , t ) = max E t p a t , p b t , 0 ≤ t ≤ T
Introduction Optimization Estimation Market maker simulations Conclusion The market maker’s objective • Maximize exponential utility � − e − γ ( X T + q T S T ) � u ( s , x , q , t ) = max E t p a t , p b t , 0 ≤ t ≤ T • Mean/variance objective E t [( X T + q T S T )] − γ v ( s , x , q , t ) = max 2 Var [( X T + q T S T )] p a t , p b t , 0 ≤ t ≤ T
Introduction Optimization Estimation Market maker simulations Conclusion The market maker’s objective • Maximize exponential utility � − e − γ ( X T + q T S T ) � u ( s , x , q , t ) = max E t p a t , p b t , 0 ≤ t ≤ T • Mean/variance objective E t [( X T + q T S T )] − γ v ( s , x , q , t ) = max 2 Var [( X T + q T S T )] p a t , p b t , 0 ≤ t ≤ T • Infinite horizon exponential utility �� ∞ � w ( x , s , q ) = max E − exp( − ω t ) exp( − γ ( X t + q t S t )) dt . p a t , p b 0 t
Introduction Optimization Estimation Market maker simulations Conclusion The market maker’s objective • Maximize exponential utility � − e − γ ( X T + q T S T ) � u ( s , x , q , t ) = max E t p a t , p b t , 0 ≤ t ≤ T • Mean/variance objective E t [( X T + q T S T )] − γ v ( s , x , q , t ) = max 2 Var [( X T + q T S T )] p a t , p b t , 0 ≤ t ≤ T • Infinite horizon exponential utility �� ∞ � w ( x , s , q ) = max E − exp( − ω t ) exp( − γ ( X t + q t S t )) dt . p a t , p b 0 t • Other objectives: minimizing shortfall risk, value at risk, etc...
Introduction Optimization Estimation Market maker simulations Conclusion The HJB equation u ( x , s , q , t ) solves u t + 1 2 σ 2 u ss + max p b λ b ( p b ) � u ( s , x − p b , q + 1 , t ) − u ( s , x , q , t ) � + max p a λ a ( p a ) [ u ( s , x + p a , q − 1 , t ) − u ( s , x , q , t )] = 0 u ( S , x , q , t ) = − exp( − γ ( x + qS )) .
Introduction Optimization Estimation Market maker simulations Conclusion The indifference or reservation prices Definition The indifference bid price r b (relative to a book of q stocks) is given implicitly by the relation u ( x − r b ( s , q , t ) , s , q + 1 , t ) = u ( x , s , q , t ) . The indifference ask price r a solves u ( x + r a ( s , q , t ) , s , q − 1 , t ) = u ( x , s , q , t ) .
Introduction Optimization Estimation Market maker simulations Conclusion The optimal quotes Theorem The optimal bid and ask prices p b and p a are given by the implicit relations � � 1 + γ λ b p b = r b − 1 γ ln ∂λ b ∂ p and � � 1 − γ λ a p a = r a + 1 γ ln . ∂λ a ∂ p
Introduction Optimization Estimation Market maker simulations Conclusion The “Frozen-Inventory” Approximation • If we assume there is no arrival of orders v ( x , s , q , t ) = E t [ − exp( − γ ( x + qS T )] � γ 2 q 2 σ 2 ( T − t ) � = − exp( − γ x ) exp( − γ qs ) exp 2
Introduction Optimization Estimation Market maker simulations Conclusion The “Frozen-Inventory” Approximation • If we assume there is no arrival of orders v ( x , s , q , t ) = E t [ − exp( − γ ( x + qS T )] � γ 2 q 2 σ 2 ( T − t ) � = − exp( − γ x ) exp( − γ qs ) exp 2 • The indifference price of a stock, given an inventory of q stocks is r ( s , q , t ) = s − q γσ 2 ( T − t )
Introduction Optimization Estimation Market maker simulations Conclusion The “Frozen-Inventory” Approximation • If we assume there is no arrival of orders v ( x , s , q , t ) = E t [ − exp( − γ ( x + qS T )] � γ 2 q 2 σ 2 ( T − t ) � = − exp( − γ x ) exp( − γ qs ) exp 2 • The indifference price of a stock, given an inventory of q stocks is r ( s , q , t ) = s − q γσ 2 ( T − t ) • This is an approximation to r a and r b for the problem with order arrivals
Introduction Optimization Estimation Market maker simulations Conclusion The “Econophysics” Approximation 1 The density of market order size is f Q ( x ) ∝ x − 1 − α Gabaix et al. (2006)
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