The Execution Game Beomsoo Park Information Systems Laboratory Electrical Engineering Stanford University Joint work with Ciamac Moallemi and Benjamin Van Roy.

Example: Société Générale Discovery of €50B position Liquidated over January Received €45.1B Market impact Triggered emergency rate cut of January 22 “cracked under the pressure of a 30 hour work week”

Basic Trading Model (Bertsimas and Lo, 1998) Initial position Trades Requirement Nominal price evolution Impact of our trades Minimize execution cost zero-mean noise amount purchased

Arbitrageur Model Initial position Trades Constraint Initial estimate Price Dynamics Minimize execution cost

Information States Arbitrageur –my position –estimate of his –decision policy Trader –my position –his position –his estimate of mine –decision policy

Perfect Bayesian Equilibrium (PBE) (  *,  * ) such that –  * is the trader’s best response to (  *,  * ) –  * is the arbitrageur’s best response to  * Trading Policy Equipartitioning policy PBE policy

Solving for PBE “Shoot first, ask questions later” Dynamic programming –Recursive computation of value functions Trader’s value function depends on –trader’s position –arbitrageur’s position –arbitrageur’s beliefs Arbitrageur’s value function depends on –arbitrageur’s position –arbitrageur’s beliefs probability distribution

No Arbitrageur trader equipartitions arbitrageur does nothing Average P&L relative volume (  0 )

Arbitrageur vs. Nonstrategic Trader relative volume (  0 ) trader equipartitions strategic arbitrageur Average P&L

Performance relative volume (  0 ) strategic trader strategic arbitrageur Average P&L

Signaling

Response to Market Activity Neutral market –Trader sells gradually –Accelerates at end of horizon Down market –Regardless of trader, arbitrageur perceives selling –Arbitrageur front runs by selling –Trader sells more evenly over time Up market –Arbitrageur perceives buying –Arbitrageur tries to front run by buying –Trader buys to front run arbitrageur –Trader profits from arbitrageur’s misunderstanding

Closing Remarks Accounting for arbitrageur activity can be important Extensions –Multiple arbitrageurs –Uncertain trader –Infinite horizon / risk aversion Microstructure Role of game theory in financial engineering