Bayesian Adaptive Trading with a Daily Cycle

1. Bayesian Adaptive Trading with a Daily Cycle CHEN RONG ZHANG WEN JUN

2. Outline • Introduction and Features • Price model including Bayesian update • Optimal trading strategies • Coding • Difficulties and Justification

3. Introduction • Presents a model for price dynamics and optimal trading that explicitly includes the daily trading cycle and the trader’s attempt to learn the targets of other market participants • Motivation: • 1. Institutional trading has an explicit daily cycle based on the assumption that at the beginning of each day each informed market participant is given a trader target exogenously. • 2. Popularity of execution algorithms that adapt to changes in price of the asset being traded, either by accelerating execution when the price moves in the trader’s favor, or conversely.

4. Features • The informed participants do not know each others’ targets, but must guess them by observing price dynamics throughout the day.We consider they will use all available information to compete with each other. • The daily cycle is an essential feature of this model. • The underlying drift factor is approximately constant throughout the day. • The trader must never sell as part of a buy program.

5. Price Model(including Bayesian update) • Price S(t) obeying an arithmetic random walk is standard Brownian motion, σ is an absolute volatility and α a drift. • Drift:

6. Bayesian Inference • At time t, stock price trajectory : • Conditional on the value of α，the distribution of S(t) is • After some calculation we can find the unconditional distribution

7. Bayesian Inference • We then use Bayes’ rule • Obtain this conditional distribution • This represents our best estimate of the true drift α

8. Trading and Price Impact • Order of X shares, begins at t=0 and completed by • Trading trajectory fn: x(t) is the shares remaining to buy with x(0)=X and x(T)=0 • Corresponding trading rate • Constraint: • Use a linear market impact function to get the actual execution price: η is the coefficient of temporary market impact

9. Trading and Price Impact • C: total cost of executing the buy program relative to the initial value • C is a random variable

10. Optimal Trading Strategies • Minimize the expectation of trading cost • Conditional on the true value of α • Our best estimate at time t for α is:

11. Optimal Trading Strategies • The expected cost of the remaining program: • Trading goal: determine such that

12. Trajectories by Calculus of Variations • Small perturbation: • Associate trade rate perturbation: • Perturbation in cost: • Here is the best available drift estimate using information at time t

13. Trajectories by Calculus of Variations Unconstrained trajectories • Optimizing satisfy the ODE: • Solution : (1) • Corresponding instantaneous rate: (2)

14. Trajectories by Calculus of Variations Constrained trajectories • There is a critical drift value such that • If , then the constraint is not binding. The solution is the one given in (1) and (2). • If , then the solution is the one of (1,2), with a shortened end time determined by

15. Trajectories by Calculus of Variations • If , then the solution is the one of (1,2), except that trading does not begin until a starting time determined by:

16. Trajectories by Calculus of Variations • The overall trade rate formula: (3) This is the Bayesian adaptive strategy: a specific formula for the instantaneous trade rate as a function of price, time, and shares remaining.

17. Illustration • Constrained Solution Starting at Time t with Shares x(t) and Drift Estimate α

18. Illustration • For , the trajectories go below the linear profile to reduce expected purchase cost. • For , the constraint is not binding (shaded region). • At the solution become tangent to line x=0 at and for larger values they hit x=0 with zero slope at • For , trading does not begin until

19. Coding • Implement the price model using Bayesian adaptive strategy by MATLAB • Mean • Standard deviation , then • Price path with volatility • Initial price • Initial shares • Impact coefficient

20. Coding Sample price path with initial price

21. Coding Trajectories according to sample price path

22. Difficulties • In this model, the random daily drift is superimposed on the price volatility caused by small random traders. • In theory, these two sources of randomness can be disentangled by measuring volatility on an intraday time scale and comparing it to daily volatility. • In practice, real price processes are far from Gaussian, so it’s difficult to do this comparison with any degree of reliability.

23. Justification • By the practical observation, some fraction of traders do express interest in using strategies similar to ours. • We provide a conceptual framework for designing optimal strategies that capture this preference. • Without any such framework it’s impossible to design algorithms except by completely special methods.

24. Q & A