Stock Market Trading Via
Stochastic Network Optimization
Current price: p1(t) = $5.10
Buy !
Current price: p2(t) = $2.48
Sell !
Michael J. Neely (University of Southern California)
Proc. IEEE Conf. on Decision and Control (CDC), Atlanta, GA, Dec. 2010
PDF of paper at: http://www-bcf.usc.edu/~mjneely/
Sponsored in part by the NSF Career CCF-0747525
Problem Setting:
•N stocks.
•Slotted time t = {0, 1, 2, …}.
•Prices p(t) = (p1(t), …, pN(t)).
•Stock shares Q(t) = (Q1(t), …, QN(t))
Current price: p1(t) = $5.10
Buy !
Current price: p2(t) = $1.57
Buy !
Current price: p3(t) = $2.03
Sell !
Goal: Make money over time while limiting the
investment risk!
Limit risk by:
•Bounding the max allowable stock level Qnmax.
•Bounding the amount buy/sell on a slot.
Decision Variables for Buying:
•An(t) = Num. shares of stock n bought on slot t.
•bn(A) = Buying transaction fee (function of A).
Current price = pn(t)
Buy !
Expensen(t) = An(t)pn (t) + bn(An (t))
Goal: Make money over time while limiting the
investment risk!
Limit risk by:
•Bounding the max allowable stock level Qnmax.
•Bounding the amount buy/sell on a slot.
Decision Variables for Selling:
•μn(t) = Num. shares of stock n sold on slot t.
•sn(μ) = Selling transaction fee (function of μ).
Current price = pn(t)
Sell !
Revenuen(t) = μn(t)pn (t) - sn(μn (t))
Control Constraints and Decision Structure:
Buying Constraints:
An(t) in {0, 1, 2, …, μnmax} for all n, t.
∑n An(t)pn(t) ≤ xmax for all t.
Selling Constraints:
μn(t) in {0, 1, 2, …, μnmax} for all n, t.
μn(t) ≤ Qn(t) for all n, t.
•Every slot t, observe prices p(t) = (p1(t), …, pN(t)).
•Make buying/selling decisions.
•Profit(t) = ∑n Revenuen(t) – ∑n Expensesn(t).
•Want to maximize time average profit subject to
the above constraints.
Talk Outline:
1) Design optimal strategy under (simplistic) assumption
that price vectors p(t) are i.i.d. over slots.
2) Use same strategy for arbitrary price vectors
(assuming only that 0 ≤ pn(t) ≤ pnmax for all t).
3) For arbitrary prices, show the strategy yields profit
arbitrarily close to that of an “ideal” algorithm with
perfect knowledge of future prices over T-slot frames
(for any integer T).
“T-slot lookahead metric”
Comparison to Related Work:
1) “Portfolio Optimization”
(Known or approximate price distributions)
[Markowitz 1952][Sharpe 1963][Samuelson 1969]
[Rudoy, Rohrs 2008] (Dynamic Programming)
2) “Universal Portfolios”
(Arbitrary price sample paths)
[Cover 1991][Cover, Ordentlich 1996][Ordentlich, Cover 1998]
[Merhav, Feder 1993]
Comparison to Related Work:
Different Structure for Prior “Universal Portfolio” work:
•
•
•
•
•
•
Optimize “growth exponent.”
No cap on queue size (more aggressive, but more risk!)
Integer constraints relaxed to real numbers.
No Transaction fees.
Based on “Universal Data Compression.” $$
Solution complexity grows with time.
Comparison to ours:
•
•
•
Our cap on buy/sell and queue size limits risk, but
also limits to only linear growth (*not exponential).
Based on “Stochastic Network Optimization.”
Simple “max-weight” solution with complexity
that is the same for all time.
*simple modifications can get back exponential growth.
time
Solution Strategy: Dynamic Queue Control
θn
Qn(t)
•Want to keep queue size high enough to take
advantage of good prices that come along.
•Don’t want queues too high (too risky).
•Lyapunov Function:
L(Q(t)) = ∑n (Qn(t) – θn)2
Control Design:
•Lyapunov Function L(Q(t))
•Lyapunov Drift Δ(t) = L(Q(t+1)) – L(Q(t))
•Every slot t, observe Q(t), p(t), and
minimize the drift-plus-penalty*:
Qn(t)
θn
Δ(t) + V[Expenses(t) – Revenue(t)]
*from stochastic network optimization theory [Georgiadis, Neely, Tassiulas 2006][Neely 2010]
Results in the following algorithm:
•(Selling) Choose μn(t) to solve:
Minimize: [θn– Qn(t) – Vpn(t)]μn(t) + Vsn(μn(t))
Subject to: μn(t) in {0, 1, 2, …, min[Qn(t), μnmax]}
Control Design:
•Lyapunov Function L(Q(t))
•Lyapunov Drift Δ(t) = L(Q(t+1)) – L(Q(t))
•Every slot t, observe Q(t), p(t), and
minimize the drift-plus-penalty*:
Qn(t)
θn
Δ(t) + V[Expenses(t) – Revenue(t)]
*from stochastic network optimization theory [Georgiadis, Neely, Tassiulas 2006][Neely 2010]
Results in the following algorithm:
•(Buying) Choose An(t) to solve: (“knapsack-like” problem)
Minimize: ∑n [Qn(t)-θn +Vpn(t)]An(t) + V∑n bn(Αn(t))
Subject to: (already stated) constraints on (An(t)).
Theorem (iid case):
Choose θn = V pnmax + 2μnmax. Then:
Qn(t)
θn
(a) μnmax ≤ Qn(t) ≤ Vpnmax + 3μnmax (for all t)
(b) For all t:
E{Time avg. Profit up to t} ≥
Profitopt – B
V
“Startup Cost”
– L(Q(0))
Vt
(c) As t  infinity, we have with prob. 1:
“Profit Gap”
limtinfinity [Time avg profit] ≥ Profitopt – B/V
Now for arbitrary (possibly non-ergodic) prices:
•Compare to an “ideal” T-slot lookahead metric.
•ΨT[r] = Optimal Profit over frame r, assuming
future stock prices over the frame are known.
Frame 0
Frame 1
Frame 2
Now for arbitrary (possibly non-ergodic) prices:
•Compare to an “ideal” T-slot lookahead metric.
•ΨT[r] = Optimal Profit over frame r, assuming
future stock prices over the frame are known.
Frame 0
Frame 1
Frame 2
Metric even allows “selling short”
Theorem (General Bounded Prices):
Under same algorithm as before:
(a) μnmax ≤ Qn(t) ≤ Vpnmax + 3μnmax (for all t)
(b) For all frame sizes T>0, all integers R>0:
Time average profit over RT slots ≥
1 R-1
CT
∑r=0 ΨΤ[r]
RT
V
- L(Q(0))
RTV
Simulations for BlockBuster Stock (BBI):
Aug. 11, 1999 – Sept. 11, 2009
Avg Profit = Avg Profit on Trades
- (1/t) x Startup Cost
+ (1/t) x Profit when sell all at end
Simulations for BlockBuster Stock (BBI):
Aug. 11, 1999 – Sept. 11, 2009
Avg Profit = Avg Profit on Trades
- (1/t) x Startup Cost
+ (1/t) x Profit when sell all at end
“Asymptotically Negligible” as t infinity
(But very significant for finite t !!!)
Parameters: μmax = 20, pmax = 30, V=1000
Compare bound For T=10 (2 weeks), T=100 (20 weeks) :
• Avg. Profit on trades ≥ “2-week ideal” – $2.0
• Avg. Profit on trades ≥ “20-week ideal” - $20.0
Simulations for BlockBuster Stock (BBI):
30
Stock Prices (Forward Time)
Cumulative Trade Profit (Forward Time)
350000
300000
25
250000
20
200000
15
150000
10
100000
5
50000
0
0
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49
Average Profit Per Transaction = Cum. Trade Profit/t = $81.52
(2-week ideal profit no more than $30.00, 20-week idea profit no more than $60.00)
Stock Prices (Backward Time)
35
Cumulative Trade Profit (Backward Time)
350000
300000
30
250000
25
200000
20
150000
15
10
100000
5
50000
0
0
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49
Average Profit Per Transaction = Cum. Trade Profit/t = $95.81
What about total profit (including startup costs)?
Forward Time:
•Trade Profit = $206,343.20
•Startup Cost = $449,097.99
•End Reward = $33,976.80
•Total Profit = -$208,778.00
Reverse Time:
•Trade Profit = $242,512.40
•Startup Cost = $35,146.80
•End Reward = $239,499.00
•Total Profit = $446,864.60
What about total profit (including startup costs)?
Forward Time:
•Trade Profit = $206,343.20
•Startup Cost = $449,097.99
•End Reward = $33,976.80
•Total Profit = -$208,778.00
Reverse Time:
•Trade Profit = $242,512.40
•Startup Cost = $35,146.80
•End Reward = $239,499.00
•Total Profit = $446,864.60
What about total profit (including startup costs)?
Forward Time:
•Trade Profit = $206,343.20
•Startup Cost = $449,097.99
•End Reward = $33,976.80
•Total Profit = -$208,778.00
Reverse Time:
•Trade Profit = $242,512.40
•Startup Cost = $35,146.80
•End Reward = $239,499.00
•Total Profit = $446,864.60
Conclusion:
•Provide a Lyapunov Optimization (“max-weight”)
approach to stock market trading.
•Can quantify risk and profit guarantees compared to
T-Slot Lookahead:
Time average profit over RT slots ≥
1 R-1
∑r=0 ΨΤ[r] - CT - L(Q(0))
RT
RTV
V
•Startup Cost is high and thus there is inherent risk, but
optimizing the trading profits mitigates this risk.
•Open question: Can other algorithms provide
provably better tradeoffs?