Optimal Trading Strategy With Optimal Horizon

Transcription

1 Optimal Trading Strategy With Optimal Horizon Financial Math Festival Florida State University March 1, 2008 Edward Qian PanAgora Asset Management

2 Trading An Integral Part of Investment Process Return forecasting Portfolio construction Trading portfolio implementation Performance attribution 1

3 Conflicting Objectives in Trading Immediacy Alpha capture Risk reduction Labor costs Opportunity costs Costs Fees Bid/Ask spread Market impact Cost 7.0% 6.0% 5.0% 4.0% 3.0% 2.0% 1.50% 1.75% 2.00% 2.25% 2.50% 2.75% 3.00% Risk 2

4 Optimal Trading Strategies Optimal trading path (sequence) with minimum costs for a given level of risk ( ) [ ] * h t t T T, 0,, is the trading horizon. Previous researches (Grinold & Kahn 1999, Almgren & Chriss 2000) used a fixed horizon T Extension to optimal trading strategy with optimal horizon (Qian 2008 JOIM, Qian, Hua, Sorensen 2007 ) ( ) * * h t, t 0, T. 3

5 Optimal Horizon - Motivation Horizon is not known in advance Single stocks versus baskets It is optimal along two dimensions h T>T* T=T* Flip-floping in optimal trading with fixed horizon t T* T 4

6 Mathematical Model - Inputs Trade weight w and trade path h( t) w, h( ) h( T ) 0 = 0 and = 1 Trade shortfall ( ) = ( ) 1 h t w w wh t Return shortfall Shortfall variance Fixed cost Market impact ( ) 1 f w h t dt 2 ( ) ( ) 2 σ w h t 1 dt c w T, c > 0 2 ( ) ( ) 2 ψ w h t dt, ψ > 0 2 5

7 Mathematical Model Objective Function * * Find path and horizon h ( t), t 0, T. that maximize T T T T J = f wh( t) 1dt λ σ ( w) 2 h( t) 1 dt c w dt ψ ( w) 2 h ( t) 2 dt Similar to MV optimization that maximizes expected return for a given level of risk 6

8 Mathematical Model Calculus of Variation Method of calculus of variation Find optimal function instead of optimal parameter Ordinary differential equation for h( t) Boundary condition for d L L dt = h h h( t) A (, ) L h h L( h, h ) h = 0 h t= T B 7

9 Mathematical Model Equations 2 nd order ODE 2 2 f w 2 2 h λσ g h = s g, with s =, g =. 2ψ 2ψ ( ) h T c w = ψ Solution consists of exponential functions with parameters s, g, and p p 8

10 Solution No Risk Aversion Three different expected returns Zero risk aversion, g=0, s = f 2ψ 1 w h g=0, s>0 s=g=0 g=0, s<0 0 t T 9

11 Solution No Risk Aversion Optimal horizon Horizon should be longer if Market impact is high Fixed cost is low Return is low (if it agrees with the trade) T = * 2 ψ c + c + f 10

12 Numerical Examples Single Stock Base parameter assumption. Optimal horizon = 0.52 day f σ 1% / day 4% / day λ 2 day / % c ψ 0.1% / day 0.5 % day s = f 2ψ 1 / day 2 g = λσ 2 2ψ 5.7 / day p = c ψ 0.45 / day 11

13 Numerical Examples Single Stock Changing parameters case I Optimal Trading Horizon f λ T*

14 Numerical Examples Single Stock Changing parameters case II Optimal Trading Horizon f c T*

15 Portfolios of Stocks Objective Function and Solution Find path and horizon, h * ( t), t 0, T * that maximize L T J = L h h dt 0 (, ) 1 h h = f c w h 1 h 1 w h 1 w h wh 2 (, ) [ ] λ [ ] [ ] A system of second order linear ODE s, which can be solved numerically 2N pit ( t) e ( t) h = a + q i= 1 14

16 Numerical Examples Two Stocks Base parameter assumption Stock A Stock B f 1% / day 1% / day σ 4% / day 4% / day λ 2 day / % 2 day / % c 0.1% / day 0.1% / day ψ 0.5 % day 2 % day s = f 2ψ 1 / day / day 2 g = λσ 2 2ψ 5.7 / day 2.8 / day p = c ψ 0.45 / day 0.22 / day * T 0.52 day 1.04 day 15

17 Numerical Examples Two Stocks Individual paths 16

18 Numerical Examples Two Stocks Combined path zero correlations 17

19 Numerical Examples Two Stocks Combined path non-zero return correlations 18

20 Numerical Examples Two Stocks Combined path non-zero impact correlations 19

21 Numerical Examples Two Stocks Combined path non-zero correlations 20

22 Numerical Examples Two Stocks Combined path non-zero correlations 21

23 Summary There is often an optimal trading horizon with optimal trading strategy Our analytic solution shows the optimal horizon depends on Expected return, stock volatility, fixed cost, market impact Correlations play a significant role for stock portfolios Further research Portfolio constraints: dollar neutral, sector neutral No reverse trading 22

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