Optimal Trading Strategy. With Optimal Horizon. Financial Math Festival. Florida
State University. March 1, 2008. Edward Qian. PanAgora Asset Management ...
Optimal Trading Strategy With Optimal Horizon Financial Math Festival Florida State University March 1, 2008 Edward Qian PanAgora Asset Management
Trading – An Integral Part of Investment Process Return forecasting Portfolio construction Trading – portfolio implementation Performance attribution
10th FMF, Florida State University, 2008
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Conflicting Objectives in Trading Immediacy
Costs Fees Bid/Ask spread Market impact
10th FMF, Florida State University, 2008
7.0%
6.0%
5.0%
Cost
Alpha capture Risk reduction Labor costs Opportunity costs
4.0%
3.0%
2.0% 1.50%
1.75%
2.00%
2.25%
Risk
2
2.50%
2.75%
3.00%
Optimal Trading Strategies Optimal trading path (sequence) with minimum costs for a given level of risk h* ( t ) , t ∈ [ 0, T ] , T 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 * .
10th FMF, Florida State University, 2008
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Optimal Horizon - Motivation Horizon is not known in advance Single stocks versus baskets
Flip-floping in optimal trading with fixed horizon
10th FMF, Florida State University, 2008
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0.8
h
It is optimal along two dimensions
1.2
0.6
T>T*
0.4
T=T* 0.2
0
t
4
T*
T
Mathematical Model - Inputs Trade weight ∆w and trade path h ( t ) ∆w, h ( 0 ) = 0 and h (T ) = 1 Trade shortfall h ( t ) ∆w − ∆w = ∆w h ( t ) − 1
Return shortfall
f ∆w h ( t ) − 1 dt
Shortfall variance σ 2 ( ∆w )2 h ( t ) − 1 dt 2
Fixed cost
c ∆w T , c > 0
Market impact
ψ ( ∆w ) h ( t ) dt ,ψ > 0
10th FMF, Florida State University, 2008
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2
5
Mathematical Model – Objective Function Find path and horizon h* ( t ) , t ∈ J=
T
0
0, T * . that
T
maximize T
T
2 2 1 2 2 2 f ∆w h ( t ) − 1 dt − λ σ ( ∆w ) h ( t ) − 1 dt − c ∆w dt −ψ ( ∆w ) h ( t ) dt 2 0 0 0
Similar to MV optimization that maximizes expected return for a given level of risk
10th FMF, Florida State University, 2008
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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 h ( t )
A
d ∂L ∂L = dt ∂h ∂h
( )
L h, h −
( )h
∂L h, h ∂h
=0
B
t =T
10th FMF, Florida State University, 2008
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Mathematical Model – Equations 2nd order ODE 2 f λσ h − g 2 h = − s − g 2 , with s = w , g 2 = . 2ψ 2ψ
h (T ) =
cw
ψ
≡p
Solution consists of exponential functions with parameters s, g, and p
10th FMF, Florida State University, 2008
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Solution – No Risk Aversion Three different expected returns Zero risk aversion, g=0, s = f w 2ψ 1 0.9 0.8 0.7
h
0.6 0.5 0.4 0.3
g=0, s>0 s=g=0
0.2
g=0, s
