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Outline Introduction Optimal Stochastic Control Model The HJB equation & Its Solution Implementation & Exercise Optimal Stochastic Control for Pairs Trading Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ March 26, 2014 Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading

Outline Introduction Optimal Stochastic Control Model The HJB equation & Its Solution Implementation & Exercise Introduction Optimal Stochastic Control Model Dynamics of Paired Stock Prices Co-integrating Vector Dynamic of Wealth Value Integrability Condition Agent’s Objective The HJB equation & Its Solution HJB Equation Particular Case δ = 0 Solution Implementation & Exercise Implement Pairs Trading in MATLAB Exercise Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading

Outline Introduction Optimal Stochastic Control Model The HJB equation & Its Solution Implementation & Exercise Introduction Pairs Trading is an investment strategy used by many Hedge Funds. Consider two co-integrated and correlated stocks which trade at some spread. For a given period, we need to maximise the agent’s terminal utility of wealth subject to budget constraints. Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading

Outline Dynamics of Paired Stock Prices Introduction Co-integrating Vector Optimal Stochastic Control Model Dynamic of Wealth Value The HJB equation & Its Solution Integrability Condition Implementation & Exercise Agent’s Objective Let S 1 and S 2 denote the co-integrated stock price which satisfies the stochastic differential equations = ( µ 1 + δ z ( t )) S 1 dt + σ 1 S 1 dB 1 , (1) dS 1 � � � 1 − ρ 2 dB 2 dS 2 = µ 2 S 2 dt + σ 2 S 2 ρ dB 1 + , (2) where B 1 and B 2 are independent Brownian Motions. Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading

Outline Dynamics of Paired Stock Prices Introduction Co-integrating Vector Optimal Stochastic Control Model Dynamic of Wealth Value The HJB equation & Its Solution Integrability Condition Implementation & Exercise Agent’s Objective The instantaneous co-integrating vector z ( t ) is defined by z ( t ) = a + ln S 1 ( t ) + β ln S 2 ( t ) . (3) The dynamics of z ( t ) is a stationary Ornstein-Uhlenbeck process dz = α ( η − z ) dt + σ β dB t , where α = − δ is the speed of mean reversion, � σ 2 1 + β 2 σ 2 σ β = 2 + 2 βσ 1 σ 2 ρ , √ 1 − ρ 2 B 1 + β σ 2 B t = σ 1 + βσ 2 ρ B 2 is a Brownian motion adapted to σ β σ β F t and µ 1 − σ 2 µ 2 − σ 2 η = − 1 � � �� 1 2 2 + β δ 2 is the equilibrium level. Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading

Outline Dynamics of Paired Stock Prices Introduction Co-integrating Vector Optimal Stochastic Control Model Dynamic of Wealth Value The HJB equation & Its Solution Integrability Condition Implementation & Exercise Agent’s Objective The dynamic of wealth value is given by dW = π 1 ( t ) dS 1 + π 2 ( t ) dS 2 . (4) Substitute equation (1) and (2) into the value of wealth (4), then we obtain the SDE as below dW = π 1 ( t ) ( µ 1 + δ z ( t )) S 1 dt + π 2 ( t ) µ 2 S 2 dt � � � 1 − ρ 2 dB 2 + π 1 ( t ) σ 1 S 1 dB 1 + π 2 ( t ) σ 2 S 2 ρ dB 1 + (5) . Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading

Outline Dynamics of Paired Stock Prices Introduction Co-integrating Vector Optimal Stochastic Control Model Dynamic of Wealth Value The HJB equation & Its Solution Integrability Condition Implementation & Exercise Agent’s Objective A pair of controls ( π 1 , π 2 ) is said to be admissible if π 1 and π 2 are real-valued, progressively measurable, are such that, (1)(2)(5) define a unique solution ( W , S 1 , S 2 ) for any t ∈ [0 , T ] and ( π 1 , π 2 , S 1 , S 2 ) satisfy the integrability condition � T ( π 1 S 1 ) 2 + ( π 2 S 2 ) 2 ds < + ∞ . E t Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading

Outline Dynamics of Paired Stock Prices Introduction Co-integrating Vector Optimal Stochastic Control Model Dynamic of Wealth Value The HJB equation & Its Solution Integrability Condition Implementation & Exercise Agent’s Objective Assume that the agent’s objective is � � �� W t , W , S 1 , S 2 J ( t , W , S 1 , S 2 ) = max (6) E U , T ( π 1 ,π 2 ) ∈A t where J ( t , W , S 1 , S 2 ) denote the value function, the agent seeks an admissible control pair ( π 1 , π 2 ) that maximizes the utility of wealth at time T . Specifying U ( W ). Now let us assume the utility function like U ( W ) = − exp( − γ W ) , (7) which is the CARA (Constant Absolute Risk Aversion) utility, where γ > 0 is constant and equal to the absolute risk aversion. Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading

Outline Introduction HJB Equation Particular Case δ = 0 Optimal Stochastic Control Model The HJB equation & Its Solution Solution Implementation & Exercise We expect the value function J ( t , W , S 1 , S 2 ) satisfy the following HJB partial differential equation: J t + max π 1 ,π 2 [ ( π 1 ( µ 1 + δ z ) S 1 + π 2 µ 2 S 2 ) J W + ( µ 1 + δ z ) S 1 J S 1 + µ 2 S 2 J S 2 + π 1 σ 2 1 S 2 1 J WS 1 + π 2 ρσ 1 σ 2 S 1 S 2 J WS 1 + π 2 σ 2 2 S 2 2 J WS 2 + π 1 ρσ 1 σ 2 S 1 S 2 J WS 2 +1 π 2 1 σ 2 1 S 2 1 + ρπ 1 π 2 σ 1 σ 2 S 1 S 2 + π 2 2 σ 2 2 S 2 � � J WW 2 2 +1 1 J S 1 S 1 + ρσ 1 σ 2 S 1 S 2 J S 1 S 2 + 1 2 σ 2 1 S 2 2 σ 2 2 S 2 2 J S 2 S 2 ] = 0 , (8) with the final condition J ( T , W , S 1 , S 2 ) = U ( W T ) = − exp( − γ W T ) . (9) Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading

Outline Introduction HJB Equation Particular Case δ = 0 Optimal Stochastic Control Model The HJB equation & Its Solution Solution Implementation & Exercise Ansatz S 1 = e x , S 2 = e y , J ( t , W , S 1 , S 2 ) = − e − γ W g ( t , x , y ) . Then the transformed HJB equation will be g t = max[ ( π 1 ( µ 1 + δ z ) S 1 + π 2 µ 2 S 2 ) γ g − ( µ 1 + δ z ) g x − µ 2 g y + π 1 σ 2 1 S 1 γ g x + π 2 ρσ 1 σ 2 S 2 γ g x + π 2 σ 2 2 S 2 γ g y + π 1 ρσ 1 σ 2 S 1 γ g y − 1 π 2 1 σ 2 1 S 2 1 + 2 π 1 π 2 ρσ 1 σ 2 S 1 S 2 + π 2 2 σ 2 2 S 2 γ 2 g � � 2 2 − 1 1 ( g xx − g x ) − 1 2 σ 2 2 σ 2 2 ( g yy − g y ) − ρσ 1 σ 2 g xy ] , (10) subject to g ( T , x , y ) = 1 . (11) Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading

Outline Introduction HJB Equation Particular Case δ = 0 Optimal Stochastic Control Model The HJB equation & Its Solution Solution Implementation & Exercise Optimal Control Weights (initial) ( µ 1 + δ z ) g x µ 2 π ∗ = + (12) − ρ , 1 γ (1 − ρ 2 ) σ 2 γ (1 − ρ 2 ) σ 1 σ 2 S 1 γ gS 1 1 S 1 µ 2 g y ( µ 1 + δ z ) π ∗ = + − ρ . (13) 2 γ (1 − ρ 2 ) σ 2 γ (1 − ρ 2 ) σ 1 σ 2 S 2 2 S 2 γ gS 2 Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading

Outline Introduction HJB Equation Particular Case δ = 0 Optimal Stochastic Control Model The HJB equation & Its Solution Solution Implementation & Exercise Particular Case δ = 0 If we set δ = 0, the value function is independent of the stocks and the amount invested in each stock are constant. Then the closed form for the value function and the amount corresponding to δ = 0 will be � µ 2 + µ 2 � � � 1 − 2 ρµ 1 µ 2 1 2 g ( t , x , y ) = exp ( T − t ) , − σ 2 σ 2 2(1 − ρ 2 ) σ 1 σ 2 1 2 µ 1 ρµ 2 π ∗ 1 S 1 = − , γ (1 − ρ 2 ) σ 2 γ (1 − ρ 2 ) σ 1 σ 2 1 µ 2 ρµ 1 π ∗ = 2 S 2 − . γ (1 − ρ 2 ) σ 2 γ (1 − ρ 2 ) σ 1 σ 2 2 Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading

Outline Introduction HJB Equation Particular Case δ = 0 Optimal Stochastic Control Model The HJB equation & Its Solution Solution Implementation & Exercise Reduce the HJB equation into a one-dimensional equation by letting X = µ 1 + δ z = µ 1 + δ ( a + x + β y ) , and by using the exponential change of variable Φ( t , X ) = − ln( g ( t , x , y )) i.e. g = exp( − Φ).Then we will obtain the linear parabolic PDE 2( X 2 + µ 2 � 1 � 1 ) − ρ µ 2 X + 1 2 2( σ 2 1 + βσ 2 Φ t = 2 )( δ Φ X ) − 1 − ρ 2 σ 2 σ 2 σ 1 σ 2 1 2 − 1 2( σ 2 1 + β 2 σ 2 2 + 2 σ 1 σ 2 βρ )( δ 2 Φ XX ) . (14) for any real number X and time 0 � t < T and is subject to the terminal condition Φ( T , X ) = 0 . (15) Hui Gong, UCL http://www.homepages.ucl.ac.uk/˜ucahgon/ Optimal Stochastic Control for Pairs Trading