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# IE680 - 1 Simulation Based Optimization for Large...

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1 Simulation Based Optimization for Large Portfolios with Transaction Costs Kumar Muthuraman and Haining Zha Speaker: Haolin Feng School of Industrial Engineering, Purdue University. 19 th April 2007.

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2 Outline of Talk Portfolio optimization with transaction costs Introduce the framework of a portfolio optimization problem Formulate the problem Argue that it is sufficient to solve a related free-boundary PDE problem Introduce a moving boundary method to solve the problem Adopt a simulation method to efficiently solve the problem with large portfolios
3 Portfolio optimization: Framework Portfolio: A collection of a risk-free (Bank) and N risky (Stock) assets. Portfolio optimization: To maintain a portfolio to maximize/minimize some objective function. Values of stock follow a Geometric Brownian Motion (GBM). Proportional transactions costs. Selling a unit value of asset i results in (1 μ i ) cash. Buying a unit value of asset i requires (1 + λ i ) cash. To maximize Long-term growth rate (Taksar et.al. (’88), Akian et.al. (’01)) lim inf t →∞ E braceleftbigg log W ( t ) t bracerightbigg .

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4 Geometric Brownian Motion A stochastic process B ( t ) is called a Standard Brownian Motion if B (0) = 0 B has independent increments, i.e. if r < s t < u , then B ( u ) B ( t ) and B ( s ) B ( r ) are independent stochastic variables. For s < t , B ( t ) B ( s ) is N (0 , t s ) . B is continuous almost surely. A stochastic process S ( t ) is called a Geometric Brownian Motion if dS ( t ) = αS ( t ) dt + σS ( t ) dB ( t ) S (0) = s for some initial value s negationslash = 0 ( > 0 for a price process).
5 Simulate GBM To simulate a sample path of S ( t ) up to time t = T , we can discretize the time interval [0 , T ] as 0 = t 0 < t 1 < . . . < t n = T , and then 1 S ( t 0 ) = s 2 Simulate independent normal random numbers w i as N (0 , t i t i 1 ) for i = 1 , . . . , n 3 S ( t i ) = S ( t i 1 ) + αS ( t i 1 ) · ( t i t i 1 ) + σS ( t i 1 ) · w i [ dS Δ S ].

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6 A Simulated Sample Path of a GBM As an example we set t i t i 1 = 0 . 05 , s 0 = 1 , α = 0 . 1 , and σ = 0 . 2 . 0 10 20 30 40 50 0 2 4 6 8 10 12 14
7 Structure of policy - No transaction costs Dynamics: Value of stock ( S i ) and bank ( S 0 ), dS i = α i S i dt + σ i S i dB i + dL i dU i dS 0 = rS 0 dt dL i + dU i ( L, U : Cumulative amt of stock bought, sold. λ, μ : Fee for buying, selling) Objective: Maximize lim inf t →∞ E braceleftbigg log W ( t ) t bracerightbigg Optimal to keep fixed fractions of wealth in each asset

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8 Structure of policy - With transaction costs Dynamics: Value of stock ( S i ) and bank ( S 0 ), dS i = α i S i dt + σ i S i dB i + dL i dU i dS 0 = rS 0 dt + summationdisplay i [ (1 + λ i ) dL i + (1 μ i ) dU i ] ( L, U : Cumulative amt of stock bought, sold. λ, μ : Fee for buying, selling) Too expensive to keep state at the Merton line Optimal policy is characterized by a No transaction region - A cone State X = S/W , fraction of wealth in each stock.
9 Changing state variable Original Problem dS 0 = r S 0 dt ( e + λ ) · dL + ( e μ ) · dU, dS = diag ( S ) [ α dt + σ dB ] + dL dU.

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IE680 - 1 Simulation Based Optimization for Large...

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