# transaction costs made tractable

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- 1. Model Results HeuristicsMethod Transaction Costs Made TractablePaolo Guasoni Stefan Gerhold Johannes Muhle-KarbeWalter SchachermayerBoston University and Dublin City UniversityStochastic Analysis in Insurance and FinanceUniversity of Michigan at Ann Arbor, May 17th , 2011

2. ModelResultsHeuristicsMethod Outline Motivation:Trading Bounds, Liquidity Premia, and Trading Volume. Model:Constant investment opportunities and risk aversion. Results:Explicit formulas. Asymptotics. Method:Shadow Price and long-run optimality. 3. ModelResults Heuristics Method Transaction Costs Classical portfolio choice: 1 Constant ratio of risky and safe assets. 2 Sharpe ratio alone determines discount factor. 3 Continuous rebalancing and innite trading volume. Transaction costs: 1 Variation in risky/safe ratio. Tradeoff between higher tracking error and higher costs. 2 Liquidity premium. Trading costs equivalent to lower expected return. 3 Finite trading volume. Understand dependence on model parameters. Tractability? 4. Model Results Heuristics MethodLiterature Magill and Constantinides (1976): the no-trade region. Constantinides (1986):no-trade region large, but liquidity premium small. Davis and Norman (1990):rigorous solution. Algorithm for trading boundaries. Taksar, Klass, and Assaf (1998), Dumas and Luciano (1991):long-run control argument. Numerical solution. Shreve and Soner (1994):Viscosity solution. Utility impact of transaction cost of order 2/3 Jane ek and Shreve (2004): cTrading boundaries of order 1/3 . Asymptotic expansion. Kallsen and Muhle-Karbe (2010),Gerhold, Muhle-Karbe and Schachermayer (2010):Logarithmic solution with shadow price. Asymptotics. 5. Model Results HeuristicsMethodThis Paper Long-run portfolio choice. No consumption. Constant relative risk aversion . Explicit formulas for:1 Trading boundaries.2 Certainty equivalent rate (expected utility).3 Trading volume (relative turnover).4 Liquidity premium.In terms of gap parameter. Expansion for gap yields asymptotics for all other quantities.Of any order. Shadow price solution. Long-run verication theorem. Shadow price also explicit. 6. ModelResults HeuristicsMethodModel Safe rate r . Ask (buying) price of risky asset: dSt = (r + )dt + dWt St Bid price (1 )St . is the spread. Investor with power utility U(x) = x 1 /(1 ). Maximize certainty equivalent rate (Dumas and Luciano, 1991): 1111 max limlog E XT T T 7. ModelResults Heuristics MethodWelfare, Liquidity Premium, TradingTheoremTrading the risky asset with transaction costs is equivalent to: investing all wealth at hypothetical safe certainty equivalent rate 2 2 CeR = r +2 2 trading a hypothetical asset, at no transaction costs, with samevolatility , but expected return decreased by the liquidity premium LiP = 2 2 . Optimal to keep risky weight within buy and sell boundaries(evaluated at buy and sell prices respectively)+ =, + = , 2 2 8. Model Results Heuristics MethodGapTheorem identied as unique value for which solution of Cauchy problem 2 +w (x) + (1 )w(x)2 + 1 w(x) =0 2 2 2 w(0) =, 2satises the terminal value condition:+u()1 (+)( 2 )w(log(u()/l())) = 2 , where l() = 1 ()(+ 2 ) . Asymptotic expansion:1/3 = 23 24 (1 )2 1/3 + O(). 9. ModelResultsHeuristicsMethod Trading VolumeTheorem Share turnover (shares traded d||||t divided by shares held |t |).1 T d t2 21 1+ShT = lim 0 |t |=2 1 2 2. T T2 1(u/l) 2 1(u/l) 12 1 Wealth turnover, (wealth traded divided by the wealth held): 1 T(1)St dTSt dWeT = limt 0 0 St0 +t (1)St +t 0 0 St0 +t StT Tt t22 (1 )+ (1+ )= 2 2 1 2 1 1 2.(u/l) 2 1 (u/l)2 1 10. ModelResultsHeuristicsMethod Asymptotics1/33 2 = (1 )2 1/3 + O(). 4 2/32 2 3 2CeR = r + (1 )22/3 + O(4/3 ). 2 224 2/3 3 2 LiP = (1 )22/3 + O(4/3 ). 224 1/32 3 2ShT =(1 )2 (1 )2 1/3 + O(1/3 )2 4 2/3 23 2WeT = (1 )21/3 + O(). 3 4 11. Model Results HeuristicsMethodImplications / 2 depends on mean-variance ratio = / 2 . Only. Trading boundaries depend only on . Certainty equivalent, liquidity premium, volume per unit variancedepend only on . Interpretation: certainty equivalent, liquidity premium, volume t2proportional to business time0 s ds. Trading strategy invariant. All results extend to St such that: dSt = (r + t )dt + t dWt St 1 Twith t independent of Wt and ergodic (limT T 0 t2 dt = 2 ). Same formulas hold,replacing / 2 with , and residual factor 2 with 2 . 12. Model Results HeuristicsMethodTrading Boundaries v. Spread0.750.700.650.600.550.50 0.000.02 0.040.06 0.080.10 = 8%, = 16%, = 5. Zero discount rate for consumption. 13. Model Results HeuristicsMethodLiquidity Premium v. Spread0.0120.0100.0080.0060.0040.002 0.01 0.02 0.030.040.05 = 8%, = 16%. = 5, 1, 0.5. 14. Model Results Heuristics MethodLiquidity Premium v. Risk Aversion0.0100.0080.0060.0040.0020.0000 2 4 6810 = 8%, = 16%. = 0.01%, 0.1%, 1%, 10%. 15. Model ResultsHeuristics Method Share Turnover v. Risk Aversion0.50.40.30.20.10.00 2 4 6 810 = 8%, = 16%, = 5. = 0.01%, 0.1%, 1%, 10%. 16. Model ResultsHeuristics Method Wealth Turnover v. Risk Aversion0.50.40.30.20.10.00 2 4 6 810 = 8%, = 16%, = 5. = 0.01%, 0.1%, 1%, 10%. 17. Model ResultsHeuristicsMethod Welfare, Volume, and Spread Liquidity premium and share turnover:LiP3= + O(5/3 )ShT4 Certainty equivalent rate and wealth turnover: 2(r + 2 ) CeR 3 = + O(5/3 ). WeT 4 Two relations, one meaning. Welfare effect proportional to spread, holding volume constant. For same welfare, spread and volume inversely proportional. Relations independent of market and preference parameters. 3/4 universal constant. 18. Model Results HeuristicsMethodWealth Dynamics Number of shares must have a.s. locally nite variation. Otherwise innite costs in nite time. Strategy: predictable process (0 , ) of nite variation. 0 units of safe asset. t shares of risky asset at time t.t t = . Shares bought minus shares sold .tt tt Self-nancing condition:StStd0 = t0 d + (1 ) 0 d tStSt Xt0 = 0 St0 , Xt = t St safe and risky wealth, at ask price St .t dXt0 =rXt0 dt St d + (1 )St d ,t t dXt =( + r )Xt dt + Xt dWt + St d St d t 19. Model ResultsHeuristics MethodControl Argument V (t, x, y ) value function. Depends on time, and on asset positions. By Its formula: 1dV (t, Xt0 , Xt ) = Vt dt + Vx dXt0 + Vy dXt + Vyy d X , X t 2 2 2= Vt + rXt0 Vx + ( + r )Xt Vy + X Vyy dt 2 t+ St (Vy Vx )d + St ((1 )Vx Vy )d + Xt dWt t t V (t, Xt0 , Xt ) supermartingale for any . , increasing, hence Vy Vx 0 and (1 )Vx Vy 0 Vx11 Vy 1 20. ModelResults HeuristicsMethodNo Trade RegionV 1 When 1 Vx 1 does not bind, drift is zero: y2 2Vx1 Vt + rXt0 Vx + ( + r )Xt Vy + Xt Vyy = 0 if 1