Equilibrium Models with Transaction Costs I

Transcription

1 Equilibrium Models with Transaction Costs I Johannes Muhle-Karbe University of Michigan Carnegie Mellon University Dresden, August 28, 217

2 Introduction Outline Introduction Partial Equilibrium General Equilibrium Summary and Outlook

3 Introduction Frictions Classical financial theory: based on frictionless markets. Arbitrary amounts can be traded immediately at no cost. Etc. Substantially simplifies the mathematical analysis. But neglects many important effects. Pedersen ( Overview of Frictional Finance ): Frictions affect asset prices, macroeconomics, monetary policy. Provide unified explanation of a wide variety of phenomena. Empirical evidence is stronger than almost any other influence on the markets. Cochrane 11: The problem is that we don t have enough math. [...] Frictions are just hard with the tools we have right now. Tractable models with frictions? E.g., trading costs?

4 Introduction Equilibrium Models Classical approach in economics: general equilibrium. Prices determined endogenously by matching supply and demand. Typically intractable beyond simple toy models. Partial equilibrium models greatly facilitate analysis. Agents still optimize, but prices are given exogenously by some estimated or calibrated model. Much more tractable. Well-suited to analyze individual optimization of small agents. But cannot address systemic effects or regulatory measures. Need for equilibrium models with frictions: Effects of a financial transaction tax, tighter margin requirements, etc.? Impact of market power on asset prices?

5 Introduction Equilibrium Models with Frictions? Equilibrium models lead to nasty fixed-point problems. Given prices, agents solve their optimization problems. Then, need to choose prices to ensure markets clear. These prices then again feed back into agents decision making. Tractable models are rare even without frictions. Problem becomes much worse with frictions. Complicated individual decision problems. Frictionless solutions typically rely on representative agent. Aggregates all market participants. No trade in equilibrium. Not suitable with frictions. Accordingly, literature on equilibrium models with frictions is rather limited.

6 Introduction General Equilibrium Models with Frictions ct d Literature on equilibrium models with frictions: Numerical solution of discrete-time tree models. Heaton/Lucas 96. Buss/Dumas 15; Buss/Vilkov/Uppal 15. Additional restrictive modeling assumptions. No risky assets (Vayanos/Vila 99, Weston 16). Constant asset prices (Lo/Mamaysky/Wang 4). Full refund of costs that is not internalized (Davila 15). Stark contrast to progress on partial equilibrium models. Starts with Magill/Constantinides 76; Dumas/Luciano 91; Davis/Norman 9, Shreve/Soner 94. Explicit asymptotic formulas now available for small costs in general settings. Soner/Touzi 14; Kallsen/M-K 14, 15; Cai/Rosenbaum/Tankov 15, 16.

7 Introduction This Lecture Recap intuition behind small-cost asymptotics for partial equilibrium models. Different frictions require different mathematical tools. Regular vs. singular vs. impulse control. We will therefore focus on the most tractable specification, the model of Almgren/Chriss 1. However, the same methods apply much more broadly. Apply to general equilibrium problems in a second step. Starting point: sufficiently regular frictionless equilibrium. Goal: understand leading-order effects of small trading costs on trading strategies and prices.

8 Introduction The Almgren/Chriss Model Unaffected price process S t. Execution price affected linearly by trade size and speed: S t + λ ϕ t t when trading ϕ t shares on [t, t + t]. Trading cost compared to frictionless model is λ( ϕt t )2 t. Wealth process in the continuous-time limit: T T x + ϕ t ds t λ ϕ 2 t dt Only finite variation strategies dϕ t = ϕ t dt are feasible.

9 Introduction The Almgren/Chriss Model ct d Quadratic costs on turnover rate: Introduced for optimal execution (Almgren/Chriss 1). Used for portfolio choice in Garleanu/ Pedersen 13, 16; Guasoni/Weber 17; Almgren/Li 16; Moreau/M-K/Soner 17. Interpretation? Price impact in a highly-resilient block-shaped order book (Roch/Soner 13, Kallsen/M-K 14). Compensation for risk market makers incur until the locate a suitable counterparty (Garleanu/Pedersen 13, 16). Stylized example of a progressive transaction tax (Subrahmanyam 98, Schied/Zhang 17). Most tractable specification among all trading costs.

10 Partial Equilibrium Goal Functional First step: fix exogenous price process S. that maximizes expected utility: Find trading rate ϕ λ t = dϕλ t dt [ ( T )] T E U x + ϕ t ds t λ ϕ 2 t dt U is increasing, concave utility function. max ϕ! Here, focus on most tractable example: U(x) = e γx. Frictionless problem (λ = ) also requires dynamic programming. Nice solution ˆϕ is starting point here. For small transaction costs λ, we expect: ϕ λ ˆϕ and λ( ϕ λ ) 2

11 Partial Equilibrium Small-Cost Expansion Approximate goal functional by (formal) Taylor expansion: E [ U ( T T x + ϕ tds t λ ϕ 2 t dt + E ( [U x + T )] E [ U ( T )] x + ˆϕ tds t ) ( T ˆϕ tds t (ϕ λ t ˆϕ t)ds t T ) λ( ϕ λ t ) 2 dt First-order condition for frictionless optimality: U (x + T ˆϕ t ds t ) is density of equivalent martingale measure Q up to normalizing constant y. Intuition: optimality implies E [ U ( x + T ˆϕ tds t )] E [ U ( T )] x + ( ˆϕ t + εϕ t)ds t ]

12 Partial Equilibrium Small-Cost Expansion ct d Taylor expansion yields E [ U ( x + T ˆϕ tds t )] E E [ [ U U ( ( T )] x + ( ˆϕ t + εϕ t)ds t T )] [ T ] x + ˆϕ tds t + εye Q ϕ tds t [ ] As a consequence, = E T Q ϕ tds t for all integrands ϕ, so that S has to be a Q-martingale. In particular: E [U ( x + T ) ( T )] ˆϕ tds t (ϕ λ t ˆϕ t)ds t = Need second-order expansion to obtain nontrivial cost-displacement tradeoff.

13 Partial Equilibrium Small-Cost Expansion ct d (Formal) second-order Taylor expansion: E [ U ( T T )] x + ϕ tds t λ ϕ 2 t dt E [ ( T )] U x + ˆϕ tds t ye Q [ T γ(x + 2 [( T ) ] ye Q λ( ϕ λ t )2 dt ( ˆϕ T ) 2 ] tds t) (ϕ λ t ˆϕ t)ds t dt Here, γ(x) = U (x)/u (x) is the risk-aversion of U. Constant for exponential utility. For more general utilities, need extension to risk-tolerance process of Kramkov/Sirbu 6, Czichowsky/Kallsen/M-K.

14 Partial Equilibrium Tracking Problem For small costs, exponential utility maximization is formally equivalent to linear-quadratic tracking problem: [ ] T ( ) E Q γ t (ϕ t ˆϕ t ) 2 + λ ϕ 2 t dt min! ϕ where γ t = γσ2 t 2 for σ 2 t = d S t dt Variants studied by Rosenbaum/Tankov 14, Almgren/Li 16, Cai/Rosenbaum/Tankov 15, 16, Bank/Soner/Voss 17. Asymptotic equivalence to exponential utility maximization is established in Ahrens 15; Cayé/Herdegen/M-K 17.

15 Partial Equilibrium Rescaled Tracking Problem Key simplification: tracking problem for small costs is myopic. Subproblems on a fine partition of [, T ] can be solved independently. Consider a partition (t i ) with meshwidth O(λ a ). Rescale displacement by λ b to obtain nontrivial limit. Rewrite local tracking problems via change of variable s = t/λ c as [ ( ti+1 E Q λ c /λ c t i /λ c γ λ c sλ 2b ( ϕλ c s ˆϕ λ c s λ b ) 2 + λ ϕ 2 λ c s Now: choose a, b, c to obtain nontrivial limit of the target ˆϕ λ c s/λ b, match orders of displacement and costs. ) ds ]

16 Partial Equilibrium Rescaled Tracking Problem ct d Small-cost limit of the rescaled target ˆϕ λ c s/λ b? If frictionless optimizer ˆϕ is an Itô process: ˆϕ λ c s λ b = λc Martingale part dominates. λ ˆb b λ c sds + λc/2 λ b ˆσ λ c sdw s If c/2 = b: converges to BM B s with volatility ˆσ ti as λ. Rescaled tracking strategy: ψ s := ϕ λ 2b s/λ b ψ s := λ b ϕ λ 2b s Different orders for pure jump processes (Rosenbaum/Tankov 14) or fbm (Czichowsky/M-K/Schelling).

17 Partial Equilibrium Rescaled Tracking Problem ct d In summary, the rescaled tracking problem is [ ] ti+1 E Q λ 2b /λ 2b ( ) γ λ 2b sλ 2b (ψ s B s ) 2 + λ 1 2b ψ s 2 ds t i /λ 2b To match displacement and trading costs: choose b = 1/4. As γ λ 2b s = γ ti + o(1), this leads to ] ti+1 /λ E Q [λ 1/2 ( ) γ ti (ψ s B s ) 2 + ψ 2 t i /λ 1/2 s ds To obtain a stationary infinite horizon problem, choose a < 1/2, so that t i+1 t i = O(λ a 1/2 ) diverges. λ 1/2

18 Partial Equilibrium Rescaled Tracking Problem ct d Approximate with ergodic tracking for Brownian motion: V ti := inf ψ E Q [ lim 1 ] ( ) γ ti (ψ s B s ) 2 + ψ s 2 ds Local tracking problem approximated by λ t i+1 t i λ 1/2 V ti. Summing over all local problems in turn gives [ ] T λ 1/2 E Q V t dt + o(λ 1/2 ) Relationship between optimal controls: ϕ λ λ 1/2 s ˆϕ λ 1/2 s λ1/4 (ψ s B s ), ϕ λ λ 1/2 s λ 1/4 ψ s

19 Partial Equilibrium Rescaled Tracking Problem ct d Convergence proof for these approximations: In probability: Cai/Rosenbaum/Tankov 14, 15, 16. In S p : Ahrens 15, Cayé/Herdegen/M-K 17. In summary: remains to solve ergodic tracking problem for Brownian motion. d ˆϕ Constant frozen volatility ˆσ = t dt. γσ Constant frozen risk aversion γ = 2 t 2 = γ d S t 2 dt. Can be tackled by different methods. Dynamic programming. Calculus of variations as in Bank/Soner/Voss 17. We will discuss the second approach in the next lecture, so let us sketch the first here.

20 Partial Equilibrium Ergodic Tracking Problem The dynamic programming principle suggests { } V t = inf γ(ϕ t B t ) 2 dt + ϕ 2 t dt + E t [V t+dt ] ϕ t In view of the quadratic structure of the problem, make the following quadratic ansatz for the value function: V t = V (ϕ t B t ) = A (T t) + A 2 (ϕ t B t ) 2 Constant growth as horizon T becomes large, deviation is only state variable because target has independent increments. Remains true for Lévy processes as in Rosenbaum/Tankov 14. Becomes more involved for fbm (Czichowsky/M-K/Schelling).

21 Partial Equilibrium Ergodic Tracking Problem ct d Recall the dynamic programming principle: { } V t = inf γ(ϕ t B t ) 2 dt + ϕ 2 t dt + E t [V t+dt ] ϕ t The ansatz V t = A (T t) + A 2 (ϕ t B t ) 2 and Itô s formula yield E t [V t+dt ] = V t + ( A + A(ϕ t B t ) ϕ t + 1 ) 2 Aˆσ2 dt Dividing by dt and sending dt, formally leads to the dynamic programming equation: = inf { γ(ϕ t B t ) 2 + ϕ 2t A + A(ϕ t B t ) ϕ t + 1 } ϕ t 2 Aˆσ2

22 Partial Equilibrium Ergodic Tracking Problem ct d The pointwise minimum is ϕ t = A 2 (B t ϕ t ). To pin down A, plug this back into the dynamic programming equation: A = 2 γ, A = 1 2 ˆσ2 A = Therefore, the optimal trading rate is: ϕ t = γ(b t ϕ t ) γ 2 ˆσ2 Deviation B t ϕ λ t is Ornstein-Uhlenbeck process with volatility ˆσ and mean-reversion speed γ. A describes long-run growth rate as T.

23 Partial Equilibrium Solution of Original Tracking Problem We have solved the ergodic tracking problem. Its solution in turn pins down the asymptotical expansion of the original tracking problem. Indeed, the corresponding approximate value becomes T E Q λγσ 2 t 2 d ˆϕ t The corresponding optimal trading rate is the rescaled feedback control: ϕ λ γσt 2 t = 2λ ( ˆϕ t ϕ t )

24 General Equilibrium To Do Now, we want to apply the asymptotic results to a general equilibrium model. Start from frictionless equilibrium. Then study the effect of a small trading cost. Need to model several agents. Trading friction not visible through representative agent. Crucial: provide motive to trade. Here: risk sharing. Agents trade risky asset to hedge correlated endowments. Alternative: heterogenous beliefs. For simplicity: focus on two agents i = 1, 2. Receive exogenous endowment streams (yt) i t [,T ].

25 General Equilibrium: Model Endowments and Preferences Agents i = 1, 2 trade two assets with dynamics to be determined in equilibrium: A bank account, in zero net supply. A stock, in unit net supply, that gives right to an exogenous dividend process. Both agents have constant absolute risk aversion γ i and a deterministic impatience rate (β i t) t [,T ]. Choose (excess) consumption rates (ct) i t [,T ] to maximize expected utility: [ T ] E e t βi u du e γi (yt i +ci t ) dt max!

26 General Equilibrium: Model Trading Friction Exogenous quadratic cost λ for buying and selling the stock. Transaction tax imposed by the government. Fees charged by the trading platform. Agents bargain how the fee is split between buyers and sellers. But where does the fee go? Should not disappear in general equilibrium. Our proposal: Model the entity receiving the fees. Also receives exogenous endowment (yt 3 ) t [,T ]. Additionally receives endogenous payments from other agents. Solves corresponding optimal consumption problem. Does not trade the stock.

27 General Equilibrium: Model Radner Equilibrium Quantities to be determined in equilibrium: Interest rate. Initial value, drift, and volatility of the stock. Agents optimal strategies and consumption rates. Bid-ask split λ 1 + λ 2 = λ of the total transaction cost. Equilibrium conditions: All agents behave optimally. Markets clear. Asymptotic equilibrium: Only ask for asymptotic optimality for small costs in individual optimization problems. Similar to notion of ε-equilibrium in game theory.

28 General Equilibrium: Model Asymptotic Perspective Start from frictionless equilibrium. Suppose all corresponding quantities are known and nice. In particular: frictionless interest rate is deterministic. Lo/Mamaysky/Wang 4: diffusive strategies, but constant stock price. Vayanos 98, Vayanos/Vila 99, Christensen/Larsen 14: deterministic, smooth trading strategies. Christensen/Larsen 14: diffusive stock prices and strategies, but stochastic interest rates. Herdegen/M-K 16: example with deterministic interest rates but diffusive prices and trading strategies. What changes with the introduction of the trading cost?

29 General Equilibrium: Results Stock Market Clearing Recall: asymptotically optimal trading rates ϕ λ,i t = γ i σ 2 t 2λ i ( ˆϕ i t ϕ λ,i t ) := l i t( ˆϕ i t ϕ λ,i t ) where ˆϕ i are the frictionless optimal strategies. Suppose that both agents have the same risk aversion as in Lo/Mamasky/Wang 4 and pay the same transaction cost. Then, l 1 t = l 2 t = l t and the frictionless equilibrium dynamics still clear the market: Frictionless stock market clearing implies ˆϕ 1 t + ˆϕ 2 t = 1 and in turn d(ϕ λ,1 t + ϕ λ,2 t ) = (1 (ϕ λ,1 t + ϕ λ,2 t ))dt If the initial allocation clears the market, this indeed has the unique solution ϕ λ,1 t + ϕ λ,2 t = 1.

30 General Equilibrium: Results Stock Market Clearing ct d For heterogenous risk aversions, the same argument still works if λ 1 /γ 1 = λ 2 /γ 2 Stronger risk-sharing motive makes higher costs acceptable: λ 1 = γ1 γ 1 + γ 2 λ, λ2 = γ2 γ 1 + γ 2 λ Ensures stock market clearing holds for the frictionless equilibrium dynamics. But what about the market for the consumption good?

31 General Equilibrium: Results Goods Market Clearing Crucial point: transaction costs shift money from Agents 1,2 to Agent 3. Do asymptotically optimal consumption rates still clear the goods market? Second main ingredient: asymptotic analysis of optimal consumption problems for small perturbations of the endowment (Herdegen/M-K 15). Similar asymptotic perspective: Start from baseline endowment for which solution is well understood. Perform sensitivity analysis for general but small perturbation. Application here?

32 General Equilibrium: Results Goods Market Clearing Asymptotic consumption adjustments (Herdegen/M-K 15): Second-order adjustment for displacement of order O(ε 1/3 ). Mean zero. Effect of order O(ε 2/3 ). Total adjustment is zero as aggregate displacement is zero. First-order optimal adjustment for transaction cost loss of order O(ε 2/3 ). Effect of order O(ε 2/3 ). First-order adjustment for Agent 3. Again of order O(ε 2/3 ). Total consumption change can be chosen to be zero, as total endowment adjustment is zero. In summary: Frictionless equilibria can still support market clearing even with small trading frictions. Prices and price dynamics need not change due to introduction of a small transaction tax. The more risk averse agent pays most of the costs.

33 Summary and Outlook Scope of Robustness Result Robustness of equilibria with respect to small trading costs. Not an envelope theorem: strategies change, but not prices. Straightforward extensions: Other trading costs. No third agent, transaction costs disappear from model. Crucial prerequisites: Only two agents. Extensions with market maker? Constant absolute risk aversions. Deterministic impatience rates. Deterministic frictionless interest rates. Justified if time horizon is not too long. Formal alternative: refunds like in the finance literature (Buss/Dumas 15, Davilo 15). Equal risk aversions or costs split accordingly. Will be relaxed in a simpler model in the next lecture.