On Using Shadow Prices in Portfolio optimization with Transaction Costs

Transcription

1 On Using Shadow Prices in Portfolio optimization with Transaction Costs Johannes Muhle-Karbe Universität Wien Joint work with Jan Kallsen Universidad de Murcia

2 Outline The Merton problem The Merton problem with transaction costs Shadow prices Application to Merton problem with transaction costs

3 The Merton problem Basic setting Bond normalized to S 0 = 1 Stock modelled as ds t = S t α t dt + S t σ t dw t Trading strategy (ϕ 0, ϕ), consumption rate c Self-financing condition: dϕ 0 t = S t dϕ t c t dt Admissibility condition: ϕ 0 t + ϕ t S t 0

4 The Merton problem Optimization problem Goal: Maximize expected utility from consumption ( ) E e δt log(c t )dt 0 over all admissible (ϕ 0, ϕ, c) Impatience rate δ log(c t ) measures utility from consumption at time t Infinite planning horizon Already solved by Merton (1971) What does the solution look like?

5 The Merton problem Solution Goal: Maximize over all admissible (ϕ 0, ϕ, c) ds t /S t = α t dt + σ t dw t ( ) E e δt log(c t )dt 0 Consume contant fraction c t = δ(ϕ 0 t + ϕ t S t ) of wealth Invest myopic fraction of wealth into stocks π t = ϕ ts t ϕ 0 t + ϕ t S t = α t σ 2 t For Black-Scholes model: α, σ and hence π are constant

6 The Merton problem Solution ct d Optimal strategy in the Black-Schloles model: Invest constant fraction πt = ϕ ts t ϕ 0 = α t + ϕ t S t σ 2 into stocks Buy stocks when prices go down, sell when they move up Consequence: Continuous trading necessary due to fluctuation of the Brownian motion W Strategy leads to instant ruin for transaction costs How to formalize this? How does the optimal policy change?

7 The Merton problem with transaction costs Basic setting Bond S 0 = 1, stock ds t = S t µ t dt + S t σ t dw t Can buy stocks only at higher ask price S t = (1 + λ)s t Can sell them only at lower bid price S t = (1 µ)s t Self-financing condition: dϕ 0 t = S t dϕ t S t ϕ t ds t c t dt Admissibility condition: ϕ 0 t + (ϕ t ) + S t (ϕ t ) S t 0

8 The Merton problem with transaction costs Optimization problem Goal: As before, maximize ( E over all admissible (ϕ 0, ϕ, c) 0 ) e δt log(c t )dt Problem does not have to be changed Only notion of admissibility has to be adapted But now, solution is much harder Results only available for Black-Scholes with constant α, σ Structure of the solution?

9 The Merton problem with transaction costs Results Remember: Without transaction costs (Merton (1971)) Fixed fraction π of wealth in stock (e.g. 31%) Consumption rate is fixed proportion of wealth Both numbers explicitly known With transaction costs (Magill & Constantinidis (1976), Davis & Norman (1990), Shreve and Soner (1994)): Minimal trading to keep fraction of wealth in stock in fixed corridor [π, π] (e.g %) Consumption rate is function of wealth in cash and stock Corridor known only as solution to free boundary problem Method: Stochastic control, PDEs. Here: Different approach

10 Shadow Prices A general principle s Optimal portfolio with transaction costs? t

11 Shadow Prices A general principle bid Optimal portfolio with transaction costs? t

12 Shadow Prices A general principle bid Optimal portfolio with transaction costs Optimal portfolio without transaction costs for shadow price t

13 Shadow prices A general principle Idea: Problem with transaction costs as problem without transaction costs for different price process Shadow price at boundary when optimal strategy transacts Min-Max theorem: sup ϕ ( ) inf Utility = S [S,S] ( ) inf sup Utility S [S,S] ϕ Similiar to concept of consistent price systems in W. Schachermayer s talk yesterday But does this really hold? Under what conditions?

14 Shadow prices A general principle? Existence of a shadow price S? Partial positive results for continuous processes in Karatzas & Cvitanić (1996), Loewenstein (2002) Kallsen & M-K (2009): Always holds, if Ω < Elementary proof, S constructed from Lagrange multipliers General theorem is still missing, current work in progress with W. Schachermayer, J. Kallsen and M. Owen Other structural results in different areas But can this be used for computations?

15 Application to Merton problem with transaction costs Using shadow prices? If d S = γ t dt + ɛ t dw t were known things would be easy: Consume contant fraction ct = δ(ϕ 0 t + ϕ S t t ) Invest constant fraction πt = γ t /ɛ 2 t into stocks Wealth now measured in terms of S instead of S But: Even if it exists, S is not known a priori Hence: Must be determined simulatneously with π and c!

16 Application to Merton problem with transaction costs Price processes Real price processes: Stock price: ds t /S t = αd t + σdw t Bid price: (1 µ)s t Ask price: (1 + λ)s t Shadow price process S [(1 µ)s, (1 + λ)s]: S t = exp(c t )S t C t = log( S t /S t ) deviation from real price C t [log(1 µ), log(1 + λ)] C moves in bounded interval. How to model such a process?

17 Application to Merton problem with transaction costs Ansatz for the shadow price How to model process C [log(1 µ), log(1 + λ)]? Naive approach: dc t = α(c t )dt + σ(c t )dw t Diffusion of order dt, drift of order dt, need drift at the boundary

18 Application to Merton problem with transaction costs Ansatz for the shadow price How to model process C [log(1 µ), log(1 + λ)]? Naive approach: dc t = α(c t )dt + σ(c t )dw t + local time Diffusion of order dt, drift of order dt, need drift at the boundary

19 Application to Merton problem with transaction costs Ansatz for the shadow price How to model process C [log(1 µ), log(1 + λ)]? Naive approach: dc t = α(c t )dt + σ(c t )dw t + local time Diffusion of order dt, drift of order dt, need drift at the boundary But: Optimal fraction Drift/Diffusion 2 would be infinite This is not a good idea with transaction costs! Different approach?

20 Application to Merton problem with transaction costs Ansatz for the shadow price ct d How to model process C [log(1 µ), log(1 + λ)]? Refined approach: dc t = α(c t )dt + σ(c t )dw t Diffusion of order dt, drift of order dt Need to have σ(c t ) 0 when approaching the boundary Analogous to square-root process for e.g. interest rates: dr t = (κ λr t )dt + r t dw t

21 Application to Merton problem with transaction costs Ansatz for the shadow price ct d Itô process dc t = α(c t )dt + σ(c t )dw t d S t / S t = Drift(C t )d t + Diffusion(C t )dw t Remember: Optimal strategy (without transaction costs): Consumption: δṽt Fraction of stocks: π(c t ) = Drift(Ct) Diffusion(C t) 2 Use transformation 1 1+exp( f (C = π(c t)) t) f (C t ) = log( π(ct) 1 π(c ) t) Need to determine 3 functions: α, σ, f f (log(1 µ)), f (log(1 + λ)) determine corridor

22 Application to Merton problem with transaction costs Conditions for the shadow price Optimality: exp( f ) = Drift Diffusion 2 No trading within bounds: dϕ t = 0 for optimal ϕ Itô s formula: (I) dϕ t = somefunction(f, f, f, α, σ)dt + anotherfunction(f, f, α, σ)dw t Hence 0 = somefunction, (II) 0 = anotherfunction (III) 3 conditions

23 Application to Merton problem with transaction costs Conditions for the shadow price ct d Solution to Equations I-III: σ σ = f 1 ( f α = α + σ 2 ) ( ) 1 f e f f satisfies the ODE ( ) f (x) = 2δ (1 + e f (x) ) σ 2 + ( 4α σ 2 + ( 2α σ 2 ) 1 4δ (1 + e f (x) ) f (x) σ 2 ) + 2 2δ (1 + e f (x) 1 e f (x) ) + (f (x)) 2 σ 2 1+e f (x) + ( 2α σ e f (x) ) (f (x)) 3 Still missing: Boundary conditions for x = log(1 µ) and x = log(1 + λ)?

24 Application to Merton problem with transaction costs Heuristics for boundary conditions Remember: has to stay in [log(1 µ, 1 + λ] dc t = α(c t )dt + σ(c t )dw t Consequence: Need σ 0 at the boundary σ = σ f 1 f = at the boundary If C = log(1 µ): Shadow price = Bid price higher sell boundary If C = log(1 + λ): Shadow price = Ask price lower buy boundary Hence: f is decreasing, f = at the boundary

25 Application to Merton problem with transaction costs The decisive ODE Have to solve second-order ODE s.t. f (log(1 µ)) = log f (x) = somefunction(f (x)) ( ) ( ) π π, f (log(1 + λ)) = log 1 π 1 π and f (log(1 µ)) =, f (log(1 + λ)) = Same number of conditions and degrees of freedom But f = is difficult both for existence proof and numerics Way out: Consider g = f 1 instead

26 Application to Merton problem with transaction costs The decisive free boundary problem g (y) = ( ) 1 e y 1+e + 1 2α y σ 2 + ( 4α + σ 2 ( 2α σ e y 1+e y 2δ ( 2δ σ 2 (1 + e y ) + 1 4δ (1 + e y ) σ ) 2 (g (y)) 3 ) (1 + e y ) σ 2 ) (g (y)) 2 g (y) s.t. ( ( )) ( ( )) π π g log = log(1 µ), g log = log(1 + λ) 1 π 1 π and ( )) ( ( )) π π g (log = 0, g log = 0 1 π 1 π Boundaries determine no-trade region

27 Application to Merton problem with transaction costs Numerical solution g (y) = somefunction(y) s.t. and g g ( ( )) ( ( )) π π log = log(1 µ), g log = 0 1 π 1 π ( ( )) ( ( )) π π log = log(1 + λ) g log = 0 1 π 1 π Numerically compute solution g to initial value proble for given boundary, find next zero of g Adjust boundary to get right value of g there This is also the basis for the existence proof

28 Application to Merton problem with transaction costs Numerical solution ct d 0.01 The function g The function f

29 Application to Merton problem with transaction costs Simulation 0.4 Optimal fraction of wealth held in stocks Shadow price/real price

30 Application to Merton problem with transaction costs Simulation ct d Value in stocks Value of the portfolio

31 Summary Computation of conditions: 1. Optimality without transaction costs, 2. Constant trading strategy within bounds, 3. Boundary conditions via Itô process assumption. Verification: 1. Prove existence of a solution to free boundary problem. 2. Prove existence of corresponding processes S etc. 3. Show that optimal investment in S trades only at boundary.

32 References This talk: Kallsen, J. and J. Muhle-Karbe (2008). On using shadow prices in portfolio optimization with transaction costs. The Annals of Applied Probability. To appear. Kallsen, J. and J. Muhle-Karbe (2009). On the existence of shadow prices in finite discrete time. Preprint. Portfolio optimization with transaction costs: Magill, M. and G. Constantinidis (1976) Portfolio selection with transaction costs. Journal of Economic Theory Davis, M. and A. Norman (1990). Portfolio selection with transaction costs. Mathematics of Operations Research Shreve, S. and M. Soner (1994). Optimal investment and consumption with transaction costs. The Annals of Applied Probability