Portfolio optimization with transaction costs

Transcription

1 Portfolio optimization with transaction costs Jan Kallsen Johannes Muhle-Karbe HVB Stiftungsinstitut für Finanzmathematik TU München AMaMeF Mid-Term Conference, , Wien

2 Outline The Merton problem with transaction costs A general principle Application to Merton problem with transaction costs References

3 The Merton problem with transaction costs Goal: Maximize expected utility from consumption ( ) E e δt u(c t )dt 0 Here: u(x) = log(x) c admissible consumption rate (no debts) Bank account: 1 (no interest paid) Stock price S: Modeled as geometric Brownian motion Proportional transaction costs µ, λ (e.g. 1%)

4 The Merton problem with transaction costs Without transaction costs (Merton [1971]): Fixed fraction of wealth in stock (e.g. 31%) Consumption rate is fixed proportion of wealth Both numbers explicitly known With transaction costs (Magill and Constantinides [1976], Davis and Norman [1990]): Fraction of wealth in stock in fixed corridor (e.g %) Consumption rate is function of wealth in cash and stock Corridor known only as solution to free boundary problem

5 A general principle Shadow prices s Optimal portfolio with transaction costs? t

6 A general principle Shadow prices bid Optimal portfolio with transaction costs? t

7 A general principle Shadow prices bid Optimal portfolio with transaction costs Optimal portfolio without transaction costs for shadow price t

8 A general principle Shadow prices Idea: Problem with transaction costs as problem without transaction costs for different price process Shadow price at boundary when optimal strategy transacts Appearances in various fields: Jouini and Kallal [1995]: No-arbitrage Lamberton et al. [1998]: Local risk minimization Cvitanić and Karatzas [1996], Loewenstein [2000]: Portfolio optimization Useful for computations?

9 Real price processes: Stock price(discounted): 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]: St = exp(c t )S t C t = log( S t /S t ) deviation from real price C t [log(1 µ), log(1 + λ)]

10 Real price processes: Stock price(discounted): 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]: St = exp(c t )S t C t = log( S t /S t ) deviation from real price C t [log(1 µ), log(1 + λ)]

11 Real price processes: Stock price(discounted): 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]: St = exp(c t )S t C t = log( S t /S t ) deviation from real price C t [log(1 µ), log(1 + λ)] Dynamics of C?

12 Ansatz: 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 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 t)) = π(c t) Need to determine 3 functions: α, σ, f f (log(1 µ)), f (log(1 + λ)) determine corridor

13 Ansatz: 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 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 t)) = π(c t) Need to determine 3 functions: α, σ, f f (log(1 µ)), f (log(1 + λ)) determine corridor

14 Ansatz: 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 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 t)) = π(c t) Need to determine 3 functions: α, σ, f f (log(1 µ)), f (log(1 + λ)) determine corridor

15 Optimality: exp( f ) = Drift Diffusion 2 (I) No trading within bounds: dϕ t = 0 for optimal ϕ Itô s formula: 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

16 Optimality: exp( f ) = Drift Diffusion 2 (I) No trading within bounds: dϕ t = 0 for optimal ϕ Itô s formula: 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

17 Optimality: exp( f ) = Drift Diffusion 2 (I) No trading within bounds: dϕ t = 0 for optimal ϕ Itô s formula: 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

18 Solution to Equations I-III: f satisfies the ODE σ σ = f 1 ( ) ( ) f α = α + σ 2 1 f e f f (x) = ( 2δ (1 + e f (x) ) ) + ( 2α 1 4δ (1 + e f (x) ) ) f (x) σ 2 σ 2 σ 2 + ( 4α 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)

19 Heuristics for boundary conditions: Optimal fraction π(c t ): Reflected diffusion (e.g. between 20% and 40%) local time at boundary ( ) Hence f (C t ) = log 1 π(ct) π(c t) has local time Our Ansatz: S t (and hence C t ) Itô process, i.e. no local time at boundary Intuition: otherwise infinite position optimal at boundary Consequence: Contradiction unless f = on boundary Boundary conditions f (log(1 µ)) = f (log(1 + λ)) =

20 Heuristics for boundary conditions: Optimal fraction π(c t ): Reflected diffusion (e.g. between 20% and 40%) local time at boundary ( ) Hence f (C t ) = log 1 π(ct) π(c t) has local time Our Ansatz: S t (and hence C t ) Itô process, i.e. no local time at boundary Intuition: otherwise infinite position optimal at boundary Consequence: Contradiction unless f = on boundary Boundary conditions f (log(1 µ)) = f (log(1 + λ)) =

21 Heuristics for boundary conditions: Optimal fraction π(c t ): Reflected diffusion (e.g. between 20% and 40%) local time at boundary ( ) Hence f (C t ) = log 1 π(ct) π(c t) has local time Our Ansatz: S t (and hence C t ) Itô process, i.e. no local time at boundary Intuition: otherwise infinite position optimal at boundary Consequence: Contradiction unless f = on boundary Boundary conditions f (log(1 µ)) = f (log(1 + λ)) =

22 Heuristics for boundary conditions: Optimal fraction π(c t ): Reflected diffusion (e.g. between 20% and 40%) local time at boundary ( ) Hence f (C t ) = log 1 π(ct) π(c t) has local time Our Ansatz: S t (and hence C t ) Itô process, i.e. no local time at boundary Intuition: otherwise infinite position optimal at boundary Consequence: Contradiction unless f = on boundary Boundary conditions f (log(1 µ)) = f (log(1 + λ)) =

23 Heuristics for boundary conditions: Optimal fraction π(c t ): Reflected diffusion (e.g. between 20% and 40%) local time at boundary ( ) Hence f (C t ) = log 1 π(ct) π(c t) has local time Our Ansatz: S t (and hence C t ) Itô process, i.e. no local time at boundary Intuition: otherwise infinite position optimal at boundary Consequence: Contradiction unless f = on boundary Boundary conditions f (log(1 µ)) = f (log(1 + λ)) =

24 Numerical solution: Consider g = f 1 ODE for g: g (y) = ( ) 1 e y + 1 2α 1+e y σ 2 + ( 4α σ 2 + ( 2α σ 2 Free boundary: y 1, y 2 with 2 1 e y 1+e y 2δ σ 2 (1 + e y ) + 1 4δ σ 2 (1 + e y ) ) (g (y)) 2 ( 2δ σ 2 (1 + e y ) ) (g (y)) 3 g(y 1 ) = log(1 µ), g (y 1 ) = 0 g(y 2 ) = log(1 + λ), g (y 2 ) = 0 ) g (y) Free boundaries y 1, y 2 determine corridor, g = f 1 determines dynamics of C and hence S = exp(c)s

25 Numerical solution ct d 0.01 The function g The function f

26 Simulation 0.4 Optimal fraction of wealth held in stocks Shadow price/real price

27 Simulation ct d Value in stocks Value of the portfolio

28 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.

29 References J. Cvitanić and I. Karatzas. Hedging and portfolio optimization under transaction costs: a martingale approach. Mathematical Finance, 6(2): , M. H. A. Davis and A. R. Norman. Portfolio selection with transaction costs. Mathematics of Operations Research, 15(4): , E. Jouini and H. Kallal. Martingales and arbitrage in securities markets with transaction costs. Journal of Economic Theory, 66(1): , D. Lamberton, H. Pham, and M. Schweizer. Local risk-minimization under transaction costs. Mathematics of Operations Research, 23(3): , M. Loewenstein. On optimal portfolio trading strategies for an investor facing transactions costs in a continuous trading market. Journal of Mathematical Economics, 33(2), M. J. P. Magill and G. M. Constantinides. Portfolio selection with transactions costs. Journal of Economic Theory, 13(2): , R. C. Merton. Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3: , 1971.