An Explicit Shadow Price for the Growth-Optimal Portfolio with - PowerPoint PPT Presentation

An Explicit Shadow Price for the Growth-Optimal Portfolio with Transaction Costs Johannes Muhle-Karbe Joint work with Stefan Gerhold and Walter Schachermayer Analysis, Stochastics, and Applications – A Conference in honour of Walter Schachermayer Wien, July 15, 2010

Outline Introduction Shadow Prices Growth-Optimal Portfolio under Transaction Costs Outlook

Introduction Markets with transaction costs Frictionless markets : ◮ Securities can be bought and sold for the same price ◮ Most optimizers of financial problems involve continuous trading, not possible in reality ◮ Applies to investment strategies, hedges, etc. Proportional transaction costs : ◮ Pay higher ask price when buying securities, only receive lower bid price when selling ◮ Much more realistic optimal trading strategies ◮ Connection to frictionless markets?

Shadow Prices A general principle 1.15 1.10 1.05 1.00 s 0.95 0.90 0.85 0.80 0.0 0.2 0.4 0.6 0.8 1.0 t Optimal portfolio with transaction costs ?

Shadow Prices A general principle 1.15 1.10 1.05 1.00 bid 0.95 0.90 0.85 0.80 0.0 0.2 0.4 0.6 0.8 1.0 t Optimal portfolio with transaction costs ?

Shadow Prices A general principle 1.15 1.10 1.05 1.00 bid 0.95 0.90 0.85 0.80 0.0 0.2 0.4 0.6 0.8 1.0 t Optimal portfolio without transaction costs for shadow price ⇐ ⇒ Optimal portfolio with transaction costs

Shadow Prices Literature Structural results : ◮ Jouini & Kallal (1995), various more recent articles: No-arbitrage ◮ Cvitanić, Pham & Touzi (1999): Superhedging ◮ Lamberton, Pham & Schweizer (1998): Local risk minimization ◮ Cvitanić & Karatzas (1996), Loewenstein (2000): Portfolio optimization Computations in the Black-Scholes model: ◮ Kallsen & M-K (2010): Infinite-horizon optimal consumption ◮ Gerhold, M-K & Schachermayer: Growth-optimal portfolio ⇒ Focus of this talk!

Growth-Optimal Portfolio under Transaction Costs Frictionless case ◮ Bond normalized to S 0 = 1 ◮ Stock modelled as geometric Brownian motion dS t / S t = µ dt + σ dW t ◮ Goal: Maximize asymptotic logarithmic growth-rate 1 T E [ log ( ϕ 0 lim sup T + ϕ T S T )] T →∞ over all admissible strategies ( ϕ 0 , ϕ ) . ◮ Optimal to keep fraction wealth in stocks equal to Merton proportion θ = µ/σ 2 ◮ Also holds for general Itô processes by inserting drift resp. diffusion coefficient µ t ( ω ) resp. σ t ( ω )

Growth-Optimal Portfolio under Transaction Costs With transaction costs ◮ Bond S 0 = 1 ◮ Can buy stocks only at higher ask price S t , where dS t / S t = µ dt + σ dW t ◮ Can sell them only at lower bid price ( 1 − λ ) S t , λ > 0 ◮ Goal: Maximize 1 T E [ log ( ϕ 0 T + ϕ + T ( 1 − λ ) S T − ϕ − lim sup T S T )] T →∞ ◮ Trading strategies of infinite variation are ruled out ◮ What about the growth-optimal portfolio? ◮ Studied by Taksar, Klass & Assaff (1988) using stochastic control theory

Growth-Optimal Portfolio under Transaction Costs Results Without transaction costs (Merton (1971)) ◮ Fixed fraction θ = µ/σ 2 of wealth in stock (e.g. 31%) With transaction costs (Taksar, Klass & Assaff (1988)): ◮ Minimal trading to keep fraction of wealth in stock in fixed corridor around θ = µ/σ 2 (e.g. 20-40%) ◮ Do nothing in the interior of this no-trade region ◮ Corridor determined by the zero of a deterministic function Goal here : ◮ Compute shadow price ◮ Find asymptotics of the corridor for small transaction costs

Growth-Optimal Portfolio under Transaction Costs Ansatz for the shadow price Suppose we start at time t 0 with... ◮ Ask price S t 0 = 1, ϕ 0 t 0 ≥ 0 bonds, ϕ t 0 ≥ 0 stocks such that ϕ t 0 S t 0 1 c = ϕ 0 π t 0 = = c + 1 , t 0 /ϕ t 0 , ϕ 0 t 0 + ϕ t 0 S t 0 lies on the buying boundary of the no-trade region Then: ◮ If S t increases, so does π t , until S t reaches ¯ s > 1 at time t 1 such that ϕ t 1 S t 1 1 π t 1 = = ϕ 0 t 1 + ϕ t 1 S t 1 c / ¯ s + 1 lies on the selling boundary , c constant ◮ Shadow price ˜ S = g ( S ) during this excursion!

Growth-Optimal Portfolio under Transaction Costs Ansatz for the shadow price Ansatz ˜ S t = g ( S t ) for g : [ 1 , ¯ s ] → [ 1 , ( 1 − λ )¯ s ] ◮ g ( 1 ) = 1 such that ˜ S = S at buying boundary s such that ˜ ◮ g (¯ s ) = ( 1 − λ )¯ S = ( 1 − λ ) S at selling boundary ◮ Itô’s formula: dg ( S t ) / g ( S t ) = ˜ µ t dt + ˜ σ t dW t ◮ Frictionless log-optimizer for ˜ S given by t 0 ˜ ϕ 1 S t c + g ( S t ) = ˜ g ( S t ) µ t = t 0 + ϕ t 0 ˜ σ 2 ϕ 0 ˜ S t t ◮ Yields ODE for g : g ′′ ( s ) = 2 g ′ ( s ) 2 c + g ( s ) − 2 µ g ′ ( s ) σ 2 s

Growth-Optimal Portfolio under Transaction Costs Ansatz for the shadow price Ansatz ˜ S t = g ( S t ) for g : [ 1 , ¯ s ] → [ 1 , ( 1 − λ )¯ s ] such that g ′′ ( s ) = 2 g ′ ( s ) 2 c + g ( s ) − 2 θ g ′ ( s ) s ◮ Merton proportion θ = µ/σ 2 ◮ g ( 1 ) = 1, g (¯ s ) = ( 1 − λ )¯ s ◮ ¯ s , c still unknown, two more boundary conditions? ˜ S = g ( S ) should remain in [( 1 − λ ) S , S ] ◮ Diffusion coefficient of ˜ S / S should vanish as S t → 1 or S t → ¯ s ◮ Leads to g ′ ( 1 ) = 1 and g ′ (¯ s ) = 1 − λ ◮ Free boundary value problem?

Growth-Optimal Portfolio under Transaction Costs Computing the candidate ◮ General solution to ODE with g ( 1 ) = g ′ ( 1 ) = 1: − cs + ( 2 θ − 1 + 2 c θ ) s 2 θ g ( s ) = s − ( 2 − 2 θ + c ( 2 θ − 1 )) s 2 θ . ◮ Plugging this into g (¯ s , g ′ (¯ s ) = ( 1 − λ )¯ s ) = 1 − λ yields � 1 / ( 2 θ − 1 ) . � c ¯ s = ¯ s ( c ) = ( 2 θ − 1 + 2 c θ )( 2 − 2 θ − c ( 2 θ − 1 )) and � 1 − θ 1 θ − 1 / 2 − 1 − λ ( 2 θ − 1 + 2 c θ ) 2 = 0 � c ( 2 θ − 1 + 2 c θ )( 2 − 2 θ − c ( 2 θ − 1 )

Growth-Optimal Portfolio under Transaction Costs Computing the candidate ◮ Elementary analysis: Exists unique solution c to � 1 − θ 1 θ − 1 / 2 − 1 − λ ( 2 θ − 1 + 2 c θ ) 2 = 0 � c ( 2 θ − 1 + 2 c θ )( 2 − 2 θ − c ( 2 θ − 1 ) ◮ Define � 1 / ( 2 θ − 1 ) . � c ¯ s = ¯ s ( c ) = ( 2 θ − 1 + 2 c θ )( 2 − 2 θ − c ( 2 θ − 1 )) ◮ Compute boundaries of the no trade region: 1 / ( 1 + c ) ≤ 1 / ( 1 + c / ¯ s ) ◮ But: No explicit solution for c . ◮ However: Fractional Taylor expansions!

Growth-Optimal Portfolio under Transaction Costs Fractional Taylor expansions for small λ Theorem (Gerhold, M-K, Schachermayer (2010)) For p k and q k that can be algorithmically computed : ∞ � k / 3 c = 1 − θ � 6 � λ k / 3 + q k ( θ ) θ θ ( 1 − θ ) k = 1 This yields expansions of arbitrary order for no-trade region: 4 θ 2 ( 1 − θ ) 2 � 1 / 3 λ 1 / 3 + 3 1 20 ( 2 θ 2 − 2 θ + 1 ) λ + O ( λ 4 / 3 ) � 3 1 + c = θ − 4 θ 2 ( 1 − θ ) 2 � 1 / 3 λ 1 / 3 − 1 1 � 3 20 ( 26 θ 2 − 26 θ + 3 ) λ + O ( λ 4 / 3 ) s = θ + 1 + c / ¯ ◮ Compare Janeček & Shreve (2004) for first terms with consumption

Growth-Optimal Portfolio under Transaction Costs Verification Up to now: Heuristic derivation of candidate ˜ S t = g ( S t ) 1 1 s � ◮ Only during one excursion of S t from 1 to ¯ s ◮ What happens when S hits the boundaries?

Growth-Optimal Portfolio under Transaction Costs Verification ◮ Start at the buying boundary at time t 0 with S t 0 = 1 ◮ ˜ S t 0 = g ( S t 0 ) = g ( 1 ) ◮ If S moves down, we should still have ˜ S = S ◮ If S t 0 � = 1, everything should scale with S t 0 ◮ Hence until S t reaches ¯ s > 1, let ˜ S t = m t g ( S t m t = min t 0 ≤ t S t , m t ) After S t hits ¯ s at time σ 1 : S t = m t g ( S t ˜ m t = max σ 1 ≤ t S t / ¯ s , m t ) until S t / m t ≤ 1. Continue in an obvious way.

Growth-Optimal Portfolio under Transaction Costs Verification ◮ Have defined continuous process ˜ S = mg ( S / m ) ◮ Moves between [( 1 − λ ) S , S ] ◮ But why should this be a nice process? Theorem (Gerhold, M-K, Schachermayer (2010)) ˜ S = mg ( S / m ) is an Itô process with bounded coefficients, which satisfies S t = g ′ � � 2 m t g ′′ � � d ˜ 1 S t S t dS t + d � S , S � t m t m t ◮ Frictionless log-optimal portfolio is well-known ◮ Number of stocks only increases resp. decreases when ˜ S = S resp. ˜ S = ( 1 − λ ) S by construction ◮ Hence, ˜ S is a shadow price !

Growth-Optimal Portfolio under Transaction Costs Construction and results Construction of the shadow price: ◮ Itô process with bounded coefficients ◮ Function of ask price S and its running minima resp. maxima during Brownian excursions ◮ Determined up to solution of one dimensional equation Asymptotic expansions in terms of λ 1 / 3 (for 0 < λ < λ 0 ): ◮ Expansions of arbitrary order for no-trade region ◮ Can also determine asymptotic growth rate 128 θ 2 ( 1 − θ ) 2 � 2 / 3 λ 2 / 3 + O ( λ 4 / 3 ) δ = µ 2 � 3 σ 3 2 σ 2 − √ ◮ Compare Janeček & Shreve (2004), Shreve & Soner (1994), Rogers (2004) for first term with consumption