An Explicit Example of a Shadow Price Process with Stochastic Investment Opportunity Set Christoph Czichowsky Faculty of Mathematics University of Vienna SIAM FM 12 New Developments in Optimal Portfolio Choice based on joint work with Philipp Deutsch and Walter Schachermayer Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 1 / 18
Utility maximisation under transaction costs Fix a strictly positive cádlág stock price process S = (S t ) 0 t T. Buy at ask price S. Sell at lower bid price (1 λ)s for fixed λ (0, 1). Standard problem: Maximise E [ U ( ϕ 0 T + (ϕ 1 T ) + (1 λ)s T (ϕ 1 ) T S )] T over all self-financing and admissible strategies (ϕ 0, ϕ 1 ) under transaction costs starting from initial endowment (ϕ 0 0, ϕ1 0 ) = (x, 0). Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 2 / 18
Utility maximisation under transaction costs Fix a strictly positive cádlág stock price process S = (S t ) 0 t T. Buy at ask price S. Sell at lower bid price (1 λ)s for fixed λ (0, 1). Standard problem: Maximise E [ U ( ϕ 0 T + (ϕ 1 T ) + (1 λ)s T (ϕ 1 ) T S )] T over all self-financing and admissible strategies (ϕ 0, ϕ 1 ) under transaction costs starting from initial endowment (ϕ 0 0, ϕ1 0 ) = (x, 0). How to obtain the solution to this problem? Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 2 / 18
Utility maximisation under transaction costs Fix a strictly positive cádlág stock price process S = (S t ) 0 t T. Buy at ask price S. Sell at lower bid price (1 λ)s for fixed λ (0, 1). Standard problem: Maximise E [ U ( ϕ 0 T + (ϕ 1 T ) + (1 λ)s T (ϕ 1 ) T S )] T over all self-financing and admissible strategies (ϕ 0, ϕ 1 ) under transaction costs starting from initial endowment (ϕ 0 0, ϕ1 0 ) = (x, 0). How to obtain the solution to this problem? Classically: Try to find solution by solving HJB equation. Davis and Norman (1992), Shreve and Soner (1994),... Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 2 / 18
Utility maximisation under transaction costs Fix a strictly positive cádlág stock price process S = (S t ) 0 t T. Buy at ask price S. Sell at lower bid price (1 λ)s for fixed λ (0, 1). Standard problem: Maximise E [ U ( ϕ 0 T + (ϕ 1 T ) + (1 λ)s T (ϕ 1 ) T S )] T over all self-financing and admissible strategies (ϕ 0, ϕ 1 ) under transaction costs starting from initial endowment (ϕ 0 0, ϕ1 0 ) = (x, 0). How to obtain the solution to this problem? Classically: Try to find solution by solving HJB equation. Davis and Norman (1992), Shreve and Soner (1994),... Alternatively: Try to find a shadow price. Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 2 / 18
Shadow price A shadow price Ŝ = (Ŝt) 0 t T is a price process valued in [(1 λ)s, S] such that the frictionless utility maximisation problem for that price has the same optimal strategy as the one under transaction costs. Frictionless trading at any price process S = ( S t ) 0 t T valued in the bid-ask spread [(1 λ)s, S] allows to generate higher terminal payoffs. Hence, a shadow price corresponds to the least favourable frictionless market evolving in the bid-ask spread. The optimal strategy for the shadow price only buys, if the shadow price equals the ask price, and sells, if the shadow price equals the bid price. If such a shadow price exists, obtain the optimal strategy by solving a frictionless problem. apply all the techniques and knowledge from frictionless markets. no qualitatively new effects arise due to transaction costs. Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 3 / 18
Shadow price A shadow price Ŝ = (Ŝt) 0 t T is a price process valued in [(1 λ)s, S] such that the frictionless utility maximisation problem for that price has the same optimal strategy as the one under transaction costs. Frictionless trading at any price process S = ( S t ) 0 t T valued in the bid-ask spread [(1 λ)s, S] allows to generate higher terminal payoffs. Hence, a shadow price corresponds to the least favourable frictionless market evolving in the bid-ask spread. The optimal strategy for the shadow price only buys, if the shadow price equals the ask price, and sells, if the shadow price equals the bid price. If such a shadow price exists, obtain the optimal strategy by solving a frictionless problem. apply all the techniques and knowledge from frictionless markets. no qualitatively new effects arise due to transaction costs. Do these shadow prices exist in general? Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 3 / 18
Previous literature: Shadow prices in general Cvitanić and Karatzas (1996): Basic idea. In Brownian setting, if the minimizer (Ẑ 0, Ẑ 1 ) to a suitable dual problem is a local martingale, then a shadow price exists and is given by Ẑ 1 Ẑ 0. Cvitanić and Wang (2001): This dual minimizer is so far only guaranteed to be a supermartingale. Loewenstein (2000): Existence in Brownian setting, if no assets can be sold short and a solution to the problem under transaction costs exists. Kallsen and Muhle-Karbe (2011): Existence in finite probability spaces. Benedetti, Campi, Kallsen and Muhle-Karbe (2011): Existence in a general multi-currency model (jumps, random bid-ask spreads), if no assets can be sold short and a solution exists. Talk this afternoon. Counter-example: unique candidate for shadow price admits arbitrage. Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 4 / 18
Previous literature: Shadow prices in general Cvitanić and Karatzas (1996): Basic idea. In Brownian setting, if the minimizer (Ẑ 0, Ẑ 1 ) to a suitable dual problem is a local martingale, then a shadow price exists and is given by Ẑ 1 Ẑ 0. Cvitanić and Wang (2001): This dual minimizer is so far only guaranteed to be a supermartingale. Loewenstein (2000): Existence in Brownian setting, if no assets can be sold short and a solution to the problem under transaction costs exists. Kallsen and Muhle-Karbe (2011): Existence in finite probability spaces. Benedetti, Campi, Kallsen and Muhle-Karbe (2011): Existence in a general multi-currency model (jumps, random bid-ask spreads), if no assets can be sold short and a solution exists. Talk this afternoon. Counter-example: unique candidate for shadow price admits arbitrage. Are there other conditions that ensure the existence of shadow prices? Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 4 / 18
General result Theorem (C./Schachermayer 2012) Suppose that i) S is continuous ii) S satisfies (NFLVR) xu and U : (0, ) R satisfies lim sup (x) U(x) < 1 and u(x) := sup E[U(g)] <. x g C(x) Then (Ẑ 0, Ẑ 1 ) is a local martingale and Ŝ := Ẑ 1 Quite sharp: There exist counter-examples, if i ) S is discontinuous and satisfies (NFLVR) and Ẑ 1 Ẑ 0 a shadow price process. Ẑ 0 satisfies (NFLVR). C./Muhle-Karbe/Schachermayer: Transaction Costs, Shadow Prices, and Connections to Duality, 2012. ii ) S is continuous and satisfies (CPS λ ) for all λ (0, 1) but not (NFLVR). Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 5 / 18
General result Theorem (C./Schachermayer 2012) Suppose that i) S is continuous ii) S satisfies (NFLVR) xu and U : (0, ) R satisfies lim sup (x) U(x) < 1 and u(x) := sup E[U(g)] <. x g C(x) Then (Ẑ 0, Ẑ 1 ) is a local martingale and Ŝ := Ẑ 1 Quite sharp: There exist counter-examples, if i ) S is discontinuous and satisfies (NFLVR) and Ẑ 1 Ẑ 0 a shadow price process. Ẑ 0 satisfies (NFLVR). C./Muhle-Karbe/Schachermayer: Transaction Costs, Shadow Prices, and Connections to Duality, 2012. ii ) S is continuous and satisfies (CPS λ ) for all λ (0, 1) but not (NFLVR). Do shadow prices allow to actually compute the solution in particular models? Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 5 / 18
Previous literature: Shadow prices in particular Shadow prices in Black-Scholes model: Various optimisation problems Kallsen and Muhle-Karbe (2009) Gerhold, Muhle-Karbe and Schachermayer (2011) Guasoni, Gerhold, Muhle-Karbe and Schachermayer (2011) Herczegh and Prokaj (2011) Choi, Sirbu and Zitkovic (2012)... Shadow prices for Itô processes: Kallsen and Muhle-Karbe (2012): asymptotics for exponential utility Results for general diffusion models (without shadow prices): Martin and Schöneborn (2011): local utility Martin (2012): multi-dimensional diffusions and local utility Soner and Touzi (2012): asymptotics for general utilities Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 6 / 18
Previous literature: Shadow prices in particular Shadow prices in Black-Scholes model: Various optimisation problems Kallsen and Muhle-Karbe (2009) Gerhold, Muhle-Karbe and Schachermayer (2011) Guasoni, Gerhold, Muhle-Karbe and Schachermayer (2011) Herczegh and Prokaj (2011) Choi, Sirbu and Zitkovic (2012)... Shadow prices for Itô processes: Kallsen and Muhle-Karbe (2012): asymptotics for exponential utility Results for general diffusion models (without shadow prices): Martin and Schöneborn (2011): local utility Martin (2012): multi-dimensional diffusions and local utility Soner and Touzi (2012): asymptotics for general utilities How do these shadow prices look like in a particular diffusion model? Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 6 / 18
Previous literature: Shadow prices in particular Shadow prices in Black-Scholes model: Various optimisation problems Kallsen and Muhle-Karbe (2009) Gerhold, Muhle-Karbe and Schachermayer (2011) Guasoni, Gerhold, Muhle-Karbe and Schachermayer (2011) Herczegh and Prokaj (2011) Choi, Sirbu and Zitkovic (2012)... Shadow prices for Itô processes: Kallsen and Muhle-Karbe (2012): asymptotics for exponential utility Results for general diffusion models (without shadow prices): Martin and Schöneborn (2011): local utility Martin (2012): multi-dimensional diffusions and local utility Soner and Touzi (2012): asymptotics for general utilities How do these shadow prices look like in a particular diffusion model? Do they allow us to actually compute the solution there? Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 6 / 18
Example: Shadow price for geometric OU process Ornstein-Uhlenbeck process: dx t = κ( x X t )dt + σdw t, X 0 = x 0. Stock price: S t = exp (X t ), i.e., ds t S t = (κ ( x log(s t ) ) ) + σ2 dt + σdw t =: µ(s t )dt + σdw t. 2 Stochastic investment opportunity set, i.e., random coefficients. Basic problem: Maximise the asymptotic logarithmic growth-rate lim sup T 1 T E[ log ( ϕ 0 T + (ϕ 1 T ) + (1 λ)s T (ϕ 1 ) T S )] T over all self-financing, admissible strategies (ϕ 0, ϕ 1 ) under transaction costs. Black-Scholes model: Taksar, Klass and Assaff (1988): Solving HJB equation. Gerhold, Muhle-Karbe and Schachermayer (2011): Shadow price. Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 7 / 18
Qualitative behaviour of the optimal strategy Without transaction costs: Invest fraction θ(s t ) := µ(st) σ 2 Trading in number of shares: = (κ( x log(st))+ σ 2 2 ) σ 2 of wealth in stock. dϕ 1 ds > 0 dϕ 1 ds < 0 dϕ 1 ds > 0 s 0 a 0 b 0 Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 8 / 18
Qualitative behaviour of the optimal strategy Without transaction costs: Invest fraction θ(s t ) := µ(st) σ 2 Trading in number of shares: = (κ( x log(st))+ σ 2 2 ) σ 2 of wealth in stock. dϕ 1 ds > 0 dϕ 1 ds < 0 dϕ 1 ds > 0 s 0 a 0 b 0 With transaction costs it is folklore : Do nothing in the interior of some no-trade region. Minimal trading on the boundary to stay within this region. Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 8 / 18
Qualitative behaviour of the optimal strategy Without transaction costs: Invest fraction θ(s t ) := µ(st) σ 2 Trading in number of shares: = (κ( x log(st))+ σ 2 2 ) σ 2 of wealth in stock. dϕ 1 ds > 0 dϕ 1 ds < 0 dϕ 1 ds > 0 s 0 a 0 b 0 With transaction costs it is folklore : Do nothing in the interior of some no-trade region. Minimal trading on the boundary to stay within this region. But how does this look like? Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 8 / 18
Ansatz for the shadow price Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 9 / 18
Ansatz for the shadow price Ansatz Ŝ t = g(s t ) during this excursion from S t0 = a to S t1 = b (1 λ)s g(s) s for all s between a and b g(a) = a and g (a) = 1 at buying boundary g(b) = (1 λ)b and g (b) = (1 λ) at selling boundary Itô s formula: dg(s t )/g(s t ) = ˆµ t dt + ˆσ t dw t Frictionless log-optimizer for Ŝ given by Yields ODE for g: ϕ 1 t 0 Ŝ t πg(s t ) = ϕ 0 t 0 + ϕ 1 t 0 Ŝ t (a π) + πg(s t ) = ˆµ t ˆσ t 2 g (s) = 2πg (s) 2 (a π) + πg(s) 2θ(s)g (s) s Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 10 / 18
Computing the candidate General solution to ODE with g(a) = a and g (a) = 1: ( where h(s) := exp g(s; a, π) = a κ σ ( x log(s) + σ 2 2 2κ ah(a) + (1 π)h(a, s), ah(a) πh(a, s) ) ) 2 and H(a, s) := s a h(u)du. Plugging this into g(b) = (1 λ)b, g (b) = 1 λ we obtain and H(a, b) + λbh(a) bh(a) + ah(a) π(a, b, λ) := a (a + λb b)h(a, b) F (a, b, λ) := H(a, b) 2 (λ 1) + (a + b(λ 1)) 2 h(a)h(b) = 0, which gives two equations λ 1,2 (a, b) = λ. Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 11 / 18
Computing the candidate General solution to ODE with g(a) = a and g (a) = 1: ( where h(s) := exp g(s; a, π) = a κ σ ( x log(s) + σ 2 2 2κ ah(a) + (1 π)h(a, s), ah(a) πh(a, s) ) ) 2 and H(a, s) := s a h(u)du. Plugging this into g(b) = (1 λ)b, g (b) = 1 λ we obtain and H(a, b) + λbh(a) bh(a) + ah(a) π(a, b, λ) := a (a + λb b)h(a, b) F (a, b, λ) := H(a, b) 2 (λ 1) + (a + b(λ 1)) 2 h(a)h(b) = 0, which gives two equations λ 1,2 (a, b) = λ. Only need λ 1 (a, b) = λ that can, however, not be solved explicitly. Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 11 / 18
Computing the candidate General solution to ODE with g(a) = a and g (a) = 1: ( where h(s) := exp g(s; a, π) = a κ σ ( x log(s) + σ 2 2 2κ ah(a) + (1 π)h(a, s), ah(a) πh(a, s) ) ) 2 and H(a, s) := s a h(u)du. Plugging this into g(b) = (1 λ)b, g (b) = 1 λ we obtain and H(a, b) + λbh(a) bh(a) + ah(a) π(a, b, λ) := a (a + λb b)h(a, b) F (a, b, λ) := H(a, b) 2 (λ 1) + (a + b(λ 1)) 2 h(a)h(b) = 0, which gives two equations λ 1,2 (a, b) = λ. Only need λ 1 (a, b) = λ that can, however, not be solved explicitly. For sufficiently small λ, there exists b(a, λ) with λ 1 (a, b(a, λ)) = λ for all a. Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 11 / 18
Computing the candidate Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 12 / 18
Theorem (Fractional Taylor expansions in terms of λ 1/3 ) For a, b (0, ) \ {a 0, b 0 }, we have expansions (of arbitrary order) ( 6λ b(a, λ) = a + a Γ(a) ( 3 π(a, λ) = θ(a) ) 1/3 ) 2κ2 3Γ(a) + σ ( x log(a) + a 4 ) 1/3 4 Γ(a)2 λ ) 1/3 ( 3 π(b, λ) = θ(b) + 4 Γ(b)2 λ 6 1/3 Γ(a) 5/3 λ 2/3 + O(λ), ) 2κ 2 σ ( x log(a) 4 λ 2/3 + O(λ), 6 1/3 Γ(a) 2/3 ) 2κ 2 σ ( x log(b) 4 λ 2/3 + O(λ), 6 1/3 Γ(b) 2/3 where Γ(s) denotes the sensitivity of the displacement from the optimal fraction Γ(s) = θ(s)(1 θ(s)) θ (s)s = 4κσ2 + σ 4 4κ 2 log(s) 2 + 8κ 2 x log(s) 4κ 2 x 2 4σ 4. Compare Gerhold, Muhle-Karbe and Schachermayer (2011) for the Black-Scholes model and Soner and Touzi (2012) for first terms. Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 13 / 18
Theorem (Fractional Taylor expansions in terms of λ 1/4 ) For a = a 0 and b = b 0, we have expansions (of arbitrary order) b 1,2 (a, λ) = a ± a ( 1/4 3σ 4 2 λ κ2 σ 2 (4κ + σ )) 1/4 + O(λ 1/2 ), 2 a 1,2 (b, λ) = b ± b ( 1/4 3σ 4 2 λ κ2 σ 2 (4κ + σ )) 1/4 + O(λ 1/2 ), 2 π(a, λ) = θ(a) π(b, λ) = θ(b) + ( ) 1/2 κ2 σ 2 (4κ + σ 2 ) λ 1/2 + O(λ 3/4 ), 3σ 4 ( ) 1/2 κ2 σ 2 (4κ + σ 2 ) λ 1/2 + O(λ 3/4 ). 3σ 4 Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 14 / 18
Verification Up to now: Only one excursion starting from S t0 = a. Need to define a process (A t ) 0 t< such that everything fits together. Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 15 / 18
Verification Defined continuous process Ŝ = g ( S; A, π(a, λ) ) Moves between [(1 λ)s, S] This is even a nice process. Proposition Ŝ = g ( S; A, π(a, λ) ) is an Itô process, which satisfies the SDE dŝt = g ( S t ; A t, π(a t, λ) ) ds t + 1 2 g ( S t ; A t, π(a t, λ) ) d S, S t Similar arguments as in Gerhold, Muhle-Karbe and Schachermayer (2011). Frictionless log-optimal portfolio is well-known Number of stocks only increases resp. decreases when Ŝ = S resp. Ŝ = (1 λ)s by construction Hence, Ŝ is a shadow price! Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 16 / 18
Summary Existence of shadow prices Sufficient condition: S is continuous and satisfies (NFLVR). Quite sharp: Counter-examples. Explicit construction of shadow price: Growth-optimal portfolio for geometric Ornstein-Uhlenbeck process. Sufficiently small but fixed transaction costs λ. Shadow price is an Itô process. Function of ask price S and a truncation of its running minima resp. maxima during excursions of an OU process. Explicitly determined up to one implicitly defined function b(a, λ). Asymptotic expansions of arbitrary order in terms of λ 1/3 and λ 1/4. Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 17 / 18
Thank you for your attention! http://www.mat.univie.ac.at/ czichoc2 Christoph Czichowsky (Uni Wien) Explicit example of a shadow price Minneapolis, July 11, 2012 18 / 18