Portfolio optimization for an exponential Ornstein-Uhlenbeck model with proportional transaction costs Martin Forde King s College London, May 2014 (joint work with Christoph Czichowsky, Philipp Deutsch and Hongzhong Zhang)
Outline of talk The exponential Ornstein-Uhlenbeck model with proportional transaction costs. Admissible self-financing trading strategies. Shadow price processes - definition and why we use them. Explicit construction of the shadow price process for the exponential OU model. Asymptotics for the no-trade region and the risky fraction when the transaction cost is small. Results extend the work of [GMS13], who deal with the Black-Scholes case, and show new phenomena. The verification argument, and links to excursion theory. Brief discussion on duality.
The modelling framework We work on some (Ω, F, (F t ), P) and consider a financial market with one riskless bond with a constant price equal to 1 (i.e. zero interest rates) and a risky asset S t. We assume that S t is given by the exponential of an Ornstein-Uhlenbeck process S t = e Xt, where dx t = κ(x X t )dt + σdw t with κ, x, σ > 0. By Itô s formula, S t satisfies ds t /S t = [κ(x log(s t )) + σ2 2 ]dt + σdw t =: µ(s t )dt + σdw t. We now model the bid-ask interval by [(1 λ)s t, S t ] for some λ (0, 1). The investor pays S t for each share bought, but only receives (1 λ)s t for each share sold. Let (ϕ 0 t, ϕ t ) denote our holding in the riskless and risky asset at time t. Investor wishes to maximize his expected long-term growth rate: lim inf T 1 T E[log V T (ϕ 0 T, ϕ T )] where V t ((ϕ 0 t, ϕ t )) = ϕ 0 t + ϕ + t (1 λ)s t ϕ t S t is the liquidation value of the portfolio at time t (investor has log utility).
Admissible trading strategies Assume the investor starts with x dollars in cash (x > 0). Then a pair of adapted processes (ϕ 0 t, ϕ t ) is called an admissible self-financing trading strategy if both processes are predictable, have finite variation and: (i) The self-financing condition: dϕ 0 t = (1 λ)s t dϕ t S t dϕ t (1) for all 0 t T. ϕ t has F.V. so ϕ t = ϕ t ϕ t, where ϕ t, ϕ t are two increasing processes (ii) The solvency condition: there exists an M > 0 such that the liquidation value V t (ϕ 0, ϕ) = ϕ 0 t + ϕ + t (1 λ)s t ϕ t S t M (2) a.s., for all 0 t T. The self-financing condition in (1) ensures that no funds are added or withdrawn to the portfolio, and (2) ensures that the investor cannot owe more than M dollars at any time.
Definition of a shadow price Definition. A shadow price is a continuous semimartingale S t [(1 λ)s t, S t ], such that the optimal trading strategy (ϕ 0 t, ϕ t ) for a fictitious market with price process S t and zero transaction costs exists, has finite variation and the number of stocks ϕ t only increases when S t = S t and decreases when S t = (1 λ)s t. Clearly any price process S t with zero transaction costs which lies in [(1 λ)s t, S t ] leads to more favourable terms of trade than the original market with transaction costs. But a shadow price process is a particularly unfavourable model, for which it s optimal to only buy when S t = S t, sell when S t = (1 λ)s t + do nothing in between.
Why do we use shadow prices? Proposition (Corollary 1.9 in Schachermayer et al.[gms13]). Let S t be a shadow price process whose optimal trading strategy (for zero transaction costs) is given by (ϕ 0 t, ϕ t ), with ϕ 0 t, ϕ t 0. Then under non-zero transaction costs, we have sup E[log V T ((ψ 0, ψ))] E[log V T ((ϕ 0, ϕ))] (ψ 0,ψ) E[log V T ((ψ 0, ψ))] + log(1 λ) for any admissible (ψ 0, ψ). Thus if we choose λ suff small so that log(1 λ) < ε and take the sup over all (ψ 0, ψ), we see that (ϕ 0, ϕ) is an ε-optimal trading strategy for the original problem. Or take liminf as T + sup over all admissible strategies, we obtain lim inf T 1 T E[log V T ((ϕ 0, ϕ))] = sup lim inf (ψ 0,ψ) T 1 T E[log V T ((ψ 0, ψ))]. Thus the optimal portfolio for the shadow price process is asymptotically optimal for the original problem under transaction costs, as λ 0 and/or as T.
The optimal portfolio for the frictionless case First consider the case when λ = 0, i.e. zero transaction costs, and assume S t follows a general Itô process of the form ds t = S t (µ t dt + σ t dw t ) with zero interest rates. For the frictionless case, we are looking to maximize: E[log V T ] = E[log[x + = E[log x + = E[log x + T 0 T 0 T 0 φ t ds t ] φ t S t x + t 0 φ tds t ds t S t 1 2 (π t µ t 1 2 π2 t σ 2 t dt)]. T 0 φ 2 t S 2 t σ 2 t dt (x + t 0 φ tds t ) 2 dt] Maximizing the integrand over all π t, we obtain that ˆπ t = µt, which σt 2 is known as the Merton fraction. For the Black-Scholes case, ˆπ t = ˆπ = µ/σ 2 is constant, but in general ˆπ t has infinite variation and so will ϕ t (unlike the case λ > 0). For the BS case, dv t = ˆπV t ds t /S t = V t (ˆπµdt + ˆπσdW t ) is GBM, and so is φ t = ˆπV t /S t.
Construction of the shadow price for the exp OU model for a single excursion from the buy bndry to the sell bndry Ansatz: if S increases from a to b without setting a new minimum in the meantime, then we guess that S t = g(s t ) for 0 t τ b, for some g C 2 and target value b = b(a, λ) to be determined. In general, S t = g(s t ; a t, b(a t, λ)) where a t = min 0 u t S u up to time τ b. For t τ b, we set b t = max τb u ts u, and then S t = g(s t ; a(b t, λ), b t ) until S returns to the buy boundary (possibly along a new g curve), and so on..
Explicit construction of the shadow price contd. Assume that S 0 = S 0 = a and S t = g(s t ) during an excursion from S = a to S = b (we postulate that no trading occurs until S hits b, we will then show how to choose b = b(a, λ)). From the drawing we see that: g(a) = a and g(b) = (1 λ)b. Smooth-pasting condition: g (a) = 1, g (b) = 1 λ - this ensures that the volatility of S t vanishes on both boundaries (see below). (1 λ)s g(s) s for all s [a, b]. Applying Itô s formula to the shadow price process, we obtain d S t = dg(s t ) = g (S t )ds t + 1 2 g (S t )σ 2 S 2 t dt or dg(s t )/g(s t ) = ˆµ t dt + 1 2 ˆσ tdw t, where ˆµ t = [g (S t )S t µ(s t ) + 1 2 g (S t )σ 2 S 2 t ]/g(s t ), ˆσ t = g (S t )σs t /g(s t ). But the optimal risky fraction for an investor who maximizes log-utility (with zero transaction costs) is given by ˆµ(s) ˆσ(s) 2 = (g (s)sµ(s) + 1 2 g (s)σ 2 s 2 )/g(s) (g (s) 2 σ 2 s 2 )/g(s) 2 = ϕg(s) ϕ 0 + ϕg(s). (3)
ODE for the shadow price Multiplying the numerator and denominator of the right hand side of a (3) by ϕ 0 +ϕa and setting π = aϕ ϕ 0 +ϕa we have πg(s) aϕ 0 ϕ 0 +ϕa + πg(s) = πg(s) a ϕ0 +aϕ aϕ ϕ 0 +ϕa + πg(s) Combining with (3) yields the following ODE for g: = πg(s) a (1 π) + πg(s). 1 2 σ2 s 2 g (s) = g (s) 2 σ 2 s 2 πg(s) g(s) a (1 π) + πg(s) g (s)sµ(s) = g (s) 2 σ 2 s 2 π a (1 π) + πg(s) g (s)sµ(s) which simplifies to g (s) = = where θ(s) = µ(s)/σ 2. 2 πg (s) 2 a (1 π) + πg(s) 2g (s)sµ(s) σ 2 s 2 2 πg (s) 2 a (1 π) + πg(s) 2g (s)θ(s) s (4)
Solution for the shadow price General solution to ODE in (4) with g(a) = a, g (a) = 1: g(s) = g(s; a, π) = a ah(a) + (1 π)h(a, s) ah(a) πh(a, s) where H(a, s) = s a h(u)du, h(s) = exp[ κ σ (log(s) x σ2 2 Plugging this into g(b) = (1 λ)b, g (b) = 1 λ we obtain: π = a(h(a, b) + λbh(a) bh(a) + ah(a)) (a + λb b)h(a, b) 2κ )2 ]. and F (a, b, λ) = H(a, b) 2 (λ 1) + (a + b(λ 1)) 2 h(a)h(b) = 0. Solving the quadratic for λ we find the physically meaningful solution is given by λ(a, b) = 1 a b 1 H(a, b) 2 2 b 2 h(a)h(b) H(a, b) 2b H(a, b) 2 b 2 h(a) 2 h(b) 2 + (5) 4a bh(a)h(b). For λ fixed, eq has 2 solns b = b 1/2 (a, λ) with b 1 < a < b 2. To choose the physically meaningful solution, it turns out there is a critical a = a 0 (λ) such that b = b 2 (a, λ) for a a 0 and b = b 1 (a, λ) for a < a 0.
Asymptotics Since λ is smooth, we can expand it in a Taylor series around the point b = a: λ(a, b) = Γ(a) 6a 3 (b a)3 + O((b a) 4 ), (6) where Γ(s) = θ(s)(1 θ(s)) θ (s)s. Inverting (6), for a / {a, b }, we obtain b(a, λ) = a + a( 6 Γ(a) ) 1 3 λ 1 3 + O(λ 2 3 )]. For the risky fraction π, plugging this expansion into (5) we get: π = θ(a) ( 3 4 Γ(a)2 ) 1 3 λ 1 3 + O(λ 2 3 ). For a = a 0 (λ) there are two b-values, and b 1,2 (a, λ) = a ± a 3σ 4 2 ( κσ (4κ + σ 2 ) )1/4 λ 1 4 + O( λ), κ (4κ + σ 2 ) 1 4 π = θ(a) 3 λ 1 2 + O( λ), 3 σ 2 Martin Forde King s College London, May 2014 (joint work with Christoph Portfolio2optimization 1 Czichowsky, for an exponential Philipp Ornstein-Uhlenbeck Deutsch modeland with prop H
Remarks The special value a 0 (λ) does show up for the Black-Scholes model where we do not see the λ 1 4 asymptotic behaviour. In general St = g(s t ; a t, π(a t, b(a t, λ), λ) for some continuous process a t with finite variation, which only increases/decreases when S t is on boundaries of the bid-ask cone. If S 0 = a, then before S hits b(a, λ) or a 0 (λ), a t = min 0 u t S u : in words, every time S t sets a new minimum, we need a new a-value for g(.), but when S t makes an excursion away from its minimum process, da t = 0, and S t just follows the g curve from left to right. For t τ b, we set b t = max τb u ts u, and then S t = g(s t ; a(b t, λ), b t ) until S returns to the buy boundary or bt hits the critical value b 0 (λ). The g curves change direction to the left of a 0 (λ) and to the right of b 0 (λ). At these critical values, there are two valid g curves (not one).
Remarks We can show that the optimal number of shares ϕ t evolves as dϕ t Γ(a t ; λ) = ϕ t π(a t, b(a t, λ), λ) da t a t (7) where Γ(a; λ) = a π (a) + π(a)(1 π(a)), and here π(a) is shorthand for π(a, b(a, λ), λ). Integrating (7) we get log ϕt ϕ 0 = F (a t ) := a t a 0 Γ(u;λ) π(u,b(u,λ),λ)u du. For S t to be a genuine shadow price, we have to verify that dϕ t 0 when S t = S t, and dϕ t 0 when S t = (1 λ) S t (this is the so-called verification argument).
Numerics Figure: Here we have plotted various shadow price curves g(s; a, b) as a function of s, for λ =.3 and κ = 3, σ = 1, x = 1, for which we find that a 0(λ) = 1.3914, b 0(λ) = 5.31052 and, and a 1(λ) = 1.95159, a 2(λ) = 10.5602 and b 1(λ) = 0.699707, b 2(λ) = 3.78616.
Duality Let S t [(1 λ)s t, S t ] and Zt 0 denote the density process of an ELMM Q for S t, and let Zt 1 = Zt 0 S t. Let V (y) = sup x>0 [U(x) xy] denote the Fenchel-Legendre transform of U, where U is a (concave) utility function. Then (as before) trading the shadow price process is more favourable than trading the real risky asset, so for any admissible trading strategy (ϕ 0, ϕ) we have E[U(V T (ϕ 0, ϕ)] E[U(x + ϕ t d S t )] E[V (yz 0 T ) + (x + ϕ t S t )yz 0 T )] E[V (yz 0 T ] + xy because E(Z 0 T ) = 1 and S t is a local martingale under Q. Taking sups and infs, we see that sup E[U(V T (ϕ 0, ϕ)] inf E[V (yz (ϕ 0,ϕ) (Z 0,Z 1 T 0 )] + xy.,y) If the dual optimizers (Ẑ 0, Ẑ 1, y) exist then we have equality, and U [V T (ϕ 0, ϕ)] = ŷẑ T 0. For U(x) = log x, we have V (y) = log y 1.
References [GMS13] Gerhold, S., J.Muhle-Karbe and W.Schachermayer, The Dual Optimizer for the Growth-Optimal Portfolio under Transaction Costs, Finance and Stochastics, Vol. 17 (2013), No. 2, pp. 325-354. [GGMS12] Gerhold, S., P.Guasoni, J.Muhle-Karbe, W.Schachermayer, Transaction costs, trading volume, and the liquidity premium, to appear in Finance and Stochastics. [GM13] Guasoni, P. and J.Muhle-Karbe, Long Horizons, High Risk Aversion, and Endogeneous Spreads, to appear Mathematical Finance.