Transaction costs, trading volume, and the liquidity premium

Abstract

In a market with one safe and one risky asset, an investor with a long horizon, constant investment opportunities and constant relative risk aversion trades with small proportional transaction costs. We derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume. At the first order, the liquidity premium equals the spread, times share turnover, times a universal constant. The results are robust to consumption and finite horizons. We exploit the equivalence of the transaction cost market to another frictionless market, with a shadow risky asset, in which investment opportunities are stochastic. The shadow price is also found explicitly.

Appendices

Appendix A: Explicit formulas and their properties

We now show that the candidate w for the reduced value function and the quantity λ are indeed well defined for sufficiently small spreads. The first step is to determine, for a given small λ>0, an explicit expression for the solution w of the ODE (4.6), complemented by the initial condition (4.7).

Lemma A.1

Let 0<μ/γσ ²≠1. Then for sufficiently small λ>0, the function

$$w(\lambda,y)= \begin{cases} \frac{a(\lambda)\tanh[\tanh^{-1}(b(\lambda)/a(\lambda ))-a(\lambda )y]+(\frac{\mu}{\sigma^2}-\frac{1}{2})}{\gamma-1},\\[5pt] \quad\mbox{\textit{if} } \gamma\in(0,1) \mbox{ \textit{and} } \frac {\mu}{\gamma\sigma^2}<1 \mbox{ \textit{or} } \gamma>1 \mbox{ \textit{and} } \frac {\mu}{\gamma\sigma^2}>1,\\[5pt] \frac{a(\lambda) \tan[\tan^{-1}(b(\lambda)/a(\lambda))+a(\lambda )y]+(\frac{\mu}{\sigma^2}-\frac{1}{2})}{\gamma-1},\\[5pt] \quad\mbox{\textit{if} } \gamma>1 \mbox{ \textit{and} } \frac{\mu }{\gamma\sigma^2} \in\left(\frac{1}{2}-\frac{1}{2}\sqrt{1-\frac {1}{\gamma}},\frac{1}{2}+\frac{1}{2}\sqrt{1-\frac{1}{\gamma }}\right),\\[5pt] \frac{a(\lambda)\coth[\coth^{-1}(b(\lambda)/a(\lambda))-a(\lambda )y]+(\frac{\mu}{\sigma^2}-\frac{1}{2})}{\gamma-1}, \\[5pt] \quad\mbox{\textit{otherwise}}, \end{cases} $$

with

$$a(\lambda)=\sqrt{\Big|(\gamma-1)\frac{\mu^2-\lambda^2}{\gamma \sigma ^4}-\Big(\frac{1}{2}-\frac{\mu}{\sigma^2}\Big)^2\Big|}, \qquad b(\lambda)=\frac{1}{2}-\frac{\mu}{\sigma^2}+(\gamma-1)\frac{\mu -\lambda}{\gamma\sigma^2}, $$

is a local solution of

$$ w'(y)+(1-\gamma)w^2(y)+\left(\frac{2\mu}{\sigma^2}-1\right)w(y)-\frac {\mu^2-\lambda^2}{\gamma\sigma^4}=0, \qquad w(0)=\frac{\mu-\lambda }{\gamma\sigma^2}. $$

(A.1)

Moreover, y↦w(λ,y) is increasing (resp. decreasing) for μ/γσ ²∈(0,1) (resp. μ/γσ ²>1).

Proof

The first part of the assertion is easily verified by taking derivatives, noticing that the case distinctions distinguish between the different signs of the discriminant

$$(\gamma-1)\frac{\mu^2-\lambda^2}{\gamma\sigma^4}-\left(\frac {1}{2}-\frac{\mu}{\sigma^2}\right)^2 $$

of the Riccati equation (A.1) for sufficiently small λ. Indeed, in the second case the discriminant is positive for sufficiently small λ. The first and third case correspond to a negative discriminant, as well as b(λ)/a(λ)<1 and b(λ)/a(λ)>1, respectively, for sufficiently small λ>0, so that the function w is well defined in each case.

The second part of the assertion follows by inspection of the explicit formulas. □

Next, we establish that the crucial constant λ, which determines both the no-trade region and the equivalent safe rate, is well defined.

Lemma A.2

Let 0<μ/γσ ²≠1 and w(λ,⋅) be defined as in Lemma A.1, and set

$${\ell(\lambda)}=\frac{\mu-\lambda}{\gamma\sigma^2-(\mu-\lambda )}, \qquad u(\lambda)=\frac{1}{1-\varepsilon}\frac{\mu+\lambda}{\gamma \sigma^2-(\mu +\lambda)}. $$

Then, for sufficiently small ε>0, there exists a unique solution λ of

$$ w\left(\lambda,\log\frac{u(\lambda)}{\ell(\lambda)} \right) -\frac{\mu+\lambda}{\gamma\sigma^2}=0. $$

(A.2)

As ε↓0, it has the asymptotics

Proof

The explicit expression for w in Lemma A.1 implies that w(λ,x) in Lemma A.1 is analytic in both variables at (0,0). By the initial condition in (A.1), its power series has the form

$$w(\lambda,x) = \frac{\mu-\lambda}{\gamma\sigma^2} +\sum_{i=1}^\infty\sum_{j=0}^\infty W_{ij} x^i \lambda^j, $$

where expressions for the coefficients W _ij are computed by expanding the explicit expression for w. (The leading terms are provided after this proof.) Hence, the left-hand side of the boundary condition (A.2) is an analytic function of ε and λ. Its power series expansion shows that the coefficients of ε ⁰ λ ^j vanish for j=0,1,2, so that the condition (A.2) reduces to

$$ \lambda^3 \sum_{i\geq0} A_i \lambda^i = \varepsilon\sum_{i,j\geq0} B_{ij} \varepsilon^i \lambda^j $$

(A.3)

with (computable) coefficients A _i and B _ij . This equation has to be solved for λ. Since

$$A_0 = \frac{4}{3\mu\sigma^2(\gamma\sigma^2-\mu)} \quad\text{and} \quad B_{00} = \frac{\mu(\gamma\sigma^2-\mu)}{\gamma^2\sigma^4} $$

are non-zero, divide (A.3) by ∑ _i≥0 A _i λ ⁱ, and take the third root, obtaining that, for some C _ij ,

$$\lambda= \varepsilon^{1/3} \sum_{i,j\geq0} C_{ij} \varepsilon^i \lambda^j = \varepsilon^{1/3} \sum_{i,j\geq0} C_{ij} (\varepsilon^{1/3})^{3i} \lambda^j . $$

The right-hand side is an analytic function of λ and ε ^1/3, so that the implicit function theorem [21, Theorem I.B.4] yields a unique solution λ (for ε sufficiently small), which is an analytic function of ε ^1/3. Its power series coefficients can be computed at any order. □

In the preceding proof, we needed the first coefficients of the series expansion of the analytic function on the left-hand side of (A.2). Calculating them is elementary, but rather cumbersome, and can be quickly performed with symbolic computation software. Following a referee’s suggestion, we present some expressions to aid readers who wish to check the calculations by hand, namely the derivatives of w at (λ,x)=(0,0) that are needed to calculate the Taylor coefficients of (A.2) used in the proof. Note that they are the same in all three cases of Lemma A.1, and given by

Henceforth, consider small transaction costs ε>0, and let λ denote the constant in Lemma A.2. Moreover, set w(y)=w(λ,y), a=a(λ), b=b(λ), and u=u(λ), ℓ=ℓ(λ). In all cases, the function w can be extended smoothly to an open neighborhood of [0,log(u/ℓ)] (resp. [log(u/ℓ),0] if μ/γσ ²>1). By continuity, the ODE (A.1) then also holds at 0 and log(u/ℓ); inserting the boundary conditions for w in turn readily yields the following counterparts for the derivative w′:

Lemma A.3

Let 0<μ/γσ ²≠1. Then, in all three cases,

$$w'(0)=\frac{\mu-\lambda}{\gamma\sigma^2}-\Bigg(\frac{\mu -\lambda}{\gamma \sigma^2}\Bigg)^2, \qquad w'\bigg(\log\frac{u}{\ell} \bigg)=\frac {\mu +\lambda}{\gamma\sigma^2}-\left(\frac{\mu+\lambda}{\gamma\sigma ^2}\right)^2. $$

Appendix B: Shadow prices and verification

The key to justify the heuristic arguments of Sect. 4 is to reduce the portfolio choice problem with transaction costs to another portfolio choice problem, without transaction costs. Here, the bid and ask prices are replaced by a single shadow price $\tilde{S}_{t}$, evolving within the bid-ask spread, which coincides with one of the prices at times of trading, and yields the same optimal policy and utility. Evidently, any frictionless market with values in the bid-ask spread leads to more favorable terms of trade than the original market with transaction costs. To achieve equality, the particularly unfavorable shadow price must match the trading prices whenever its optimal policy transacts.

Definition B.1

A shadow price is a frictionless price process $\tilde{S}$, evolving within the bid-ask spread ($(1-\varepsilon)S_{t} \le\tilde{S}_{t} \le S_{t}$ a.s.), such that there is an optimal strategy for $\tilde{S}$ which is of finite variation and which entails buying only when the shadow price $\tilde{S}_{t}$ equals the ask price S _t , and selling only when $\tilde{S}_{t}$ equals the bid price (1−ε)S _t .

Once a candidate for such a shadow price is identified, long-run verification results for frictionless models (cf. Guasoni and Robertson [20]) deliver the optimality of the guessed policy. Further, this method provides explicit upper and lower bounds on finite-horizon performance (cf. Lemma B.3 below), thereby allowing to check whether the long-run optimal strategy is approximately optimal for a horizon T. Put differently, it shows which horizons are long enough.

2.1 B.1 Derivation of a candidate shadow price

With a smooth candidate value function at hand, a candidate shadow price can be identified as follows. By definition, trading at the shadow price should not allow the investor to outperform the original market with transaction costs. In particular, if $\tilde{S}_{t}$ is the value of the shadow price at time t, then allowing the investor to carry out at single trade at time t at this frictionless price should not lead to an increase in utility. A trade of ν risky shares at the frictionless price $\tilde{S}_{t}$ moves the investor’s safe position X _t to $X_{t}-\nu\tilde{S}_{t}$ and her risky position (valued at the ask price S _t ) from Y _t to Y _t +νS _t . Then, recalling that the second and third arguments of the candidate value function V from Sect. 4 were precisely the investor’s safe and risky positions, the requirement that such a trade does not increase the investor’s utility is tantamount to

$$V(t,X_t-\nu\tilde{S}_t,Y_t+\nu S_t) \leq V(t,X_t,Y_t), \quad\forall \nu\in\mathbb{R}. $$

A Taylor expansion of the left-hand side for small ν then implies that we should have $-\nu\tilde{S}_{t} V_{x}+\nu S_{t} V_{y} \leq0$. Since this inequality must hold both for positive and negative values of ν, it yields

$$ \tilde{S}_t=\frac{V_y}{V_x} S_t. $$

(B.1)

That is, the multiplicative deviation of the shadow price from the ask price should be the marginal rate of substitution of risky for safe assets. In particular, this argument immediately yields a candidate shadow price, once a smooth candidate value function has been identified. For the long-run problem, we have derived in the previous section the candidate value function

$$V(t,x,y)=e^{-(1-\gamma)(r+\beta)t}x^{1-\gamma} e^{(1-\gamma)\int _0^{\log(y/\ell x)}w(y)\,dy}. $$

Using this equality to calculate the partial derivatives in (B.1), the candidate shadow price becomes

$$ \tilde{S}_t=\frac{w(\varUpsilon_t)}{\ell e^{\varUpsilon _t}(1-w(\varUpsilon_t))}S_t, $$

(B.2)

where ϒ _t =log(Y _t /ℓX _t ) denotes the logarithm of the risky-safe ratio, centered at its value at the lower buying boundary ℓ. If this candidate is indeed the right one, then its optimal strategy and value should coincide with their frictional counterparts derived heuristically above. In particular, the optimal risky fraction $\tilde{\pi}_{t}$ should correspond to the same numbers $\varphi^{0}_{t}$ and φ _t of safe and risky shares, if measured in terms of $\tilde {S}_{t}$ instead of the ask price S _t . As a consequence,

$$ \tilde{\pi}_t=\frac{\varphi_t \tilde{S}_t}{\varphi ^0_tS^0_t+\varphi _t\tilde{S}_t} =\frac{\varphi_t S_t \frac{w(\varUpsilon_t)}{\ell e^{\varUpsilon _t}(1-w(\varUpsilon_t))}}{\varphi^0_t S^0_t +\varphi_t S_t \frac{w(\varUpsilon_t)}{\ell e^{\varUpsilon_t}(1-w(\varUpsilon_t))}} =\frac{\frac{w(\varUpsilon_t)}{1-w(\varUpsilon_t)}}{1+\frac {w(\varUpsilon_t)}{1-w(\varUpsilon_t)}}=w(\varUpsilon_t), $$

where for the third equality we have used the fact that the risky-safe ratio $\varphi_{t} S_{t}/\varphi^{0}_{t} S^{0}_{t}$ can be written as $\ell e^{\varUpsilon_{t}}$ by the definition of ϒ _t .

We now turn to the corresponding frictionless value function $\tilde{V}$. By the definition of a shadow price, it should coincide with its frictional counterpart V. In the frictionless case, it is more convenient to factor out the total wealth $\tilde{X}_{t}=\varphi^{0}_{t} S^{0}_{t}+\varphi_{t} \tilde{S}_{t}$ (in terms of the frictionless risky price $\tilde{S}_{t}$) instead of the safe position $X_{t}=\varphi^{0}_{t} S^{0}_{t}$, giving

$$\tilde{V}(t,\tilde{X}_t,\varUpsilon_t)=V(t,X_t,Y_t)=e^{-(1-\gamma )(r+\beta)t} \tilde{X}_t^{1-\gamma} \left(\frac{X_t}{\tilde {X}_t}\right)^{1-\gamma} e^{(1-\gamma)\int_0^{\varUpsilon_t}w(y)\,dy}. $$

Since $X_{t}/\tilde{X}_{t}=1-w(\varUpsilon_{t})$ by the definitions of $\tilde {S}_{t}$ and ϒ _t , one can rewrite the last two factors as

Then, setting $\tilde{w}=w-\frac{w'}{1-w}$, the candidate long-run value function for $\tilde{S}$ becomes

$$\tilde{V}(t,\tilde{x},\tilde{y})=e^{-(1-\gamma)(r+\beta)t} \tilde {x}^{1-\gamma} e^{(1-\gamma)\int_0^{\tilde{y}}\tilde{w}(y)\,dy}\big (1-w(0)\big)^{\gamma-1}. $$

Starting from the candidate value function and optimal policy for $\tilde{S}$, we can now proceed to verify that they are indeed optimal for $\tilde{S}$, by adapting the argument from [20]. But before we do that, we have to construct the respective processes.

2.2 B.2 Construction of the shadow price

The above heuristic arguments suggest that the optimal ratio $Y_{t}/X_{t}=\varphi_{t} S_{t}/\varphi^{0}_{t} S^{0}_{t}$ should take values in the interval [ℓ,u]. As a result, ϒ _t =log(Y _t /ℓX _t ) should be [0,log(u/ℓ)]-valued if the lower trading boundary ℓ for the ratio Y _t /X _t is positive. If the investor shorts the safe asset to leverage her risky position, the ratio becomes negative. In the frictionless case, and also for small transaction costs, this happens if the risky weight μ/γσ ² is bigger than 1. Then, the trading boundaries ℓ≤u are both negative, so that the centered log-ratio ϒ _t should take values in [log(u/ℓ),0]. In both cases, trading should only take place when the risky-safe ratio reaches the boundaries of this region. Hence, the numbers of safe and risky units $\varphi^{0}_{t}$ and φ _t should remain constant, and $\varUpsilon_{t}=\log(\varphi_{t}/\ell\varphi ^{0}_{t})+\log (S_{t}/S^{0}_{t})$ should follow a Brownian motion with drift as long as ϒ _t moves in (0,log(u/ℓ)) (resp. in (log(u/ℓ),0) if μ/γσ ²>1). This argument motivates the definition of the process ϒ as reflected Brownian motion, i.e.,

$$ d\varUpsilon_t=(\mu-\sigma^2/2)\,dt+\sigma \,dW_t+dL_t-dU_t, \qquad \varUpsilon_0 \in[0,\log(u/\ell)], $$

(B.3)

for continuous, adapted minimal processes L and U which are nondecreasing (resp. non-increasing if μ/γσ ²>1) and increase (resp. decrease if μ/γσ ²>1) only on the sets {ϒ=0} and {ϒ=log(u/ℓ)}, respectively. Starting from this process, the existence of which is a classical result of [36], the process $\tilde{S}$ is defined in accordance with (B.2).

Lemma B.2

Define

$$ \varUpsilon_0= \begin{cases} 0, &\mbox{\textit{if} }\ell\xi^0S^0_0 \geq\xi S_0,\\[4pt] \log (u/\ell ), &\mbox{\textit{if} } u\xi^0 S^0_0 \leq\xi S_0,\\[4pt] \log[(\xi S_0/\xi^0 S^0_0)/\ell], &\mbox{\textit{otherwise},} \end{cases} $$

(B.4)

and let ϒ be defined as in (B.3), starting at ϒ ₀. Then $\tilde{S} = S \frac{w(\varUpsilon)}{\ell e^{\varUpsilon} (1-w(\varUpsilon))}$, with w as in Lemma A.1, has the dynamics

$$ d\tilde{S}_t/\tilde{S}_t = \big(\tilde{\mu}(\varUpsilon_t)+r\big)\,d t+ \tilde{\sigma}(\varUpsilon_t)\,d W_t, $$

where $\tilde{\mu}(\cdot)$ and $\tilde{\sigma}(\cdot)$ are defined as

Moreover, the process $\tilde{S}$ takes values within the bid-ask spread [(1−ε)S,S].

Note that the first two cases in (B.4) arise if the initial risky-safe ratio $\xi S_{0}/(\xi^{0} S_{0}^{0})$ lies outside of the interval [ℓ,u]. Then we need to jump from the initial position $(\varphi_{0-}^{0}, \varphi_{0-}) = (\xi^{0},\xi)$ to the nearest boundary value of [ℓ,u]. This transfer requires the purchase resp. sale of the risky asset and hence the initial price $\tilde{S} _{0}$ is defined to match the buying resp. selling price of the risky asset.

Proof of Lemma B.2

The dynamics of $\tilde{S}$ result from Itô’s formula, the dynamics of ϒ, and the identity

$$ w''(y) = 2(\gamma-1)w'(y) w(y)- (2\mu/\sigma^2-1) w'(y), $$

(B.5)

obtained by differentiating the ODE (A.1) for w with respect to y. Therefore it remains to show that $\tilde{S}_{t}$ indeed takes values in the bid-ask spread [(1−ε)S _t ,S _t ]. To this end, notice that in view of the ODE (A.1) for w, the derivative of the function g(y):=w(y)/ℓe ^y(1−w(y)) is given by

$$g'(y)=\frac{w'(y)-w(y)+w^2(y)}{\ell e^y (1-w(y))^2}=\frac{\gamma (w^2-2\frac{\mu}{\gamma\sigma^2} w)+(\mu^2-\lambda^2)/\gamma \sigma ^4}{\ell e^y (1-w(y))^2}. $$

Due to the boundary conditions for w, the function g′ vanishes at 0 and log(u/ℓ). Differentiating its numerator gives $2\gamma w'(y)(w(y)-\frac{\mu}{\gamma\sigma^{2}})$. For $\frac{\mu}{\gamma \sigma ^{2}} \in(0,1)$ (resp. $\frac{\mu}{\gamma\sigma^{2}}>1$), w is increasing from $\frac{\mu-\lambda}{\gamma\sigma^{2}}<\frac{\mu }{\gamma \sigma^{2}}$ to $\frac{\mu+\lambda}{\gamma\sigma^{2}}>\frac{\mu }{\gamma \sigma^{2}}$ on [0,log(u/ℓ)] (resp. decreasing from $\frac{\mu +\lambda}{\gamma\sigma^{2}}$ to $\frac{\mu-\lambda}{\gamma\sigma ^{2}}$ on [log(u/ℓ),0]); hence, w′ is nonnegative (resp. non-positive). Moreover, g′ starts at zero for y=0 (resp. log(u/ℓ)), then decreases (resp. increases), and eventually starts increasing (resp. decreasing) again, until it reaches level zero again for y=log(u/ℓ) (resp. y=0). In particular, g′ is non-positive (resp. nonnegative), so that g is decreasing on [0,log(u/ℓ)] (resp. increasing on [log(u/ℓ),0] for $\frac{\mu}{\gamma\sigma^{2}}>1$). Taking into account that g(0)=1 and g(log(u/ℓ))=1−ε, by the boundary conditions for w and the definition of u and ℓ in Lemma A.2, the proof is now complete. □

2.3 B.3 Verification

The long-run optimal portfolio in the frictionless “shadow market” with price process $\tilde{S}$ can now be determined by adapting the argument in Guasoni and Robertson [20]. The first step is to determine finite-horizon bounds, which provide upper and lower estimates for the maximal expected utility on any finite horizon T.

Lemma B.3

For a fixed time horizon T>0, let $\beta= \frac{\mu^{2}-\lambda ^{2}}{2\gamma\sigma^{2}}$ and let the function w be defined as in Lemma A.1. Then, for the shadow payoff $\tilde{X}_{T}$ corresponding to the risky fraction $\tilde{\pi}(\varUpsilon_{t}) = w(\varUpsilon_{t})$ and the shadow discount factor $\tilde{M}_{T}=e^{-rT}\mathcal{E}(-\int_{0}^{\cdot}\frac{{\tilde{\mu}}}{ {\tilde{\sigma}}}\,dW)_{T}$, the following bounds hold true:

(B.6)

where $\tilde{q} (y) := \int_{0}^{y} (w(z)-\frac{w'(z)}{1-w(z)}) dz$ and $\hat{E} \left[\cdot\right]$ denotes the expectation with respect to the myopic probability $\hat{P}$, defined by

Proof

First note that ${\tilde{\mu}}, {\tilde{\sigma}}$ and w are functions of ϒ _t , but the argument is omitted throughout to ease notation. Now, to prove (B.6), notice that the frictionless shadow wealth process $\tilde{X}$ with dynamics $\frac{d\tilde {X}_{t}}{\tilde{X}_{t}}=w \frac{d\tilde{S}_{t}}{\tilde{S}_{t}}+(1-w)\frac {dS^{0}_{t}}{S^{0}_{t}}$ satisfies

$$ \tilde{X}_T^{1-\gamma}= \tilde{X}_0^{1-\gamma} e^{(1-\gamma)\int_0^T (r+{\tilde{\mu}}w -\frac {{\tilde{\sigma}}^2}{2}w^2) \,dt +(1-\gamma)\int_0^T {\tilde{\sigma}}w \,dW_t}. $$

Hence we get

$$\tilde{X}_T^{1-\gamma} = \tilde{X}_0^{1-\gamma}\frac{d\hat{P}}{dP} e^{\int_0^T ((1-\gamma)(r+{\tilde{\mu}}w -\frac{{\tilde{\sigma}}^2}{2}w^2) +\frac{1}{2}(-\frac{{\tilde{\mu}}}{ {\tilde{\sigma}}}+{\tilde{\sigma}}w)^2)\,dt+\int_0^T ((1-\gamma ){\tilde{\sigma}}w-(-\frac{{\tilde{\mu}}}{ {\tilde{\sigma}}}+{\tilde{\sigma}}w)) \,dW_t}. $$

After inserting the definitions of ${\tilde{\mu}}$ and ${\tilde{\sigma}}$, respectively, the second integrand simplifies to $(1-\gamma)\sigma (\frac {w'}{1-w}-w)$. Similarly, the first integrand reduces to

$$(1-\gamma)\biggl(r+\frac{\sigma^2}{2}\biggl(\frac{w'}{1-w}\biggr)^2-(1-\gamma)\sigma^2 \frac{w' w}{1-w} +(1-\gamma)\frac{\sigma^2}{2}w^2\biggr). $$

In summary,

(B.7)

The boundary conditions for w and w′ imply

$$w(0)-\frac{w'(0)}{1-w(0)}=w\big(\log(u/\ell)\big)-\frac{w'(\log (u/\ell ))}{1-w(\log(u/\ell))}=0; $$

hence, Itô’s formula yields the result that the minimal nondecreasing terms vanish in the dynamics of $\tilde{q}(\varUpsilon_{t})$, so that

(B.8)

because w−w′/(1−w) vanishes on the sets where the processes L and U increase. Substituting the second derivative w″ according to the ODE (B.5) and using the resulting identity to replace the stochastic integral in (B.7) yields

$$\tilde{X}_T^{1-\gamma} = \tilde{X}_0^{1-\gamma}\frac{d\hat{P}}{dP} e^{(1-\gamma)\int_0^T (r+\frac{\sigma^2}{2}w'+(1-\gamma)\frac {\sigma^2}{2}w^2+(\mu-\frac{\sigma^2}{2})w)\,dt} e^{(1-\gamma)(\tilde {q}(\varUpsilon_0)-\tilde{q}(\varUpsilon_T))}. $$

After inserting the ODE (A.1) for w, the first bound thus follows by taking the expectation.

The argument for the second bound is similar. Plugging in the definitions of ${\tilde{\mu}}$ and ${\tilde{\sigma}}$, the shadow discount factor $\tilde{M}_{T}=e^{-rT}\mathcal{E}(-\int_{0}^{\cdot}\frac{{\tilde{\mu}}}{ {\tilde{\sigma}}}\,dW)_{T}$ and the myopic probability $\hat{P}$ yields

Again replace the stochastic integral using (B.8) and the ODE (B.5), obtaining

$$\tilde{M}_T^{1-\frac{1}{\gamma}}=\frac{d\hat{P}}{dP} e^{\frac {1-\gamma }{\gamma}\int_0^T (r+\frac{\sigma^2}{2}w'+(1-\gamma)\frac{\sigma ^2}{2}w^2+(\mu-\frac{\sigma^2}{2})w)\,dt}e^{\frac{1-\gamma}{\gamma }(\tilde {q}(\varUpsilon_0)-\tilde{q}(\varUpsilon_T))}. $$

By inserting the ODE (A.1) for w, taking the expectation, and raising it to power γ, the second bound follows. □

With the finite-horizon bounds at hand, it is now straightforward to establish that the policy $\tilde{\pi}(\varUpsilon)$ is indeed long-run optimal in the frictionless market with price $\tilde{S}$.

Lemma B.4

Let 0<μ/γσ ²≠1 and let w be defined as in Lemma A.1. Then the risky weight $\tilde{\pi}(\varUpsilon _{t})=w(\varUpsilon_{t})$ is long-run optimal with equivalent safe rate r+β in the frictionless market with price process $\tilde{S}$. The corresponding wealth process (in terms of $\tilde{S}_{t}$), and the numbers of safe and risky units are given by

Proof

The formulas for the wealth process and the corresponding numbers of safe and risky units follow directly from the standard frictionless definitions. Now let $\tilde{M}$ be the shadow discount factor from Lemma B.3. Then standard duality arguments for power utility (cf. Lemma 5 in Guasoni and Robertson [20]) imply that the shadow payoff $\tilde{X}_{T}^{\phi}$ corresponding to any admissible strategy ϕ satisfies the inequality

$$ E \big[(\tilde{X}^\phi_T)^{1-\gamma} \big]^{\frac{1}{1-\gamma}} \le E \Big[\tilde{M}_T^{\frac{\gamma-1}{\gamma}}\Big]^{\frac{\gamma}{1-\gamma}} . $$

This inequality in turn yields for any admissible strategy ϕ in the frictionless market with shadow price $\tilde{S}$ the upper bound

$$ \liminf_{T \to\infty} \frac{1}{(1-\gamma)T}\log E\big[(\tilde {X}^\phi _T)^{1-\gamma}\big] \le\liminf_{T\rightarrow\infty} \frac{\gamma}{(1-\gamma)T} \log E\Big[{\tilde{M}_T^{\frac{\gamma -1}{\gamma}}}\Big]. $$

Since the function $\tilde{q}$ is bounded on the compact support of ϒ _t , the second bound in Lemma B.3 implies that the right-hand side equals r+β. Likewise, the first bound in the same lemma implies that the shadow payoff $\tilde{X}_{T}$ (corresponding to the policy φ) attains this upper bound, concluding the proof. □

The next lemma establishes that the candidate $\tilde{S}$ is indeed a shadow price.

Lemma B.5

Let 0<μ/γσ ²≠1. Then the number of shares $\varphi _{t}=w(\varUpsilon_{t})\tilde{X}_{t}/\tilde{S}_{t}$ in the portfolio  $\tilde {\pi }(\varUpsilon_{t})$ in Lemma B.4 has the dynamics

$$ \frac{d\varphi_t}{\varphi_t}=\left(1-\frac{\mu-\lambda}{\gamma \sigma ^2}\right)dL_t-\left(1-\frac{\mu+\lambda}{\gamma\sigma^2}\right)dU_t. $$

(B.9)

Thus φ _t increases only when ϒ _t =0, that is, when $\tilde{S}_{t}$ equals the ask price, and decreases only when ϒ _t =log(u/ℓ), that is, when $\tilde{S}_{t}$ equals the bid price.

Proof

Itô’s formula and the ODE (B.5) yield

$$dw(\varUpsilon_t)=-(1-\gamma)\sigma^2 w'(\varUpsilon _t)w(\varUpsilon _t)\,dt+\sigma w'(\varUpsilon_t)\,dW_t+w'(\varUpsilon_t)(dL_t-dU_t). $$

Integrating $\varphi_{t}=w(\varUpsilon_{t})\tilde{X}_{t}/\tilde{S}_{t}$ by parts twice, inserting the dynamics of w(ϒ _t ), $\tilde {X}_{t}$, $\tilde{S}_{t}$, and simplifying yields

$$\frac{d\varphi_t}{\varphi_t}=\frac{w'(\varUpsilon _t)}{w(\varUpsilon _t)}\,d(L_t-U_t). $$

Since L _t and U _t only increase (resp. decrease when μ/γσ ²>1) on {ϒ _t =0} and {ϒ _t =log(u/ℓ)}, respectively, the assertion now follows from the boundary conditions for w and w′. □

The optimal growth rate for any frictionless price within the bid-ask spread must be greater than or equal as in the original market with bid-ask process ((1−ε)S,S), because the investor trades at more favorable prices. For a shadow price, there is an optimal strategy that only entails buying (resp. selling) stocks when $\tilde{S}_{t}$ coincides with the ask resp. bid price. Hence, this strategy yields the same payoff when executed at bid-ask prices, and thus is also optimal in the original model with transaction costs. The corresponding equivalent safe rate must also be the same, since the difference due to the liquidation costs vanishes as the horizon grows in (2.2).

Proposition B.6

For a sufficiently small spread ε, the strategy (φ ⁰,φ) from Lemma B.4 is also long-run optimal in the original market with transaction costs, with the same equivalent safe rate.

Proof

As φ _t only increases (resp. decreases) when $\tilde{S}_{t}=S_{t}$ (resp. $\tilde{S}_{t}=(1-\varepsilon)S_{t}$), the strategy (φ ⁰,φ) is also self-financing for the bid-ask process ((1−ε)S,S). Since $S_{t} \geq\tilde{S}_{t} \geq (1-\varepsilon )S_{t}$ and the number φ _t of risky shares is always positive, it follows that

(B.10)

The shadow risky fraction $\tilde{\pi}(\varUpsilon_{t})=w(\varUpsilon_{t})$ is bounded from above by the term (μ+λ)/γσ ²=μ/γσ ²+O(ε ^1/3). For a sufficiently small spread ε, the strategy (φ ⁰,φ) is therefore also admissible for ((1−ε)S,S). Moreover, (B.10) then also yields

(B.11)

that is, (φ ⁰,φ) has the same growth rate either with $\tilde{S}$ or with ((1−ε)S,S).

For any admissible strategy (ψ ⁰,ψ) for the bid-ask spread [(1−ε)S,S], set $\tilde{\psi}_{t}^{0}=\psi^{0}_{0-}-\int_{0}^{t} \tilde{S}_{s}/S^{0}_{s} \,d\psi_{s}$. Then $(\tilde{\psi} ^{0},\psi)$ is a self-financing trading strategy for $\tilde{S}$ with $\tilde{\psi} ^{0} \geq\psi^{0}$. Together with $\tilde{S}_{t} \in[(1-\varepsilon )S_{t},S_{t}]$, the long-run optimality of (φ ⁰,φ) for $\tilde{S}$ and (B.11), it follows that

Hence (φ ⁰,φ) is also long-run optimal for ((1−ε)S,S). □

The main result now follows by putting together the above statements.

Theorem B.7

For ε>0 small and 0<μ/γσ ²≠1, the process $\tilde{S}$ in Lemma B.2 is a shadow price. A long-run optimal policy—both for the frictionless market with price $\tilde{S}$ and in the market with bid-ask prices (1−ε)S,S—is to keep the risky weight $\tilde{\pi}_{t}$ (in terms of $\tilde {S}_{t}$) in the no-trade region

$$[\pi_-,\pi_+]=\left[\frac{\mu-\lambda}{\gamma\sigma^2},\frac {\mu+\lambda }{\gamma\sigma^2}\right]. $$

As ε↓0, its boundaries have the asymptotics

The corresponding equivalent safe rate is

If μ/γσ ²=1, then $\tilde{S} =S$ is a shadow price, and it is optimal to invest all the wealth in the risky asset at time t=0 and never to trade afterwards. In this case, the equivalent safe rate is given by the frictionless value r+β=r+μ ²/2γσ ²=r+μ/2.

Proof

First let 0<μ/γσ ²≠1. Optimality with equivalent safe rate r+β of the strategy (φ ⁰,φ) associated to $\tilde{\pi}(\varUpsilon)$ for $\tilde{S}$ has been shown in Lemma B.4. The asymptotic expansions are an immediate consequence of the fractional power series for λ (cf. Lemma A.2) and Taylor expansion.

Next, Lemma B.5 shows that $\tilde{S}$ is a shadow price process in the sense of Definition B.1. In view of the asymptotic expansions for π _±, Proposition B.6 shows that for small transaction costs ε, the same policy is also optimal, with the same equivalent safe rate, in the original market with bid-ask prices (1−ε)S,S.

Consider now the degenerate case μ/γσ ²=1. Then the optimal strategy in the frictionless model $\tilde{S} =S$ transfers all wealth to the risky asset at time t=0, never to trade afterwards ($\varphi^{0}_{t}=0$ and $\varphi_{t}=\xi+\xi^{0} S^{0}_{0}/S_{0}$ for all t≥0). Hence it is of finite variation, and the number of shares never decreases, and increases only at time t=0, where the shadow price coincides with the ask price. Thus $\tilde{S} =S$ is a shadow price. For small ε, the remaining assertions then follow as in Proposition B.6 above. □

Next is the proof of Theorem 3.1, which establishes asymptotic finite-horizon bounds. In fact, the proof yields exact bounds in terms of λ, from which the expansions in the theorem are obtained.

Proof of Theorem 3.1

Let (ϕ ⁰,ϕ) be any admissible strategy starting from the initial position $(\varphi^{0}_{0-},\varphi_{0-})$. Then as in the proof of Proposition B.6, we have $\varXi^{\phi}_{T} \leq\tilde {X}^{\phi}_{T}$ for the corresponding shadow payoff, that is, the terminal value of the wealth process $\tilde{X}^{\phi}=\phi^{0}_{0}+\phi_{0} \tilde {S}_{0}+\int_{0}^{\cdot}\phi_{s}\, d\tilde{S}_{s}$ corresponding to trading with ϕ in the frictionless market with price process $\tilde{S}$. Hence Lemma 5 in Guasoni and Robertson [20] and the second bound in Lemma B.3 imply that

(B.12)

For the strategy (φ ⁰,φ) from Lemma B.5, we have $\varXi^{\varphi}_{T} \geq(1-\frac{\varepsilon}{1-\varepsilon }\frac {\mu+\lambda}{\gamma\sigma^{2}})\tilde{X}^{\varphi}_{T}$ by the proof of Proposition B.6. Hence the first bound in Lemma B.3 yields

(B.13)

To determine explicit estimates for these bounds, we first analyze the sign of the function $\tilde{w} =w-\frac{w'}{1-w}$ and hence the monotonicity of $\tilde{q}(y)=\int_{0}^{y} \tilde{w}(z)\,dz$. Whenever $\tilde{w}=0$, i.e., w′=w(1−w), the derivative of $\tilde{w}$ is

where we have used the ODE (B.5) for the second equality. Since $\tilde{w}$ vanishes at 0 and log(u/ℓ) by the boundary conditions for w and w′, this shows that the behavior of $\tilde {w}$ depends on whether the investor’s position is leveraged or not. In the absence of leverage, μ/γσ ²∈(0,1), $\tilde {w}$ is defined on [0,log(u/ℓ)]. It vanishes at the left boundary 0 and then increases since its derivative is initially positive by the initial condition for w. Once the function w has increased to level μ/γσ ², the derivative of $\tilde{w}$ starts to become negative; as a result, $\tilde{w}$ begins to decrease until it reaches level zero again at log(u/ℓ). In particular, $\tilde{w}$ is nonnegative for μ/γσ ²∈(0,1).

In the leverage case μ/γσ ²>1, the situation is reversed. Then, $\tilde{w}$ is defined on [log(u/ℓ),0] and, by the boundary condition for w at log(u/ℓ), therefore starts to decrease after starting from zero at log(u/ℓ). Once w has decreased to level μ/γσ ², $\tilde{w}$ starts increasing until it reaches level zero again at 0. Hence $\tilde{w}$ is non-positive for μ/γσ ²>1.

Now consider case 2 of Lemma A.1; the calculations for the other cases follow along the same lines with minor modifications. Then μ/γσ ²∈(0,1) and $\tilde{q}$ is positive and increasing. Hence,

$$ \frac{\gamma}{(1-\gamma)T}\log\hat{E}\big[e^{(\frac{1}{\gamma }-1)(\tilde {q}(\varUpsilon_0)-\tilde{q}(\varUpsilon_T))}\big]\leq\frac {1}{T}\int _0^{\log(u/\ell)}\tilde{w}(y)\,dy $$

(B.14)

and likewise

$$ \frac{1}{(1-\gamma)T}\log\hat{E}\big[e^{(1-\gamma)(\tilde {q}(\varUpsilon _0)-\tilde{q}(\varUpsilon_T))}\big] \geq-\frac{1}{T}\int_0^{\log (u/\ell )} \tilde{w}(y)\,dy. $$

(B.15)

Since $\tilde{w}(y)=w(y)-w'/(1-w)$, the boundary conditions for w imply

$$ \int_0^{\log(u/\ell)} \tilde{w}(y)\,dy=\int_0^{\log(u/\ell)}w(y)\,dy -\log \frac{\mu-\lambda-\gamma\sigma^2}{\mu+\lambda-\gamma\sigma^2} . $$

(B.16)

By elementary integration of the explicit formula in Lemma A.1 and using the boundary conditions from Lemma A.3 for the evaluation of the result at 0 resp. log(u/ℓ), the integral of w can also be computed in closed form to give

(B.17)

As ε↓0, a Taylor expansion and the power series for λ then yield

$$\int_0^{\log(u/\ell)} \tilde{w}(y)\,dy=\frac{\mu}{\gamma\sigma ^2}\varepsilon+O(\varepsilon^{4/3}). $$

Likewise,

$$\log\left(1-\frac{\varepsilon}{1-\varepsilon}\frac{\mu-\lambda }{\gamma \sigma^2}\right)=-\frac{\mu}{\gamma\sigma^2}\varepsilon +O(\varepsilon^{4/3}), $$

as well as

$$\log(\varphi^0_{0-}+\varphi_{0-}\tilde{S}_0) \geq\log(\varphi ^0_{0-}+\varphi_{0-}S_0)- \frac{\varphi_{0-}S_0}{\varphi ^0_{0-}+\varphi _{0-}S_0}\varepsilon+O(\varepsilon^2). $$

The claimed bounds then follow from (B.12) and (B.14), resp. (B.13) and (B.15). □

Appendix C: Trading volume

As above, let $\varphi_{t}=\varphi_{t}^{\uparrow}-\varphi_{t}^{\downarrow}$ denote the number of risky units at time t, written as the difference of the cumulated numbers of shares bought resp. sold until t. Relative share turnover, defined as the measure $d\|\varphi\| _{t}/|\varphi_{t}|=d\varphi_{t}^{\uparrow}/|\varphi_{t}|+d\varphi ^{\downarrow }_{t}/|\varphi_{t}|$, is a scale-invariant indicator of trading volume (cf. Lo and Wang [26]). The long-term average share turnover is defined as

$$\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^T \frac{d\|\varphi\| _t}{|\varphi_t|}. $$

Similarly, relative wealth turnover is defined as the amount of wealth transacted divided by current wealth,

$$(1-\varepsilon)S_t\,d\varphi^{\downarrow}_t\big{/}\big(\varphi^0_t S^0_t+\varphi _t (1-\varepsilon)S_t\big)+S_t\, d\varphi^{\uparrow}_t\big{/}(\varphi^0_t S^0_t+\varphi_t S_t), $$

where both quantities are evaluated in terms of the bid price (1−ε)S _t when selling shares resp. in terms of the ask price S _t when purchasing them. As above, the long-term average wealth turnover is then defined as

$$\lim_{T\rightarrow\infty}\frac{1}{T}\left(\int_0^T \frac {(1-\varepsilon )S_t\,d\varphi^{\downarrow}_t}{\varphi^0_t S^0_t+\varphi_t (1-\varepsilon )S_t}+\int_0^T\frac{S_t\, d\varphi^{\uparrow}_t}{\varphi^0_t S^0_t+\varphi _t S_t}\right). $$

Both of these limits admit explicit formulas in terms of the gap, which yield asymptotic expansions for ε↓0. The analysis starts with a preparatory result (cf. Janeček and Shreve [23, Remark 4] for the case of driftless Brownian motion).

Lemma C.1

Let ϒ be a diffusion on an interval [ℓ,u], 0<ℓ<u, reflected at the boundaries, i.e.,

$$ d\varUpsilon_t = b(\varUpsilon_t) \,dt + a(\varUpsilon_t)^{1/2} \,dW_t + dL_t - dU_t, $$

where the mappings a(y)>0 and b(y) are both continuous, and the continuous, minimal nondecreasing processes L and U satisfy L ₀=U ₀=0 and only increase on {L=ℓ} and {U=u}, respectively. Denoting by ν(y) the invariant density of ϒ, we have the almost sure limits

$$ \lim_{T\rightarrow\infty} \frac{L_T}{T} = \frac{a(\ell) \nu(\ell)}{2}, \qquad \lim_{T\rightarrow\infty} \frac{U_T}{T} = \frac{a(u) \nu(u)}{2}. $$

Proof

For f∈C ²([ℓ,u]), write $\mathcal{L} f(y):=b(y) f'(y)+a(y)f''(y)/2$. Then, by Itô’s formula,

Now, take f such that f′(ℓ)=1 and f′(u)=0, and pass to the limit T→∞. The left-hand side vanishes because f is bounded; the stochastic integral also vanishes by the Dambis–Dubins–Schwarz theorem, the law of the iterated logarithm, and the boundedness of f′. Thus, the ergodic theorem [5, II.35 and II.36] implies that

$$\lim_{T\rightarrow\infty} \frac{L_T}{T} = -\int_\ell^u \mathcal{L}f(y) \nu(y)\, dy. $$

Now, the self-adjoint representation [33, VII.3.12] $\mathcal{L}f = (a f' \nu)'/2 \nu$ yields

$$\lim_{T\rightarrow\infty} \frac{L_T}{T} = -\frac{1}{2}\int_\ell^u (af'\nu)'(y) \,dy = \frac{a(\ell)\nu(\ell)f'(\ell)}{2} - \frac{a(u)\nu(u)f'(u)}{2} = \frac {a(\ell)\nu(\ell)}{2}. $$

The other limit follows from the same argument, using f such that f′(ℓ)=0 and f′(u)=1. □

Lemma C.2

Let 0<μ/γσ ²≠1 and, as in (B.3), let

$$\varUpsilon_t=\left(\mu-\frac{\sigma^2}{2}\right)t+\sigma W_t +L_t-U_t $$

be Brownian motion with drift, reflected at 0 and log(u/ℓ). Then if μ≠σ ²/2, we have the almost sure limits

$$\lim_{T \to\infty} \frac{L_T}{T}= \frac{\sigma^2}{2}\left(\frac {\frac {2\mu}{\sigma^2}-1}{(u/\ell)^{\frac{2\mu}{\sigma^2}-1}-1}\right) \quad\mbox{\textit{and}} \quad \lim_{T \to\infty} \frac{U_T}{T}=\frac{\sigma^2}{2}\left(\frac {1-\frac {2\mu}{\sigma^2}}{(u/\ell)^{1-\frac{2\mu}{\sigma^2}}-1}\right) . $$

If μ=σ ²/2, then lim _T→∞ L _T /T=lim _T→∞ U _T /T=σ ²/(2log(u/ℓ)) a.s.

Proof

First let μ≠σ ²/2. Moreover, suppose that μ/γσ ²∈(0,1). Then the scale function and the speed measure of the diffusion ϒ are

The invariant distribution of ϒ is the normalized speed measure

$$\nu(dy)=\frac{m(dy)}{m([0,\log(u/\ell)])}=1_{[0,\log(u/\ell )]}(y)\frac {\frac{2\mu}{\sigma^2}-1}{(u/\ell)^{\frac{2\mu}{\sigma ^2}-1}-1}e^{(\frac {2\mu}{\sigma^2}-1)y}\,dy. $$

For μ/γσ ²>1, the endpoints 0 and log(u/ℓ) exchange their roles, and the result is the same, up to replacing [0,log(u/ℓ)] with [log(u/ℓ),0] and multiplying the formula by −1. Then the claim follows from Lemma C.1. In the case μ=σ ²/2 of driftless Brownian motion, ϒ has uniform stationary distribution on [0,log(u/ℓ)] (resp. on [log(u/ℓ),0] if μ/γσ ²>1), and the claim again follows by Lemma C.1. □

Lemma C.2 and the formula for φ _t from Lemma B.5 yield the long-term average trading volumes. The asymptotic expansions then follow from the power series for λ (cf. Lemma A.2).

Corollary C.3

If μ/γσ ²≠1, the long-term average share turnover is

and the long-term average wealth turnover is

If μ/γσ ²=1, the long-term average share and wealth turnover both vanish.