Optimal long-term investment in illiquid markets when prices have negative memory

In a discrete-time financial market model with instantaneous price impact, we find an asymptotically optimal strategy for an investor maximizing her expected wealth. The asset price is assumed to follow a process with negative memory. We determine how the optimal growth rate depends on the impact parameter and on the covariance decay rate of the price.


Introduction
Fractional Brownian motions (FBMs) with various Hurst parameters H ∈ (0,1) have been enticing researchers of financial mathematics for a long time, since the appearance of [4]. In idealistic models of trading, however, FBMs do not provide admissible models since they generate arbitrage opportunities (for H = 1/2), see [5]. In the presence of market frictions arbitrage disappears and FBMs become interesting candidates for describing prices.
In markets with instantaneous price impact the first analysis of long-term investment has been carried out in [2]: the optimal growth rate of expected portfolio wealth has been found and an asymptotically optimal strategy has been exhibited. The robustness of such results was the next natural question: is the particular structure of FBMs needed for these conclusions? In [2] a larger class of Gaussian processes could also be treated where future increments are positively correlated to the past and the covariance structure is similar to that of FBMs with H > 1/2. The question of extending the case of FBMs with H < 1/2 remained open.
The current paper provides such an extension, more involved than in the positively correlated case. For simplicity, we stay in a discrete-time setting. We derive the same conclusions as [2] did in the case of FBMs with H < 1/2 but for a larger class of Gaussian processes.

Market model
Let (Ω,F , P) be a probability space equipped with a filtration F t , t ∈ Z. Let E[X ] denote the expectation of a real-valued random variable X (when exists). Consider a financial market where the price of a risky asset follows a process S t , t ∈ N, adapted to F t , t ∈ N.
We will present a model where trading takes place with a temporary, nonlinear price impact, along the lines of [3] but in discrete time. For some T ∈ N the class of feasible strategies up to terminal time T is defined as As we will see, φ t represents the change in the investor's position in the given asset. Let z = (z 0 , z 1 ) ∈ R 2 be a deterministic initial endowment where z 0 is in cash and and z 1 is in the risky asset. For a feasible strategy φ ∈ S (T), at any time t ≥ 0, the number of shares in the risky asset is equal to We will shortly derive a similar formula for the cash position of the investor. In classical, frictionless models of trading, cash at time T + 1 equals Algebraic manipulation of (3) yields We assume that price impact is a superlinear power function of the "trading speed" φ so we augment the above with a term that implements the effect of friction: where we assume α > 1 and λ > 0. We wish to utilize only those portfolios where the risky asset is liquidated by the end of the trading period so we define We finally get that, for φ ∈ G (T), the position in the riskless asset at time T + 1 is given by For simplicity, we also assume z 0 = z 1 = 0 from now on, i.e. portfolios start from nothing.
To investigate the potential of realizing monetary profits, we focus on a riskneutral objective: a linear utility function. Let x − := max{−x,0} for x ∈ R. Define, for T ∈ N, the class of strategies starting from a zero initial position in both assets and ending at time T + 1 in a cash only position with expected value greater than −∞. The value of the problem we will consider is thus The investors's objective is to find φ which, at least asymptotically as T → ∞, achieves the same growth rate as u(T).

Asymptotically optimal investment
First we introduce assumptions on the price process and its dependence structure.
Assumption 3.1. Let Z t , t ∈ Z be an adapted, real-valued, zero-mean stationary Gaussian process which will represent price increments. Let r(t) := cov(Z 0 , Z t ), t ∈ Z denote its covariance function. We assume that there exists T 0 > 0 and J 1 , is satisfied for some parameter H ∈ 0, 1 2 . Furthermore, t∈Z r(t) = 0.
Let us introduce the adapted price process defined by S 0 = 0 and S t = S t−1 + Z t , t ≥ 1. (5) and (6) express that Z is a process with negative memory, see Definition 1.1.1 on page 1 of [1]. When Z t , t ∈ Z are the increments of a FBM with Hurst parameter H < 1/2, then (5) is satisfied. This is the motivation for choosing H for parametrization (and not 2H − 2).

Remark 3.2. Properties
The next theorem is our main result: it provides the explicit form of an (asymptotically) optimal strategy and determines its expected asymptotic growth rate.
where T runs through multiples of 6 everywhere.

General bounds for variance and covariance
First we make some useful preliminary observations. Using stationarity of the increments of the process S, we have Furthermore, for s > t we similarly have Observe also that we can write Turning to the variances, we first obtain a convenient expression for them. Using (10) and (12), we have and algebraic manipulation of the summation operation −2t t−1 where the last line is only a reordering of terms. Setting C 1 = Now we are ready to present three lemmas, providing a lower and an upper bound for the variance and an upper bound for the covariance. Proof. Using properties induced by the choice of T 0 in Assumption 3.1 first note that Using these and (13) var The threshold T 1 and the constant B 1 can be explicitly calculated in terms of the constants present in the above expression. This completes the proof.

Lemma 4.2.
There exist T 2 ∈ N and B 2 > 0 such that for all t ≥ T 2 we have Proof. First note that algebraic manipulation of the operation −2t t−1 To proceed observe that, using the asymptotics in Assumption 3.1, for t > 2 we have These results yield for t > max(2, T 0 ), using again (13), that The threshold T 2 and the constant B 2 could again be explicitly given. The proof is complete.
We proceed with the lemma controlling the covariance cov(S s − S t , S t ). For a fixed v > 1, define There exists K > 1 and T 4 ∈ N such that Proof. Let us set and define C 6 = C 4 − C 5 . Note that, for each t ∈ N, Since proving the first statement of the lemma. Now, for all v > 1 the property s t > vtogether with the previous constraint of t > T 0 -further implies Obviously, for large enough t the bound becomes strictly negative, proving the second statement. Now, assuming This shows that K can be chosen so large that C 6 − C 7 K 2H − 1 < 0 and then, since 2H − 1 < 0, a threshold T 4 -depending on K -for the variable t can be specified so that whenever t exceeds the threshold, proving the third statement, completing the proof of the lemma.
Proof of Theorem 3.3. First we determine the maximal expected growth rate of portfolios. Let us define Let G(x) := λ|x| α , x ∈ R and denote its Fenchel-Legendre conjugate By definition of G * , for all φ ∈ G (T), for some C > 0 and hence Note that this bound is independent of φ. Using Lemma 4.2 it holds that Thus the maximal expected profit grows as T H 1+ 1 α−1 +1 with the power of the horizon, this proves (7). With the strategy defined in (8), the dynamics takes the form In the above expression let us denote the four terms by I 1 (T), I 2 (T), I 3 (T), I 4 (T), respectively, so that The upper bound constructed in (21) for Q(T) right away gives us an upper estimate for EI 1 (T) as EI 1 (T) = Q(T/2). Using Lemma 4.1, we likewise present a lower estimate as To treat the terms I 2 (T) and I 4 (T), note that with α > 1 the function x → |x| α is convex, thus applying Jensen's inequality Controlling term I 3 (T) is done via exploiting a specific property of Gaussian processes, namely that S s for s > t can be decomposed as S s = ρ(s, t)S t + W s,t , where W s,t is independent of S t and zero mean. With this, observe that The above, using (23), boils down to Using (21) and (22), with λ < ε/3, dividing through with T H(1+ 1 α−1 )+1 proves the statement in (8), and the proof of Theorem 3.3 is complete.