Limit order trading with a mean reverting reference price

2.1. Settings

We assume that the reference price S t of the risky asset follows an Ornstein–Uhlenbeck (OU) process (1) d S t = α ( μ − S t ) d t + σ d B t , where α is the mean-reverting rate, μ is the long-term mean, and σ is the volatility. Note that we are not considering any feedback effect of traders’ behavior on the reference price here.

The portfolio of the limit trader consists of two parts: cash and the risky asset. We denote the cash process by X t and the inventory process of the risky asset by Q t . The process Q t can be expressed as the difference of ask and bid limit orders fulfilled up to time t, denoted by Q t a and Q t b : (2) Q t = Q t b − Q t a + q 0 , assuming that the trader only post limit orders and q 0 is the initial inventory. The portfolio is self-financing, so (3) d X t = p t a d Q t a − p t b d Q t b , where p t a and p t b are the ask and bid limit prices respectively. Gathering (1), (2) and (3), the dynamics of variables in our model are d Q t = d Q t b − d Q t a , (4) d X t = p t a d Q t a − p t b d Q t b , d S t = α ( μ − S t ) d t + σ d B t . Note that p t a and p t b are the controls of the limit order trader, while the processes of the fulfilled limit orders Q t a and Q t b may be affected by those limit order prices as well as the reference price S t .

Combining empirical results from econophysics in [7,8,13,16,19], Avellaneda and Stoikov proposed that the process of the fulfilled limit orders follows a doubly stochastic Poisson process with intensity A e − κ δ t , where δ t is the spread of the limit order at time t, and A and κ are positive constants characterizing statistically the liquidity of the asset. Namely (5) Q t a ∼ Poi ( A e − κ δ t a ) , Q t b ∼ Poi ( A e − κ δ t b ) , where δ t a = p t a − S t and δ t b = S t − p t b are the spread of ask and bid limit orders posted at time t.

The trader aims to solve the optimal control problem (6) sup δ a , δ b E [ − e − γ W T ] , where W t = X t + Q t S t is the process of total wealth.

The parameters in our model are (7) A : the magnitude of market order flow ; κ : dictating the shape of order book ; γ : risk-aversion factor ; α : the mean reverting rate of the α : reference price ; σ : the volatility of the reference price ; T : the length of the trading period .

2.2. Dynamic programming

Consider the value function u ( t , q , x , s ) = sup δ a , δ b E [ − e − γ W T | Q t = q , (8) X t = x , S t = s ] . The HJB equation for the optimal control problem specified in (4) (5) and (6) is u t + σ 2 2 u s s + α ( μ − s ) u s + sup δ a { [ u ( t , q − 1 , x + s + δ a , s ) − u ( t , q , x , s ) ] A e − κ δ a } + sup δ b { [ u ( t , q + 1 , x − s + δ b , s ) (9) − u ( t , q , x , s ) ] A e − κ δ b } = 0 with the terminal condition u ( T , q , x , s ) = − e − γ ( x + q s ) .

Because of the special form of the terminal utility, namely the CARA1 utility, it is known from the studies in Zhang [20] and Gueant, Lehalle, and Fernandez-Tapia [10] that the ansatz u ( t , q , x , s ) = − e − γ ( x + v ( t , q , s ) ) can reduce (9) to v t − σ 2 2 ( γ v s 2 − v s s ) + α ( μ − s ) v s + 1 γ sup δ a { [ 1 − e − γ ( s + δ a + v ( t , q − 1 , s ) − v ( t , q , s ) ) ] × A e − κ δ a } + 1 γ sup δ b { [ 1 − e − γ ( − s + δ b + v ( t , q + 1 , s ) − v ( t , q , s ) ) ] (10) × A e − κ δ b } = 0 with terminal condition (11) v ( T , q , s ) = q s . To find the optimal feedback control, we only need to maximize (12) F a ( δ a ) = [ 1 − e − γ ( s + δ a + v ( t , q − 1 , s ) − v ( t , q , s ) ) ] × A e − κ δ a , F b ( δ b ) = [ 1 − e − γ ( − s + δ b + v ( t , q + 1 , s ) − v ( t , q , s ) ) ] × A e − κ δ b separately. Both F a and F b have a unique global maximum which yields the optimal feedback spreads (13) δ a ∗ ( t , q , s ) = 1 γ log ( 1 + γ κ ) − s δ a ∗ ( t , q , s ) = − v ( t , q − 1 , s ) δ a ∗ ( t , q , s ) = + v ( t , q , s ) , δ b ∗ ( t , q , s ) = 1 γ log ( 1 + γ κ ) + s δ b ∗ ( t , q , s ) = − v ( t , q + 1 , s ) δ b ∗ ( t , q , s ) = + v ( t , q , s ) . Therefore the problem is reduced to solving the HJB equation v t − σ 2 2 ( γ v s 2 − v s s ) + α ( μ − s ) v s + 1 γ [ 1 − e − γ ( s + δ a ∗ + v ( t , q − 1 , s ) − v ( t , q , s ) ) ] × A e − κ δ a ∗ + 1 γ [ 1 − e − γ ( − s + δ b ∗ + v ( t , q + 1 , s ) − v ( t , q , s ) ) ] (14) × A e − κ δ b ∗ = 0 with the terminal condition in (11) and the optimal controls in (13).

We make a change of time τ : = T − t in (14), and define v ˜ ( τ , q , s ) = v ( T − t , q , s ) . We abuse the notation by still using v instead of v ˜ . Plugging the optimal controls to (13), we have v τ = σ 2 2 ( v s s − γ v s 2 ) + α ( μ − s ) v s + A κ + γ ( 1 + γ κ ) − κ γ × e − κ ( − s − v ( τ , q − 1 , s ) + v ( τ , q , s ) ) + A κ + γ ( 1 + γ κ ) − κ γ (15) × e − κ ( s − v ( τ , q + 1 , s ) + v ( τ , q , s ) ) with the initial condition v ( 0 , q , s ) = q s .

Note that (15) is highly nonlinear because of the appearance of value function v in the exponent. Moreover, this equation involves both continuous variables, t and s, and a discrete variable q. There is no available theory on its well-posedness. On the other hand, for the case α = 0 , this equation can be transformed to an ODE system, which, under the assumption of finite inventory limits, is finite-dimensional and can be solved explicitly. See Zhang [20] or Gueant, Lehalle and Fernandez-Tapia [10] for detail.

2.3. Scaling

We use two scalings for our model, one on time and another one on price: t ˜ = t / α and p ˜ = γ p . We define new variables in our model Q ˜ t = Q t ˜ / α , X ˜ t = γ X t ˜ / α , S ˜ t = γ S t ˜ / α , δ ˜ t a = γ δ t ˜ / α a , δ ˜ t b = γ δ t ˜ / α b and new parameters accordingly (16) A ˜ = A α , σ ˜ = γ σ α , μ ˜ = γ μ , κ ˜ = κ γ , T ˜ = α T . We will use those variables and parameters from now and abuse the notations by dropping all the tildes. With those new variables and parameters, the model is equivalent to that described in Section 2 with α = γ = 1 , and the HJB equation (15) becomes v τ = σ 2 2 ( v s s − v s 2 ) + ( μ − s ) v s + A κ + 1 ( 1 + 1 κ ) − κ × e − κ ( − s − v ( τ , q − 1 , s ) + v ( τ , q , s ) ) + A κ + 1 ( 1 + 1 κ ) − κ (17) × e − κ ( s − v ( τ , q + 1 , s ) + v ( τ , q , s ) ) with the initial condition v ( 0 , q , s ) = q s .

The optimal feedback controls are given by (18) δ a ∗ ( t , q , s ) = log ( 1 + 1 κ ) − s − v ( t , q − 1 , s ) + v ( t , q , s ) , δ b ∗ ( t , q , s ) = log ( 1 + 1 κ ) + s − v ( t , q + 1 , s ) + v ( t , q , s ) . From now on, we will only consider the scaled model, the HJB equation in (17), and optimal feedback controls in (18). However, when showing our numerical simulation results, we would use the prices before the price-scaling, which are directly observable from the market, instead of the dimensionless ones after the scaling. For the parameters used in the numerical simulations, we may also choose the ones before the price-scaling, since they are easier to reason and are practically easier to calibrate to market data.

We point out that:

4. Long time behavior

Studying standard Avellaneda-Stoikov model, Gueant, Lehalle, and Fernandez-Tapia [10] observed a long-term stationary behavior of the optimal spreads δ t a ∗ and δ t b ∗ : (25) lim T − t → ∞ δ a ∗ ( t , q ) = δ ∞ a ∗ ( q ) , lim T − t → ∞ δ b ∗ ( t , q ) = δ ∞ b ∗ ( q ) .

In our model, we observe a long-time behavior of the optimal limit order prices p t a ∗ = S t + δ t a ∗ and p t b ∗ = S t − δ t b ∗ instead of that of the optimal spreads δ t a ∗ and δ t b ∗ .

Our numerical simulations presented in Section 5 indicate that the optimal feedback limit order prices given by (26) p a ∗ ( τ , q , s ) = log ( 1 + 1 κ ) − v ( τ , q − 1 , s ) + v ( τ , q , s ) , p b ∗ ( τ , q , s ) = − log ( 1 + 1 κ ) + v ( τ , q + 1 , s ) − v ( τ , q , s ) converge to constants (27) p ∞ a ∗ = μ + log ( 1 + 1 κ ) , p ∞ b ∗ = μ − log ( 1 + 1 κ ) when τ → ∞ . Or equivalently, (28) lim τ → ∞ v ( τ , q , s ) − v ( τ , q − 1 , s ) = μ .

We have not yet developed an analytical proof of the convergence in (28) as this is work in progress. Note that v ( 0 , q , s ) = q s , so v can be intuitively viewed as the value of the asset held by the trader at time T − τ . The limiting property in (28) suggests that, in the long run, the value of each share of asset is just μ, the long term mean of the reference price. Moreover, the convergence suggests that when we are sufficiently far away from the terminal time, it is better to post constant limit prices than to track the reference price closely. One heuristic explanation is that the trading period is so long compared to the mean reversion time that plenty of rebalancing is guaranteed. Therefore, as long as the trader can gain the premium from rebalancing by using the constant limit ask and bid prices, he/she does not need to track the reference price.

In the rest of this section, we discuss three closely related models that can be solved analytically and compare the limit of the optimal limit prices in those models with the ones in (27).

4.2. Analysis of small κ

Both Fodra and Labadie [4] and Zhang [20] considered approximations of (15) with linearization. We briefly state Zhang’s results here.

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Fig. 1.

We plot optimal ask limit order prices at different times from the model with small κ ( κ = 1 ) and the model with medium κ ( κ = 6 ). The black line is the ask price in Zhang’s small κ analysis for comparison. To make the prices from different models comparable, we subtract the optimal feedback ask price at q = 0 from each feedback ask price function. Note that after this normalization, the limit of Zhang’s small κ approximation from two models coincides with each other. When time is sufficiently far away from the terminal time, the optimal ask limit order prices from both models become significantly different from the limit in Zhang’s small κ analysis and converge a constant. Moreover, the limit order price in the model with small κ tends to a constant more slowly than the one in the model with medium κ.

After a linearization of the exponential terms, Zhang shows analytically that the value function v becomes independent from s exponentially fast v ( τ , s , q ) − C 0 τ ⟶ θ 1 + μ · q − σ 2 4 q 2 , (30) τ ⟶ ∞ , where C 0 and θ 1 are two constants. Accordingly, the optimal prices converge to the following limits: (31) p ∞ a ∗ ( q ) = log ( 1 + 1 κ ) + μ p ∞ a ∗ ( q ) = − σ 2 4 ( 2 q − 1 ) , p ∞ b ∗ ( q ) = − log ( 1 + 1 κ ) + μ p ∞ b ∗ ( q ) = − σ 2 4 ( 2 q + 1 ) , where the slope with respect to q relies only on the scaled σ. In order for the linearization to work well, the terms κ δ a ∗ and κ δ b ∗ are required to be small. According to (31), the terms δ a ∗ and δ b ∗ will be linear with respect to q when time is far away from terminal, so at least in this time regime, to keep κ δ a ∗ and κ δ b ∗ small, this result is only valid for sufficiently small q.

We compare the optimal feedback ask limit prices computed by our numerical methods to those in the limit of Zhang’s approximation2 in Figure 1. We plot the feedback ask limit prices as functions of inventory q as they have already become insensitive to the reference price s. Translation is applied on those feedback optimal prices to make them comparable. We refer to Figure 1 for more detail.

5. Numerical results

We apply the numerical methods described in Section 3 to solve the HJB equation (17) for the value function and optimal controls in our model.

5.1. Evolution of optimal feedback limit order prices

We are interested in how the optimal feedback controls in our model, namely the optimal limit order prices, evolve as a function of the inventory and reference price. As stated in Section 4, we observe that the optimal prices converge to constants in (27) when time is away from terminal. In addition, as shown in Figure 2, we observe that the optimal limit order prices become insensitive to the reference price much faster than to the inventory, which leads to an “intermediate” regime where the optimal limit order prices only respond to the change of inventory.

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Fig. 2.

Feedback optimal ask limit order prices, from top to bottom, corresponding to 0, 1, 4 and 800 mean reversion cycles from the terminal time. The prices are the ones before the price-scaling described in Section 2.3 instead of the dimensionless ones after the scaling. Each line, as a function of reference price, corresponds to a value of inventory. The optimal ask prices have already become independent from the reference price at 4 mean reversion cycles from the terminal time (the 3rd plot from top), while it took 800 mean reversion cycles (backwards in time) to become independent from the inventory as well (the bottom plot). Here the 3rd plot from top corresponds to the intermediate regime and the bottom plot corresponds to the far-away-from-terminal regime.

For a model with unscaled parameters A = 10 , σ = 0.05 , γ = 0.005 , κ = 5.0 , μ = 1 and α = 1 , Figure 2 shows the optimal feedback ask prices at following time:

(1)

the terminal time;

(2)

near-terminal regime: 1 mean reversion cycle of the reference price from the terminal time;

(3)

intermediate regime: 4 mean reversion cycles from the terminal time;

(4)

limit regime: 800 mean reversion cycles from the terminal time.

We can see that the near-terminal regime is short and very quickly, backwards in time, trading gets into the intermediate regime where the optimal prices become insensitive to the reference price. In the intermediate regime, the trader updates his limit order only according to the inventory. Doing so, he could keep the variance of inventory low which reflects his risk aversion rooted in the terminal utility.

In Figure 2, it takes 800 mean reversion cycles to observe the insensitivity of the optimal prices to the inventory, as shown in the bottom plot. Namely for a large portion of a trading period, the trader would post limit orders with prices only affected by the change of his own inventory ignoring the fluctuation of the reference.

In some parameter ranges, the intermediate regime can be very long and so we observe insensitivity of optimal limit prices to the reference price but do observe dependence on inventory. That is, in the very beginning of the trading period the intermediate regime is already valid, in which case we would not observe the limit regime at all. To illustrate this, we choose two sets of parameters and show the corresponding simulation results in the next section where within the trading period we can only observe the intermediate and near-terminal regimes but not the limit regime.

Note that even though in this paper we do not calibrate our parameters to real data, there is literature on how to do this. For the liquidity parameters A and κ, their calibration is studied in Chapter 4 of [3]. A calibration framework is presented, which can be extended to the model with a mean reverting reference price. For the parameters α and σ characterizing the mean reverting reference price, which we assume is observed, we can calibrate them by a maximal likelihood estimation (MLE) studied in [14,18].

5.2. Simulation results

We show some simulation results of our trading models in Figure 3 and Figure 4. The unscaled parameters used in Figure 3 are A = 2 , σ = 0.4 , γ = 2 , κ = 1.5 , μ = 1 and α = 1 ; the ones used in Figure 4 are the same except that A = 6 . We choose those parameters so that the intermediate regime can be observed clearly. For instance, we choose a large γ so that a jump of the optimal prices due to a change in inventory is evident. In those two figures, the trading period is not long enough to observe the limit regime.

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Fig. 3.

Simulation results for limit order prices, inventory, and spreads for 10 mean reversion cycles of the underlying reference price. The pattern clearly shows that near the terminal time, the trader tracks the reference price closely whereas in the intermediate regime, the optimal limit prices effectively only respond to the change of inventory.

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Fig. 4.

Similar plots to Figure 3, but with larger parameter A representing greater volume of incoming market orders. When there is a trend in the reference price, for instance, between time 2 and 4, there will also be a trend in optimal prices in the same direction but with a lag. The trend in optimal prices is a result of the trend in the inventory formed during a trend of the reference price when the volume of incoming market orders is large.

Figure 3 shows a simulation result for 10 mean reversion cycles of reference price. Between time 0 and 8, the trading is in the intermediate regime in the sense that the optimal limit order prices will remain almost constant when no limit order is taken and will jump when the inventory changes.

In Figure 4, the model has the same parameters except that the market-order-volume parameter A is greater. In the top plot of Figure 4, while it seems that the optimal prices are tracking the reference price, however, a closer look shows that the pattern in this plot is essentially the same as the pattern in the top plot of Figure 3. That is, the limit order prices effectively respond only to the change of inventory and ignore the fluctuation of the reference price, which suggests that we are in the “intermediate regime.” For instance, between time 2 and 3 in the top plot of Figure 4, there is a significant drop of the reference price, but the limit prices does not drop accordingly. They begin to decrease only after the inventory increases. In this case, parameter A is sufficiently large that enough limit orders will be taken in one trend of price, which builds up a trend in the inventory and in turn creates a trend in the optimal limit order prices. This explains why on first sight, the limit order prices follow the same trend as the reference price, and why there is a lag between the trend of the reference price and that of the limit order prices.

When the model moves from the near-terminal regime to the intermediate regime, the sensitivity of the optimal prices to the inventory is mainly affected by the scaled σ. We observed that the greater the scaled σ is, the greater the jump size of the optimal prices is when the inventory changes by one unit. The magnitude of a jump decays to 0 as time goes backwards, with the decay rate affected by A and κ; for greater values of A and κ, the jump size decays faster. Note that, larger values of A and κ means a larger market order flow and a shallower order book respectively. These properties signify higher liquidity in the market. So one insight we can gain from this model is that, for a limit order trader trading a liquid asset with mean-reverting price, his optimal limit prices converge faster backwards in time than they do in the case where he trades a less liquid asset, and therefore his optimal limit prices are less sensitive to the change of inventory.

Recall that here the parameters are the ones after scalings described in Section 2.3, so A, κ, and σ 2 are in fact A α , κ γ , and γ 2 σ 2 α in terms of the parameters before scalings. In contrast, the prices in the figures shown in this section are those before the price-scaling described in Section 2.3. That is, they are the observable prices instead of the dimensionless ones.

To check whether our numerical solution of the system in (17) is still valid even when time is far away from terminal, we analytically consider the equilibrium of that system and compare the result with our numerical solution of the time-dependent system.

As described in Section 4, our conjecture is that, for a solution v of the PDE (17) and any q and s, v ( τ , q , s ) − v ( τ , q − 1 , s ) → μ , (36) as τ → ∞ . Thus, for the equilibrium, we expect v ( τ , q , s ) − C τ → θ 0 + θ ( s ) + μ q , (37) as τ → ∞ , where θ 0 is a constant and θ ( s ) satisfies the equilibrium HJB equation (38) 0 = C + σ 2 2 ( θ s 2 − θ s s ) − ( μ − s ) θ s 0 = − M [ e − κ ( − s + μ ) + e − κ ( s − μ ) ] , θ ( μ ) = 0 with M = A κ + 1 ( 1 + 1 κ ) − κ > 0 . Therefore, when τ is large, we expect the “s-dependent” part of v, defined as v ( τ , q , s ) − v ( τ , q , μ ) , to be close to θ ( s ) , the solution to (38). We will transform (38) to a Schrödinger eigenvalue problem, then solve it and compare the result to the numerical solution of v when time is away from terminal.

Moreover, recall that Zhang [20] has obtained a closed form solution of the linearized model with small κ. We can compare the limit of a value function in our model in (37) and the one in Zhang’s small κ analysis in (30). In both equations, when τ = T − t is large, the derivative of the value function with respect to τ is a constant: C in (37) and C 0 in (30). It is shown in [17] that lim κ → 0 C = lim κ → 0 C 0 . That is, the constants in the two convergence results are consistent when κ is small. We refer to [17] for more detail.

Note that the limit in (37) is indeed a solution of the HJB equation in (17), but it does not satisfy the initial condition. Now we analyze equation (38) to gain some insight into constant C and solution θ.

6.2. Numerical results on the equilibrium equation

We would like to solve the equation (40) to find the constant C ˆ that yields a solution which vanishes at infinity and does not change its sign. Recall that C ˆ = C 2 σ 2 + 1 σ 2 , where the constant C can be approximated by v τ with large τ as shown in equation (37). We search in the neighborhood of 2 σ 2 v τ + 1 σ 2 to find the desired constant C ˆ . After we have the constant C ˆ , we compute the solution of the following system, which is derived from (40), (42) m ′ = n , n ′ = [ s 2 σ 4 + 2 M σ 2 ( e κ s + e − κ s ) − C ˆ ] · m .

We need to specify the initial conditions m ( 0 ) and n ( 0 ) . Since equation (40) is homogeneous, m ( 0 ) could be arbitrary, so we set m ( 0 ) = 1 for simplicity. Additionally, we set n ( 0 ) = 0 due to the symmetry of the equation.

Consider two models with shared parameters (43) A = 0.9 , σ = 0.3 , γ = 0.01 , μ = 1.0 and different values of κ: (44) κ = 0.3 in the first model , 0.01 in the second model , where the parameters are the ones before the price-scaling.

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Fig. 5.

We compare θ ( s ) in the Schrödinger equation (38) and v Limit ( s ) − v Limit ( μ ) , namely the “s-dependent” part of the function v from the numerical result of the time dependent system in equation (17) when time is far away from terminal. Here θ ( s ) and v Limit ( s ) − v Limit ( μ ) match very well for both large κ ( κ = 0.3 ) and the small κ ( κ = 0.01 ). When κ is small, both θ ( s ) and v Limit ( s ) − v Limit ( μ ) are flat, which is consistent with the result in Zhang’s small κ analysis.

We numerically solve (42) for m ( s ) , then in turn compute θ ( s ) with θ ( μ ) = 0 in (38). We compare it in Figure 5 to the “s-dependent” part of v which is defined as (45) v Limit ( s ) − v Limit ( μ ) , where v Limit ( s ) ≜ v ( τ , q = 0 , s ) for large τ. We can see that for each model, θ ( s ) is very close to the “s-dependent” part of v, and that for the model with smaller κ, θ ( s ) is flatter. Note that the result in Section 4.2, which states that the limits of v calculated via a small κ expansion does not depend on s, can be viewed as the limit when κ goes to 0.

In this paper, we consider the limit order book model of Avellaneda and Stoikov [1] with a mean reverting underlying price. Our main result is that when time is far from terminal, it is optimal to post constant limit order prices instead of tracking the underlying price. We use two different numerical methods to solve the HJB equation, and both of them confirm the long-time behavior. This result implies that when the underlying price is mean reverting then, when time is far from terminal, it is optimal to focus on the mean price and ignore the fluctuations around it. This observation, admittedly from a stylized model, confirms what limit order traders might expect.

The numerical results also show that between the time regime where constant limit order prices are optimal and the one close to the terminal time, there is an intermediate time period where limit order prices are influenced by the inventory of outstanding orders. The duration of this intermediate period depends on the parameters A and κ that quantify the liquidity of the market.

We also study the equilibrium of the optimal control problem. The equilibrium of the HJB equation can be transformed to a Schrödinger equation, as an eigenvalue problem. The solution agrees with the long-time limit of our numerical result of the time-dependent model, which confirms the validity and accuracy of our numerical methods, even for long time. When the liquidity parameter κ is small, the numerical solutions also match the analysis in Zhang [20].

Even though the numerical calculations strongly suggest convergence of the optimal limit order prices, the proof remains open and needs further study.