42 Order flow in limit order books
Y2 When the spread is small, the price advantage of a limit order compared to a market order is small. However, market orders are executed immediately, without risk of non-execution.
Hence traders cancel limit orders stored on the order book (and convert them to market orders).
Y3 When there is more volume on the best bid than on the best ask, there is an excess of supply of buy limit orders. Patient traders place more limit orders on the ask side of the order book.
Y4 When there is more volume on the best ask than on the best bid, there is an excess of supply of ask limit orders. Impatient traders cancel limit orders on the ask side of the order book.
Note that effects induced by the spread are the same for the bid side XB, YB, whereas the effects induced by volume-imbalance are exactly opposite.
3.2 Choice of order flow processes 43 X4: Similarly, when the spread is small (compared to the reference valueµ), the term−κ(µ−
s(t))dt adds a negative drift.
X5: Suppose we have a zero spread (s = 0) and the external drift (d1 +d2z) = 0, then dXA(t) =−VA(t)κµ2 dt <0, and no limit orders are placed inside the spread.
X6: When the volume-imbalancez is positive,−d2z(t)dtinduces a negative drift term.
X7: When the volume-imbalancez is negative,−d2z(t)dtinduces a positive drift term.
In addition, there might be an extra drift induced by external market events and which does not depend on order book parameters. This external drift is represented by d1.
Next, we check that dYA as defined in equation (3.3) satisfies (Y1)-(Y4):
Y1: When the spread is large (i.e. aboveµ),−δ(µ−s(t))dtadds a positive drift-term.
Y2: When the spread is small (i.e. belowµ),−δ(µ−s(t))dt adds a negative drift-term.
Y3: When the volume imbalance is high (compared to the reference valueν), there is an excess of buy limit orders and the positive drift term−d3(ν−z(t))dtincreases the volume of ask limit orders. As we will see, ν can be interpreted as the long-time mean of the volume imbalance.
Y4: When the volume imbalance is small (compared to the reference valueν), there is an excess of ask limit orders (compared to average) and the negative drift term −d3(ν −z(t))dt decreases the volume of ask limit orders.
Thus the state-dependent behaviour of the order flow which can expressed in terms of the model key parameters that was noticed in theoretical models as well as empirical studies, and which we synthesized in (X1)-(X7) and (Y1)-(Y4) is reflected in our choice of order flow processes defined above.
Figure 3.1 shows a sample paths of (α, β) and (VA, VB) for t ∈ [0,10] with parameters d1 = 0.01, d2 = 0.2, d3 =d4 = 0.1, κ= 0.1, δ= 0.1, µ= 0.03, σ0 = 0.1, σ1 = 0.01, σ2 = 0.1 and initial values (α(0), β(0), VA(0), VB(0)) = (27.11,27.09,500,400).
44 Order flow in limit order books
(a) Sample path of best askα(red) and best bidβ(blue)
(b) Sample path of ask volumeVA (red) and bid volumeVB (blue)
Figure 3.1: Sample path of order book with dynamics given as in (3.1)- (3.4). Note that (i) the best ask is always above the best bid, so the spread is always positive, (ii) one clearly notes the drift induced by volume-imbalance; when there is more volume on the best bid (and thus more demand for the stock), the price is driven up.
Chapter 4
Analysis of order book
We will now analyse the order book constructed in chapter 3. We start with the basic properties, such as existence and consistency of the order book and moments of best ask/bid and volume imbalance. We will also give economic interpretations of each term appearing in the model equations.
Then we will move on to the analysis of the time-to-fill, i.e. the time it takes for a limit order to be executed. It will be analysed using a Dirichlet problem formulation, asymptotic analysis and via Monte Carlo methods.
4.1 Basic order book properties
Let us start by translating the order flow processes (3.1) - (3.4) in terms of dynamics for best ask/bid and volume. As we are working in the block-shape orderbook model, we can plug (3.1) - (3.4) into (2.25) and (2.26), and obtain the following SDEs for the key parameters (α, β, VA, VB):
dα(t) = κ
2(µ−s(t))dt+1
2(d1+d2z(t))dt+σ1
2
ps(t)dW1(t) +σ0
2 dW0(t), (4.1) dβ(t) =−κ
2(µ−s(t))dt+1
2(d1+d2z(t))dt−σ1 2
ps(t)dW2(t) +σ0
2 dW0(t), (4.2) dVA(t) =VA(t) (d3z(t)dt−δ(µ−s(t))dt+σ2dW3(t)), (4.3) dVB(t) =VB(t) (−d4z(t)dt−δ(µ−s(t))dt+σ2dW4(t)). (4.4) By theorem 2.1, we know that the orderbook exists. The block-shape order book model is then given by
A(t, p) =VA(t)(p−α(t)), B(t, p) =VB(t)(β(t)−p).
Moreover, the proposition below summarizes the basic properties of the orderbook model.
Proposition 4.1 (Properties of order book). Let α, β, VA, VB be given by (4.1) - (4.4) and initial values β(0)≤α(0), VA(0)>0, VB(0)>0. Then there exist four independent standard Brownian motions B0, B1, B2, B3 such that
45
46 Analysis of order book
(i) The midquote price satisfies the SDE
dm(t) = 1
2(d1+d2z(t))dt+ σ1
2√ 2
ps(t)dB2(t) +σ0
2 dB0(t). (4.5)
(ii) The spread is a mean-reverting Cox-Ingersoll-Ross (CIR) process satisfying the SDE
ds(t) =κ(µ−s(t))dt+ σ1
√2
ps(t)dB1(t). (4.6)
(iii) The volume-imbalance is a mean-reverting Ornstein-Uhlenbeck (OU) process satisfying the SDE
dz(t) =ρ(ν−z(t))dt+ σ2
√
2dB3(t), (4.7)
where ρ=d3+d4≥0.
(iv) There exist unique strong solutions forα, β, VA, VB.
(v) The model is consistent, that is a.s. for all t≥0,α(t)≥β(t). Moreover, if 4κµ≥σ12, then a.s. for all t≥0, α(t)> β(t).
(vi) We have closed-form expressions for mean and variance of best ask/bid, spread, midquote-price and volume imbalance:
E[α(t)] = α(0) +β(0)
2 +1
2d1t+ d2
2ρ (z(0)−ν) 1−e−ρt
+νρt +e−κtα(0)−β(0)
2 +µ
2 1−e−κt E[β(t)] = α(0) +β(0)
2 +1
2d1t+ d2
2ρ (z(0)−ν) 1−e−ρt
+νρt
−e−κtα(0)−β(0)
2 −µ
2 1−e−κt Var(α(t)) = Var(β(t))
= d22σ22
16ρ2 2ρt−3 + 4e−ρt−e−2ρt + σ12
8κ tκµ+ (s(0)−µ)(1−e−tκ) + σ12
8κs(0) e−κt−e−2κt +µσ21
16κ 1−2e−κt+e−2κt + σ20
4 t
4.1 Basic order book properties 47 E[s(t)] =e−κts(0) +µ 1−e−κt
Var(s(t)) = σ12
2κs(0) e−κt−e−2κt +µσ12
4κ 1−2e−κt+e−2κt E[m(t)] =m(0) + 1
2d1t+ d2
2ρ (z(0)−ν) 1−e−ρt
+νρt Var(m(t)) = d22σ22
16ρ2 2ρt−3 + 4e−ρt−e−2ρt +σ20
4 t + σ12
8κ tκµ+ (s(0)−µ)(1−e−tκ) E[z(t)] =e−ρtz(0) +ν(1−e−ρt)
Var(z(t)) = σ22
4ρ 1−e−2ρt
Let us consider the different components that drive the dynamics of the best ask:
dα(t) = κ
2(µ−s(t))dt
| {z }
(D1)
+1
2d2z(t)dt
| {z }
(D2)
+1 2d1dt
| {z }
(D3)
+σ1
2
ps(t)dW1(t)
| {z }
(V1)
+σ0
2 dW0(t)
| {z }
(V2)
The five components can be interpreted in the following way
(D1) is a microstructure drift term, that is endogenously given by the spread of the stock. A large spread pulls the best ask down, whereas a small spread pushes it up. The mean-reversion property of the spread ensures that the sign of this drift term alternates con-stantly. It depends on the current state of the order book, but not on external events.
(D2) is another microstructure drift term that results from volume-imbalance. A positive vol-ume imbalance pushes the best ask up, since there is an excess of supply of buy offers.
The mean-reversion property of the volume imbalance ensures that the sign of this drift term alternates constantly. As (D1), it depends on the current state of the order book, but not on external events.
(D3) is an exogenously given drift term which describes the general market trend. It is influ-enced by external events and news, and does not depend on the current microstructure of the order book.
(V1) is a microstructure volatility term, that is endogenously given by the spread of the stock.
The larger the spread, the larger the volatility. A zero spread corresponds to no volatility contribution from the microstructure of the order book.
(V2) is an exogenously given volatility term. It captures the volatility which does not depend on the current state of the order book, but comes from external risk sources.
The effect of (D1) and (D2) were already noted in the discussion at the beginning of section 3.1.
It is clear that the price drift cannot depend solely on endogenous terms (spread, volume imbalance) of the order book microstructure, but also depends on external events and market trends. This motives the existence of (D3).
It has long been noticed that a high (total) volatility of an asset results in higher spreads: Harris (2003) notes that
48 Analysis of order book
since volatility increases limit order option values, traders widen their spreads, when trading volatile instruments to minimize the value of the timing option.
Bollerslev and Melvin (1994) examine the relationship between bid-ask spreads for exchange-rate quotes and the volatility of the underlying exchange-exchange-rate process. They find a positive relationship between latent volatility and observed spreads in the Deutschemark/dollar foreign exchange market. This is captured by the microstructure volatility term (V1): when a higher (total) volatility is observed, this effect must stem from a higher spread, as the exogenous volatility (V2) is assumed to be constant.
As in the drift case, the exogenous volatility (V2) is explained by external risks that are inde-pendent of the risk associated to the order book microstructure (e.g. the risk of crossing a large spread, instead of waiting for the limit order).
Note that the midquote price dm(t) = 1
2d2z(t)dt
| {z }
(D2)
+1 2d1dt
| {z }
(D3)
+ σ1 2√ 2
ps(t)dB2(t)
| {z }
(V1)
+σ0 2 dB0(t)
| {z }
(V2)
has the same components as the best ask, except for the microstructure drift term stemming from the spread. This is due to the assumption that best bid and ask have exactly opposite spread behaviors which neutralize each other and thus vanish in the midquote price. Moreover the volatility components (V1) and (V2) are now driven by other Browian motions, which are correlated with the Brownian motions driving the best ask.