Bid-ask spreads and quote depths link to price volatility from the liquidity per- spective. Classic information-based MMS theory (e.g. Copeland and Galai (1983), Easley and O’Hara (1992)) predicts a positive relationship between the bid-ask spread and return volatility, due to the arrival of private information, which increases the degree of information asymmetry in the market. Kyle (1985) and Rahman, Krish- namurti, and Lee (2005) assert that market depths are negatively affected by the information asymmetry, and for larger depths, it is less likely that orders can ‘walk the book’, and therefore are associated with lower return volatility. Parlour (1998) suggests that, when the bid (ask) side of the limit order book has excess liquidity over the ask (bid) side, traders will submit market buy (sell) orders instead of limit buy (sell) orders for prompt execution, which will subsequently move up (down) the price and cause a contemporaneous increase in return volatility. Thus, these theories suggest that bid-ask spread and the difference between bid depth and ask depth should move in the same direction as price volatility, while the quote depths per se will move in the opposite direction. These theoretical findings are supported by many empirical studies, for instance Bollerslev and Melvin (1994), Handa and Schwartz (1996), Foucault (1999), Ahn, Bae, and Chan (2001), Næs and Skjeltorp (2006), Nolte (2008) and Hussain (2011), among others.
Concluding from above, all MMS variables contain information about price volatility. We expect a positive relationship between volume, order imbalance, order flow, bid- ask spread and volatility, and a negative relationship between quote depth difference with volatility. The relationship between number of trades and volatility is mixed, but the results in Engle and Russell (1998) are very close to our approach, and suggest a positive relationship. In this chapter, we provide a clear picture of the relationship between these variables and local volatility in a high-frequency setting, and assess their relative informativeness in volatility estimation.
2.3
Price Duration and Volatility Estimation
We follow the general framework of point process-based volatility estimation in Engle and Russell (1998), Gerhard and Hautsch (2002), Tse and Yang (2012) and Nolte, Taylor, and Zhao (2018). Let P(t) denote an observed price process, and suppose a decision maker in need of a risk measure is concerned about the size of a significant price change,δ. Construct the absolute price change point process (Engle and Russell, 1998) as follows:
Definition 2.1 (The Absolute Price Change Point Process). Based on a realization of the observed price process in level P(t), we construct a point process by:
1. Set t0(δ)=0 and choose a threshold δ.
2. For i=1,2,· · ·, compute the first exit time, ti(δ), of P(ti(−δ1)) through the double barrier [P(ti(−δ1))−δ,P(ti(−δ1)) +δ] as:
ti(δ)= inf
t>t(i−δ)1
{|P(t)−P(ti(−δ)1)| ≥δ}.
Iterate until the sample is depleted.
The point process {ti(δ)} describes the arrival times of transactions that result in a price change of at least δ. We will refer to the arrival times {ti(δ)} as the δ-related price events, or price events for short. Intuitively, for a given price series, more frequent arrivals of price events in a given time frame can be translated into higher price volatility.
To construct a volatility measure from the absolute price change point process, we introduce three related concepts that are considered as equivalent representations of the point process.
Definition 2.2 (Counting Representation). For the absolute price change point process defined by the arrival times of price events {ti(δ)}, the counting process of this point process is defined by:
N(δ)(t)≡ ∞
∑
i=1
1l{t≥ti}. (2.1) Definition 2.3 (Price Duration Representation). The price duration process of{ti(δ)} is defined by:
x(iδ)≡ti(δ)−ti(−δ)1. (2.2) Definition 2.4 (Conditional Intensity Representation). Let Ft denote the natural
filtration of the point process {ti(δ)}. The Ft-conditional intensity process of {ti(δ)} is
defined by: λ(δ)(t|Ft)≡lim ∆↓0 1 ∆E[N (δ)(t+ ∆)−N(δ)(t)|Ft]. (2.3) Definitions 2.2 to 2.4 are three equivalent characterizations of the point process
{ti(δ)}. Specially, the Ft-conditional intensity process can be interpreted as the expected number of price events for the next instant, which has a close connection to an instantaneous volatility measure as given by the formula below (Hautsch, 2012):
σ(2δ)(t) =λ(δ)(t|Ft)
δ
P(t)
2
2.3 Price Duration and Volatility Estimation | 57
Intuitively, each price event is associated with δ P(t)
2
amount of price volatility. The above formula simply uses the expected number of price events multiplied by the price volatility contribution of each price event as an instantaneous volatility measure. To model the latent object λ(δ)(t|Ft) using the observable price durations x
(δ) i , we introduce an alternative definition of the conditional intensity (Daley and Vere- Jones, 2003): λ(δ)(t|Ft) = fx(t−ti(−δ)1|F ti(−δ1)) 1−Fx(t−ti(−δ1)|F ti(−δ)1) , t ∈(ti(−δ1),ti(δ)],i=1,2, . . . (2.5) in which fx(·|F
ti(−δ)1) and Fx(·|Fti(−δ)1) are the conditional densities of x
(δ)
i conditioning on the information set F
t(i−δ)1. Thus, by modelling the conditional cumulative density function (CDF) of x(iδ), we can make inference about instantaneous volatility within the spell of a price duration.2
Usually we are more interested in the integrated conditional variance over some period (0,T). It is then natural to integrate (2.5) to obtain an estimate of the integrated conditional variance (ICV). Suppose thattI(δ)≤T is the last arrival of price events in the dataset, then an estimate of the ICV can be constructed as:
ICV(0,T)≡ Z T 0 σ(2δ)(t)dt=− I
∑
i=1 ln1−Fx(xi(δ)|F ti(−δ1)) δ P(ti(−δ1)) !2 (2.6)In practice, we will replace the conditional CDFFx(·|Ft(δ)
i−1
) in the above estimator by an empirical estimate. The performance of the ICV estimator then depends crucially on the goodness-of-fit of the model for the conditional density ofx(iδ).
In the discussion above, when we consider the conditional distribution of x(iδ), we restrict ourselves to only condition on the natural filtration F
ti(−δ1), which contains all the internal history of the point process up to time ti(−δ)1. However, as discussed in the literature section, there are various MMS covariates that are considered to be related to price volatility. Therefore, it is expected that by conditioning on an extended information set, we can model the conditional distribution of x(iδ) with better accuracy, which in return improves the quality of volatility estimates. In our
2The information about the conditional intensity is updated upon arrival of every price events,
study, we model the following conditional density by extending the information set to:
Fx(x(iδ)|F
ti(−δ1)∪Gti(δ)), (2.7) in which G
ti(δ) is the information set of some MMS covariates up to timet
(δ)
i . We will use Ff
ti(δ) =Fti(−δ)1∪Gti(δ) to denote the extended information set. Note that we are essentially using contemporaneous information in other MMS covariates to fit the conditional density of x(iδ). As our main interest is to provide a precise ex post price duration-based volatility estimator, exploiting information in the contemporaneous covariates can to a large extent improve the goodness-of-fit of the duration density, which in turn yields a more precise volatility estimator. We would like to stress that although contemporaneous information is not permitted in a forecasting setting, here it does not does not contradict our method. In the same way as RV estimates, we focus on the construction of an input for a forecasting model rather than the specification of the forecasting model, and volatility estimates obtained from our model can always be used in volatility forecasting specifications such as the HAR model (Corsi, 2009).
The use of Ff
ti(δ) in volatility estimation is also motivated by the fact that we can analyse the interactions between contemporaneous MMS variables and price volatility. To see this, firstly note that under correct specification of Fx(x(iδ)|Ff
t(iδ)), the following holds:
−ln1−Fx(x(iδ)|Ff
ti(δ))
∼i.i.d.exp(1). (2.8)
The above relationship is known as the exponential probability integral transform. Let us define ICVi≡ICV(ti(−δ1),ti(δ)), then from (2.6) and (2.8) it is immediate that ICVi is a process of i.i.d. exponential random variables. If we define the average instantaneous volatility within the i-th price duration as:
σ2i =ICVi/x(iδ), (2.9) then by taking logarithm on both sides of (2.9), it follows that:
E[lnσ2i] +E[lnxi(δ)] =lnC+γ, (2.10) Cov(lnσ2i,lnx(iδ)) = π 2 12− V[lnσ2i] +V[lnx(iδ)] 2 , (2.11)
2.3 Price Duration and Volatility Estimation | 59
where C=δ2/P(ti(−δ1))2 and γ ≈0.5772 is the Euler-Mascheroni constant.3 Since empirically Cov(lnσ2i,lnx(iδ)) is almost always negative due to the large variations in the price durations (in our data, V[lnx(iδ)] is around 2 for all securities), we can interpret price durations as an inverse measure of the average instantaneous volatility within the duration. Intuitively, the longer the price duration, the longer it takes for the price to change by δ amount, and consequently the lower is the average instantaneous volatility therein.
By examining the correlation between price durations and other MMS covariates, we can therefore infer the correlation between other MMS covariates and the average instantaneous volatility. For example, if the trading volume per second in a price duration is negatively correlated with price duration, then we would expect it to be positively correlated with the average instantaneous volatility, and vice versa. However, there are some caveats in the interpretation of the correlation between covariates and price durations. Since a price duration is also a measure of time, if an included MMS covariate also scales up with time, then the correlation between that covariate and price duration will be strongly positive. These MMS covariates include trading volume, number of trades and total quote depths submitted within a price duration. To ensure that we can translate the relationship between covariates and price durations into the relationship between covariates and average instantaneous volatility, we use a per second measure of these variables by dividing them by the length of the corresponding price duration.
The (price) duration-based volatility estimator has an intrinsic link to the pop- ular RV approach. The RV approach typically relies on the assumption that the log-price process follows a jump-diffusion process with observation error (discretiza- tion, MMS noise4, etc.), and estimates the integrated variance of the log-price process non-parametrically. Thus, based on a specific log-price model, asymptotic properties of the RV-type estimators can be derived, and biases introduced by observation error and jumps can be corrected. On the contrary, the price-duration based volatility estimator does not impose any assumptions on the log-price process, and simply treats the price events as a proxy of volatility. This approach is apparently less vulnerable to specification error of the log-price model, but no theoretical results can be derived without any assumptions about the underlying price process. Li,
3Note that to derive the above relationships, we used the fact that if
εi is i.i.d. unit exponential,
thenE[lnεi] =γ andV[lnεi] =π
2
12.
4In the RV literature, MMS noise or market microstructure noise refers to the deviation of
observed price from the latent efficient price process caused by microstructure effects such as the bid-ask bounce effect, strategic trading and imperfections in the trading mechanisms, etc.
Nolte, and Nolte (2018a) prove that if we assume that the log-price process follows a jump-diffusion process, then the ICV estimator consistently estimates integrated variance of the jump-diffusion process with a much higher efficiency compared to the RV-type estimators, if the conditional intensity process is known.5 This re- sult establishes the theoretical foundation of the ICV estimator, and suggests that the quality of the volatility estimates of the ICV estimator depends crucially on the goodness-of-fit of the parametric model of the absolute price change point process. We would like to conclude our discussion of the duration-based volatility estimator by summarizing the advantages of the ICV estimator over the RV estimator. Firstly, a parametric structure can be specified for the price durations or the conditional intensity process. This allows us to include more data outside the window for which volatility needs to be estimated. For example, we could use monthly data to construct daily volatility estimators. This can lead to efficiency gains as the RV-type approach is confined to the data within the estimation window of volatility. Moreover, this parametric structure allows the inclusion of other MMS covariates, which can poten- tially improve the accuracy of intraday volatility estimates. Also, intraday volatility estimates or even instantaneous volatility estimates can be constructed based on the price durations or conditional intensity process. This can be particularly challenging for RV-type estimators as smaller estimation windows results in a much smaller amount of data, which greatly affects the performance of the RV-type estimators.