The market microstructure invariance paper by Kyle and Obizhaeva (2016) suggests that the
distribution of risk transfers (bets) and transaction costs are constant
in the crosssection and timeseries when scaled per unit of business
time. These constants are structual estimates and allow to infer the
arrival rate of bets (market velocity), the bet size distribution and
transaction costs.
Business time or market velocity measures the rate at which bets are submitted to the market. Consequently, for activly traded assets or large asstes with high trading volume and high volatility buisness time passes at a greater pace than for small assets with low trading volume and low volatility. The expected arrival rate of bets is defined as \(\gamma_{i, t}\), which follows a Poisson process and its unit is bets per calender day. Market velocity allows us to scale market charactersitics in order to interpret them across different assets. For instance, if the arrival rate rises by a factor of 9, then bet volatility decreases by a factor of 3 as \(\bar{\sigma}_{i, t} \cdot \gamma_{i, t}^{-1/2}\).
The transfer of risk is the decisions to buy or sell a long-term
position in a specific asset of certain size. In this transfer, or as
Treynor would call it, in this game informed traders, noise traders and
market makers trade with each other. The bets themselves are assumed to
be approximately independently distributed and thus are random
variables. On the other hand, short term trading by liquidity providers,
(ultra-) high frequency traders or arbitrageurs is not considered as bet
but as intermediary that clears the price of a bet. For instance,
restructuring a portfolio is a single bet, even if the trade is splitted
over days due to its size. Another example for a single bet based on
several orders is when multiple customers of a specific analyst invest
in the same recommendation. To summarize trading games the signed size
is introduced. It is the number of shares of a bet, which is also a
random variable and denoted by \(\tilde{Q}_{i,
t}\) with \(\mathop {\mathbb
E}[\tilde{Q}_{i, t}] = 0\).
The trading volume \(V_{i, t}\) are all shares traded per day of an asset and can be decomposed in bet volume \(\zeta_{i, t} \cdot V_{i, t}\) and intermediary volume \((\zeta_{i, t}-1) \cdot V_{i, t}\). If no intermediaries participate in the market, then \(\zeta_{i, t}=1\). Considering a monopolistic intermediary, then \(\zeta_{i, t}=2\), such that the factor increases linearly in intermediaries. Therefore expected bet volume can be defined as: \[\bar{V}_{i, t} := \gamma_{i, t} \cdot \mathop {\mathbb E}[|\tilde{Q}_{i, t}|]= \frac{2}{\zeta_{i, t}} \cdot V_{i, t}\]
Similarly, return volatility \(\sigma_{i, t}\) is the percentage standard deviation of daily returns per asset \(i\), which can be again decomposed in bet volatility \[\bar{\sigma}_{i, t} := \psi_{i, t} \cdot \sigma_{i, t}\] and intermediary volatility. Then the dollar volatility of a bet is the product of the asset price \(P_{i, t}\) and the bet volatility.
From these definitions we can already derive the signed standard deviation: \[\tilde{I}_{i, t} :=\tilde{Q}_{i, t} \cdot P_{i, t} \frac{\bar{\sigma}_{i, t}}{\gamma_{i, t}^{1/2}} \overset{d}{=} \tilde{I}\] This measure can be interpreted as the dollar risk transfer per unit of business time. The signed std. dev. is negative for short and positive for long positions. One advantage of using dollar volume is that the measure is unaffected by stock splits. Furthermore, it is immune to leverage effects as an increase in leverage would decrease the price at approximately the same rate as it would increase volatility. Kyle and Obizhaeva (2016) hypothesizes that the signed std. dev. is constant in distribution across assets and time and thus one should find the same value per quantile for different stocks or timepoints.
Also the aggregate dollar risk transferred by all traded orders \(W_{i, t}=\sigma_{i, t}P_{i, t}V_{i, t}\) can be broken into an intermediary and a bet part. The bet trading activity is simply derived and lead to several expressions: \[\bar{W}_{i, t}=\bar{\sigma}_{i, t}P_{i, t}\bar{V}_{i, t}= \bar{\sigma}_{i, t}P_{i, t}\gamma_{i, t} \cdot \mathop {\mathbb E}[|\tilde{Q}_{i, t}|] = \gamma_{i, t}^{3/2} \mathop {\mathbb E}[|\tilde{I}|]= \psi_{i, t} \cdot \sigma_{i, t}P_{i, t}\frac{2}{\zeta_{i, t}} \cdot V_{i, t} = W_{i, t} \cdot \frac{2\psi_{i, t}}{\zeta_{i, t}}\]
From the level of bet trading activity they infer the unobservable market velocity (middle term from above) as well as the expected size of bets (first bet volume term): \[\gamma_{i, t}=\bar{W}_{i, t}^{2/3}\cdot \mathop {\mathbb E}[|\tilde{I}|]^{-2/3} \qquad \text{ with } \qquad \frac{\partial\gamma_{i, t}}{\partial\bar{W}_{i, t}}=\frac{2}{3}\bar{W}_{i, t}^{-1/3}\cdot \mathop {\mathbb E}[|\tilde{I}|]^{-2/3}\] and \[\mathop {\mathbb E}[|\tilde{Q}_{i, t}|]=\bar{V}_{i, t} \cdot \bar{W}_{i, t}^{-2/3} \mathop {\mathbb E}[|\tilde{I}|]^{2/3} =\bar{W}_{i, t}^{1/3} \cdot \frac{\mathop {\mathbb E}[|\tilde{I}|]^{2/3}}{\bar{\sigma}_{i, t}P_{i, t}}\] Which immediately leads to: \[\frac{\tilde{Q}_{i, t}}{\bar{V}_{i, t} }\overset{d}{=} \bar{W}_{i, t}^{-2/3} \cdot \tilde{I} \cdot\mathop {\mathbb E}[|\tilde{I}|]^{-1/3}\] The above finding implies that the expected arrival rate of bets changes by 2/3 whenever the trading activity changes by 1% and the distribution of bet size increases by 1/3 of 1%. Another example for interpretaion could be to observe in the market an increase in market velocity by a factor of 9, as mentioned above. If the invariance hypothesis is correct, it is necessary that the bet size \(\tilde{Q}_{i, t}\) rises by 3.