The implied intra-day probability of informed trading

Abstract

The paper develops a methodology for estimating the intra-day probability of informed trading for NYSE stocks, implied by the specialist’s quotes and depths. The time series pattern of our measure (PROBINF) in an intra-day analysis around earnings announcements is consistent with previous findings and with expectations regarding informed trading. Moreover, we find that PROBINF exhibits a strong and robust relationship with PIN, the level of insider trading and with measures of the price impact of trades. Our methodology complements the one developed in Easley et al. (J Financ 51(3):811–833, 1996a, J Financ 51(4):1405–1436, b), as it can be used to measure short term changes in informed trading and information asymmetry around events such as merger and acquisition announcements, share repurchases, stock splits, dividend announcements and index additions and deletions.

Appendix: Proof of Eq. (7)

As described in the paper, the dealer’s expected revenues from liquidity traders can be expressed as follows:

$$ (1 - p_{I} )[p_{BL} \cdot D_{A} \cdot (A - S_{0} ) + p_{SL} \cdot D_{B} \cdot (S_{0} - B) + p{}_{NL} \cdot 0] $$

(3)

At the same time, the dealer’s losses to informed traders can be expressed as follows:

$$ p_{I} [D_{A} \cdot C(S_{0} ,A) + D_{B} \cdot P(S_{0} ,B)] $$

(4)

Finally, the dealer’s order-processing and inventory costs can be expressed as follows:

$$ \begin{aligned} & p_{I} [0.5 \cdot D_{A} \cdot (OP + INVsh) + 0.5 \cdot D_{B} \cdot (OP + INVsh)] \\ & \quad + (1 - p_{I} )[D_{A} \cdot p_{BL} \cdot (OP + INVsh) + D_{B} \cdot p_{SL} \cdot (OP + INVsh) + p_{NL} \cdot 0] \\ \end{aligned} $$

(5)

where B is the bid price, A is the ask price, D _B is the depth at the bid, D _A is the depth at the ask, S ₀ is the dealer’s estimate of the “true” value of the stock (B < S ₀ < A), p _I is the probability that the next trade originates from an informed trader (PROBINF), p _BL , p _SL and p _NL are the conditional probabilities that the next liquidity trader will buy, sell or not trade when he faces the market maker, C(S ₀ , A) is the value of an American call option on the stock with a strike price of A, P(S ₀ , B) is the value of an American put option on the stock with a strike price of B, OP represents the average order processing cost per traded share, while INVsh represents the average inventory cost per traded share.

Consistent with Copeland and Galai (1983), we make the additional assumption that all liquidity traders will trade (p _NL  = 0), and that a liquidity trader can be a buyer or a seller with the same probability (p _BL  = p _SL  = 0.5). Moreover, setting the dealer expected losses equal to expected revenues, we obtain:

$$ (1 - p_{I} ) \cdot \frac{{(A - B) \cdot (D_{A} + D_{B} )}}{4} = p_{I} [D_{A} \cdot C(S_{0} ,A) + D_{B} \cdot P(S_{0} ,B)] + \frac{{(OP + INVsh) \cdot (D_{A} + D_{B} )}}{2} $$

Solving for PROBINF, we obtain:

$$ PROBINF = \frac{{0.25 \cdot (A - B - INV - 2 \cdot OP) \cdot (D_{A} + D_{B} )}}{{D_{A} \cdot C(S_{0} ,A) + D_{B} \cdot P(S_{0} ,B) + 0.25 \cdot (A - B) \cdot (D_{A} + D_{B} )}} $$

(7)

where INV represents the inventory component of the spread, and it is equal to twice the inventory component per share (INVsh).