Insider Trading, Stochastic Liquidity, and Equilibrium Prices

Abstract

FOS S1. DETERMINISTIC GROWTH RATE OF NOISE TRADER VOLATILITYIN GENERAL, WHEN NOISE TRADING VOLATILITY IS STOCHASTIC (ν = 0) and there is predictability (m t = 0), then price impact is stochastic and negatively correlated (in changes) with noise trading volatility. However, price volatility and the posterior variance of the fundamental value (Σ t ) are both deterministic and only depend on the unconditional expected path of noise trading volatility. For illustration, Figure S1(a) plots the paths of the posterior variance (Σ t ) for three cases of constant growth rate m = 0 5, m = 0, and m = −0 5. It is remarkable that irrespective of ν t (and thus of realized shocks to noise trading, i.e., of realized volume), private information is revealed following a deterministic path, which only depends on the expected rate of change in noise trading volatility, despite the fact that the strategy of the insider is stochastic. This is, of course, the result of the offsetting effect noise trading volatility has on price impact. If the level of noise trading variance increases (decreases) unexpectedly, then the insider trades more (less) aggressively, but price impact decreases (increases) one for one, making price dynamics and information arrival rate independent of the volatility level. Instead, if noise trading variance is expected to increase on average because m > 0, for example, then the insider is expected to scale back his trading initially and to trade more aggressively later on. As Figure S1(a) shows, this leads to private information getting into prices more slowly initially, and then faster later on. So posterior variance follows a deterministic concave path if noise trading volatility is expected to increase, but a convex path if it is expected to decrease. As a result, the equilibrium price process exhibits deterministic time-varying volatility. Price volatility increases (decreases) exponentially if noise trading volatility is expected to increase (decrease).In Figure S1(b), we plot the expected optimal trading rate of the insider normalized by the initial undervaluation (E[θ t |v F 0 ]/(v − P 0 ) from equation (27) in the paper) for different levels of constant m. As we can see, when noise trading volatility is unpredictable (m = 0), then we expect the insider to trade at a constant rate as in the Kyle model. Instead, if noise trading volatility is expected to increase (m > 0), then, unconditionally, we expect the insider to trade more aggressively on average in the future when more noise trading will occur. In fact, when m > 0, the insider initially scales back his expected trading relative to the m = 0 case despite the fact that he starts out with the same level of noise trader volatility (σ 0 ) in both cases and indeed, despite the fact that the

Equilibrium Price and Optimal Insider Trading Strategy Under Stochastic Liquidity with Long Memory

“…It should be remarked here that if the uncertain term were general Brownian motion without memories, then the model would degenerate to the one studied by Collin-Duersne and Fos [11]. Furthermore, if the noise trading σ t were constant, then the model would become exactly the same as the classical Kyle-Back model [22].…”

“…Here, the second equality uses the fact that σ t is independent of the asset value in the order flow, the third equality is obtained from the fact that the expected change in the order flow is zero for the conjectured policy θ t , and the last equality follows from (11).…”

“…This completes the proof. [11], where the insider trader has an opportunity to wait for a better liquidity to trade. Indeed, the price impact must increase on average to encourage the insider to trade early and give up his opportunity to wait for better liquidity states under the framework of stochastic liquidity.…”

“…Such an assumption on the noise trading volatility is still not appropriate since it gives rise to the deterministic price impact and further leads to the situation where expected execution costs of liquidity traders do not depend on the timing of their trading. A more recent extension to the Kyle model was proposed by Collin-Dufresne and Fos [11], who considered stochastic noise trading volatility, and showed that this is the only guarantee that insider's liquidity timing option can generate a stochastic price volatility as well as a nonzero correlation among the price volatility, market depth, and uninformed trading volume.Despite all the advantages of the model proposed in [11], it has not taken into consideration that the volatility is characterized by long memory, which is a consensus among financial econometricians. In particular, Bollerslev and Jubinski [7] presented the evidence of long memory in the volatility and investigated the extent to which the volume and volatility share common long-run dependencies.…”

“…Such an assumption on the noise trading volatility is still not appropriate since it gives rise to the deterministic price impact and further leads to the situation where expected execution costs of liquidity traders do not depend on the timing of their trading. A more recent extension to the Kyle model was proposed by Collin-Dufresne and Fos [11], who considered stochastic noise trading volatility, and showed that this is the only guarantee that insider's liquidity timing option can generate a stochastic price volatility as well as a nonzero correlation among the price volatility, market depth, and uninformed trading volume.…”