# Online Appendix. Supplemental Material for Insider Trading, Stochastic Liquidity and. Equilibrium Prices. by Pierre Collin-Dufresne and Vyacheslav Fos

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3 matters for the rate at which information will flow into prices ex-post and consequently for future price volatility. Even though price impact is stochastic, price volatility is not stochastic, and the model cannot generate any contemporaneous relation between changes in volume and price volatility or between price impact and price volatility. To generate such relations we need a stochastic growth rate of noise trading volatility. The next section presents a framework which generates both true stochastic volatility in prices and a meaningful correlation between price volatility and volume. 2. Mean-reverting noise trading volatility Here we consider an example where noise trading volatility follows a diffusion process with mean-reversion. Specifically, we consider the case where x t = logσ t follows a meanreverting Ornstein-Uhlenbeck process: dx t = ( ν2 2 κx tdt+νdw t. (1 We parametrize the drift of x t so that, when κ = 0, volatility is a martingale: dσ t σ t = κx t dt+νdw t. (2 As a result, we can focus on the impact of mean-reversion alone, and use a series expansion in κ around the known solution when κ = 0 (derived in example 2. The following result characterizes the solution. Theorem 4. If the log of noise trading volatility follows a mean reverting process as given in equation (2, then the process G(t admits the solution: G(t = σ 2 ta(t t,x t,κ 2, (3 3

4 where the function A(τ,x,κ can be approximated by a series expansion: A(τ,x,κ = T t ( 1+ n i i=1( kτ i( j=0 c ijk t +O(κ k n+1, (4 x j i j k=0 where the c ijk are positive constants that depend only on ν 2 and can be solved explicitly. 11 In that case, private information enters prices at a stochastic rate that depends on the level of noise trading volatility: dσ t 1 = Σ t A(T t,x t,κ 2dt. (5 Market depth is stochastic and given by: λ t = The trading strategy of the insider is: Σt σ t A(T t,x t,κ. (6 θ t = σ t Σt A(T t,x t,κ (v P t. (7 Stock price dynamics follow a three factor (P, x, Σ Markov process with stochastic volatility given by: dp t = (v P t Σt A(T t,x t,κ 2dt+ A(T t,x t,κ dz t. (8 In particular, stock price volatility is a stochastic and tends to be higher when noise trading volatility is higher. The unconditional expected profit at time zero of the insider is Tσ v σ 0 A 0 T. Proof 10. To prove this result, we observe that m t = κx t and that x t has following 11 We provide in the section below the fifth order solution. Higher order expansions can be obtained easily using Mathematica (program available upon request. 4

5 dynamics under the P measure: dx t = ( ν2 2 κx tdt+νd W t, (9 where by Girsanov s theorem we have defined W t = W t ν 2 t a standard P-measure Brownian motion. Thus x t is a one-factor Markov process under P. Using the Markov property for conditional expectations, we guess that the solution to equation (?? is given by a function A(t,x t. As shown in the proof of lemma 4, this function satisfies: Ẽ t [da(t,x t /dt+m t A(t,x t + ] 1 = 0. (10 2A(t,x t Using Itô s lemma we obtain the following non-linear PDE for A(T t,x (where we change variables to τ = T t and drop the argument of the function for simplicity: ν 2 2 (A xx +A x + κx(a x +A A τ + 1 2A = 0 (11 subject to boundary conditions A(0,x = 0. When κ = 0, the solution is simply A(τ,x;κ = 0 = τ. Assuming the solution is analytic in its arguments, we seek an series expansion solution of the form given in equation (4 above. Plugging this guess into the left hand side of the PDE and Taylor expanding in κ, we find that each term in the series expansion can be set to zero by an appropriate choice of the constants c ijk. We can thus recursively solve for these constants and obtain an approximate solution to the PDE. In figures 2 below we plot the 0th, 1st, 2nd and 5th order expansion solution for ν = 0.7 T = 1,κ = 0.25 and for three values of x 0 = { 0.3;0;+0.3}. The first term in the series expansion of the A(τ,x,κ function is instructive. Indeed, we find: A(τ,x,κ = τ(1 κ 2 τ(ν2 τ 6 +x+o(κ2. (12 This confirms that we need κ to be different from zero for uncertainty about future noise 5

7 3. Expansion solution The fifth order expansion of the A function (with v = ν 2. A(τ,x,κ = ( ( vt t 1 κt 12 + x ( 13v +κ 2 t 2 2 t 2 ( vt x vt x2 24 ( 89v κ 3 t 3 3 t 3 ( 3v x t vt v2 t 2 ( 59vt x ( 1237v +κ 4 t 4 4 t v3 t 3 ( 71v 2 t x vt v2 t ( 17v 3 t 3 + x v2 t vt ( vt +x vt 720 ( 601v 2 t 2 κ 5( t 5( 6299v 5 t v4 t v3 t 3 ( 4241v x 2 3 t v2 t vt x 4 ( 431vt vt x x3 +x 360 +O(κ 6 + 3vt x x v2 t vt ( 287v 4 t v3 t v2 t vt We illustrate the convergence of the expansion in the following figures. A t,x,κ th 1st 2nd 5th t Figure 2: A function expansion solution given in equation (4 for different order (0,1,2,5 of the expansion for x 0 = 0. Other parameter values are κ = 0.25,ν = 0.7,T = 1. 7

8 A t,x,κ th 1st 2nd 5th t Figure 3: A function expansion solution given in equation (4 for different order (0,1,2,5 of the expansion for x 0 = Other parameter values are κ = 0.25,ν = 0.7,T = 1. A t,x,κ th 1st 2nd 5th t Figure 4: A function expansion solution given in equation (4 for different order (0,1,2,5 of the expansion for x 0 = 0.7. Other parameter values are κ = 0.25,ν = 0.7,T = 1. 8

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