In the continuous-time Kyle model, assume logv˜ is normally distributed instead of v˜ being normally distributed. Denote the mean of logv˜ by μ and the variance of log v˜ by σ2. Set λ = σv/σz. Show that the strategies P0 = eμ+ 1 2 σ2 v

dPt = λPt dYt dXt = (logv˜ − μ)/λ− Yt 1− t dt form an equilibrium by showing the following:

(a) Define Wt = Yt/σz. Show that, conditional on v˜, W is a Brownian bridge on [0,1] with terminal value (logv˜ −μ)/σv. Use this fact to show that P satisfies P1 = ˜v and is a martingale relative to the market makers’

information.

(b) Forv > 0 and p > 0, define J(t, p) = p −v+ v(log v−log p)

λ +

1 2

σvσz(1− t)v.

Prove the verification theorem. (The intuition for this J is the same as that described in Section 24.5—take θ = 0 and then trade at the end to the point that p = v.)