A General Solution Method for Insider Problems

Abstract

We develop a flexible approach to solve a continuous-time, multi-asset/multi-option Kyle-Back model of informed trading under very general assumptions, including on the distribution of the belief about the fundamental, and the noise process. The main insight is to postulate the pricing rule of the market maker at maturity as an optimal transport map. The optimal control of the informed trader reduces to the computation of a conjugate convex function, explicit in some cases, and otherwise easily obtainable using fast numerical algorithms. To illustrate the power of our method, we apply it to a long-standing problem: how are informed investors splitting trades between a spot asset and its options? Our method allows to i) prove the existence of an equilibrium and characterize the informed trader's trading strategy in the spot and the option markets, even for non-Gaussian price priors (e.g., lognormal); ii) show there can be cross-market price impact between the spot market and multiple options even when their noise trading is independent; and iii) compare our pricing results to a simple Black-Scholes model and quantify the price distortion of the option due to strategic trading. In particular, we show that a Black-Scholes implied volatility (IV) smile/smirk can emerge because of the market marker's adaptation to asymmetric information.

Keywords: Kyle's model, options, informed trading, optimal transport, asymmetric information, price impact

JEL Classification: G12, G13, G14