In this talk, we consider a continuous-time Kyle-Back model which is a game between an insider and a market maker. The existing literature typically focuses on constructing equilibria with a PDE approach, which requires certain Markovian structures. We characterize all equilibria through a coupled system of forward-backward SDEs. In particular, when the time duration is small, we show that the FBSDE is well-posed, and therefore the game has a unique equilibrium. Moreover, this unique equilibrium may be non-Markovian and thus not attainable via the PDE approach. We next study the set value of the game, which roughly speaking is the set of insider's values over all equilibria and thus is by nature unique. Finally, we characterize the set value through a level set of a certain standard HJB equation.
In the second part of the talk, we apply the new approach to the Kyle-Back model with dynamic legal risk. In the presence of an active regulator and large numbers of noise traders, the insider chooses a strategy that conceals his identity within a large volume of surrounding trades and concentrates on medium-sized trades. We establish an intensity-based mathematical framework for explaining the interconnections between insider trading and stealth trading. It turns out that when the number of noise traders becomes large, the price impact of the insider asymptotically vanishes, and consequently the stochastic game becomes deterministic optimizations, which also carry implications for regulatory investigations and sanctions.
The results above include the joint work with Weixuan Xia and with Jianfeng Zhang.