Stochastic control approach to derivative pricing and hedging in incomplete markets modeled by general It么 SDE systems: an overview and an application in FX derivatives

摘要

In a recent series of works the author has established a complete theory of simultaneous ("neutral") pricing of multiple types of (liquid) tradable financial derivative contracts under multidimensionality of risks in incomplete markets, including markets with non-hedgable interest rate risks. The non-hedgable risk premium is determined by the selection of the investor''s risk aversion parameter, and characterized via an additional non-linear PDE. The derived pricing PDE system may possibly be viewed as the "ultimate" extension of the famous Black-Scholes PDE. Moreover, the hedging formula of same generality was derived as well. Both, the general pricing PDE system, and the general (most conservative) hedging formula, are derived as consequences of two (different) optimal portfolio problems, i.e., as consequences of two stochastic control problems. Furthermore, both results are derived as corollaries of the discovered formula for a matrix inverse, therefore called the "fundamental matrix of derivatives pricing and hedging". This note is a short overview of the established general theory, with an example presented as well.