Optimal Momentum Hedging via Hypoelliptic Reduced Monge--Ampère PDE

Abstract

The celebrated optimal portfolio theory of Merton was successfully extended by the author to assets that do not obey Log-Normal price dynamics in [S. Stojanovic, Computational Financial Mathematics Using Mathematica®: Optimal Trading in Stocks and Options, Birkhäuser Boston, Boston, 2003]. Namely, a general one-factor model was solved and applied in the case of appreciation-rate reversing market dynamics. Here, we extend a general methodology to solve the stochastic control problem of optimal portfolio hedging under momentum market dynamics: the corresponding HJB PDE is transformed into the associated Monge--Ampère PDE, which is, utilizing the special structure of the problem, further reduced to a lower-dimensional Monge--Ampère PDE, which is then finally solved numerically. The present problem, in addition to being a two-factor model, has a substantive difficulty due to the degeneracy of the underlying Markov process, yielding the hypoellipticity of its infinitesimal generator, and the corresponding degeneracy of all the fully nonlinear PDEs derived. Furthermore, we solve the problem of optimal hedging and pricing of European and American options in momentum markets, derive a hypoelliptic Black--Scholes PDE/obstacle problem, and introduce a notion of options trading opportunity.