Optimal Momentum Hedging via Hypoelliptic Reduced Monge–Ampère PDEs

The celebrated optimal portfolio theory of R. C. 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, 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 be 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 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.