TITLE:
Super-Diffusive Noise Source in Asset Dynamics
AUTHORS:
Max-Olivier Hongler
KEYWORDS:
Black-Scholes Dynamics; Non-Gaussian Volatility; Optimal Stopping; Adaptive Optimal Control; Exact Solutions
JOURNAL NAME:
Journal of Mathematical Finance,
Vol.3 No.1,
February
26,
2013
ABSTRACT: Given an asset with value St, we revisit the Black and Scholes dynamicswhen the driving noise ξtis a non-Gaussian super-diffusive stochastic process with variance of the type . This super-diffusive quadratic variance behavior, synthesizes a ballistic component which would occur in strongly fluctuating environments. When , the assets can, with high probability, be driven towards the bankruptcy . This extra dynamic feature significantly affects the management of an optimal portfolio. In this context, we focus on basic decisions like: 1) determine the optimal level to sell the asset; 2) determine how to balance a portfolio which incorporates such a high volatility asset; and 3) when facing incertitudes on the asset’s growth rate μ, construct an optimal adaptive portfolio control. In all mentioned cases and despite the presence of this highly non-Gaussian noise source, we are able to deliver simple exact and fully explicit optimal control rules.