Aspects of market completion and stochastic volatility

Abstract

This work sets out to give a rigorous formulation to market completion as an attempt to investigate the “dynamics of the volatility surface". The center stage in the model is occupied by a latent factor process whose natural filtration is the market filtration and whose cardinality is the multiplicity of the spanning set. Given a strong solution of the purely diffusive SDE instantiating the state, market completeness is predicated on martingale representation: a weights process is identified which replicates any contingent claim(s) observed in the market by a combination of the forward process and a set of options. A rich but tractable class of factor processes whose stochastic variance is a functional of the Wishart family has been singled out. The search for two properties has driven its choice: (i) affinity of the characteristic function and (ii) an adequate geometry of the state space. The decisive factor for (i) has been the availability of a well-understood theory that reduces Backward equations to simple ODEs plus the availability of fast algorithms. The idea behind (ii) has been to pick one that by construction could allow for richer factor cross-correlation. Imposing some additional structure on the functions characterizing the factor process and the boundaries, the check that a particular choice of spanning securities does indeed complete the market has been reformulated as a simple computable condition given in terms of the solution set of the Riccati equation characterizing the solution of the factor process. After some numerics, whereby the robustness of the model is investigated against real market data, the dynamic hedging problem under scrutiny has finally been rephrased into a non-linear tracking problem whereby the discrete-time market prices are interpreted as the small noise observation process whose non-linear relationship with the signal - the state vector- is taken care of by a variation on the Extended Kalman Filter.