Complete And Incomplete Market Under The Greeks Formula

This paper computes Greeks (i.e. price sensitivities) of European options in both complete and incomplete financial markets using ideas and methods in stochastic geometry and Malliavin calculus. In complete market, which means the volatility (?) satisfies the ellipticity condition, we propose to use elliptic diffusion on general complete and noncompact Lie group to describe stocks'prices, noticing that the geometric Brownian motion St in the Black-Scholes model is indeed a special exam-ple of elliptic diffusion processes on the Lie group R*. When the volatility satisfies ellipticity condition but not uniform ellipticity condition, we get Greeks formulaes of European options which are the extension of the results of P. L. Lions and his collaborators in [9]. In incomplete market, we suppose stocks'prices are hypoelliptic diffusion processes on complete Riemannian manifold. Under this condition, using the invertibility of the Malliavin covariance, we establish the Greeks formulae with respect to the initial stocks'prices and get the gradient formulas of the heat kernel in hypoelliptic case, which improve and simplify the results getting by S. Kusuoka and D. Stroock in [19].