Market completion with derivative securities

• Published in: Finance and stochastics Vol. 21; no. 1; pp. 263 - 284
• Main Author: Schwarz, Daniel C.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2017; Springer Nature B.V
• Subjects: Assets; Derivatives; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Endogenous; Equilibrium; Finance; Fundamental frequency; Insurance; Management; Mathematical models; Mathematics; Mathematics and Statistics; Prices; Probability Theory and Stochastic Processes; Put & call options; Quantitative Finance; Random variables; Securities markets; Statistics for Business; Stochastic models; Studies; Volatility; 60G44; Analytic functions; 35K90; G10; 35K15; Derivatives; Parabolic equations; Completeness; 91G20; Diffusion; 60H05; Integral representation; Martingales; Jacobian determinant
• ISSN: 0949-2984; 1432-1122
• DOI: 10.1007/s00780-016-0317-z

Let S F be a ℙ-martingale representing the price of a primitive asset in an incomplete market framework. We present easily verifiable conditions on the model coefficients which guarantee the completeness of the market in which in addition to the primitive asset, one may also trade a derivative contract S B . Both S F and S B are defined in terms of the solution X to a two-dimensional stochastic differential equation: S t F = f ( X t ) and S t B : = E [ g ( X 1 ) | F t ] . From a purely mathematical point of view, we prove that every local martingale under ℙ can be represented as a stochastic integral with respect to the ℙ-martingale S : = ( S F , S B ) . Notably, in contrast to recent results on the endogenous completeness of equilibria markets, our conditions allow the Jacobian matrix of ( f , g ) to be singular everywhere on R 2 . Hence they cover as a special case the prominent example of a stochastic volatility model being completed with a European call (or put) option.