Market completion with derivative securities

Abstract

Let SF 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 SB. Both SF and SB are defined in terms of the solution X to a two-dimensional stochastic differential equation: StF=f(Xt) and StB:=E[g(X1)|Ft]. 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: = (SF, SB). 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 R2. Hence they cover as a special case the prominent example of a stochastic volatility model being completed with a European call (or put) option.