Market Completion with Derivative Securities

Let $S^F$ be a $\mathbb{P}$-martingale representing the price of a primitive asset in an incomplete market framework. We present easily verifiable conditions on 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 $2$-dimensional stochastic differential equation: $S^F_t = f(X_t)$ and $S^B_t:=\mathbb{E}[g(X_1) | \mathcal{F}_t]$. From a purely mathematical point of view we prove that every local martingale under $\mathbb{P}$ can be represented as a stochastic integral with respect to the $\mathbb{P}$-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 $\mathbf{R}^2$. Hence they cover, as a special case, the prominent example of a stochastic volatility model being completed with a European call (or put) option.