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Abstract
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^{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 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 mathbb{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.
Highlights
Let (, F, P) be a probability space, consider a fixed time horizon equal to one and let F = (Ft )t∈[0,1] be a filtration satisfying the usual conditions with F0 being UK D.C
Let S = (Stj ) be a d-dimensional stochastic process describing the evolution of the discounted prices of liquidly traded securities in a financial market and with the property that S is a martingale under the measure P
The second fundamental theorem of asset pricing asserts that the above statements are equivalent to P being the unique martingale measure for S in the class of equivalent measures
Summary
The differences in the financial setup are twofold: first, in the classical setting of a Radner equilibrium, all security prices are defined in backward form, whereas the problem of market completion with derivative securities requires some security prices to be defined in forward form, leading to a forward–backward setup; second, in an equilibrium setting, the martingale measure for the price process S is determined endogenously in terms of the utility functions of individual agents, whereas in our setting the measure P is given exogenously. Throughout the text, N > 0 denotes a constant whose value may vary from line to line
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