Market completion with derivative securities

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• 作者：Daniel C. Schwarz
• 关键词：Completeness ; Derivatives ; Integral representation ; Diffusion ; Martingales ; Parabolic equations ; Analytic functions ; Jacobian determinant
• 刊名：Finance and Stochastics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：21
• 期：1
• 页码：263-284
• 全文大小：829KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Quantitative Finance; Finance, general; Statistics for Business/Economics/Mathematical Finance/Insurance; Economic Theory/Quantitative Economics/Mathematical Methods; Probability Theory and Stochastic
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1122
• 卷排序：21

文摘

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.