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Dynamic programming and pricing of contingent claims in an incomplete market. (English) Zbl 0831.90010
In a complete financial market, the pricing of contingent claims is well understood. If the stochastic process X describes discounted stock prices, ‘complete’ means that every sufficiently integrable random variable can be represented as the sum of a constant and a stochastic integral of X satisfying suitable integrability conditions. An (up to technical conditions) equivalent formulation is the statement that there exists a unique equivalent martingale measure P * for X. The price of the claim (random variable) H is then E * [H] (see [J. M. Harrison and S. R. Pliska, Stochastic Processes Appl. 11, 215-260 (1981; Zbl 0482.60097)]). In an incomplete market, there are many equivalent martingale measures Q, leading to a range of possible prices between inf Q E Q [H] and sup Q E Q [H]. Assuming that X is a multidimensional Itô process, the authors show that sup Q E Q [H] is the smallest initial wealth w for which there exists a suitable integrand ϑ such that w+ 0 T ϑ s dX s H. A similar result holds for inf Q E Q [H]. This means that one needs at least the amount sup Q E Q [H] for the construction of a self-financing hedging strategy which covers the claim H, and thus implies that sup Q E Q [H] and inf Q E Q [H] can be viewed as fair selling and buying prices for H, respectively. Actually, the authors prove a dynamical version of the above theorem. Analogous results for a general semimartingale X have subsequently been obtained by D. O. Kramkov [‘Optimal decomposition of supermartingales and hedging contingent claims in incomplete security markets’ (1994), submitted to Probability Theory and Related Fields], using a different approach.

91G20Derivative securities
90C39Dynamic programming
93E25Computational methods in stochastic control