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 \(\vartheta\) such that \(w+ \int^T_0 \vartheta_s dX_s\geq H\). 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.


91G20 Derivative securities (option pricing, hedging, etc.)
90C39 Dynamic programming
93E25 Computational methods in stochastic control (MSC2010)


Zbl 0482.60097
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