## 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.

### MSC:

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

Zbl 0482.60097
Full Text: