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Abstract
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This work attempts to extend the complete market option pricing theory to incomplete markets in the direction of risk control. Instead of eliminating the risk by a perfect hedging portfolio as in complete markets, partial hedging will be adopted and some residual risk at expiration will be tolerated. In the spirit of the utility indifference principle, the risk measure (or risk indifference) prices charged for buying or selling an option are associated to the capital required for dynamic hedging so that the risk exposure will not increase. The associated optimal hedging portfolio is decided by minimizing a convex measure of risk. The general framework (definition of risk measure prices and risk-efficient options and the existence of optimal partial hedging portfolios) will be established with convex risk measures, or particularly, coherent risk measures. It will be confirmed that options evaluated by risk measure pricing rules are indeed risk-efficient. Relationships to utility indifference pricing and pricing by valuation and stress measures will be discussed. Examples using the shortfall risk measure and average Value-at-Risk will be shown. |
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