Private Valuation of Contingent Claims in a Discrete Time/State Model
The importance of the risk-neutral measure as a pricing operator generates interest in the calibration problem: given observed market prices, retrieve a pricing measure compatible with them. It can be established through convex duality arguments that the Arbitrage Pricing Theory calibration problem, i.e., calibration to a set of benchmark securities, is equivalent to a certain portfolio optimization problem. We present an explicit duality argument to extend this result: every portfolio optimization problem in a market that allows liquid trading is equivalent to a pricing measure retrieval problem, and vice versa. The resulting pricing measures reflect private valuation of investors’ positions – a consequence of considering portfolio optimization problem from the point of view of a hedger endowed with specific goals and liabilities. Such an investor operates in a “submarket” of available financial securities, which he must see as arbitrage-free, but in which he may be a buyer or seller of particular contingent claims depending on his particular liabilities and goals. This perspective shifts our vision of arbitrage pricing from a “global” to a “local” market, i.e., to the one seen by an investor through the prism of his particular portfolio. It also allows us to use microeconomic considerations when choosing a calibration utility.
KeywordsRelative Entropy Portfolio Optimization Contingent Claim Relative Risk Aversion Terminal Wealth
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