The numéraire portfolio in semimartingale financial models
The authors consider a general semimartingale model without further mathematical assumptions. On the economic side, it is assumed that assets have their prices determined exogenously and can be traded without ``frictions'': transaction costs are non-existent or negligible. The main concern is a problem of dynamic stochastic optimization: to find a trading strategy whose wealth appears ``better'' when compared to the wealth generated by any other strategy, in the sense that the ratio of the two processes is a supermartingale. If such a strategy exists, it is essentially unique and is called numéraire portfolio. Necessary and sufficient conditions for the numéraire portfolio to exist are derived, in terms of the triplet of predictable characteristics of the stock-price returns. A bare-hands approach is used, the assumption of finite expected log-utility is dropped entirely; no normative assumptions is imposed. It turns out that the numéraire portfolio can exist even when the classical No Arbitrage condition fails.
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