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Additional resources for Discrete time finance
Maximize subject to The real problem lies here in the set of constraints, determining the set Wx . We know already, that in general, in an incomplete market model there exists an infinite number of risk neutral measures, leading to an infinite number of constraints. The Lagrange multiplier method which we used to solve the optimal portfolio problem before, however only works for a finite number of constraints. We therefore have to reduce the number of constraints to a finite number. This is possible as the following lemma shows.
What is our measure for the performance of a trading strategy. In order to define such a performance measure three fundamental characteristics of financial markets, or more precisely of the agents trading on the market, have to be included : 37 1. Agents prefer higher payoffs to less payoffs This idea is so innate to financial markets, that it doesn’t need any discussion. However we cannot judge trading strategies purely by this characteristic. As we saw in the previous sections, the payoffs of financial assets are generally modeled as random variables.
2 2 The reason why the agent would prefer the first alternative is, that he is risk averse. His idea is, that 10 million will bring him financial safety and the possibility of an upper class lifestyle and that in fact for him 10 million or 20 million wouldn’t be such a big difference, but 10 million or nothing would make a huge difference. We therefore have found our third and last characteristic, which we want to include into our model. E(X 1 ) = 10 million = 3. Agents are risk averse The concept of risk aversion can be included in our model by the use of utility functions.