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Extra info for Encyclopaedie der mathematischen Wissenschaften und Anwendungen. Geometrie
1 Arbitrage Before getting into the mathematical details, let us present this notion through an example. Let us suppose that the real estate return is greater than the fixed income rate. The trader will borrow money and invest in the real estate. If the real estate return remains the same, the trader earn money. We say that there is an arbitrage situation as money can be earned without any risk. Let us suppose now that as we apply this winning strategy and start earning a lot of money, others who observe our successful strategy will start doing the same thing.
So in order to have no-arbitrage, we should impose that µ = r. More generally, we will see in the following that the no-arbitrage condition imposes that the drift of traded assets in our market model is fixed to the instantaneous rate in a wellchosen measure P (not necessarily unique) called the risk-neutral measure. To define precisely the meaning of no-arbitrage, we introduce a class of strategies that could generate arbitrage. In this context, we introduce the concept of a self-financing portfolio.
This is formalized by the notion of conditional expectation. 1 Conditional expectation Let X ∈ L1 (Ω, F, P) and let G be a sub σ-algebra of F. Then the conditional expectation of X given G, denoted EP [X|G], is defined as follows: 1. ) 2. v. Y . It can be shown that the map X → EP [X|G] is linear. v. X and Y admitting a probability density, the conditional expectation of X ∈ L1 conditional to Y = y can be computed as follows: The probability to have X ∈ [x, x + dx] and Y ∈ [y, y + dy] is by definition p(x, y)dxdy.