Using the Mizar system [], [], we start to show, that the Call-Option, the Put-Option and the Straddle (more generally defined as in the literature) are random variables ([], p. 15), see (Def. 1) and (Def. 2). Next we construct and prove the simple random variables ([], p. 14) in (Def. 8).
In the third section, we introduce the definition of arbitrage opportunity, see (Def. 12). Next we show, that this definition can be characterized in a different way (Lemma 1.3. in [], p. 5), see (17). In our formalization for Lemma 1.3 we make the assumption that ϕ is a sequence of real numbers (there are only finitely many valued of interest, the values of ϕ in R^{d}). For the definition of almost sure with probability 1 see p. 6 in []. Last we introduce the risk-neutral probability (Definition 1.4, p. 6 in []), here see (Def. 16).
We give an example in real world: Suppose you have some assets like bonds (riskless assets). Then we can fix our price for these bonds with x for today and x · (1 + r) for tomorrow, r is the interest rate. So we simply assume, that in every possible market evolution of tomorrow we have a determinated value. Then every probability measure of Ω_{fut}_{1} is a risk-neutral measure, see (21). This example shows the existence of some risk-neutral measure. If you find more than one of them, you can determine – with an additional conidition to the probability measures – whether a market model is arbitrage free or not (see Theorem 1.6. in [], p. 6.)
A short graph for (21):
Suppose we have a portfolio with many (in this example infinitely many) assets. For asset d we have the price π(d) for today, and the price π(d) (1 + r) for tomorrow with some interest rate r > 0.
Let G be a sequence of random variables on Ω_{fut}_{1}, Borel sets. So you have many functions f_{k} : {1, 2, 3, 4}→ R with G(k) = f_{k} and f_{k} is a random variable of Ω_{fut}_{1}, Borel sets. For every f_{k} we have f_{k}(w) = π(k)·(1+r) for w {1, 2, 3, 4}.
Here, every probability measure of Ω_{fut}_{1} is a risk-neutral measure.
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