Introduction to Stochastic Finance: Random Variables and Arbitrage Theory

摘要

Using the Mizar system [ 1 ], [ 5 ], we start to show, that the Call-Option, the Put-Option and the Straddle (more generally defined as in the literature) are random variables ([ 4 ], p. 15), see (Def. 1) and (Def. 2). Next we construct and prove the simple random variables ([ 2 ], 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 [ 4 ], p. 5), see (17). In our formalization for Lemma 1.3 we make the assumption that ? is a sequence of real numbers (there are only finitely many valued of interest, the values of ? in Rsupd/sup). For the definition of almost sure with probability 1 see p. 6 in [ 2 ]. Last we introduce the risk-neutral probability (Definition 1.4, p. 6 in [ 4 ]), 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 ?subfut/subsub1/sub 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 [ 4 ], 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 ?subfut/subsub1/sub, Borel sets. So you have many functions fsubk/sub : {1, 2, 3, 4}→ R with G(k) = fsubk/sub and fsubk/sub is a random variable of ?subfut/subsub1/sub, Borel sets. For every fsubk/sub we have fsubk/sub(w) = π(k)·(1+r) for w {1, 2, 3, 4}. Today Tomorrow only one scenario { w 21 = { 1 , 2 } w 22 = { 3 , 4 } for all d ∈ ?? holds π ( d ) { f d ( w ) = G ( d ) ( w ) = π ( d ) ? ( 1 + r ) , w ∈ w 21 or w ∈ w 22 , r 0 is the interest rate . Here, every probability measure of ?subfut/subsub1/sub is a risk-neutral measure.