@MISC{FINANCE6.MIZ,
  AUTHOR = {Jaeger, Peter},
  TITLE = {Introduction to Stochastic Finance: Random-Variables
    and Arbitrage Theory},
  SECTION1 = {Put-Option, Call-Option and Straddle are Random-variables},
  SECTION2 = {Simple-random-variables},
  SECTION3 = {Arbitrage Theory: Definition and Alternative Representation},
  SECTION4 = {Risk-neutral Probability Measure},
  DAY = {27},
  MONTH = {March},
  YEAR = {2018},
  ADDRESS1 = {},
  SUMMARY = {We start to show, that the Call-Option, the Put-Option and the
             Straddle (more generally defined as in the literature) are
             random-variables (\cite{follmerschied:2004}, p.~15).
             Next we construct and prove the simple random-variables
             (\cite{heinz:2002}, p.~14).\par
             In the second part, we introduce the definition
             of arbitrage opportunity.
             Next we show, that this definition can be characterized in
             a different way (Lemma 1.3. in \cite{follmerschied:2004}, p.~5).
             In our formalization for Lemma 1.3 we make the assumption that phi
             is a Real_Sequence (there are only finitley many valued of
             interest, the values of phi in $R^d$).
             For the definition of almost sure with Probability 1 see
             \cite{heinz:2002}, p.~6.
             Last we introduce the risk-neutral Probability
             (Definition 1.4. in \cite{follmerschied:2004}, p.~6).
             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 \verb!Omega_fut1! is a risk-neutral
             measure (see FINANCE7:13). 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 \cite{follmerschied:2004}, p.~6.)\par
             A short graph for FINANCE7:13:
             \par
             Suppose we have a portfolio with many (in this example infinitely
             many) assets. For asset d we have the price jpi.d for today, and
             the price jpi.d*(1+r) for tomorrow with some interest rate r>0.
             \begin{verbatim}
             Let G be a sequence of random-variables on Omega_fut1,Borel_Sets.
             So you have many functios f_k:{1,2,3,4}->R with G.k=f_k and
              f_k is random-variable of Omega_fut1,Borel_Sets.
             For every f_k we have f_k(w)=jpi.k*(1+r) for w in {1,2,3,4}.

                            Today                         Tomorrow

                                                          w_21={1,2}
                       only one scenario
                                                          w_22={3,4}

                            Today                        Tomorrow

for all d in NAT holds:     jpi.d                  f_d(w)=(G.d)(w)=jpi.d*(1+r)
                                                    w in w_w21 or w in w_22,
                                                    r>0 is the interest rate,
             \end{verbatim}

         Here, every Probability-measure of Omega_fut1
         is a Risk-neutral-measure.
}}

@book{follmerschied:2004,
  author={F\"ollmer, Hans and Schied, Alexander},
  title={Stochastic Finance: An Introduction in Discrete Time},
  edition={2nd},
  volume={27},
  series={Studies in Mathematics},
  publisher={de Gruyter},
  year={2004},
  ADDRESS={Berlin}}

@book{heinz:2002,
author={Heinz, Bauer},
title={Wahrscheinlichkeitstheorie},
publisher={de Gruyter-Verlag},
year={2002},
ADDRESS={Berlin, New York}}

