{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T01:13:52Z","timestamp":1783905232429,"version":"3.55.0"},"reference-count":7,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2018,4,1]],"date-time":"2018-04-01T00:00:00Z","timestamp":1522540800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,4,1]]},"abstract":"Summary<\/jats:title>\n 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).<\/jats:p>\n \n 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\n \u03d5<\/jats:italic>\n is a sequence of real numbers (there are only finitely many valued of interest, the values of\n \u03d5<\/jats:italic>\n in\n \n R\n d<\/jats:sup>\n <\/jats:italic>\n ). 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).\n <\/jats:p>\n \n 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\n x<\/jats:italic>\n for today and\n x<\/jats:italic>\n \u00b7 (1 +\n r<\/jats:italic>\n ) for tomorrow,\n r<\/jats:italic>\n 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 \u03a9\n \n fut<\/jats:sub>\n <\/jats:italic>\n 1<\/jats:sub>\n 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 \u2013 with an additional conidition to the probability measures \u2013 whether a market model is arbitrage free or not (see Theorem 1.6. in [4], p. 6.)\n <\/jats:p>\n A short graph for (21):<\/jats:p>\n \n Suppose we have a portfolio with many (in this example infinitely many) assets. For asset\n d<\/jats:italic>\n we have the price\n \u03c0<\/jats:italic>\n (\n d<\/jats:italic>\n ) for today, and the price\n \u03c0<\/jats:italic>\n (\n d<\/jats:italic>\n ) (1 +\n r<\/jats:italic>\n ) for tomorrow with some interest rate\n r ><\/jats:italic>\n 0.\n <\/jats:p>\n \n Let G be a sequence of random variables on \u03a9\n \n fut<\/jats:sub>\n <\/jats:italic>\n 1<\/jats:sub>\n , Borel sets. So you have many functions\n \n f\n k<\/jats:sub>\n <\/jats:italic>\n : {1, 2, 3, 4}\u2192\n R<\/jats:italic>\n with\n G<\/jats:italic>\n (\n k<\/jats:italic>\n ) =\n \n f\n k<\/jats:sub>\n <\/jats:italic>\n and\n \n f\n k<\/jats:sub>\n <\/jats:italic>\n is a random variable of \u03a9\n \n fut<\/jats:sub>\n <\/jats:italic>\n 1<\/jats:sub>\n , Borel sets. For every\n \n f\n k<\/jats:sub>\n <\/jats:italic>\n we have\n \n f\n k<\/jats:sub>\n <\/jats:italic>\n (\n w<\/jats:italic>\n ) =\n \u03c0<\/jats:italic>\n (\n k<\/jats:italic>\n )\u00b7(1+\n r<\/jats:italic>\n ) for\n w<\/jats:italic>\n {1, 2, 3, 4}.\n <\/jats:p>\n \n \n \n \n \n \n \n \n \n \n Today<\/m:mi>\n <\/m:mrow>\n <\/m:mtd>\n \n \n Tomorrow<\/m:mi>\n <\/m:mrow>\n <\/m:mtd>\n <\/m:mtr>\n \n \n \n only<\/m:mtext>\n \u2009<\/m:mi>\n one<\/m:mtext>\n \u2009<\/m:mi>\n scenario<\/m:mtext>\n <\/m:mrow>\n <\/m:mtd>\n \n \n \n {<\/m:mo>\n \n \n \n \n \n \n w<\/m:mi>\n \n 21<\/m:mn>\n <\/m:mrow>\n <\/m:msub>\n =<\/m:mo>\n \n {<\/m:mo>\n \n 1<\/m:mn>\n ,<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mtd>\n <\/m:mtr>\n \n \n \n \n w<\/m:mi>\n \n 22<\/m:mn>\n <\/m:mrow>\n <\/m:msub>\n =<\/m:mo>\n \n {<\/m:mo>\n \n 3<\/m:mn>\n ,<\/m:mo>\n 4<\/m:mn>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mtd>\n <\/m:mtr>\n <\/m:mtable>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mtd>\n <\/m:mtr>\n \n \n \n for<\/m:mtext>\n \u2009<\/m:mi>\n all<\/m:mtext>\n \u2009<\/m:mi>\n d<\/m:mi>\n \u2208<\/m:mo>\n \ud835\udd45<\/m:mi>\n \u2009<\/m:mi>\n holds<\/m:mtext>\n \u2009<\/m:mi>\n \u03c0<\/m:mi>\n \n (<\/m:mo>\n d<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mtd>\n \n \n \n {<\/m:mo>\n \n \n \n \n \n \n f<\/m:mi>\n d<\/m:mi>\n <\/m:msub>\n \n (<\/m:mo>\n w<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n =<\/m:mo>\n G<\/m:mi>\n \n (<\/m:mo>\n d<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n \n (<\/m:mo>\n w<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n =<\/m:mo>\n \u03c0<\/m:mi>\n \n (<\/m:mo>\n d<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n \u22c5<\/m:mo>\n \n (<\/m:mo>\n \n 1<\/m:mn>\n +<\/m:mo>\n r<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n ,<\/m:mo>\n <\/m:mrow>\n <\/m:mtd>\n <\/m:mtr>\n \n \n \n w<\/m:mi>\n \u2208<\/m:mo>\n \n w<\/m:mi>\n \n 21<\/m:mn>\n <\/m:mrow>\n <\/m:msub>\n \u2009<\/m:mi>\n or<\/m:mi>\n \u2009<\/m:mi>\n w<\/m:mi>\n \u2208<\/m:mo>\n \n w<\/m:mi>\n \n 22<\/m:mn>\n <\/m:mrow>\n <\/m:msub>\n ,<\/m:mo>\n <\/m:mrow>\n <\/m:mtd>\n <\/m:mtr>\n \n \n \n r<\/m:mi>\n ><\/m:mo>\n 0<\/m:mn>\n \u2009<\/m:mi>\n is<\/m:mtext>\n \u2009<\/m:mi>\n the<\/m:mtext>\n \u2009<\/m:mi>\n interest<\/m:mtext>\n \u2009<\/m:mi>\n rate<\/m:mtext>\n .<\/m:mo>\n <\/m:mrow>\n <\/m:mtd>\n <\/m:mtr>\n <\/m:mtable>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mtd>\n <\/m:mtr>\n <\/m:mtable>\n <\/m:mrow>\n <\/m:math>\n $$\\matrix{ {Today} & {Tomorrow} \\cr {{\\rm{only}}\\,{\\rm{one}}\\,{\\rm{scenario}}} & {\\left\\{ {\\matrix{ {w_{21} = \\left\\{ {1,2} \\right\\}} \\hfill \\cr {w_{22} = \\left\\{ {3,4} \\right\\}} \\hfill \\cr } } \\right.} \\cr {{\\rm{for}}\\,{\\rm{all}}\\,d \\in N\\,{\\rm{holds}}\\,\\pi \\left( d \\right)} & {\\left\\{ {\\matrix{ {f_d \\left( w \\right) = G\\left( d \\right)\\left( w \\right) = \\pi \\left( d \\right) \\cdot \\left( {1 + r} \\right),} \\hfill \\cr {w \\in w_{21} \\,or\\,w \\in w_{22} ,} \\hfill \\cr {r > 0\\,{\\rm{is}}\\,{\\rm{the}}\\,{\\rm{interest}}\\,{\\rm{rate}}.} \\hfill \\cr } } \\right.} \\cr }$$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:disp-formula>\n <\/jats:p>\n \n Here, every probability measure of \u03a9\n \n fut<\/jats:sub>\n <\/jats:italic>\n 1<\/jats:sub>\n is a risk-neutral measure.\n <\/jats:p>","DOI":"10.2478\/forma-2018-0001","type":"journal-article","created":{"date-parts":[[2018,8,2]],"date-time":"2018-08-02T20:32:23Z","timestamp":1533241943000},"page":"1-9","source":"Crossref","is-referenced-by-count":0,"title":["Introduction to Stochastic Finance: Random Variables and Arbitrage Theory"],"prefix":"10.2478","volume":"26","author":[{"given":"Peter","family":"Jaeger","sequence":"first","affiliation":[{"name":"Siegmund-Schacky-Str. 18a, 80993 Munich , Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2018,7,28]]},"reference":[{"key":"2026071213090520598_j_forma-2018-0001_ref_001_w2aab3b7ab1b6b1ab1ab1Aa","doi-asserted-by":"crossref","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, and Josef Urban. 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