Introduction to Stochastic Finance: Random Variables and Arbitrage Theory

Peter Jaeger 1
  • 1 , 80993, Munich, Germany

Summary

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 Rd). 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 Ωfut1 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 Ωfut1, Borel sets. So you have many functions fk : {1, 2, 3, 4}→ R with G(k) = fk and fk is a random variable of Ωfut1, Borel sets. For every fk we have fk(w) = π(k)·(1+r) for w {1, 2, 3, 4}.

TodayTomorrowonlyonescenario{w21={1,2}w22={3,4}foralld𝕅holdsπ(d){fd(w)=G(d)(w)=π(d)(1+r),ww21orww22,r>0istheinterestrate.

Here, every probability measure of Ωfut1 is a risk-neutral measure.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.

  • [2] Heinz Bauer. Wahrscheinlichkeitstheorie. de Gruyter-Verlag, Berlin, New York, 2002.

  • [3] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. The measurability of extended real valued functions. Formalized Mathematics, 9(3):525–529, 2001.

  • [4] Hans Föllmer and Alexander Schied. Stochastic Finance: An Introduction in Discrete Time, volume 27 of Studies in Mathematics. de Gruyter, Berlin, 2nd edition, 2004.

  • [5] Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.

  • [6] Peter Jaeger. Elementary introduction to stochastic finance in discrete time. Formalized Mathematics, 20(1):1–5, 2012. doi:10.2478/v10037-012-0001-5.

  • [7] Peter Jaeger. Modelling real world using stochastic processes and filtration. Formalized Mathematics, 24(1):1–16, 2016. doi:10.1515/forma-2016-0001.

OPEN ACCESS

Journal + Issues

Search