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StochProcess..Basics.slides.html
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StochProcess..Basics.slides.html
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
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<h2 id="Stochastische-Prozesse-und-eine-kurze-Einführung-in-Monte-Carlo-Simulationen">Stochastische Prozesse und eine kurze Einführung in Monte-Carlo Simulationen<a class="anchor-link" href="#Stochastische-Prozesse-und-eine-kurze-Einführung-in-Monte-Carlo-Simulationen">¶</a></h2>
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<h2 id="Was-ist-ist-ein-stochastischer-Prozess?">Was ist ist ein stochastischer Prozess?<a class="anchor-link" href="#Was-ist-ist-ein-stochastischer-Prozess?">¶</a></h2><p>Wikipedia beschreibt einen stochastischen Prozess als</p>
<p>die mathematische Beschreibung von zeitlich geordneten, zufälligen Vorgängen.</p>
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<p><strong>"Defintion":</strong></p>
<p>Sei $(\Omega, P, F)$ ein Wahrscheinlichkeitsraum, $T$ eine Indexmenge (üblicherweise T=N oder T=R).
Eine Folge von von Zufallsvariablen $X_t$ mit $t \in T$ mit $X_t : \Omega \to R$ einen stochastischen Prozess.</p>
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<p><strong>Beispiele</strong></p>
<ul>
<li><p>Aktienkurse, Zinsen, Fx-Kurse</p>
</li>
<li><p>Warteschlange an der Supermarktkasse</p>
</li>
<li><p>Arbeitslosenzahlen</p>
</li>
<li><p>...</p>
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<p>Eine Realisierung / Beobachtung eines Stochastischen Prozess nennt man auch <strong>Pfad</strong>.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">dt</span> <span class="o">=</span> <span class="mf">1.</span><span class="o">/</span><span class="mf">100.</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">dt</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">):</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">dt</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">T</span><span class="p">,</span><span class="n">X</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">"T"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">"X_t"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">"Pfade eines stochastischen Prozess"</span><span class="p">)</span>
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<pre><matplotlib.text.Text at 0x73286d8></pre>
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<h3 id="Wiener-Prozess">Wiener Prozess<a class="anchor-link" href="#Wiener-Prozess">¶</a></h3><p>Eine besondere Klasse ist der Wiener Prozess. Er hat folgende Eigenschaften:</p>
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<li>$W(0) = 0$</li>
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<li>$W(t_2) - W(t_1) \sim N(0,t_2-t_1)$</li>
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<li>Differenzen sind i.i.d</li>
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<h2 id="Stochastische-Differentialgleichungen">Stochastische Differentialgleichungen<a class="anchor-link" href="#Stochastische-Differentialgleichungen">¶</a></h2><p>In der Bewertung von Finanzinstrumente spielen stochastiche Prozesse eine wichtige Rolle.</p>
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<h3 id="Beispiel-Call-Option">Beispiel Call Option<a class="anchor-link" href="#Beispiel-Call-Option">¶</a></h3><p>Wir haben eine Aktie, den Aktienkurs den wir beobachten folgt einem stochastischen Prozess $S_t$.</p>
<p>Eine Call-Option gibt dem Optionshalter das Recht zum Zeitpunkt $T$ (OptionExpiry) die Aktie zu einem festgelegten Kurs $K$ (Strike) zu kaufen.</p>
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<p>Zum Ausübungszeitpunkt ist der Wert der Option mit $V(T) = \max(0.0, S_T-K)$ gegeben.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">K</span> <span class="o">=</span> <span class="mi">130</span>
<span class="n">S_T</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="mi">160</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">S_T</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">fmax</span><span class="p">(</span><span class="n">S_T</span><span class="o">-</span><span class="n">K</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="p">)</span>
</pre></div>
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<div class="output_area"><div class="prompt output_prompt">Out[3]:</div>
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<pre>[<matplotlib.lines.Line2D at 0x7ac8898>]</pre>
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