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<h1 class="header-title">Autocallable Bonds</h1>
<p class="header-date"> <a href="https://bspeice.github.io/author/bradlee-speice.html">Bradlee Speice</a>, Fri 27 November 2015, <a href="https://bspeice.github.io/category/blog.html">Blog</a></p>
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<a href="https://bspeice.github.io/tag/finance.html">finance</a>, <a href="https://bspeice.github.io/tag/monte-carlo.html">monte carlo</a>, <a href="https://bspeice.github.io/tag/simulation.html">simulation</a> </p>
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<p>My only non-core class this semester has been in Structure Products. We've been surveying a wide variety of products, and the final project was to pick one to report on.
Because these are all very similar, we decided to demonstrate all 3 products at once.</p>
<p>What follows below is a notebook demonstrating the usage of <a href="http://julialang.com">Julia</a> for Monte-Carlo simulation of some exotic products.</p>
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<div class=" highlight hl-julia"><pre><span class="k">using</span> <span class="n">Gadfly</span>
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<h1 id="Athena/Phoenix-Simulation">Athena/Phoenix Simulation<a class="anchor-link" href="#Athena/Phoenix-Simulation">&#182;</a></h1><h2 id="Underlying-simulation">Underlying simulation<a class="anchor-link" href="#Underlying-simulation">&#182;</a></h2><p>In order to price the autocallable bonds, we need to simulate the underlying assets. Let's go ahead and set up the simulation first, as this lays the foundation for what we're trying to do. We're going to use <a href="http://finance.yahoo.com/q?s=jnj">JNJ</a> as the basis for our simulation. This implies the following parameters:</p>
<ul>
<li>$S_0$ = \$102.2 (as of time of writing)</li>
<li>$q$ = 2.84%</li>
<li>$r$ = [.49, .9, 1.21, 1.45, 1.69] (term structure as of time of writing, linear interpolation)</li>
<li>$\mu$ = $r - q$ (note that this implies a negative drift because of current low rates)</li>
<li>$\sigma$ = $\sigma_{imp}$ = 15.62% (from VIX implied volatility)</li>
</ul>
<p>We additionally define some parameters for simulation:</p>
<ul>
<li><code>T</code>: The number of years to simulate</li>
<li><code>m</code>: The number of paths to simulate</li>
<li><code>n</code>: The number of steps to simulate in a year</li>
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<div class=" highlight hl-julia"><pre><span class="n">S0</span> <span class="o">=</span> <span class="mf">102.2</span>
<span class="n">nominal</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">q</span> <span class="o">=</span> <span class="mf">2.84</span> <span class="o">/</span> <span class="mi">100</span>
<span class="n">σ</span> <span class="o">=</span> <span class="mf">15.37</span> <span class="o">/</span> <span class="mi">100</span>
<span class="n">term</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="o">.</span><span class="mi">49</span><span class="p">,</span> <span class="o">.</span><span class="mi">9</span><span class="p">,</span> <span class="mf">1.21</span><span class="p">,</span> <span class="mf">1.45</span><span class="p">,</span> <span class="mf">1.69</span><span class="p">]</span> <span class="o">/</span> <span class="mi">100</span> <span class="o">+</span> <span class="mi">1</span>
<span class="c">###</span>
<span class="c"># Potential: Based on PEP</span>
<span class="c"># S0 = 100.6</span>
<span class="c"># σ = 14.86</span>
<span class="c"># q = 2.7</span>
<span class="c">###</span>
<span class="c"># Simulation parameters</span>
<span class="n">T</span> <span class="o">=</span> <span class="mi">5</span> <span class="c"># Using years as the unit of time</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">250</span> <span class="c"># simulations per year</span>
<span class="n">m</span> <span class="o">=</span> <span class="mi">100000</span> <span class="c"># paths</span>
<span class="n">num_simulations</span> <span class="o">=</span> <span class="mi">5</span><span class="p">;</span> <span class="c"># simulation rounds per price</span>
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<pre>5</pre>
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<h3 id="Defining-the-simulation">Defining the simulation<a class="anchor-link" href="#Defining-the-simulation">&#182;</a></h3><p>To make things simpler, we simulate a single year at a time. This allows us to easily add in a dividend policy without too much difficulty, and update the simulation every year to match the term structure. The underlying uses GBM for simulation between years.</p>
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<div class=" highlight hl-julia"><pre><span class="n">simulate_gbm</span> <span class="o">=</span> <span class="n">function</span><span class="p">(</span><span class="n">S0</span><span class="p">,</span> <span class="n">μ</span><span class="p">,</span> <span class="n">σ</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="c"># Set the initial state</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">length</span><span class="p">(</span><span class="n">S0</span><span class="p">)</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">T</span> <span class="o">/</span> <span class="n">n</span>
<span class="n">motion</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="n">motion</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">S0</span>
<span class="c"># Build out all states</span>
<span class="k">for</span> <span class="n">i</span><span class="o">=</span><span class="mi">1</span><span class="p">:(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">motion</span><span class="p">[:,</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">motion</span><span class="p">[:,</span><span class="n">i</span><span class="p">]</span> <span class="o">.*</span> <span class="n">exp</span><span class="p">((</span><span class="n">μ</span> <span class="o">-</span> <span class="n">σ</span><span class="o">^</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="p">)</span> <span class="o">.*</span> <span class="n">exp</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">*</span> <span class="n">σ</span> <span class="o">.*</span> <span class="n">randn</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">motion</span>
<span class="k">end</span>
<span class="k">function</span><span class="nf"> display_motion</span><span class="p">(</span><span class="n">motion</span><span class="p">,</span> <span class="n">T</span><span class="p">)</span>
<span class="c"># Given a matrix of paths, display the motion</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">length</span><span class="p">(</span><span class="n">motion</span><span class="p">[</span><span class="mi">1</span><span class="p">,:])</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">length</span><span class="p">(</span><span class="n">motion</span><span class="p">[:,</span><span class="mi">1</span><span class="p">])</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">repmat</span><span class="p">(</span><span class="mi">1</span><span class="p">:</span><span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
<span class="c"># Calculate the ticks we&#39;re going to use. We&#39;d like to</span>
<span class="c"># have an xtick every month, so calculate where those</span>
<span class="c"># ticks will actually be at.</span>
<span class="k">if</span> <span class="p">(</span><span class="n">T</span> <span class="o">&gt;</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">num_ticks</span> <span class="o">=</span> <span class="n">T</span>
<span class="n">xlabel</span> <span class="o">=</span> <span class="s">&quot;Years&quot;</span>
<span class="k">else</span>
<span class="n">num_ticks</span> <span class="o">=</span> <span class="n">T</span> <span class="o">*</span> <span class="mi">12</span>
<span class="n">xlabel</span> <span class="o">=</span> <span class="s">&quot;Months&quot;</span>
<span class="k">end</span>
<span class="n">tick_width</span> <span class="o">=</span> <span class="n">n</span> <span class="o">/</span> <span class="n">num_ticks</span>
<span class="n">x_ticks</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span><span class="o">=</span><span class="mi">1</span><span class="p">:</span><span class="n">round</span><span class="p">(</span><span class="n">num_ticks</span><span class="p">)</span>
<span class="n">x_ticks</span> <span class="o">=</span> <span class="n">vcat</span><span class="p">(</span><span class="n">x_ticks</span><span class="p">,</span> <span class="n">i</span><span class="o">*</span><span class="n">tick_width</span><span class="p">)</span>
<span class="k">end</span>
<span class="c"># Use one color for each path. I&#39;m not sure if there&#39;s</span>
<span class="c"># a better way to do this without going through DataFrames</span>
<span class="n">colors</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="o">=</span> <span class="mi">1</span><span class="p">:</span><span class="n">m</span>
<span class="n">colors</span> <span class="o">=</span> <span class="n">vcat</span><span class="p">(</span><span class="n">colors</span><span class="p">,</span> <span class="n">ones</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">i</span><span class="p">)</span>
<span class="k">end</span>
<span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="n">motion</span><span class="o">&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">,</span> <span class="n">Geom</span><span class="o">.</span><span class="n">line</span><span class="p">,</span>
<span class="n">Guide</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">ticks</span><span class="o">=</span><span class="n">x_ticks</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">false</span><span class="p">),</span>
<span class="n">Guide</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="n">xlabel</span><span class="p">),</span>
<span class="n">Guide</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">&quot;Value&quot;</span><span class="p">))</span>
<span class="k">end</span><span class="p">;</span>
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<h3 id="Example-simulation">Example simulation<a class="anchor-link" href="#Example-simulation">&#182;</a></h3><p>Let's go ahead and run a sample simulation to see what the functions got us!</p>
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<div class=" highlight hl-julia"><pre><span class="n">initial</span> <span class="o">=</span> <span class="n">ones</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">*</span> <span class="n">S0</span>
<span class="c"># Using μ=0, T=.25 for now, we&#39;ll use the proper values later</span>
<span class="n">motion</span> <span class="o">=</span> <span class="n">simulate_gbm</span><span class="p">(</span><span class="n">initial</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">σ</span><span class="p">,</span> <span class="o">.</span><span class="mi">25</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
<span class="n">display_motion</span><span class="p">(</span><span class="n">motion</span><span class="p">,</span> <span class="o">.</span><span class="mi">25</span><span class="p">)</span>
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a)].toFixed(4)};n.toString=function(){return"matrix("+[this.get(0),this.get(1),this.get(2),this.get(3),this.get(4),this.get(5)].join()+")"};n.offset=function(){return[this.e.toFixed(4),this.f.toFixed(4)]};n.determinant=function(){return this.a*this.d-this.b*this.c};n.split=function(){var b={};b.dx=this.e;b.dy=this.f;var d=[[this.a,this.c],[this.b,this.d] ];b.scalex=f.sqrt(k(d[0]));p(d[0]);b.shear=d[0][0]*d[1][0]+d[0][1]*d[1][1];d[1]=[d[1][0]-d[0][0]*b.shear,d[1][1]-d[0][1]