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<!doctype html><html lang=en dir=ltr class="blog-wrapper blog-post-page plugin-blog plugin-id-default" data-has-hydrated=false><meta charset=UTF-8><meta name=generator content="Docusaurus v3.6.0"><title data-rh=true>Autocallable Bonds | The Old Speice Guy</title><meta data-rh=true name=viewport content="width=device-width,initial-scale=1.0"><meta data-rh=true name=twitter:card content=summary_large_image><meta data-rh=true property=og:url content=https://speice.io/2015/11/autocallable><meta data-rh=true property=og:locale content=en><meta data-rh=true name=docusaurus_locale content=en><meta data-rh=true name=docusaurus_tag content=default><meta data-rh=true name=docsearch:language content=en><meta data-rh=true name=docsearch:docusaurus_tag content=default><meta data-rh=true property=og:title content="Autocallable Bonds | The Old Speice Guy"><meta data-rh=true name=description content="For a final project, my group was tasked with understanding three exotic derivatives: The Athena, Phoenix without memory, and Phoenix with memory autocallable products."><meta data-rh=true property=og:description content="For a final project, my group was tasked with understanding three exotic derivatives: The Athena, Phoenix without memory, and Phoenix with memory autocallable products."><meta data-rh=true property=og:type content=article><meta data-rh=true property=article:published_time content=2015-11-27T12:00:00.000Z><link data-rh=true rel=icon href=/img/favicon.ico><link data-rh=true rel=canonical href=https://speice.io/2015/11/autocallable><link data-rh=true rel=alternate href=https://speice.io/2015/11/autocallable hreflang=en><link data-rh=true rel=alternate href=https://speice.io/2015/11/autocallable hreflang=x-default><script data-rh=true type=application/ld+json>{"@context":"https://schema.org","@id":"https://speice.io/2015/11/autocallable","@type":"BlogPosting","author":{"@type":"Person","name":"Bradlee Speice"},"dateModified":"2024-11-03T23:57:32.000Z","datePublished":"2015-11-27T12:00:00.000Z","description":"For a final project, my group was tasked with understanding three exotic derivatives: The Athena, Phoenix without memory, and Phoenix with memory autocallable products.","headline":"Autocallable Bonds","isPartOf":{"@id":"https://speice.io/","@type":"Blog","name":"Blog"},"keywords":[],"mainEntityOfPage":"https://speice.io/2015/11/autocallable","name":"Autocallable Bonds","url":"https://speice.io/2015/11/autocallable"}</script><link rel=alternate type=application/rss+xml href=/rss.xml title="The Old Speice Guy RSS Feed"><link rel=alternate type=application/atom+xml href=/atom.xml title="The Old Speice Guy Atom Feed"><link rel=stylesheet href=https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css integrity=sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM crossorigin><link rel=stylesheet href=/assets/css/styles.ae6ff4a3.css><script src=/assets/js/runtime~main.751b419d.js defer></script><script src=/assets/js/main.62ce6156.js defer></script><body class=navigation-with-keyboard><script>!function(){var t,e=function(){try{return new URLSearchParams(window.location.search).get("docusaurus-theme")}catch(t){}}()||function(){try{return window.localStorage.getItem("theme")}catch(t){}}();t=null!==e?e:"light",document.documentElement.setAttribute("data-theme",t)}(),function(){try{for(var[t,e]of new URLSearchParams(window.location.search).entries())if(t.startsWith("docusaurus-data-")){var a=t.replace("docusaurus-data-","data-");document.documentElement.setAttribute(a,e)}}catch(t){}}()</script><div id=__docusaurus><div role=region aria-label="Skip to main content"><a class=skipToContent_fXgn href=#__docusaurus_skipToContent_fallback>Skip to main content</a></div><nav aria-label=Main class="navbar navbar--fixed-top"><div class=navbar__inner><div class=navbar__items><button aria-label="Toggle navigation bar" aria-expanded=false class="navbar__toggle clean-btn" type=button><svg width=30 height=30 viewBox="0 0 30 30" aria-hidden=true><path stroke=currentColor stroke-linecap=round stroke-miterlimit=10 stroke-width
<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 target=_blank rel="noopener noreferrer">Julia</a> for Monte-Carlo simulation of some exotic products.</p>
<hr>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token keyword" style="color:hsl(301, 63%, 40%)">using</span><span class="token plain"> Gadfly</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<h1>Athena/Phoenix Simulation</h1>
<h2 class="anchor anchorWithStickyNavbar_LWe7" id=underlying-simulation>Underlying simulation<a href=#underlying-simulation class=hash-link aria-label="Direct link to Underlying simulation" title="Direct link to Underlying simulation">​</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" target=_blank rel="noopener noreferrer">JNJ</a> as the basis for our simulation. This implies the following parameters:</p>
<ul>
<li><span class=katex><span class=katex-mathml><math><semantics><mrow><msub><mi>S</mi><mn>0</mn></msub></mrow><annotation encoding=application/x-tex>S_0</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.8333em;vertical-align:-0.15em></span><span class=mord><span class="mord mathnormal" style=margin-right:0.05764em>S</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3011em><span style=top:-2.55em;margin-left:-0.0576em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class=vlist-s>​</span></span><span class=vlist-r><span class=vlist style=height:0.15em><span></span></span></span></span></span></span></span></span></span> = $102.2 (as of time of writing)</li>
<li><span class=katex><span class=katex-mathml><math><semantics><mrow><mi>q</mi></mrow><annotation encoding=application/x-tex>q</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.625em;vertical-align:-0.1944em></span><span class="mord mathnormal" style=margin-right:0.03588em>q</span></span></span></span> = 2.84%</li>
<li><span class=katex><span class=katex-mathml><math><semantics><mrow><mi>r</mi></mrow><annotation encoding=application/x-tex>r</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.4306em></span><span class="mord mathnormal" style=margin-right:0.02778em>r</span></span></span></span> = [.49, .9, 1.21, 1.45, 1.69] (term structure as of time of writing, linear interpolation)</li>
<li><span class=katex><span class=katex-mathml><math><semantics><mrow><mi>μ</mi></mrow><annotation encoding=application/x-tex>\mu</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.625em;vertical-align:-0.1944em></span><span class="mord mathnormal">μ</span></span></span></span> = <span class=katex><span class=katex-mathml><math><semantics><mrow><mi>r</mi><mo>−</mo><mi>q</mi></mrow><annotation encoding=application/x-tex>r - q</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.6667em;vertical-align:-0.0833em></span><span class="mord mathnormal" style=margin-right:0.02778em>r</span><span class=mspace style=margin-right:0.2222em></span><span class=mbin>−</span><span class=mspace style=margin-right:0.2222em></span></span><span class=base><span class=strut style=height:0.625em;vertical-align:-0.1944em></span><span class="mord mathnormal" style=margin-right:0.03588em>q</span></span></span></span> (note that this implies a negative drift because of current low rates)</li>
<li><span class=katex><span class=katex-mathml><math><semantics><mrow><mi>σ</mi></mrow><annotation encoding=application/x-tex>\sigma</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.4306em></span><span class="mord mathnormal" style=margin-right:0.03588em>σ</span></span></span></span> = <span class=katex><span class=katex-mathml><math><semantics><mrow><msub><mi>σ</mi><mrow><mi>i</mi><mi>m</mi><mi>p</mi></mrow></msub></mrow><annotation encoding=application/x-tex>\sigma_{imp}</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.7167em;vertical-align:-0.2861em></span><span class=mord><span class="mord mathnormal" style=margin-right:0.03588em>σ</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:-0.0359em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">im</span><span class="mord mathnormal mtight">p</span></span></span></span></span><span class=vlist-s>​</span></span><span class=vlist-r><span class=vlist style=height:0.2861em><span></span></span></span></span></span></span></span></span></span> = 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>
</ul>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">S0 </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">102.2</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">nominal </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">100</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">q </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">2.84</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">/</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">100</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">σ </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">15.37</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">/</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">100</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">term </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">.49</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">.9</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1.21</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1.45</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1.69</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">/</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">100</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token comment" style="color:hsl(230, 4%, 64%)">###</span><span class="token plain"></span><br></span>
<h3 class="anchor anchorWithStickyNavbar_LWe7" id=defining-the-simulation>Defining the simulation<a href=#defining-the-simulation class=hash-link aria-label="Direct link to Defining the simulation" title="Direct link to Defining the simulation">​</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>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">simulate_gbm </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">function</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">S0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> μ</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> σ</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> T</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> n</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)"># Set the initial state</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> m </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">S0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> t </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> T </span><span class="token operator" style="color:hsl(221, 87%, 60%)">/</span><span class="token plain"> n</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> motion </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> zeros</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">m</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> n</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> motion</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> S0</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)"># Build out all states</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">for</span><span class="token plain"> i</span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span cla
<h3 class="anchor anchorWithStickyNavbar_LWe7" id=example-simulation>Example simulation<a href=#example-simulation class=hash-link aria-label="Direct link to Example simulation" title="Direct link to Example simulation">​</a></h3>
<p>Let's go ahead and run a sample simulation to see what the functions got us!</p>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">initial </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> ones</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token number" style="color:hsl(35, 99%, 36%)">5</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> S0</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token comment" style="color:hsl(230, 4%, 64%)"># Using μ=0, T=.25 for now, we'll use the proper values later</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">motion </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> simulate_gbm</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">initial</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> σ</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">.25</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">200</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">display_motion</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">motion</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">.25</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<p><img decoding=async loading=lazy src=/assets/images/_notebook_6_0-46e660e38189a411644eac491e9b35ad.svg width=535 height=378 class=img_ev3q></p>
<h3 class="anchor anchorWithStickyNavbar_LWe7" id=computing-the-term-structure>Computing the term structure<a href=#computing-the-term-structure class=hash-link aria-label="Direct link to Computing the term structure" title="Direct link to Computing the term structure">​</a></h3>
<p>Now that we've got the basic motion set up, let's start making things a bit more sophisticated for the model. We're going to assume that the drift of the stock is the difference between the implied forward rate and the quarterly dividend rate.</p>
<p>We're given the yearly term structure, and need to calculate the quarterly forward rate to match this structure. The term structure is assumed to follow:</p>
<p><span class=katex><span class=katex-mathml><math><semantics><mrow><mi>d</mi><mo stretchy=false>(</mo><mn>0</mn><mo separator=true>,</mo><mi>t</mi><mo stretchy=false>)</mo><mo>=</mo><mi>d</mi><mo stretchy=false>(</mo><mn>0</mn><mo separator=true>,</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo stretchy=false>)</mo><mo>⋅</mo><msub><mi>f</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo separator=true>,</mo><mi>i</mi></mrow></msub></mrow><annotation encoding=application/x-tex>d(0, t) = d(0,t-1)\cdot f_{i-1, i}</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:1em;vertical-align:-0.25em></span><span class="mord mathnormal">d</span><span class=mopen>(</span><span class=mord>0</span><span class=mpunct>,</span><span class=mspace style=margin-right:0.1667em></span><span class="mord mathnormal">t</span><span class=mclose>)</span><span class=mspace style=margin-right:0.2778em></span><span class=mrel>=</span><span class=mspace style=margin-right:0.2778em></span></span><span class=base><span class=strut style=height:1em;vertical-align:-0.25em></span><span class="mord mathnormal">d</span><span class=mopen>(</span><span class=mord>0</span><span class=mpunct>,</span><span class=mspace style=margin-right:0.1667em></span><span class="mord mathnormal">t</span><span class=mspace style=margin-right:0.2222em></span><span class=mbin>−</span><span class=mspace style=margin-right:0.2222em></span></span><span class=base><span class=strut style=height:1em;vertical-align:-0.25em></span><span class=mord>1</span><span class=mclose>)</span><span class=mspace style=margin-right:0.2222em></span><span class=mbin>⋅</span><span class=mspace style=margin-right:0.2222em></span></span><span class=base><span class=strut style=height:0.9805em;vertical-align:-0.2861em></span><span class=mord><span class="mord mathnormal" style=margin-right:0.10764em>f</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:-0.1076em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class=vlist-s>​</span></span><span class=vlist-r><span class=vlist style=height:0.2861em><span></span></span></span></span></span></span></span></span></span></p>
<p>Where <span class=katex><span class=katex-mathml><math><semantics><mrow><msub><mi>f</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn><mo separator=true>,</mo><mi>i</mi></mrow></msub></mrow><annotation encoding=application/x-tex>f_{i-1, i}</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.9805em;vertical-align:-0.2861em></span><span class=mord><span class="mord mathnormal" style=margin-right:0.10764em>f</span><span class=msupsub><span class="vlist-t vlist-t2"><span class=vlist-r><span class=vlist style=height:0.3117em><span style=top:-2.55em;margin-left:-0.1076em;margin-right:0.05em><span class=pstrut style=height:2.7em></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class=vlist-s>​</span></span><span class=vlist-r><span class=vlist style=height:0.2861em><span></span></span></span></span></span></span></span></span></span> is the quarterly forward rate.</p>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">forward_term </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">function</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">yearly_term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)"># It is assumed that we have a yearly term structure passed in, and starts at year 0</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)"># This implies a nominal rate above 0 for the first year!</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> years </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)"># because we start at 0</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> structure </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">i</span><span class="token operator" style="color:hsl(221, 87%, 60%)">+</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">/</span><span class="token plain"> term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">i</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">for</span><span class="token plain"> i</span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token plain">years</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token keyword" style="color:hsl(301, 63%, 40%)">end</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">;</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn>
<h3 class="anchor anchorWithStickyNavbar_LWe7" id=illustrating-the-term-structure>Illustrating the term structure<a href=#illustrating-the-term-structure class=hash-link aria-label="Direct link to Illustrating the term structure" title="Direct link to Illustrating the term structure">​</a></h3>
<p>Now that we've got our term structure, let's validate that we're getting the correct results! If we've done this correctly, then:</p>
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">term[2] == term[1] * structure[1]</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token comment" style="color:hsl(230, 4%, 64%)"># Example term structure taken from:</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token comment" style="color:hsl(230, 4%, 64%)"># http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token comment" style="color:hsl(230, 4%, 64%)"># Linear interpolation used years in-between periods, assuming real-dollar</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token comment" style="color:hsl(230, 4%, 64%)"># interest rates</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">forward_yield </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> forward_term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">calculated_term2 </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> forward_yield</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token keyword" style="color:hsl(301, 63%, 40%)">println</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token string" style="color:hsl(119, 34%, 47%)">"Actual term[2]: $(term[2]); Calculated term[2]: $(calculated_term2)"</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Actual term[2]: 1.0049; Calculated term[2]: 1.0049</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<h3 class="anchor anchorWithStickyNavbar_LWe7" id=the-full-underlying-simulation>The full underlying simulation<a href=#the-full-underlying-simulation class=hash-link aria-label="Direct link to The full underlying simulation" title="Direct link to The full underlying simulation">​</a></h3>
<p>Now that we have the term structure set up, we can actually start doing some real simulation! Let's construct some paths through the full 5-year time frame. In order to do this, we will simulate 1 year at a time, and use the forward rates at those times to compute the drift. Thus, there will be 5 total simulations batched together.</p>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">full_motion </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> ones</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token number" style="color:hsl(35, 99%, 36%)">5</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> S0</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">full_term </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> vcat</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> forward_yield</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"></span><span class="token keyword" style="color:hsl(301, 63%, 40%)">for</span><span class="token plain"> i</span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token plain">T</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> μ </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">full_term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">i</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">-</span><span class="token plain"> q</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> year_motion </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> simulate_gbm</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">full_motion</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token keyword" style="color:hsl(301, 63%, 40%)">end</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> μ</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> σ</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</spa
<p><img decoding=async loading=lazy src=/assets/images/_notebook_12_0-1106bdfe947224ae55d7227b5a631d0e.svg width=535 height=378 class=img_ev3q></p>
<h3 class="anchor anchorWithStickyNavbar_LWe7" id=final-simulation>Final simulation<a href=#final-simulation class=hash-link aria-label="Direct link to Final simulation" title="Direct link to Final simulation">​</a></h3>
<p>We're now going to actually build out the full motion that we'll use for computing the pricing of our autocallable products. It will be largely the same, but we will use far more sample paths for the simulation.</p>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">full_simulation </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">function</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">S0</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> T</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> n</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> m</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> forward </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> vcat</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">]</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> forward_term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">term</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token comment" style="color:hsl(230, 4%, 64%)"># And an S0 to kick things off.</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> final_motion </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> ones</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">m</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"> </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> S0</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">for</span><span class="token plain"> i</span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token number" style="color:hsl(35, 99%, 36%)">1</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">:</span><span class="token plain">T</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> μ </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">forward</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">[</span><span class="token plain">i</span><span class="token punctuation" style="color:hsl(119, 34%
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Time to run simulation: 5.34s</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<h2 class="anchor anchorWithStickyNavbar_LWe7" id=athena-simulation>Athena Simulation<a href=#athena-simulation class=hash-link aria-label="Direct link to Athena Simulation" title="Direct link to Athena Simulation">​</a></h2>
<p>Now that we've defined our underlying simulation, let's actually try and price an Athena note. Athena has the following characteristics:</p>
<ul>
<li>Automatically called if the underlying is above the <strong>call barrier</strong> at observation</li>
<li>Accelerated coupon paid if the underlying is above the <strong>call barrier</strong> at observation<!-- -->
<ul>
<li>The coupon paid is <span class=katex><span class=katex-mathml><math><semantics><mrow><mi>c</mi><mo>⋅</mo><mi>i</mi></mrow><annotation encoding=application/x-tex>c \cdot i</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.4445em></span><span class="mord mathnormal">c</span><span class=mspace style=margin-right:0.2222em></span><span class=mbin>⋅</span><span class=mspace style=margin-right:0.2222em></span></span><span class=base><span class=strut style=height:0.6595em></span><span class="mord mathnormal">i</span></span></span></span> with <span class=katex><span class=katex-mathml><math><semantics><mrow><mi>i</mi></mrow><annotation encoding=application/x-tex>i</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.6595em></span><span class="mord mathnormal">i</span></span></span></span> as the current year, and <span class=katex><span class=katex-mathml><math><semantics><mrow><mi>c</mi></mrow><annotation encoding=application/x-tex>c</annotation></semantics></math></span><span class=katex-html aria-hidden=true><span class=base><span class=strut style=height:0.4306em></span><span class="mord mathnormal">c</span></span></span></span> the coupon rate</li>
</ul>
</li>
<li>Principle protection up until a <strong>protection barrier</strong> at observation; All principle at risk if this barrier not met</li>
<li>Observed yearly</li>
</ul>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">call_barrier </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> S0</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">strike </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> S0</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">protection_barrier </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> S0 </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">.6</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">coupon </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> nominal </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">.07</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">price_athena </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">function</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">initial_price</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> year_prices</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> call_barrier</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> protection_barrier</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> coupon</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> forward_structure</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> total_coupons </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> </span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> t </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">year_prices</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></s
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 1: $103.2805; Simulation time: 5.59s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 2: $103.3796; Simulation time: 5.05s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 3: $103.4752; Simulation time: 5.18s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 4: $103.4099; Simulation time: 5.37s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 5: $103.3260; Simulation time: 5.32s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean over 5 simulations: 103.37421610015554</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Present value of Athena note: $95.00, notional: $100.00</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<h2 class="anchor anchorWithStickyNavbar_LWe7" id=phoenix-without-memory-simulation>Phoenix without Memory Simulation<a href=#phoenix-without-memory-simulation class=hash-link aria-label="Direct link to Phoenix without Memory Simulation" title="Direct link to Phoenix without Memory Simulation">​</a></h2>
<p>Let's move into pricing a Phoenix without memory. It's very similar to the Athena production, with the exception that we introduce a coupon barrier so coupons are paid even when the underlying is below the initial price.</p>
<p>The Phoenix product has the following characteristics (example <a href=https://www.rbccm.com/usstructurednotes/file-780079.pdf target=_blank rel="noopener noreferrer">here</a>):</p>
<ul>
<li>Automatically called if the underlying is above the <strong>call barrier</strong> at observation</li>
<li>Coupon paid if the underlying is above a <strong>coupon barrier</strong> at observation</li>
<li>Principle protection up until a <strong>protection barrier</strong> at observation; All principle at risk if this barrier not met</li>
<li>Observed yearly</li>
</ul>
<p>Some example paths (all assume that a call barrier of the current price, and coupon barrier some level below that):</p>
<ul>
<li>At the end of year 1, the stock is above the call barrier; the note is called and you receive the value of the stock plus the coupon being paid.</li>
<li>At the end of year 1, the stock is above the coupon barrier, but not the call barrier; you receive the coupon. At the end of year 2, the stock is below the coupon barrier; you receive nothing. At the end of year 3, the stock is above the call barrier; the note is called and you receive the value of the stock plus a coupon for year 3.</li>
</ul>
<p>We're going to re-use the same simulation, with the following parameters:</p>
<ul>
<li>Call barrier: 100%</li>
<li>Coupon barrier: 70%</li>
<li>Coupon: 6%</li>
<li>Capital protection until 70% (at maturity)</li>
</ul>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">call_barrier </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> S0</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">coupon_barrier </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> S0 </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">.8</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">protection_barrier </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> S0 </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">.6</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">coupon </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> nominal </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">.06</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">price_phoenix_no_memory </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">function</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">initial_price</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> year_prices</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> call_barrier</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> coupon_barrier</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> protection_barrier</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> coupon</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> forward_structure</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> total_coupons </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> t </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> length</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">yea
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 1: $106.0562; Simulation time: 5.72s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 2: $106.0071; Simulation time: 5.85s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 3: $105.9959; Simulation time: 5.87s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 4: $106.0665; Simulation time: 5.93s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 5: $106.0168; Simulation time: 5.81s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean over 5 simulations: 106.02850857209883</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Present value of Phoenix without memory note: $97.44</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div>
<h2 class="anchor anchorWithStickyNavbar_LWe7" id=phoenix-with-memory-simulation>Phoenix with Memory Simulation<a href=#phoenix-with-memory-simulation class=hash-link aria-label="Direct link to Phoenix with Memory Simulation" title="Direct link to Phoenix with Memory Simulation">​</a></h2>
<p>The Phoenix with Memory structure is very similar to the Phoenix, but as the name implies, has a special "memory" property: <strong>It remembers any coupons that haven't been paid at prior observation times, and pays them all if the underlying crosses the coupon barrier</strong>. For example:</p>
<ul>
<li>Note issued with 100% call barrier, 70% coupon barrier. At year 1, the underlying is at 50%, so no coupons are paid. At year 2, the underlying is at 80%, so coupons for both year 1 and 2 are paid, resulting in a double coupon.</li>
</ul>
<p>You can also find an example <a href=https://www.rbccm.com/usstructurednotes/file-781232.pdf target=_blank rel="noopener noreferrer">here</a>.</p>
<p>Let's go ahead and set up the simulation! The parameters will be the same, but we can expect that the value will go up because of the memory attribute</p>
<div class="language-julia codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-julia codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">call_barrier </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> S0</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">coupon_barrier </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> S0 </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">.8</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">protection_barrier </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> S0 </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">.6</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">coupon </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> nominal </span><span class="token operator" style="color:hsl(221, 87%, 60%)">*</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">.07</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain">price_phoenix_with_memory </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token keyword" style="color:hsl(301, 63%, 40%)">function</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">(</span><span class="token plain">initial_price</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> year_prices</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> call_barrier</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> coupon_barrier</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> protection_barrier</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> coupon</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">,</span><span class="token plain"> forward_structure</span><span class="token punctuation" style="color:hsl(119, 34%, 47%)">)</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain" style=display:inline-block></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> last_coupon </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token plain"></span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> total_coupons </span><span class="token operator" style="color:hsl(221, 87%, 60%)">=</span><span class="token plain"> </span><span class="token number" style="color:hsl(35, 99%, 36%)">0</span><span class="token plain"></span>
<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-background-color:hsl(230, 1%, 98%);--prism-color:hsl(230, 8%, 24%)"><div class=codeBlockContent_biex><pre tabindex=0 class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="background-color:hsl(230, 1%, 98%);color:hsl(230, 8%, 24%)"><code class=codeBlockLines_e6Vv><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 1: $108.8612; Simulation time: 5.89s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 2: $109.0226; Simulation time: 5.90s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 3: $108.9175; Simulation time: 5.92s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 4: $108.9426; Simulation time: 5.94s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean of simulation 5: $108.8087; Simulation time: 6.06s</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Mean over 5 simulations: 108.91052564051816</span><br></span><span class=token-line style="color:hsl(230, 8%, 24%)"><span class="token plain"> Present value of Phoenix with memory note: $100.09</span><br></span></code></pre><div class=buttonGroup__atx><button type=button aria-label="Copy code to clipboard" title=Copy class=clean-btn><span class=copyButtonIcons_eSgA aria-hidden=true><svg viewBox="0 0 24 24" class=copyButtonIcon_y97N><path fill=currentColor d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"/></svg><svg viewBox="0 0 24 24" class=copyButtonSuccessIcon_LjdS><path fill=currentColor d=M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z /></svg></span></button></div></div></div></div></article><nav class="pagination-nav docusaurus-mt-lg" aria-label="Blog post page navigation"><a class="pagination-nav__link pagination-nav__link--prev" href=/2015/11/welcome><div class=pagination-nav__sublabel>Older post</div><div class=pagination-nav__label>Welcome, and an algorithm</div></a><a class="pagination-nav__link pagination-nav__link--next" href=/2015/12/testing-cramer><div class=pagination-nav__sublabel>Newer post</div><div class=pagination-nav__label>Testing Cramer</div></a></nav></main><div class="col col--2"><div class="tableOfContents_bqdL thin-scrollbar"><ul class="table-of-contents table-of-contents__left-border"><li><a href=#underlying-simulation class="table-of-contents__link toc-highlight">Underlying simulation</a><ul><li><a href=#defining-the-simulation class="table-of-contents__link toc-highlight">Defining the simulation</a><li><a href=#example-simulation class="table-of-contents__link toc-highlight">Example simulation</a><li><a href=#computing-the-term-structure class="table-of-contents__link toc-highlight">Computing the term structure</a><li><a href=#illustrating-the-term-structure class="table-of-contents__link toc-highlight">Illustrating the term structure</a><li><a href=#the-full-underlying-simulation class="table-of-contents__link toc-highlight">The full underlying simulation</a><li><a href=#final-simulation class="table-of-contents__link toc-highlight">Final simulation</a></ul><li><a href=#athena-simulation class="table-of-contents__link toc-highlight">Athena Simulation</a><li><a href=#phoenix-without-memory-simulation class="table-of-contents__link toc-highlight">Phoenix without Memory Simulation</a><li><a href=#phoenix-with-memory-simulation class="table-of-contents__link toc-highlight">Phoenix with Memory Simulation</a></ul></div></div></div></div></div><footer class=footer><div class="container container-fluid"><div class="footer__bottom text--center"><div class=footer__copyright>Copyright © 2024 Bradlee Speice</div></div></div></footer></div>
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