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MRMM-1-volatility-smiles.html
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<!doctype html>
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<head>
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<title>Market Risk Measurement and Management | Chapter 1 | Volatility Smiles</title>
<meta name="description" content="Financial Risk Manager Part 2 Study Materials">
<meta name="author" content="MacLane Wilkison">
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<section>
<h1>Chapter 1</h1>
<h3>Volatility Smiles</h3>
<p>
<small>Created for <a href="http://alchemistsacademy.com">Alchemists Academy</a> by <a href="http://alchemistsacademy.com/about">MacLane Wilkison</a></small>
</p>
</section>
<section>
<h2>Introduction</h2>
<p><em>Definition: A plot of the implied volatility of an option with a certain life as a function of its strike price</em></p>
</section>
<section>
<h2>Volatility Smile for Calls and Puts</h2>
<ul>
<li>p + S<sub>0</sub>e<sup>-qT</sup> = c + Ke<sup>-rT</sup></li>
<li>For a given strike price and maturity, the implied volatilities of a European put and call option will be equal</li>
</ul>
<aside class="notes">
'c' is the European call price. 'p' is the European put price. 'K' is the strike price. 'T' is the time to maturity. S<sub>0</sub> is the price of the underlying asset, 'r' is the risk-free rate. 'q' is the yield on the asset.
</aside>
</section>
<section>
<h2>Foreign Currency Options</h2>
<ul>
<li>In practice, an option's implied distribution has heavier tails and a higher peak than the lognormal distribution assumed under Black-Scholes</li>
<li>Why?</li>
<ol>
<li>The volatility of exchange rates are not constant</li>
<li>Assets exhibit frequent "jumps" in price</li>
</ol>
</ul>
<aside class="notes">
Lognormal distributions require that (1) the volatility of the asset is constant and (2) the price of the asset changes smoothly with no jumps
</aside>
</section>
<section>
<h2>Equity Options</h2>
<ul>
<li>Volatility skew</li>
<li>Volatility decreases as strike price increases</li>
</ul>
<aside class="notes">
The implied distribution has a heavier left tail and a less heavy right tail than the lognormal distribution
</aside>
</section>
<section>
<h2>Volatility Term Structure</h2>
<ul>
<li>Low short-dated volatility -> implied volatility increases with maturity</li>
<li>High short-dated volatility -> implied volatility decreases with maturity</li>
<li>Illustrative volatility surface table:</li>
</ul>
<img src="images/MRMM1/volatility-surface.png" alt="illustrative volatility surface table" />
<aside class="notes">
When short-dated volatility is low, implied volatility is an increasing function of maturity (volatility is expected to rise). When short-dated volatility is high, implied volatility is a decreasing function of maturity (volatility is expected to fall). The volatility surface is simply an analysis combining volatility smiles and volatility term structures.
</aside>
</section>
<section>
<h1>THE END</h1>
<h3><a href="http://alchemistsacademy.com">AlchemistsAcademy.com</a></h3>
</section>
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