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MRMM-6-the-art-of-term-structure-models-volatility-and-distribution.html
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<!doctype html>
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<title>Market Risk Measurement and Management | Chapter 6 | The Art of Term Structure Models: Volatility and Distribution</title>
<meta name="description" content="Financial Risk Manager Part 2 Study Materials">
<meta name="author" content="MacLane Wilkison">
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<h1>Chapter 6</h1>
<h3>The Art of Term Structure Models: Volatility and Distribution</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>Model 3: Time-Dependent Volatility</h2>
<p>dr = λ<sub>t</sub>dt + σ<sub>t</sub>dw = λ<sub>t</sub>dt + σe<sup>-αt</sup>dw</p>
<ul>
<li>α is the decay rate</li>
</ul>
<aside class="notes">
Like the time-dependent volatility model, this model is used to quote fixed income options prices that are not easily observable.
</aside>
</section>
<section>
<h2>Lognormal Models</h2>
<ul>
<li>Cox-Ingersoll-Ross: dr = k(θ-r)dt + σ√(r)dw</li>
<li>Courtadon: dr = k(θ-r)dt + σrdw</li>
<li>Model 4: dr = ardt + σrdw
<li>Black-Karasinski: dr = k<sub>t</sub>[ln(θ<sub>t</sub>)-ln(r)]dt + σ<sub>t</sub>rdw</li>
</ul>
<aside class="notes">
These models specify the basis-point volatility of the short rate as an increasing function of the short rate. The economic justification for this is that, during periods of high inflation and high short-term interest rates, the basis-point volatility of the short-rate tends to be high due to the inherenet instability of the macro-environment. The Black-Karasinski model allows volatility, mean reversion, and the central tendency of the short rate to be time-dependent.
</aside>
</section>
<section>
<h1>THE END</h1>
<h3><a href="http://alchemistsacademy.com">AlchemistsAcademy.com</a></h3>
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