{"categories":["Credit"],"contentHtml":"<h2>Dependence cases</h2>\n<p>Relevant variables for these credits</p>\n<p><strong>Page 4</strong></p>\n<p>Let’s look at, let’s say, minimum dependence:</p>\n<p>In what situation might we have such minimum dependence?</p>\n<p>The k where we have the lowest probability that we have both names defaulting.</p>\n<p>Both of these credits have an idiosyncratic component and a market exposure z.</p>\n<p>So one way we can achieve minimum dependence is to have an opposite exposure to z.</p>\n<p>If we have a large negative exposure to z, we’ll say beta is large. For credit j, we can change the exposure to market variable z, and instead of having a large negative number, it becomes a large positive numbers, and we lower the probability of default.</p>\n<p>For minimum dependence, if we take it in the <strong>limit</strong>, for β_i = -β_j</p>\n<p>In the limit, we can take β_i = 1, β_j = -1</p>\n<p>One has 100% exposure to z, the other has negative exposure to z, and both of them have zero idiosyncratic component.</p>\n<p>So correlation is inverse, ρ = -1.</p>\n<p>In this case, such probability is dependent on the survival probability of the names, \\((1 - Q_i(T) - Q_j(T))_+\\)</p>\n<p>The joint probability of default is zero as long as Q_i(T) + Q_j(T) &gt; 1</p>\n<p>Independence means correlation is 0. And to do that, we have to remove market exposure. So in the limit, we take</p>\n<p>β_i = β_j = 0, which means the correlation is zero, it’s entirely idiosyncratic.</p>\n<p>Maximum Dependence is when β_i = β_j = 1, maximum market exposure. It’s the minimum of either 1 - Q_i(T), 1 - Q_j(T).</p>\n<p>If you want to illustrate this,</p>\n<p><strong>Slide 5</strong></p>","contentMarkdown":"## Dependence cases\n\nRelevant variables for these credits\n\n**Page 4**\n\nLet’s look at, let’s say, minimum dependence:\n\nIn what situation might we have such minimum dependence?\n\nThe k where we have the lowest probability that we have both names defaulting.\n\nBoth of these credits have an idiosyncratic component and a market exposure z.\n\nSo one way we can achieve minimum dependence is to have an opposite exposure to z.\n\nIf we have a large negative exposure to z, we’ll say beta is large. For credit j, we can change the exposure to market variable z, and instead of having a large negative number, it becomes a large positive numbers, and we lower the probability of default.\n\nFor minimum dependence, if we take it in the **limit**, for β\\_i \\= \\-β\\_j\n\nIn the limit, we can take β\\_i \\= 1, β\\_j \\= \\-1\n\nOne has 100% exposure to z, the other has negative exposure to z, and both of them have zero idiosyncratic component.\n\nSo correlation is inverse, ρ \\= \\-1.\n\nIn this case, such probability is dependent on the survival probability of the names, \\((1 - Q_i(T) - Q_j(T))_+\\)\n\nThe joint probability of default is zero as long as Q\\_i(T) \\+ Q\\_j(T) \\> 1\n\nIndependence means correlation is 0\\. And to do that, we have to remove market exposure. So in the limit, we take\n\nβ\\_i \\= β\\_j \\= 0, which means the correlation is zero, it’s entirely idiosyncratic.\n\nMaximum Dependence is when β\\_i \\= β\\_j \\= 1, maximum market exposure. It’s the minimum of either 1 \\- Q\\_i(T), 1 \\- Q\\_j(T).\n\nIf you want to illustrate this,\n\n**Slide 5**","dataUrl":"https://sharifhsn.dev/api/posts/advanced-derivatives-week-12.json","date":"2025-04-17","datePublished":"2025-04-17","description":"Relevant variables for these credits","site":"https://sharifhsn.dev","slug":"advanced-derivatives-week-12","source":"Advanced Derivatives","sourceUrl":null,"tags":["Credit","Dependence","Copulas"],"title":"Dependence and Copulas","url":"https://sharifhsn.dev/blog/advanced-derivatives-week-12/","version":"1","wordCount":270}