1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
<meta http-equiv="Content-Type" content="text/html;charset=UTF-8">
<meta name="robots" content="none">
<title>CDO Class Reference</title>
<link rel="stylesheet" href="quantlib.css" type="text/css">
<link rel="stylesheet" href="print.css" type="text/css" media="print">
<link rel="shortcut icon" href="favicon.ico" type="image/x-icon">
<link rel="icon" href="favicon.ico" type="image/x-icon">
</head>
<body>
<div id="container">
<div id="header">
<img class="titleimage"
src="QL-title.jpg" width="185" height="50" border="0"
alt="QuantLib">
<br>
<h3 class="subtitle">A free/open-source library for quantitative finance</h3>
</div>
<div id="menu">
<h3 class="navbartitle">Version 1.2</h3>
<hr>
<h3 class="navbartitle">Getting started</h3>
<ul class="navbarlist">
<li class="navlink"><a href="index.html">Introduction</a></li>
<li class="navlink"><a href="where.html">Where to get QuantLib</a></li>
<li class="navlink"><a href="install.html">Installation</a></li>
<li class="navlink"><a href="config.html">Configuration</a></li>
<li class="navlink"><a href="usage.html">Usage</a></li>
<li class="navlink"><a href="history.html">Version history</a></li>
<li class="navlink"><a href="resources.html">Additional resources</a></li>
<li class="navlink"><a href="group.html">The QuantLib group</a></li>
<li class="navlink"><a href="license.html">Copyright and license</a></li>
</ul>
<hr>
<h3 class="navbartitle">Reference manual</h3>
<ul class="navbarlist">
<li class="navlink"><a href="modules.html">Modules</a></li>
<li class="navlink"><a href="hierarchy.html">Class Hierarchy</a></li>
<li class="navlink"><a href="annotated.html">Compound List</a></li>
<li class="navlink"><a href="files.html">File List</a></li>
<li class="navlink"><a href="functions.html">Compound Members</a></li>
<li class="navlink"><a href="globals.html">File Members</a></li>
<li class="navlink"><a href="todo.html">Todo List</a></li>
<li class="navlink"><a href="bug.html">Known Bugs</a></li>
<li class="navlink"><a href="caveats.html">Caveats</a></li>
<li class="navlink"><a href="test.html">Test Suite</a></li>
<li class="navlink"><a href="examples.html">Examples</a></li>
</ul>
</div>
<div id="content">
<!--Doxygen-generated content-->
<!-- Generated by Doxygen 1.7.6.1 -->
<div id="nav-path" class="navpath">
<ul>
<li class="navelem"><b>QuantLib</b> </li>
<li class="navelem"><a class="el" href="class_quant_lib_1_1_c_d_o.html">CDO</a> </li>
</ul>
</div>
</div>
<div class="header">
<div class="summary">
<a href="#pub-methods">Public Member Functions</a> </div>
<div class="headertitle">
<div class="title">CDO Class Reference</div> </div>
</div><!--header-->
<div class="contents">
<!-- doxytag: class="QuantLib::CDO" --><!-- doxytag: inherits="QuantLib::Instrument" -->
<p>collateralized debt obligation
<a href="class_quant_lib_1_1_c_d_o.html#details">More...</a></p>
<p><code>#include <ql/experimental/credit/cdo.hpp></code></p>
<div class="dynheader">
Inheritance diagram for CDO:</div>
<div class="dyncontent">
<div class="center"><img src="class_quant_lib_1_1_c_d_o__inherit__graph.png" border="0" usemap="#_c_d_o_inherit__map" alt="Inheritance graph"/></div>
<map name="_c_d_o_inherit__map" id="_c_d_o_inherit__map">
<area shape="rect" id="node2" href="class_quant_lib_1_1_instrument.html" title="Abstract instrument class." alt="" coords="5,6,88,37"/></map>
<center><span class="legend">[<a href="graph_legend.html">legend</a>]</span></center></div>
<p><a href="class_quant_lib_1_1_c_d_o-members.html">List of all members.</a></p>
<table class="memberdecls">
<tr><td colspan="2"><h2><a name="pub-methods"></a>
Public Member Functions</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_c_d_o.html#abf537615f823c0c8b1201e35af0abbc0">CDO</a> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> attachment, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> detachment, const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > &nominals, const std::vector< <a class="el" href="class_quant_lib_1_1_handle.html">Handle</a>< <a class="el" href="class_quant_lib_1_1_default_probability_term_structure.html">DefaultProbabilityTermStructure</a> > > &basket, const <a class="el" href="class_quant_lib_1_1_handle.html">Handle</a>< <a class="el" href="class_quant_lib_1_1_one_factor_copula.html">OneFactorCopula</a> > &copula, bool protectionSeller, const <a class="el" href="class_quant_lib_1_1_schedule.html">Schedule</a> &premiumSchedule, <a class="el" href="group__types.html#gaede435af51236692b1107d7639581d39">Rate</a> premiumRate, const <a class="el" href="class_quant_lib_1_1_day_counter.html">DayCounter</a> &dayCounter, <a class="el" href="group__types.html#gaede435af51236692b1107d7639581d39">Rate</a> recoveryRate, <a class="el" href="group__types.html#gaede435af51236692b1107d7639581d39">Rate</a> upfrontPremiumRate, const <a class="el" href="class_quant_lib_1_1_handle.html">Handle</a>< <a class="el" href="class_quant_lib_1_1_yield_term_structure.html">YieldTermStructure</a> > &yieldTS, <a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> nBuckets, const <a class="el" href="class_quant_lib_1_1_period.html">Period</a> &integrationStep=<a class="el" href="class_quant_lib_1_1_period.html">Period</a>(10, Years))</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a29092d9d6550ef55d3acdaadc5672b49"></a><!-- doxytag: member="QuantLib::CDO::nominal" ref="a29092d9d6550ef55d3acdaadc5672b49" args="()" -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>nominal</b> ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="ab1e1ba3b401ff50b64f835425a53259f"></a><!-- doxytag: member="QuantLib::CDO::lgd" ref="ab1e1ba3b401ff50b64f835425a53259f" args="()" -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>lgd</b> ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a9f640384e88e9a3f579443135bc90d84"></a><!-- doxytag: member="QuantLib::CDO::attachment" ref="a9f640384e88e9a3f579443135bc90d84" args="()" -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>attachment</b> ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a4700a808d9f4ab060d54fbe7ffc15f6b"></a><!-- doxytag: member="QuantLib::CDO::detachment" ref="a4700a808d9f4ab060d54fbe7ffc15f6b" args="()" -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>detachment</b> ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a90837b8aa099984148ccd2c6f2f483bd"></a><!-- doxytag: member="QuantLib::CDO::nominals" ref="a90837b8aa099984148ccd2c6f2f483bd" args="()" -->
std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > </td><td class="memItemRight" valign="bottom"><b>nominals</b> ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a9a4c32403766cd16ca4e3a5479f075bd"></a><!-- doxytag: member="QuantLib::CDO::size" ref="a9a4c32403766cd16ca4e3a5479f075bd" args="()" -->
<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> </td><td class="memItemRight" valign="bottom"><b>size</b> ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a274c03751addc5c2ea63cc23d14a0bfe"></a><!-- doxytag: member="QuantLib::CDO::isExpired" ref="a274c03751addc5c2ea63cc23d14a0bfe" args="() const " -->
bool </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_c_d_o.html#a274c03751addc5c2ea63cc23d14a0bfe">isExpired</a> () const </td></tr>
<tr><td class="mdescLeft"> </td><td class="mdescRight">returns whether the instrument might have value greater than zero. <br/></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a439b4331763c1772af92bec6b182c62c"></a><!-- doxytag: member="QuantLib::CDO::fairPremium" ref="a439b4331763c1772af92bec6b182c62c" args="() const " -->
<a class="el" href="group__types.html#gaede435af51236692b1107d7639581d39">Rate</a> </td><td class="memItemRight" valign="bottom"><b>fairPremium</b> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a8729d80e43b77566cbdf0cfe0b9b93dd"></a><!-- doxytag: member="QuantLib::CDO::premiumValue" ref="a8729d80e43b77566cbdf0cfe0b9b93dd" args="() const " -->
<a class="el" href="group__types.html#gaede435af51236692b1107d7639581d39">Rate</a> </td><td class="memItemRight" valign="bottom"><b>premiumValue</b> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="abb77e8f956fe2e7458681bef8f73f9d0"></a><!-- doxytag: member="QuantLib::CDO::protectionValue" ref="abb77e8f956fe2e7458681bef8f73f9d0" args="() const " -->
<a class="el" href="group__types.html#gaede435af51236692b1107d7639581d39">Rate</a> </td><td class="memItemRight" valign="bottom"><b>protectionValue</b> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="afca3e3cf414a77cea53374f408cf8e0a"></a><!-- doxytag: member="QuantLib::CDO::error" ref="afca3e3cf414a77cea53374f408cf8e0a" args="() const " -->
<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> </td><td class="memItemRight" valign="bottom"><b>error</b> () const </td></tr>
</table>
<hr/><a name="details" id="details"></a><h2>Detailed Description</h2>
<div class="textblock"><p>collateralized debt obligation </p>
<p>The instrument prices a mezzanine <a class="el" href="class_quant_lib_1_1_c_d_o.html" title="collateralized debt obligation">CDO</a> tranche with loss given default between attachment point <img class="formulaInl" alt="$ D_1$" src="form_66.png"/> and detachment point <img class="formulaInl" alt="$ D_2 > D_1 $" src="form_67.png"/>.</p>
<p>For purchased protection, the instrument value is given by the difference of the protection value <img class="formulaInl" alt="$ V_1 $" src="form_68.png"/> and premium value <img class="formulaInl" alt="$ V_2 $" src="form_69.png"/>,</p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ V = V_1 - V_2. \]" src="form_70.png"/>
</p>
<p>The protection leg is priced as follows:</p>
<ul>
<li>Build the probability distribution for volume of defaults <img class="formulaInl" alt="$ L $" src="form_46.png"/> (before recovery) or Loss Given Default <img class="formulaInl" alt="$ LGD = (1-r)\,L $" src="form_71.png"/> at times/dates <img class="formulaInl" alt="$ t_i, i=1, ..., N$" src="form_72.png"/> (premium schedule times with intermediate steps)</li>
<li>Determine the expected value <img class="formulaInl" alt="$ E_i = E_{t_i}\,\left[Pay(LGD)\right] $" src="form_73.png"/> of the protection payoff <img class="formulaInl" alt="$ Pay(LGD) $" src="form_74.png"/> at each time <img class="formulaInl" alt="$ t_i$" src="form_75.png"/> where <p class="formulaDsp">
<img class="formulaDsp" alt="\[ Pay(L) = min (D_1, LGD) - min (D_2, LGD) = \left\{ \begin{array}{lcl} \displaystyle 0 &;& LGD < D_1 \\ \displaystyle LGD - D_1 &;& D_1 \leq LGD \leq D_2 \\ \displaystyle D_2 - D_1 &;& LGD > D_2 \end{array} \right. \]" src="form_76.png"/>
</p>
</li>
<li>The protection value is then calculated as <p class="formulaDsp">
<img class="formulaDsp" alt="\[ V_1 \:=\: \sum_{i=1}^N (E_i - E_{i-1}) \cdot d_i \]" src="form_77.png"/>
</p>
where <img class="formulaInl" alt="$ d_i$" src="form_78.png"/> is the discount factor at time/date <img class="formulaInl" alt="$ t_i $" src="form_57.png"/></li>
</ul>
<p>The premium is paid on the protected notional amount, initially <img class="formulaInl" alt="$ D_2 - D_1. $" src="form_79.png"/> This notional amount is reduced by the expected protection payments <img class="formulaInl" alt="$ E_i $" src="form_80.png"/> at times <img class="formulaInl" alt="$ t_i, $" src="form_81.png"/> so that the premium value is calculated as</p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ V_2 = m \, \cdot \sum_{i=1}^N \,(D_2 - D_1 - E_i) \cdot \Delta_{i-1,i}\,d_i \]" src="form_82.png"/>
</p>
<p>where <img class="formulaInl" alt="$ m $" src="form_39.png"/> is the premium rate, <img class="formulaInl" alt="$ \Delta_{i-1, i}$" src="form_83.png"/> is the day count fraction between date/time <img class="formulaInl" alt="$ t_{i-1}$" src="form_84.png"/> and <img class="formulaInl" alt="$ t_i.$" src="form_85.png"/></p>
<p>The construction of the portfolio loss distribution <img class="formulaInl" alt="$ E_i $" src="form_80.png"/> is based on the probability bucketing algorithm described in</p>
<p><b> John Hull and Alan White, "Valuation of a CDO and nth to default CDS
without Monte Carlo simulation", Journal of Derivatives 12, 2, 2004 </b></p>
<p>The pricing algorithm allows for varying notional amounts and default termstructures of the underlyings.</p>
<dl class="todo"><dt><b><a class="el" href="todo.html#_todo000006">Possible enhancements:</a></b></dt><dd>Investigate and fix cases <img class="formulaInl" alt="$ E_{i+1} < E_i. $" src="form_86.png"/> </dd></dl>
</div><hr/><h2>Constructor & Destructor Documentation</h2>
<a class="anchor" id="abf537615f823c0c8b1201e35af0abbc0"></a><!-- doxytag: member="QuantLib::CDO::CDO" ref="abf537615f823c0c8b1201e35af0abbc0" args="(Real attachment, Real detachment, const std::vector< Real > &nominals, const std::vector< Handle< DefaultProbabilityTermStructure > > &basket, const Handle< OneFactorCopula > &copula, bool protectionSeller, const Schedule &premiumSchedule, Rate premiumRate, const DayCounter &dayCounter, Rate recoveryRate, Rate upfrontPremiumRate, const Handle< YieldTermStructure > &yieldTS, Size nBuckets, const Period &integrationStep=Period(10, Years))" -->
<div class="memitem">
<div class="memproto">
<table class="memname">
<tr>
<td class="memname"><a class="el" href="class_quant_lib_1_1_c_d_o.html">CDO</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>attachment</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>detachment</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > & </td>
<td class="paramname"><em>nominals</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const std::vector< <a class="el" href="class_quant_lib_1_1_handle.html">Handle</a>< <a class="el" href="class_quant_lib_1_1_default_probability_term_structure.html">DefaultProbabilityTermStructure</a> > > & </td>
<td class="paramname"><em>basket</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const <a class="el" href="class_quant_lib_1_1_handle.html">Handle</a>< <a class="el" href="class_quant_lib_1_1_one_factor_copula.html">OneFactorCopula</a> > & </td>
<td class="paramname"><em>copula</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">bool </td>
<td class="paramname"><em>protectionSeller</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const <a class="el" href="class_quant_lib_1_1_schedule.html">Schedule</a> & </td>
<td class="paramname"><em>premiumSchedule</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="el" href="group__types.html#gaede435af51236692b1107d7639581d39">Rate</a> </td>
<td class="paramname"><em>premiumRate</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const <a class="el" href="class_quant_lib_1_1_day_counter.html">DayCounter</a> & </td>
<td class="paramname"><em>dayCounter</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="el" href="group__types.html#gaede435af51236692b1107d7639581d39">Rate</a> </td>
<td class="paramname"><em>recoveryRate</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="el" href="group__types.html#gaede435af51236692b1107d7639581d39">Rate</a> </td>
<td class="paramname"><em>upfrontPremiumRate</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const <a class="el" href="class_quant_lib_1_1_handle.html">Handle</a>< <a class="el" href="class_quant_lib_1_1_yield_term_structure.html">YieldTermStructure</a> > & </td>
<td class="paramname"><em>yieldTS</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> </td>
<td class="paramname"><em>nBuckets</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const <a class="el" href="class_quant_lib_1_1_period.html">Period</a> & </td>
<td class="paramname"><em>integrationStep</em> = <code><a class="el" href="class_quant_lib_1_1_period.html">Period</a>(10, Years)</code> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div>
<div class="memdoc">
<dl class="params"><dt><b>Parameters:</b></dt><dd>
<table class="params">
<tr><td class="paramname">attachment</td><td>fraction of the LGD where protection starts </td></tr>
<tr><td class="paramname">detachment</td><td>fraction of the LGD where protection ends </td></tr>
<tr><td class="paramname">nominals</td><td>vector of basket nominal amounts </td></tr>
<tr><td class="paramname">basket</td><td>default basket represented by a vector of default term structures that allow computing single name default probabilities depending on time </td></tr>
<tr><td class="paramname">copula</td><td>one-factor copula </td></tr>
<tr><td class="paramname">protectionSeller</td><td>sold protection if set to true, purchased otherwise </td></tr>
<tr><td class="paramname">premiumSchedule</td><td>schedule for premium payments </td></tr>
<tr><td class="paramname">premiumRate</td><td>annual premium rate, e.g. 0.05 for 5% p.a. </td></tr>
<tr><td class="paramname">dayCounter</td><td>day count convention for the premium rate </td></tr>
<tr><td class="paramname">recoveryRate</td><td>recovery rate as a fraction </td></tr>
<tr><td class="paramname">upfrontPremiumRate</td><td>premium as a tranche notional fraction </td></tr>
<tr><td class="paramname">yieldTS</td><td>yield term structure handle </td></tr>
<tr><td class="paramname">nBuckets</td><td>number of distribution buckets </td></tr>
<tr><td class="paramname">integrationStep</td><td>time step for integrating over one premium period; if larger than premium period length, a single step is taken </td></tr>
</table>
</dd>
</dl>
</div>
</div>
</div><!-- contents -->
</div>
<div class="footer">
<div class="endmatter">
Documentation generated by
<a href="http://www.doxygen.org">Doxygen</a> 1.7.6.1
</div>
</div>
</div>
</body>
</html>
|
