LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Copula function inmultivariate dependency analysisJSFHIgor HlivkaMUFJ Securities International, LONDONLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Copula is a constructor function for multivariate distribution from univariate marginals. It is a method to link univariate samples, not necessarily from identical distributions, into joint multivariate distributions. In this way, copulas are more generic and flexible functions to study dependency arising from multivariate distributions. Conceptually, copulas are based on transformation of the underlying marginal into new derived variable with uniform distribution. Consequently, any multivariate distribution can be expressed in the form of copula function. If each marginal is continuous then copula is unique. Sklar in 1959 was the first to point this out.Copulas represent a broad set of functions and they generally differ by (i) number of dependency factors and (ii) construction complexity. The choose of copula depends on the nature of the multivariate study and fitting objectives to an underlying data.Copulas have been originally applied in reliability and actuarial analysis. For the past 10 years dependency feature in copulas attracted much attention in the financial risk modeling and they were heavily used in valuation of now defunct collateralised debt obligations (CDO). Heavy dependence on one particular copula class - Gaussian - is now frequently cited as a reason for the credit-led financial crisis in the past several years. Nevertheless, copulas have remained a useful tool for dependency analysis and have been recently re-introduced into finance industry for other asset classes and derivative products.JSFH
<Text-field style="Heading 1" layout="Heading 1">Definition of copula</Text-field>For the n uniform random variables LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+.....,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRicvRjZRJ25vcm1hbEYn the joint distribution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= - or copula - can be defined as:
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where \316\270 is dependency parameter that links marginal into full joint distribution.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Copulas linkage to marginal distribution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1YkdGJDYlLUYsNiVRInhGJ0YvRjItRiM2JS1GLDYlUSJpRidGL0YyRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0YzUSdub3JtYWxGJ0ZIRkg= can be expressed as: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYvThis shows that copula can be expressed an multivariate distribution function with univariate marginals.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Let's define two random variables: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=and define their joint distribution function as Copula: 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We can derive their joint PDF 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 can be obtained by integrating joint PDF over the respective region:
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, Conditional PDF can be obtained from the Copula function as: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for 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
<Text-field style="Heading 1" layout="Heading 1">Dependency factor</Text-field>Dependency factor in multivariate distributions is frequently associated with ccorrelation. Standard (Pearson's) correlation between two variables LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiWUYnRi9GMkY5 is defined as:
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=However:correlation may not always exists (when variance becomes infinite), and it depends on marginal distribution. Copulas, on the other hand, define joint distribution, and therefore contain all information on dependence.There are other dependency measures associated with copula function. Most frequently, this is limited to two common cases:Spearman's rank correlation: 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Kendall tau: \317\204(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%;Both dependency measures are therefore defined in terms of copula function and it shows that copula determines the dependency.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Examples of copulas</Text-field>In this section we present several examples of bivariate copulas with varying underlyings. Let's consider a simple case of two RV:u \342\211\210 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v\342\211\210 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoVW5pZm9ybUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY7NiRRIjFGJ0Y+Rj5GPkY+and linked them into joint distribution function with the following copula function: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(); # input 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print(); # input 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print(); # input placeholderwhich, as expected, is the Standard Normal PDFLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjTUZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GLzYlUSJ2RidGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVy1GLzYlUSlzaW1wbGlmeUYnRjJGNS1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYvNiVRJGludEYnRjJGNS1GaG42JC1GIzYqLUYvNiVRKUpvaW50UERGRidGMkY1LUZBNi1RIixGJ0ZERkYvRkpGNEZLRk1GT0ZRRlMvRlZRJjAuMGVtRicvRllRLDAuMzMzMzMzM2VtRictRi82JVEidUYnRjJGNS1GQTYtUSI9RidGREZGRklGS0ZNRk9GUUZTRlVGWC1GQTYtUSomdW1pbnVzMDtGJ0ZERkZGSUZLRk1GT0ZRRlMvRlZRLDAuMjIyMjIyMmVtRicvRllGaHAtRi82JVEpaW5maW5pdHlGJ0YyRjUtRkE2LVEjLi5GJ0ZERkZGSUZLRk1GT0ZRRlNGZ3AvRllGW3BGanBGREZELUZBNi1RIjtGJ0ZERkZGaW9GS0ZNRk9GUUZTRmpvRlg=print(); # input placeholderwhich is the uniform PDF on [0,1] 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(); # input placeholderwhich is conditional PDF derived from Copula function.
<Text-field style="Heading 2" layout="Heading 2">Independent Copula</Text-field>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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 2" layout="Heading 2">Frank Copula</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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(); # input 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print(); # input 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(); # input placeholderJSFH
Graph of the bivariate 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%;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%;JSFH
Graph of the bivariate 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%;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEncGxvdDNkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNjwtRiw2JVElZXZhbEYnRi9GMi1GNjYkLUYjNiUtRiw2JVEjQ01GJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY2NiYtRiM2Ni1GLDYlUSJhRidGL0YyLUZFNi1RIj1GJ0ZIRkovRk5GTEZPRlFGU0ZVRlcvRlpRLDAuMjc3Nzc3OGVtRicvRmduRmVvLUkjbW5HRiQ2JFEkMC41RidGSEZELUYsNiVRImJGJ0YvRjJGYG8tRmhvNiRRJDEuOUYnRkhGRC1GLDYlUSJjRidGL0YyRmBvLUZobzYkUSIxRidGSEZELUYsNiVRImRGJ0YvRjJGYG8tRmhvNiRRJTAuNDVGJ0ZIRkQtRkU2LVEifkYnRkhGSkZjb0ZPRlFGU0ZVRldGWS9GZ25GZW4tRiw2JVEoJmFscGhhO0YnL0YwRkxGSEZgby1GaG82JFElMC4yNUYnRkhGSC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZIRkQtRiw2JVEidUYnRmRxRkhGYG8tRkU2LVEqJnVtaW51czA7RidGSEZKRmNvRk9GUUZTRlVGVy9GWlEsMC4yMjIyMjIyZW1GJy9GZ25GZXItRmhvNiRRIjNGJ0ZILUZFNi1RIy4uRidGSEZKRmNvRk9GUUZTRlVGV0ZkckZgcUZnckZELUYsNiVRInZGJ0ZkcUZIRmBvLUZobzYkUSIwRidGSEZqci1GaG82JFEiOEYnRkhGRC1GLDYlUSVheGVzRidGZHFGSEZgby1GLDYlUSdmcmFtZWRGJ0ZkcUZIRkRGXXEtRiw2JVEmdGl0bGVGJ0ZkcUZIRl1xRmBvRl1xLUY2NiYtRiM2KC1GLDYjUSFGJy1JI21zR0YkNiNRVkJpdmFyaWF0ZX5Ob3JtYWwmbWludXM7TG9nTm9ybWFsfk1vcmdlcnN0ZXJufn5Db3B1bGF+RidGRC1GLDYlUSVmb250RidGZHFGSEZgby1GNjYmLUYjNictRiw2JVEmVElNRVNGJ0ZkcUZIRkQtRiw2JVElQk9MREYnRmRxRkhGRC1GaG82JFEjMTJGJ0ZIRkhGaHFGW3JGSEZocUZbckZILUZFNi1RIjtGJ0ZIRkpGTUZPRlFGU0ZVRldGWUZmbw==%;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Graph of the bivariate 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%;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%;JSFH
Graph of the bivariate 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%;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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%;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Graph of the bivariate 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%;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%;JSFH
JSFH
<Text-field style="Heading 2" layout="Heading 2">Gaussian Copula</Text-field>Normal or Gaussian copula is simply the multivariate normal distribution with correlation matrix \316\243.For demonstration purposes we will consider here a bivariate case onlyWe will proceed in few steps:Define Bivariate Normal PDFConstruct Bivariate Normal 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print(); # input placeholderDefine the CDF of the first marginal distribution => eg. LogNormal distributionDefine the CDF of the second marginal distribution => Gumbel distributionFind the Normal inverse of each CDFs above: 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Create the Normal Copula => 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5MzQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 = Bivariate Normal CDF
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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYzLUkjbWlHRiQ2JVEmSlBEQk5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSlzaW1wbGlmeUYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJWRpZmZGJ0YvRjItRlM2JC1GIzYnLUYsNiVRJENCTkYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRIlRGJy9GMEY9RjlGW28tRiw2JVEiVUYnRi9GMkY5RjktRjY2LVEiO0YnRjlGO0Zeb0ZARkJGREZGRkhGX29GTS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0Zgby8lJmRlcHRoR0ZicC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYlUSdKUERCTjJGJ0YvRjJGNS1GLDYlUSVzdWJzRidGL0YyLUZTNiQtRiM2JS1GUzYmLUYjNiktRiw2JUZlb0YvRjItRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tSSVtc3ViR0YkNiUtRiw2JVElSW52RkYnRi9GMi1GIzYjLUYsNiVRInVGJ0YvRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Zbb0Znb0ZqcS1GXnI2JUZgci1GIzYjLUYsNiVRInZGJ0YvRjJGaHJGOS8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGW29GK0Y5RmpvRl1wLUYsNiVRJUNCTjJGJ0YvRjJGNUZdcS1GUzYkLUYjNiVGZHFGW29GaG5GOUZqbw==print(); # input placeholderJSFH
Graph of the bivariate 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%;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%;JSFH
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Generation of random numbers from copulas</Text-field>In this section we show how one can generate random samples from a given Copula function.The objective is to generate uniformly-distributed random variables 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such that their joint distribution function is Copula. This can be achieved in the following steps (for demonstration purposes we will be working in bivariate space)Generate the pair of independent uniform RV 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"Condition" the Copula function of v for the given value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ1RidGL0YyRjk=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Obtain new "transformed" uniform variable v such that 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We now have the correct dependency structure in the marginal space 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To get the required RV, we will need to complete the following steps:Build 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 - the CDF of two RV we want to link with a given copula functionObtain the desired RV by inverting each CDF 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 pair LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ5RidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ is the sample from the copula function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYpLUklbXN1YkdGJDYlLUYsNiVRJyYjOTM0O0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1GIzYnLUYsNiVRInVGJ0YvRjJGLy8lK2V4ZWN1dGFibGVHRkEvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUY2NiQtRiM2JS1GLDYlUSJ4RidGL0YyRklGQkZCLUkjbW9HRiQ2LVEiLEYnRkIvJSZmZW5jZUdGQS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GOzYlRj0tRiM2Jy1GLDYlUSJ2RidGL0YyRi9GSUZLRjJGTi1GNjYkLUYjNiVGXnBGSUZCRkJGSUZCRkJGSUZCWe will use the Morgenstern Copula to demonstrate the example: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print(); # input 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print(); # input 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print(); # input placeholderWe need 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 so we pick the second 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print(); # input placeholderWe build the CDFs of two RVs we want to link with the above copulaRV1 ~ Normal (a,b)RV2 ~ LogNormal(c,d)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print(); # input placeholderFinally, we will obtain the desired RV by substituting numerical values in the inverses of each RV: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print(); # input placeholderNumerical 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(); # input 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%;
<Text-field style="Heading 1" layout="Heading 1">Application in Finance</Text-field>Given the procedure defined above, we will show how copulas can be used to price financial options with dependent underlying input
<Text-field style="Heading 2" layout="Heading 2">Call on Max of two dependent assets</Text-field>We want to price a call option on the maximum of two stocks (S1,S2) minus strike K:
Max [0, Max(S1,S2)-K]
whereS1 ~ LogNormal (mu1,sigma1)S2 ~ LogNormal(mu2,sigma2)S2 ~ LogNormal(mu3,sigma3)Assume that all three stocks share a joint PDF generated from Clayton copulaWe first define the underlying processes, copula function and generate independent uniform 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print(); # input placeholderQyQ+SSRDdXZHNiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkSSJIR0YlSSJ1R0YlIiIiprint(); # input placeholderSince we want 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we select the second solution
QyY+SSVDaW52RzYiLUkmc29sdmVHRiU2JC9JJEN1dkdGJUkiekdGJUkidkdGJSIiIj5JJEN1ekdGJS1JKHVuYXBwbHlHRiU2JiZGJDYjIiIjSSJ1R0YlRitJJmFscGhhR0YlRi0=print(); # input placeholderWe solve for the inverse of LogNormal distribution to obtain the distribution of dependent sample from Clayton copula
S1dist, S2Dist are effectively future stock prices samples we need as underlyings for option pricing
Qyo+SSZGMUludkc2Ii1JKWFzc3VtaW5nR0koX3N5c2xpYkdGJTYkNyMtSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0YoNiQvJkkiRkdGJTYjIiIiSSJ3R0YlSSJhR0YlNyMtSSI+R0YuNiRJInNHRiUiIiEhIiI+SSZGMkludkdGJS1GJzYkNyMtRiw2JC8mRjI2IyIiI0Y1SSJiR0YlNyMtRjk2JEkidUdGJUY8Rj0+SSdTMWRpc3RHRiUtSSh1bmFwcGx5R0YtNiZGJEkkbXUxR0YlSSdzaWdtYTFHRiVGNUY0PkknUzJkaXN0R0YlLUZRNiZGP0kkbXUyR0YlSSdzaWdtYTJHRiVGNUY0print(); # input placeholderWe set the sample size n and copula dependency parameter \316\261Qy4+SSJuRzYiIiUrNSEiIj5JJmFscGhhR0YlJCIkdiIhIiNGJy1JKlJhbmRvbWl6ZUdGJUYlRic+SSNBMUdGJS1JJ1NhbXBsZUdGJTYkJkkiVUdGJTYjIiIiRiRGJz5JI0EyR0YlLUYyNiQmRjU2IyIiI0YkRic+SSNCMUdGJTcjLUkkc2VxRyUqcHJvdGVjdGVkRzYkLUkkQ3V6R0YlNiUmRjA2I0kiaUdGJSZGOUZKRikvRks7RjdGJEYnprint(); # input placeholderHere LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQTFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is the uniform variable sample = CDF of S1 distribution and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is the copula-transformed uniform sample = CDF of S2 distribution.We supply numerical values for each stock's mean, std deviation and strike price to get the option value numerically:
Let's recall that Call option premium = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JUYrLUklbXN1cEdGJDYlLUkjbW9HRiQ2LVEvJkV4cG9uZW50aWFsRTtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictRiM2Ky1GNTYtUSomdW1pbnVzMDtGJ0Y4L0Y8USZmYWxzZUYnL0Y/RlYvRkFGVi9GQ0ZWL0ZFRlYvRkdGVi9GSUZWL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmhuLUYsNiVRInJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GOVEnaXRhbGljRictRjU2LVEnJnNkb3Q7RidGOEZVRldGWEZZRlpGZW5GZm5GSi9GTkZMLUYsNiVRInRGJ0Zdb0Zgb0Zdby8lK2ZvcmVncm91bmRHUStbMCwxNjAsODBdRicvJSxwbGFjZWhvbGRlckdGX28vJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRl9vRmBvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y4RitGOA==\356\224\274[LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRInRGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn-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
where in our case 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2ljRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEifkYnL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZXRkQ= is the i-the sample from from the Copula function.We want to price 1-year European call option on the max of two stocks where their current values / strike are defined as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEjMTBGJy9GNlEnbm9ybWFsRidGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlRi4tRiM2JS1GOzYkUSMyMEYnRj5GMkY1RkBGQy1GRDYtUSJ+RidGPkZHL0ZLRklGTEZORlBGUkZURlYvRlpGWC1GLzYlUSJLRidGMkY1Rj4=Please note that mean and std deviation of each stock price process is defined as 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%;Once the future stock prices samples from the given Copula are known, we can price the option premium as the discounted average:
print(); # input 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%;JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=