The book begins by giving a very quick overview of derivatives pricing, stochastic vol models, risk neutral pricing and ends the first chapter by giving couple of motivating examples where loosely coupled joined distributions given marginals , are used to price options .
Chapter 2: Bivariate Copulas
Copula is a type of function . But a very special function which maps marginals to joint. Hence one needs to be aware about the nature of the function and also its limitations. In that sense, this chapter develops copula function from scratch by explaining in detail the various conditions for a function to be called a copula. Boundary conditions of a generic copula are introduced using Frechet bounds. The good thing about this book is that it tries to give a real life application at the right stages of the book so that a reader can immediately relate the concept. For example the Frechet bounds are technically introduced in most of the books(Nelson). However the authors in this book cite an example of an exotic option where Frechet bounds are used to calculate the minimum and maximum value of the option. Various types of copulae are introduced such as subcopula, minimum copula, maximum copula, product copula, Survival copula, Joint Survival copula for uniform variates etc. The highlight of this chapter is explain Sklar’s theorem in a detailed manner.
Absolutely continous condition and Singularity conditions are discussed in the context of a bivariate copula . Also mentioned is the canonical representation which links the density of the joint distribution, density of marginals and the copula.
Various examples showing the application of copula to option pricing / credit risk(this area has made copula notorious, thanks to the article in wired titled –“The Formula that killed the Wall Street”.These examples would make any reader curious about the inner math of copula
Chapter 3 : Market Comovements
This chapter is extremely important as it introduced nonparametric measures of dependence like Kendall’s tau and Spearman Rho’s in the context of Copulas. One will develop a healthy dose of scepticism towards the simple correlation estimates that one sees in our daily lives often. One can clearly see that linear correlation that most of us know , is so narrowly defined .The book will make the reader pause and think of the ton of assumptions behind the all pervasive metric.
The second part of this chapter introduces various forms of copulae such as gaussian, student t, frechet family, Archimedean family by giving the connections between the marginals and the joint.
Chapter 4: Multivariate Copulas
This chapter extends bivariate copulas to multi dimensional case. Basically all the theorems used in the bivariate case are written in multivariate world.
One important aspect mentioned in this chapter is Density and Canonical representation of a multidimensional copula. This is extremely important when one needs to find out conditional distributions of copulas for generating random numbers for various copulae.
Chapter 5: Estimation and Calibration from Market Data
One of the challenging aspects of estimating a copula model from a set of securities is the sheer number of parameters that needs to be estimated and the way to compare fitness statistics across copulas. Saviour is good old MLE. However MLE also becomes unstable if an estimation is made all at once. Hence the estimation is generally split in two steps.
First Step involves estimating parameters for the assumed marginals, which can be assumed as normal/ t / gamma / GARCH etc . Second Step involves estimating the copula function given these parameters for the marginals. This procedure is called Inference functions for the Margins(IFM).
The second method described is a bit more non parametric. The method is called CML , i.e Canonical Maximum Likelihood. The key step in this process is to estimate the empirical distribution function for the margins and use it in the second step of estimating copula parameters.
Where can one use this estimation procedure ? What if you don’t want copula in your strategies/ algos / modeling ? In that case, copula can serve as a superb diagnostic tool of your assumptions. For example lets say you are interesting in creating a portfolio of a few stocks from a specific sector. So, you estimate basic moments of each of the assets and then make a ballpark estimate of risk using correlation matrix. Can you trust this correlation matrix ? Well one way to answer this question is let’s say you fit a Student t marginal to the stocks in your portfolio and fit a gaussian copula. The dependence structure of the fitted copula will give tons of insight in to the assumed correlation. Should you use robust estimators ? Should you stick to the age old product moment correlation matrix ? All these questions can be answered by fitting various marginals to the gaussian copula. Agreed we are in the end fitting a gaussian copula which has no tail dependence, but atleast you are introducing symmetric tail dependence in the individual securities.
Another application is fitting a stochastic process to the individual securities and then joining them with an appropriate copula. This kind of loose coupling of marginals and joints give a modeller tremendous amount of flexibility. One can also think of a time varying copula function.
Chapter 6 :Simulation of Market Scenarios
Typically in any market of n securities, each security follows a stochastic path. The most generic form of modeling all the variables together is to Fit N different marginals to the n different securities and then connect the N marginals with a time varying Copula function. Each of these marginals can in turn be a conditional marginal which will complicate the things. So, any quant exercise which involves data analysis/modeling/simulation with more than one security obviously needs to have this general framework in mind and one has to clearly state the assumptions that are made.
Surprisingly this generic framework is not clearly stated in most of books and one typically finds a situation where margins are normal and copula is guassian normal copula, meaning the joint distribution is multivariate normal. Whenever one does modeling one should always keep in mind that the Gaussian framework is a strawman , which is to be taken as example101 for the generic framework.
For any portfolio strategy involving multiple assets, backtesting a strategy necessitates simulation of alternate worlds. While sampling with replacement is a wonderful way to test a strategy, at the same time, it is sometimes very harsh and one ends up simulating multivariate normal which I think is too lenient. The former might give rise to Type II error while latter can give rise to Type I error. In that context, this chapter is extremely useful to simulate various market scenarios. The usage of copulas gives the flexibility of modeling tail dependence. Methods of generating random variables for various copula like Gaussian copula, Archimedean Copulae such as Gumbel, Clayton and Frank are described in great detail.
For generating an n variate Gaussian copula, the procedure is similar to generating a multivariate random variable with a given mean vector and covariance vector. With a few additional steps to the multivariate normal sample, one can easily generate a Gaussian copula.
For generating an n variate Student T copula, the procedure is again straightforward , starting with generating a multivariate random number and then dividing by average chi-square random number.However generating n dim random numbers for any of the Archimedean Copula is not straightforward. Iterative procedures on conditional distributions are used. In some cases like Clayton and Frank Copula, there are no closed forms that form the part of iterative procedure. However for Gumbel there is no closed form solution for evaluating the various random numbers in the iterative process.
The last 2 chapters cover a range of applications in the credit world and exotic option pricing world.
Any modeller deals with more than one variable in his work. This book will help one see the powerful use of joint distributions. Also, It is definitely the most accessible introduction to Copulas which will help a reader be more comfortable with reading and understanding the applications of Copulas in various fields.
Just because Guassian Copula was used in pricing CDS and CDO and they all went bust, doesn’t mean Copulas as a concept is flawed. Well, what is flawed is substituting a million correlations amongst mortgages that needs to be fed in to the model , with ONE Single correlation . That is a plain misapplication of the concept.
My takeaway is that Copulas are excellent diagnostic tools for analyzing joint behaviour of variables and they must be a part of Quant’s arsenal.