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# Pricing: CDS, CDO, Copula funtion

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### Pricing: CDS, CDO, Copula funtion

1. 1. Giulio Laudani Cod. 20263 Scheme on CDS and CopulaEmpirical data features: ____________________________________________________________________ 2 Univariate world:________________________________________________________________________________ 2 Multivariate world: ______________________________________________________________________________ 2Multivariate insight: ______________________________________________________________________ 2 Measure and test _______________________________________________________________________________ 2 Dependence Measure: ____________________________________________________________________________________ 2 Test on distribution: ______________________________________________________________________________________ 4 Gaussian multivariate correlation __________________________________________________________________ 4 Other multivariate specification ____________________________________________________________________ 4Copula function __________________________________________________________________________ 5 What is a copula e when it is suitable _______________________________________________________________ 5 Sklar’s theorem [1959] ___________________________________________________________________________ 6 How to generate a Copula: ________________________________________________________________________ 6 Type of copula function __________________________________________________________________________ 7 Implicit types: ___________________________________________________________________________________________ 7 Explicit types: ___________________________________________________________________________________________ 7 Meta-distribution:________________________________________________________________________________________ 7Risk modeling ____________________________________________________________________________ 8 Goals and features: ______________________________________________________________________________ 8 Approaches: ____________________________________________________________________________________ 8 Comments and further details: _____________________________________________________________________ 9Pricing Multi-asset derivatives _____________________________________________________________ 10 The BL Formula _________________________________________________________________________________________ 10 The Practitioners’ Approach: ______________________________________________________________________________ 10 The copula approach: ____________________________________________________________________________________ 10 Credit Derivatives focus: _________________________________________________________________________ 10 Market practice: ________________________________________________________________________________________ 11 Pricing a CDS: __________________________________________________________________________________________ 11 CDS typologies and CDO financial product: ___________________________________________________________________ 14 1
2. 2. Giulio Laudani Cod. 20263Empirical data features:We are going to use as an explanatory model of return evolution the one that is assuming the presence of two compo-nent: the permanent information and the temporary noisy one. The second one prevails in the short term high fre-quency observation, while the first emerges for longer horizon. This property implies: In general, predictability of returns increases with the horizon. The best estimate for the mean of returns at high frequency is zero, but a slowly evolving time varying mean of returns at long-horizons could and should be mod- eled. [there is a strong evidence of correlation between industrialized countries stocks] If we run a regression we are expecting high statistical significance for the parameters for log horizon The presence of the noise component in returns causes volatility to be time-varying and persistent, and the an- nualized volatility of returns decrease with horizon Data shows non normality behavior, unconditional distribution with higher tails Non-linearity is also a feature of returns at high frequency: a natural approach to capture nonlinearity is to diffe- rentiate alternative regimes of the world that govern alternative description of dynamics (such as level of volatil- ity in the market), for example Markov chain.Univariate world:Those time series have little serial correlation, high absolute return correlation, excepted conditional return are closeto zero, volatility appears to change over time, leptokurtosis and skewness are main features and extremeevent ap-pear in cluster (thus it can predict), lastly long term interval converge more to Gaussian hpMultivariate world:Little correlation across time, except for contemporaneous returns, strong correlation for absolute returns, correla-tion[to compute the correlation we should fit different models for changing correlation and make a statistical compari-son] vary over time (and clustering effect), extreme returns in one series are correlated with other returns (extremedependence)Multivariate insight:This topic is relevant for pricing. Asset allocation and risk management issues. When we dealt with multi securities weneed to model not only the dynamics of each securities, but also the joint evolution, i.e. their dependence structure.Before starting to describe the first approach we need to provide some notation that will be used here after: Consider a d-dimensional Vector X the joint distribution is The marginal distribution will be If the [marginal distribution] is continuous we can obtain the densities by computing the first derivatives, so the joint densities will be the non-negative function Note that the existence of a joint density implies the existence of all the marginal densities but not vice-versa, unless the components are orthogonal one to the otherMeasureand testDependence Measure:The Linear correlation is the natural dependence measure only for multivariate normal (it will coincide with the maxi-mum likelihood estimators, hence it will satisfied all the desirable properties) and when the variance is (must be) a finitenumber (not obvious, insurance business). It depends on marginal distribution (since it is computed with moments, 2
3. 3. Giulio Laudani Cod. 20263Hoffding formula )), hence it is not invariant under more general transformation (itis still invariant under liner transformation) and it wouldn’t capture the correlation dynamic under more generic func-tion.The Rank Correlations are dependence measure that depend only on the copula and not on marginal, so it is invariantunder general increasing transformations. To compute the rank correlations we need to know the ordering1 of each va-riable, not the actual numerical value ( we are assuming continuous margins for simplicity) : there exist two measures: Kendall’s tau, we compute the number of “c” concordant and “d” discordant pair between two variables with n variables. sample version. The population version simply consider the probability of concordance minus the probability of discordance , where the two are vectors with iid distribution Spearman’s rho [sample version], we need to compute the rank variables (the same time series are ordered) of the two variables and then compute the correlation between these new variables . The popula- tion version is the correlation of the marginal valueHere in the following we will provide a proof on the dependence of the estimators only on copula distribution function: The Kendall’s rho , since the sum of the two is one, we can write the one as a function of the other . We know that since the two variable are i.i.d. this last term is the joint distribution, which can be expressed as an integral form thanks to the Sklars’ theorem we can rewrite the previously equation as function of copula distribution (by change the integral interval with 0,1). hence we have proven that the estimator depends on . The Sperman’stao population estimators is . The covariance formula is , hence we can rewrite it as a function of copula . The so if we put together all this findings we ends up withThere exists also a multivariate version of the rank correlation measure, where instead of taking the expected value(population version) we will take the “cov” function or correlation matrix depending if we were using the Kendall orSpearman measure, hence the estimator will be at least semi positive definite. Both the measure can assume the valuein the interval [-1,1] and 0 is independent.The tail dependence measure has been set to judge the extreme dependence between pair of variable (it is hard to ge-neralize to the d-dimensional case), the idea behind is to limit conditional probabilities of quantileexecedances. The up-per measure is given by and the value are in the interval [0,1], the opposite forthe lower tail . Those formula can be written s function of copula (this measuredepends only on copula, hence it is a proper dependence measure), by applying the Bayes theorem1 When we refer to the ordering of a variable we need to introduce the concept of Concordance and Discordance. In the first case itis meaningful to say that one pair of observations is bigger than the other (without use the Probability distribution) 3
4. 4. Giulio Laudani Cod. 20263Some example are provided: the following Gaussian is asymptotically independent in both tails, t-copula has a symme-tric (thus not good for real data features) dependence , where higher value of v degree of free-dom will bring an higher tail dependence. The Gumbel copula has upper tail dependence , and the Clayton copulahas lower tail dependence isTest on distribution:To test univariate distributionhp we can use the QQ plot technique to have a graphical sense of the validity of the hp.To have a numerical sense we can use the Jarque-Bera test which is based on the joint check of the skewness and kurto-sis of the sample.To test multivariate distribution we need some other tools, specifically set up. The first is based on a quadratic formwhere we will compute for each observation the , which should be distributed according toa chi-square with d degree of freedom , which can be seen graphically with a QQ plot or numerically with the MardaTest(by using the D, third and fourth moments, the first distribute as a chi-square with degree of freedom [ ],the second follow a standard Gaussian) which is basically a ri-proposition of the jarque-Bera idea, however it is not ajoint test, since we test the skewness and kurtosis separately.Gaussian multivariate correlationThe first and most trivial measure of dependence is the correlation or equivalently the variance covariance matrix. It isat least a semi-definite matrix, and if it is positive-definite it is possible to apply the Cholesky decomposition. It is usuallycomputed with the sampling estimators2.The Gaussian distribution is the most simple hypothesis, it is characterized by the first two moments, which will entirelydescribe the dependence structure, linear combination remains normally distributed and it is easy to aggregate differentdistributions (we just need to know the correlation between them).There exist several closed formula under the geometric Brownian motion in the derivatives pricing and it is simple tomanage the risk by using the VaR approach.The Gaussian world is unfeasible with the empirical evidence, it has a poorfit with real data3, hence we need to develop a new set of model which allow for fat-tail and more flexibility in definingthe dependence structure. Note that a multivariate set of variable to be jointly Gaussian, each of the univariate distribu-tion must be Gaussian if self.Other multivariate specificationThe first attempt to solve the problem was to work with the conditional distribution, such as GARCH model ,where we are hoping that the rescaled return by volatility is normal. Even if those model performs better than the clas-sical one, we still need something more complete model, furthermore it is a really demanding specification for multiva-riate needs4. The second attempt was to use different distribution with fatter tails, such as the t-Student’s2 We are implicitly assuming the each observation is independent and identically distributed, in fact only in this case this estimatorwill coincide with the maximum estimators and have all the desired properties, furthermore this estimator will depend on the truemultivariate distribution3 We observe fat tail both in the marginal distribution and in the joint distribution, furthermore the volatility seems to move in clus-ter, while return are less time dependent, and correlation vary over time4 We need elements for each securities in the matrix to be modeled, where d is the number of coefficient in the algorithmsspecification. The same problem for the ETV 4
5. 5. Giulio Laudani Cod. 20263tion5(note that each marginal distribution must be t-student with same characteristic) or more advance parametric spe-cification: Normal Inverse or Levy process. The third possibility was to model just the extreme distribution behavior EVT,which is very popular and useful in the risk metrics word, less when we need to price (in those cases we need the wholedistribution)A totally different approach to the problem was to the dimension reduction techniques, the general idea is based onthe empirical observation that a limited number of common factors explain most of the dynamics. The simplest way toachieve such a solution is the PCA (here the Cholesky decomposition plays the role).To conclude all of those proposal model a joint distribution without any insight on the marginal distribution (or a leastdo not allow them to be different form the one accepted by the joint distribution chosen), hence it isn’t a flexible ap-proach. It will by far more interesting to try to put together different marginal distribution which will properly fit theempirical data and then set the joint distribution, like in a building up approach; this is the Copula method which will betreated in the following section.Copula functionThis approach is preferable for its higher flexibility: the bottom-approach allows to independently control/specify an ap-propriate distribution for each factor and the joint dependence structure. Other important properties are independentto scaling effect since it works on quantile, it is invariant to strictly positive transformation.What is a copula e when it is suitableA d-dimensional copula is a distribution function on the unit hypercube with standard uniform marginal distribu-tions. We denote copulas as and it must follows those properties (otherwise we cannotspecify the copula): 1. is increasing in each component.to be proven you have to perform the first derivatives 2. If at least one component u=0 the Copula function must be equal 0 3. meaning that if each other marginal function is realized the copula function result will the i- esimomarginal distribution. To prove, change the other variable with 1 and let the variable to change. 4. The rectangle inequality for all with o In a bivariate case it simply means that for every a1 < b1 and a2 < b2 we have: C(a1, a2) − C(a1, b2) − C(b1, a2) + C(b1, b2) > 0Properties (1) and (3) are required to have a multivariate distribution function, and to ensure that the marginal distribu-tions are uniform. Property (4) denotes that the copula function is d-increasing. If a function C fulfills these properties, itis a copula6.5 Since it is an elliptical distribution it enjoys most of the Gaussian properties. The most important which is shared among all the el-liptical distribution family is that the normalize value is distributed v= degree of freedom and d= # securities, i.e. dependsonly those two values. Furthermore it is closed under convolution (if two variables has the same dispersion matrix we can add themto obtain a new distribution, by assuming independency) Note that to fit fat tail distribution we may ends up with few degree offreedom66 Also, for 2 6 k < d, the k-dimensional margins are themselves copulas. 5
6. 6. Giulio Laudani Cod. 20263Sklar’s theorem [1959]This famous and import theorem states that Copula can be extracted from a joint distribution with known margins andthat the copula uniqueness is granted if marginal distribution are continuous, vice versathe space of existence of thecopula is the multiplication of the marginal distribution range. The proof relies on the generalized inverse theorem.The first part of the proof is, by assuming the existence of the joint distribution function with margins , then exist a Copula , where (by assuming continuous margin), by substituting we have .The second part given a Copula (unique) with given margins, we want to prove the existence of joint distribution withthe previously defined margins. 1. We take the vector U with the same distribution of the Copula 2. Then we compute the variable X defined as , where the F are the inverse marginal distri- bution (used to build the Copula). 3. We then write the distribution function of the variable X this last term shows the equivalency of the distribution X (joint distribution) to the Copula, if the margin are assumed to continuous.Another important result of the theorem is that Copula could be extracted by multivariate jointtion .Copula is bounded between the Fréchet limit: countermonotocity( ) [it is a copula only if d<2] and comonotocity , which are two copula function itself.How to generate a Copula:Structural model proposes as correlation the asset correlation, basically the equity correlation. Intensity based approachpropose to use for a given time horizon , unfortunately not enough data available for joint obser-vation. Thus has been proposed a MC algorithm called Li’s model (2000), where by using the hazard rate bootstrappedfrom CDS term structure and a given correlation taken from asset correlation, however it is computationally intensesince default events are rare.If we start form a given copula distribution (the first example will be a Gaussian one) We need to decompose via Cho-lesky decomposition the correlation matrix (we need to specify a procedure to produce this matrix). We needto run d-random variable Y and multiply them with A to obtain the variable Z (where A will contribute to give the corre-lation structure) we will use this variable Z to compute our distribution given a chosen Gaussian copula. To run a t-copula we need to add a step, basically we need to multiply the variable Z by where v are the degree of free-dom and s is a random variable generated by a chi-square distribution with “v” degree.How to fit a copula on empirical data:A different problems to fit empirical data into a proper copula dependence structure specification, there exist severalmethodologies available: Full maximum likelihood where we will estimates both the copula and margins parameter all together, this me- thod is of course the more accurate and unbiased, however it is tremendously intense. IFM or inference function for margins, where we specify a parameter marginal distribution, which will be used to compute the cumulative probability of the empirical data set, then we will used those probability to run a 6
7. 7. Giulio Laudani Cod. 20263 maximum likelihood to estimate the copula parameters. This method highlydepends on the marginsspecifica- tion, hence it is quite instable Canonical maximum likelihood is similar to the previous one, however we are going to use empirical marginal distribution, we are neutral on this side, while we are more focus on the joint dynamics Method of moments (used by MatLab), basically we are going to use the empirical estimates for some rank cor- relation measure, which is assumed to be our lead indicator/driverType of copula functionAll those copula function can be expressed in term of survival probability, they are named survival copula. An index ofthe flexibility of the copula dependence structure is the number of parameter in the algorithms, the most flexible pre-sented is the t-copula (degree of freedom and the correlation)Implicit types:In this class of copula belongs all those classes which do not have an explicit formula. The Gaussian [], where the inde-pendent and the comonoticity , defined only for the bivariate caseare special case of the Gaus-sian for correlation value equal to 0 or 1. It depends on the correlation variable only, which is a matrix.The t-student is another example. Note that in this case we do not require the margin to follow a specified distributionExplicit types:Closed formula are available for this class of copula. The Fréchet copula is the combination of the three fundamentalcopula: independent and the two limits (each of those is an explicit copula). It is a sort of average with coefficients β,α.We can obtain copula as a linear combination of this fundamental copula as follow 1,…+ ( 1,…), the beta and Alfa describe the dependence structureBelonging to the Archimedean class we have:Gumbel Copula defined by the equation Where the variable is the only parameter (not really flexible), it is bounded between 1 and infinite. Atextreme it converge to the independent and to comonotonicity copula. Clayton Copuladefined by the equation Where the variable is the only parameter (not really flexible), it is bounded between 0 and infinite.At extreme it converge to comonotonicity the and to independent copulaMeta-distribution:In this case we not directly apply the Skalar’s theorem. At first we will generate the random variable U (cumulativeprobability) form a specified copula function (which is the one that we want to use). We will use those generate va-riables into a specified marginal inverse distribution to generate our random variables. Basically we have run the applythe inverse procedure form the top to the bottomAn example is the Li’s model where Gaussian copula is used to joint together exponential margin to obtain a model forthe default times of copula when these default time are considered to be correlated. 7
8. 8. Giulio Laudani Cod. 20263Risk modelingGoals and features:As a risk manager we are interested in the loss distribution 7function defined (in general) as wherethe randomness come into with the variable “t”, basically the loss is function of time. Generally speaking we are interestin the function which defined the capital absorbed (regulators).Those numbers may be used for: Management tool, capital requirement and performance measuring adjusted by risk. toperform those tasks we need a model to predict asset value evolution, the most used one is a factors model mapped onspecific securities’ features. However this approach has the limit to assume a constant composition of the portfolio andthe prediction is made by a linearization approximation, which might work only for small factor changes and a negligiblesecond derivatives effect.Risk management may use unconditional or conditional distribution, where the later are preferable to assess a dynamicenvironment, where new information come into through time. The unconditional distribution are used for time intervalgreater than four months, since the time dependence of the return start to be negligible. It is a crucial task to under-stand the data, needs and tools available in the daily RM routine.Approaches:Notional is the most used by regulators, since is very simple. The general idea is to weight the entity asset by specific risklevel. This method has severe limits since there is no diversification or netting benefits and to value derivatives positionit is wrong to account the national value since the transaction is anchor to a specific payoff.Factors model follows a delta-gamma approach. It is not comparable across different asset class, it doesn’t allow to con-sider overall risk (you need to model the correlation of each factors, no sense in put together delta and Vega effect),however it performs well when judging well specified eventThe most used Loss distributionmethod is the VaR approach.For a given portfolio, confidence level and time horizon,VaR is defined as a threshold value such that the probability that the mark-to-market loss on the portfolio over the giventime horizon exceeds this value (assuming normal markets and no trading in the portfolio) is the given probability level.The limits are: we do not have a clear and fully agreed parameters calibration 8confidence level an time horizon, that’swhy the regulators force you to use a one day @ 0.99 parameters), it is not always sub additive, this problem get moreand more severe when we dealt with non-elliptical distribution, it didn’t give us any insight on the tail (after) possibleworst outcome lossThe ES will overcome those issues (It is a coherent risk measure), however is tough to be computed, one solution is toslice the tail distribution into equal probability occurrence, compute the VaR and make the mean, however for non-parametric distribution the observation might be two few. This method can be inferred from parametric, historical andMC distribution, however they are intensive on the computational side. Market risk Into the credit risk definition we have the Credit risk itself, migration risk and o Riskmetrics, where we will generate a whole migration matrix to be used7 Our interest in loss is related both to the job performed by risk management, and to the relevance and well defined academicmodeling 8
9. 9. Giulio Laudani Cod. 20263 o Credit view idea is to use macroeconomic prospective to correct the risk measurement, note that if the time horizon used is a point in time there is no need to adjust by economic cycle, in this case there would be a double countingScenario based approach suffer of misspecification, it is good only for well-defined problem (factors) and it is not easyto compare with multi asset securities.Variance was used as a risk measure, however it highly depends on the hp distribution (only symmetric, since deviationare equally weighted), should exist. Possible solution are partial moments, which is the base idea of expected shortfall.Comments and further details:Backtesting is really important. The idea is to check how many time the measure fail over time, it is a pointless/dauntingtask to run Backtesting on ES, since errors on VaR are few and average reduce the effect.The scaling problem is undertake with overlapping or with separated time interval. The first will allow to do not lose da-ta point, but it introduces serial correlation on the data, the second reduce the data that can be used. The convention torescaling by multiplying with a time factor is correct only if the data are iid, that is not the caseThe stress test must always be undertaken by risk manager this technique is a perfect complementary tool to track theportfolio risk dynamics.Any risk measure should fit the coherent risk measureproperties:Some comments on the major characteristic of credit models: Default-mode versus multinomial one, where only credit risk + belongs to the first Future values vs. loss rate, meaning that the model can be based on the distribution of possible value or possi- ble future loss, the first use as input the spread curve by maturity, while in the second the spread it is not neces- sary to be known. Creditmetrics is a typical market value model, while credit risk + loss one Conditional vs. un-Conditional, portfolio views belong to the first, however this distinction is useful only if the model works through the cycle Monte Carlo vs analytical solution Asset correlation vs. default correlation, it’s less important than other, in fact they are close to each other. Cre- ditmetrics is belongs to asset correlation, while credit + to the second one. 9
10. 10. Giulio Laudani Cod. 20263Pricing Multi-asset derivativesThe finance cornerstone in pricing any derivative is , where our goal is to com-pute a meaningfully and arbitrage free measure of the function f(x).The multi-asset derivatives family is the one where the payoff depends on more than one underlying. CDS by itself is nota multi-asset derivatives, however there exist in the market some example of derivatives protection against default ofmore than one underlying. The most used multi-asset payoff types are: Best of, Worst of and Basket option. There existthree possible framework to be used to price those derivatives, in the following section there will be a brief descriptionof all of them, however our focus will be set on the Copula one.The BL FormulaIn this set of hypothesis the motion used is the multivariate geometric Brownian one, where the correlation betweenasset is modeled into the stochastic (diffusion) component by the following relationship . In thisframework we can obtain a closed formula, however the distribution is lognormal and the parameters are assumed tobe constant, hence the risk-neutral density for the log returns is normal, not really feasible.The Practitioners’ Approach:It is usually introduce some more sophistication into the pricing formula, such as a stochastic volatility and jump diffu-sion process for individual random variable. At first we need to calibrate the models’ parameter for each asset and thenwe will estimate the correlation matric via historical data. The last stage is to use a Monte Carlo simulation techniques tocompute the price.This approach is really flexible and consistent (Rosenberg states that it is meaningful to mix an objective dependencestructure with univariate distribution) with univariate pricing, however the use of the linear correlation estimators is apoor fit to describe dependence structure.The copula approach:Given a copula function we can easily compute the joint density function aswhere the right term is simplythe joint derivatives of the copula function . This approach is an extension of thepreviously one, since we are modeling more sophisticated dependence structure via copula.With those information we can recall the pricing formula of an European derivatives with general payoff , by recalling that the joint density is a function of the marginal densities (easilycomputed form market data) and risk neutral copula (not easy to extract form market data), however it has been proventhat historical one is equal to risk neutral one under fairly general hp, which is to use affine transformation.This approach is really flexible, suitable for both parametric and non-specification and furthermore can be extended toadd more sophisticated dynamics. Note that we are still assuming a static approach, where the parameters are not let tochange over time, and the information derived from historical behavior.Credit Derivatives focus:Within this class of derivatives securities belong any instrument that enables the trading/management of credit risk inisolation from the other risk associated with the underlying. Those instruments are traded over the counter, hence it isvery flexible, however they have been standardized in the recent year to enhance the liquidity. 10