Frameworks for modeling risks and prices of complex structures
                                         2009




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Frameworks for modeling risks and prices of complex structures
                                                  2009

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Frameworks for modeling risks and prices of complex structures
                                                  2009

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Frameworks for modeling risks and prices of complex structures
                                                   2009

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Frameworks for modeling risks and prices of complex structures
                                                 2009

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Frameworks for modeling risks and prices of complex structures
                                                  2009

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                                                      2009
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Frameworks for modeling risks and prices of complex structures
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                                              2009

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Modelling Risk and Return of Complex Financial Instruments

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Published in MDI Monetrix Finetune 2nd Edition, 2009

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Modelling Risk and Return of Complex Financial Instruments

  1. 1. Frameworks for modeling risks and prices of complex structures 2009 Frameworks for modeling risks and prices of complex structures Team Name: Finesse College: NMIMS Team Members: Jigar Jani Chetan Ganatra Prasad Shahane 9869559899 9920682059 9833515715 1 Page
  2. 2. Frameworks for modeling risks and prices of complex structures 2009 Structured products like CDO and CDS, But mortgage pools are complex than most once the darlings of the Wall Street, are now bonds. There's no guaranteed interest rate, being primarily blamed for the meltdown of since the amount of money homeowners the financial system worldwide. How did collectively pay back every month is a these financial instruments manage to pack a function of how many have refinanced and punch so strong as to knock most of the how many have defaulted. Repayment world’s strongest economies off their feet? occurs in irregular patterns as people pay Why were the risks associated with such down their mortgages at unpredictable products not identified? Could the pricing of times—for instance, when they decide to such complex structured products be done in sell their house. And there is no easy way to a more efficient way? Let us try and find the assign a default probability to such an answers to the above questions. instrument. Investment bankers solved many of these problems through a process called ‘The name is Bond’ tranching, which divides a pool and allows Let us start by looking at the multi-trillion for the creation of different tranches with the dollar bond markets. Bonds are nothing but first tranche being the first to be paid-off and a promise to repay the money borrowed with hence would have a risk-free triple-A credit the interest. Investors usually look at the rating. The next tranche may get only a financials of a company issuing the bonds to double-A credit rating but investors will be make sure that they have the ability to pay able to charge a higher interest rate for back. The higher the perceived risk, higher bearing the slightly higher chance of default. the interest rate that is going to be charged However tranching still didn’t solve the by the investor to the issuer. Investors problem of mortgage-pool risk like falling usually assign a probability to the default of house prices wherein everybody’s house the issuer and ask for a premium for the equity drops simultaneously. If home values default risk that they are taking depending in a location decline and a mortgage taker on the probability. may lose some of his equity and there's a Enter Mortgage Backed Securities (MBS) good chance his neighbors will lose theirs as where bond investors invest in pools of well. If, as a result, he defaults on his 2 Page hundreds or even thousands of mortgages. mortgage, there's a higher probability they
  3. 3. Frameworks for modeling risks and prices of complex structures 2009 will default, too. That's called correlation— premium leg; the contingent payment that the degree to which one variable moves in might have to be made by the protection line with another—and measuring it is an seller is called the protection leg. important part of determining how risky Li argued the investors could either lend mortgage bonds are. As a consequence, directly to the borrower or sell credit default bond investors and mortgage lenders badly swaps for the borrower which is nothing but want to be able to compute, model, and price insurance against those same borrowers correlation. defaulting. Either way, you get a steady ‘The fuel that started the fire’ income stream which may be made up of interest payments or insurance payments or The missing link of finding the correlations if the borrower defaults, the protection seller seem to have been filled when David X. Li , or investor loses a lot of money. The returns a star mathematician while working at JP on both strategies are nearly identical, but Morgan in 2000 published a paper that because an indefinite number of credit appeared in The Journal of Fixed Income default swaps can be sold against each titled "On Default Correlation: A Copula borrower, the CDS market went into Function Approach." David Li in this exponential growth. As the price of the CDS ingenious paper made use of the Gaussian goes up it indicates that the default risk has copula function to model default correlation risen which Li uses for by seeing if the CDS by not looking at historical default rates data prices of two separate borrowers seem to but by using the prices of CDSs. move in the same direction indicating a In a credit default swap, the protection buyer strong correlation between their defaults. pays a fee, the swap premium, to the ‘Taking a look at the magic formula’ protection seller in return for the right to receive a payment conditional upon the Now let’s take a deeper look to the single default of the reference obligation or the factor Gaussian copula function and try and reference entity. Collectively, the payments understand its various components made by the protection buyer are called the P [Tx<1, Ty<1] = φ2 (φ-1(Fx (1)), φ-1(Fy (1)), γ) 3 Page
  4. 4. Frameworks for modeling risks and prices of complex structures 2009 P represents the joint default probability of equation blowing up. One of them was that two members of a pool of loans say X and Y Li did not take into account the infinite defaulting at the same time which is what relationships between the different types of the investor is always looking for. loans that make up the pool. Li also did not mention what would happen when you mix φ2 stands for the copula function. A copula negative correlations with positive ones. The is an Latin term for ‘bond’ or ‘tie’ hence this equality sign made it an exact model leaving function tries to tie the individual no room for mistakes. probabilities associated with X and Y to come up with a single number. ‘Take-Off for the Securitization market’ Tx and Ty are survival times, the amount of The effect of Li’s formula on the time remaining till X and Y can be expected securitization market was nothing short of to default. Li got this idea from actuarial astonishing. Rating agencies like Moody’s science in which he studied what happens to and later S & P adopted this method for the life expectancy of a person whose rating tranches which meant that they did spouse dies. A similar analogy was drawn to not have to ponder over the underlying find out what happens to expected time to securities which make up the tranche all default of company Y if company X they need to know is the single correlation defaults. number which would indicating the default risk of the tranche. Fx and Fy are the distribution functions giving the probabilities of how long X and Y A further enhancement in the securitization are expected to survive. market came in the form Collateralized Debt Obligation (CDO) a collection of various γ is the all powerful correlation parameter bonds which might include corporate bonds, will reduced correlation to a constant scalar bank loans, MBS, all held together in a quantity. basket which could be tranched and the top The above equation looked simple, beautiful tranche rated as Triple-A all though none of and powerful but below its aesthetic beauty the individual components may be Triple-A. lied a clutch of ugly assumptions which if 4 The growth in the CDS and CDO market Page did not hold true would lead to the entire was phenomenal were the CDS market went
  5. 5. Frameworks for modeling risks and prices of complex structures 2009 from 1 trillion dollars in 2001 to 62 trillion such a house price depreciation because if dollars in 2007 and the CDO market went they did they would not be able to rate any from 275 billion dollars in 2001 to 4.7 of these securities Triple A. trillion dollars in 2007 and at the engine of ‘Correlation between assets was assumed to this growth was Li’s formula. be a scalar quantity’ ‘Fall of the Phoenix’ Li tried to establish a relationship between If the formula was so great what went two assets using a scalar quantity which is wrong? Let us try and dissect a few issues never possible. Consider two umbrella which contributed to the mess. makers, when the market for umbrellas is growing both the companies grow and the ‘CDS prices were used from a time period correlation between them is positive. where house prices only appreciated’ However when one of the companies uses a Li’s copula function used CDS prices to celebrity backed marketing campaign and calculate correlation. Hence it invariably starts to acquire more market share, the was deriving data from a period of time prices of the two stocks tend to move in when house prices soared. Naturally, default opposite direction thereby exhibiting correlations were rock bottom in those negative correlation. However during years. But when the mortgage bubble burst summers the sales of both the companies and real estate prices started falling across drop thereby exhibiting positive correlation the nation, correlations soared. yet again. Bankers and rating agencies are also too Also correlations in the real world are hard blame for the mess because they knew that to determine. Consider a case where we the model was highly sensitive to a have to decide the correlation of bonds nationwide depreciation in housing prices issued by a Polish sneaker maker and an and if such a situation arose the most of the Indian software company. The correlations Triple A rated securities CDO’s would blow might be considered negligible since the up. But investment bankers merrily companies belong to not only different continued on making such securities. Rating industries but also to different geographies. 5 Page agencies also did not build any cushion for However, if they have been lent money by
  6. 6. Frameworks for modeling risks and prices of complex structures 2009 the same troubled bank who is now calling over time for Dow Jones and Euro/US- its loans back then the correlation increases Dollar Exchange Rate which further drastically. provides strength to the argument that correlation cannot be factored by a constant Further evidence is provided by the charts scalar quantity. below showing the correlation and volatility Figure 1: Correlation - Dow Jones and EUR/USD rate Figure 2: Volatility Distribution Source: http://www-num.math.uni-wuppertal.de/Files/amna_06_03.pdf 6 Page
  7. 7. Frameworks for modeling risks and prices of complex structures 2009 The conclusion that can be drawn is instead of local correlation for modeling the ‘Correlation is more unstable than price. volatility’. Therefore it is wrong to assume Stochastic correlation can be modeled using that correlation is a constant quantity, rather random walk model and Brownian model correlation should be modeled as a equations. This adjustment will not only stochastic process. To understand the lead to accuracy in pricing, but is also easy concept better let’s look at the difference in to implement in presently used Gaussian the market quotes and the model quotes model, so it will reduce the efforts of (based on Gaussian model) for CDO. calibrating the model. Gaussian model in this specific instance used least square method to fit the data. ‘Tail events in Gaussian copula are asymptotically independent of each other’ Tranche Market Gaussian Probability of co-occurrence of tail events is [0-3%] 24.00% 19.30% termed as tail dependence; this is the usual [3-6%] 82.50 234.70 case in case of default contagion. In case of Gaussian copula, it is presumed that there is [6-9%] 26.50 82.00 no dependence in tail events as we move [9-12%] 14.90 32.90 further out in tail (tail events are [12-22%] 8.75 6.99 asymptotically independent). This argument [22-100%] 3.53 0.05 seems counter intuitive when the default Source: Market Quotes as of 30-Aug-05 rates in current financial crisis are observed. These quotes shows that Gaussian based This problem of underestimating the risk model underpriced the senior and the equity could have been solved by using ‘Student’s t tranche of the CDO whereas they overpriced Distribution’ or ‘t-Copula’. the mezzanine tranche. These models are t-Copula based models naturally incorporate based on local correlation which cannot be default contagion through tail dependence used to build and parameterize the skewed which is not present in Gaussian copula. An model. This issue of mispricing can be even better approach would be to use solved if we use stochastic correlation 7 skewed t-copula, for which the upper and Page lower tail dependence is not equal. The
  8. 8. Frameworks for modeling risks and prices of complex structures 2009 model based on these two copulas are not consideration, without which the model is of very difficult to implement and are now no real use. been seriously considered by financial risk ‘Decision makers did not understand the managers. risks involved and the assumptions of the ‘Linear dependence is the sufficient statistic model’ for the dependence distribution’ Another factor to be considered was that Gaussian model assumes that linear most of these results came from computer correlation is the sufficient statistic for the models hence the underlying assumptions dependence distribution. This is extension of were not known to everyone. Also the the point that was just discussed, as ‘Quants’ who knew the weaknesses of this Gaussian Copula does not considered fat approach were not the ones making the big tails it worked well with the linear money calls on asset allocations. The dependence assumption, but if we are managers lacked the mathematical skill to modeling risk using other copulas then it understand it but understood the single becomes imperative for us to see whether correlation number and hence made all asset correlation alone is sufficient or not? allocation decisions based on it wherein hidden was the problem. It is known fact that correlation is the measure that only says about degree of ‘All models are wrong but some are useful’ relationship between two variables and The above quote is made by famous industry nothing on the distribution of the two statistician George Box indicating that most variables. Correlation alone is insufficient of the models have their limitations for describing complex dependence regarding what they can do and are merely structures in finance. Most of the financial tools for making decisions and Li should not markets data are represented by fat tail be blamed much here. More to blame in this distributions like Clayton distribution, scenario are the bankers who misinterpreted Skewed t distribution, etc. Therefore it it and all the traders who adopted it. This becomes imperative for any model to be situation is a striving environment for a able to capture the distribution 8 bubble where everybody is doing the same Page characteristics of the system in thing and the bubble eventually burst
  9. 9. Frameworks for modeling risks and prices of complex structures 2009 leaving behind wreckage for bankers, dangerous part is when people believe regulators and investors worldwide. As Li everything coming out of it". himself said of his own model: "The most References: Li, D. X. (2000). On default correlation: A copula function approach. RiskMetrics Group. 27 April 2009. PPT on ‘Financial Crisis of 2008: What went wrong?’ by David Marshall, Senior VP, Fed Reserve Bank of Chicago, April 14, 2009. http://www.math.fsu.edu/~kercheva/papers/skewedt.pdf http://www.thisisthegreenroom.com/2009/deconstructing-the-gaussian-copula-part-iii/ http://www.socialresearchmethods.net/kb/statcorr.php http://www-num.math.uni-wuppertal.de/Files/amna_06_03.pdf http://www.risknet.de/typo3conf/ext/bx_elibrary/elibrarydownload.php?downloaddata=264 9 Page

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