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Derivatives
In this section, the reports deal with articles on derivatives: the instruments, the pricing models, arbitrage theory, Monte-Carlo simulations, PDEs, least-square simulation methods, volatility, skew, volatility surface, the empirical studies etc..

Economic Catastrophe Bonds

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Economic Catastrophe Bonds 

In this paper, the authors, Josh Coval, Jakob Jurek and Erik Stafford, analyze the pricing of Collateralized Debt Obligation (CDO) senior tranches. They show that the spreads paid on these tranches do not sufficiently compensate investors for the risk of economic catastrophe. By focussing narrow the expected default component of the tranche credit spreads, rating agencies and investors ignore the systematic risk exposure of these tranches.

The authors combine the Merton credit default model and the CAPM to price credit portfolios and portfolio tranches. In the Merton framework, default of a firm occurs when the firm asset value (driven by a market factor and an idiosyncratic factor) falls below the face value of its debt. The authors can then calculate the default probabilities and the expected losses at the single firm level but also at the portfolio level.They first focus on the pricing of digital tranches that pays 1 if the portfolio losses are less than a given level (the attachment point) and 0 otherwise. They can price any tranches using these digital ones.

They manage to calculate closed-form valuation formula when the number of assets is very large (with the Vasicek formula). They find that the tranche spread is increasing as the number of assets rises because the exposure to systematic risk is larger.To price the tranches, the authors use the state price densities estimated from European equity index options (5-year options on the S&P500 since they are interested in the 5-year CDX portfolio). It is well known that the risk neutral probability is equal to the discounted value of the second order derivative of a call option with respect to the strike price.

They check the validity of their calibration by examining the predicted changes in spreads and the coverage ratios (ratio of risk neutral default intensity over the actual default intensity).

They later find that the more senior tranches provide with the highest risk premium.They then price the CDX trances by simulation of the market and point out that the payoff of a tranche is very similar to the payoff of a put spread (purchase of a put and writing of another put with a higher strike price) on the market.

They calculate the price of a put spread that should replicate the index tranche in combination with a risk less position in a bond. The replicating put portfolio pays a higher yield than the equivalent tranche.

 

 Reference: 

Coval, Joshua D., Jurek, Jakub W. and Stafford, Erik, "Economic Catastrophe Bonds" (July 2007). HBS Finance Working Paper No. 07-102 Available at SSRN: http://ssrn.com/abstract=995249

See also {ln:nw:An Empirical Analysis of the Pricing of Collateralized Debt Obligations} and {ln:nw:Measuring Default Risk Premia from Default Swap Rates and EDFs}

 

Hedging Tranched Index Products: Illustration of the Model Dependency

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In this paper, the authors analyze delta-hedging of liquid CDO tranches based on four different models:

-          Gaussian one-factor model with base correlation

-          Double-t one-factor model with base correlation

-          Double Normal Inverse Gaussian (NIG, from the same authors) with base correlation

-          NIG with historical correlation 

In the Gaussian one-factor model, default of a single entity occurs when its asset value falls below a threshold. The asset value is a combination of a market factor and a firm-specific factor.  

More precisely, the asset value of the ith firm is V(i) 

V(i)=rho(i) * Z + epsilon(i) * sqrt(1-rho(i)^2) 

Z is the market factorepsilon(i) is the idiosyncratic risk of the ith firmrho(i) is the loading on the market factor 

The epsilon(i) and the market factor are assumed to be independent Gaussian random variables. 

The correlation between the ith firm and the jth firm is the product rho(i)* rho(j) 

The correlation is assumed to be the same and the correlation that prices a tranche spread is the tranche implied correlation.

The implied correlation curve (implied correlation on y-axis vs. detachment points on x-axis) will usually have a smile. 

The tranche spreads create a sequence of base tranches, equity tranches of same detachment points as the initial tranches (the first base tranche is the equity tranche, the second base tranche is the equity tranche in addition to the first mezzanine tranche etc.). The base correlations are then the implied correlations that price the base correlations. Each base correlation will have its base correlation. The base correlation curve will usually have a skew (upward slopping). 

The double-t one-factor model with base correlation is similar to the Gaussian one-factor model with base correlation but the two factors follow instead t-distribution with fatter tails. 

The double Normal Inverse Gaussian (NIG, from the same authors) model takes a different approach. It takes the correlation as given and fit a NIG distribution on the two factors. The NIG distribution is flexible enough to produce skewness and kurtosis. 

The main contribution of the paper is the comparison of hedge ratios of the equity tranche by the first mezzanine tranche of Itraxx 5 Years and 10 Years Series 3 and 4 (from 21 March 2005 to 20 March 2006) calculated by the different models with the historical hedge ratio. The authors find some non-negligible differences between the hedge ratios and conclude that hedging activity still incurs a lot of model risk. Users beware!

Houdain, Julien and Guegan, Dominique, "Hedging Tranched Index Products: Illustration of the Model Dependency" (April 2006). Available at SSRN: http://ssrn.com/abstract=900526 

 

Option-Implied Correlations and the Price of Correlation Risk

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In this article, the authors study the correlation risk premium embedded in the equity option market, in particular the S&P100 index options. 

They compare the variance premium, the difference between the implied variance and the realized variance, in index options and the individual variance risk premia from single-name options over the period 1996 to 2003 (optionmetrics provided the option data, stock returns data come from CRSP).  

Their measures of implied-volatility does not come from Black-Scholes but are model-free, coming directly from the option prices. 

They find that the index premium is positive and statistically different from zero whereas the individual premia are close to zero. They take this finding as evidence that there is a correlation risk premium. 

The strategy of selling index variance and buying individual variances has a high Sharpe ratio and is very attractive for investors with CRRA (constant relative risk aversion) preferences. 

Furthermore, they extend their analysis to correlation risk premium since the index risk premium is constituted of the individual security risks and their correlation. The correlation risk premium is negative (-6.6 interpreted as the Sharpe ratio of a variance swap). Assets (such as index options) that increase in value when correlation is high should earn a lower return following the logic of Merton’s ICAPM. 

The authors also find that the correlation risk is priced in the cross-section of stock returns in addition to the Fama and French factors and the momentum factor. 

Driessen, Joost, Maenhout, Pascal J. and Vilkov, Grigory, "Option-Implied Correlations and the Price of Correlation Risk" (December 2006). EFA 2005 Moscow Meetings Available at SSRN: http://ssrn.com/abstract=673425

 

The Pricing of Correlated Default Risk: Evidence from the Credit Derivatives

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The Pricing of Correlated Default Risk:Evidence from the Credit Derivatives Market

In this report, the authors compare the prices of the liquid credit portfolio tranches to the prices implied by the single name credit default swaps and extract a correlation risk premium. They focus on the tranches (0-3%, 3-7%, 7-10% and 10-15%) of CDX.IG.5Y from November 13, 2003 to March 18, 2005.  JP Morgan provided the CDX data. 

They estimate the single name default correlations from the time series of default intensities extracted from credit default swaps of the CDX constituents (136 entities over the sample, data from Markit Partners).

They use a simple Merton formula and some simplifying assumptions to calculate the correlations (from daily observations over 6 months). 

They then use the default probabilities and the default correlations to simulate default by Monte Carlo and price the different tranches (predicted spreads) every day. 

They compare the market spreads and the predicted spreads and find that equity market spread is usually tighter than the predicted equity spread but that the senior spread is wider than predicted. The physical correlation is responsible for the error terms between the spreads. 

They also estimate the correlation risk premium, the difference between the risk-neutral correlation (from fitting the same correlation to all the four tranches) and the cross-sectional average of pair-wise physical correlations (from the single name correlation). The correlation risk premium is always positive in the sample. 

The correlation risk premium decreases with the physical correlation and is not greatly affected by the default probabilities. An increase in correlation risk premium however increases future physical correlation. 

Tarashev, Nikola A. and Zhu, Haibin, "The Pricing of Correlated Default Risk: Evidence from the Credit Derivatives Market" (February 2007). Available at SSRN: http://ssrn.com/abstract=967330 

 

An Empirical Analysis of the Pricing of Collateralized Debt Obligations

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In this paper, the authors estimate a three-factor CDO model based on firm-specific, industry-specific and economy-wide default intensities on historical synthetic credit index and tranche data (CDX).  

Synthetic CDO tranches are swap contracts that give exposure to the credit risk of a tranche of a credit portfolio.

The credit portfolio can be a standard portfolio such as the CDX NA IG (CDX North America Investment Grade, that represents 125 investment-grade names) or a bespoke portfolio. 

The most junior tranche of the CDX is the equity and represents the first 3% of the credit portfolio. Default losses that amount to less than 3% of the portfolio notional are limited to the equity tranche. Beyond 3%, defaults eat in the first mezzanine tranche (3%-7%), then the second mezzanine (7%-10%), third mezzanine tranche (10%-15%) and the senior tranche (15%-30%).  

The authors model the total portfolio loss as a random variable. Three stochastic jump processes drive the loss: a firms-specific jump, a sector-wide jump and an economy-wide jump.  

They assume that: 

L is the loss 

dL/(1-L)=gamma1 * dN1 + gamma2 * dN2 + gamma2 * dN2 

The gammi are the jump sizes and dNi are the jump processes with intensity lambdai. 

The jump intensities (probabilities of jumping) are themselves stochastic and follow a process: 

dlambdait=sigmai * sqrt(lambdait) * dZit,  dZit is a Brownian motion 

They estimate the coefficients gammai, sigmai and the fitted default intensities lambdait in time series and cross-section by iteration.  

They find that the jump sizes are consistent with the firm, sector and economy-wide interpretations of their jump processes. For instance the size of a firm-specific jump is smaller than the notional of a single firm. From the jump intensities, they find that on average a firm-specific default occurs in 1.16 year, a sector-wide default occurs in 41.5 years and a market-wide default in 763 years (they however correspond to risk-neutral defaults not actual defaults). The CDX spread presents mostly firm specific default risk (58%), then sector (33%) and market-wide risk (9%). 

They evaluate the fit of the model in terms of RMSE (root mean squared errors) and pricing errors (actual spreads vs. fitted spreads) and conclude that their model performed well with the exception of the first two CDX series when the market was probably less efficient.   

Longstaff, Francis A. and Rajan, Arvind, "An Empirical Analysis of the Pricing of Collateralized Debt Obligations" (April 2006). NBER Working Paper No. W12210 Available at SSRN: http://ssrn.com/abstract=902562

 



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