A decade of CDO pricing


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A decade of CDO pricing

  1. 1. A Decade of CDO Pricing World Congress on Computational Finance Jon Gregory [email_address] March 26 th 2007 trant: Cover Option 2
  2. 2. Growth of Structured Credit Products Note: Notional excludes asset swaps Source: British Bankers Association Credit Derivatives Report 2006; Barclays Capital Credit Research 0 5 10 15 20 25 30 35 1996 1998 1999 2000 2001 2002 2003 2004 2006 2008E $ Trillion <ul><ul><ul><li>Credit Derivative Notional Outstanding </li></ul></ul></ul>Cash Flow Structured Finance CDOs Synthetic Balance Sheet CDOs; N th -to-Default Baskets Single Tranche CDOs; Managed CDOs; CDS Indices Bespoke Managed CDOs; Equity Default Swaps; Constant Maturity Default Swaps; Interest Rate Hybrids Options; Capital Structure Arbitrage; CDO 2 Synthetic Arbitrage CDOs Recovery Swaps; Dow Jones CDX/iTraxx Tranche Trades Leveraged super senior, CPPI and CPDO
  3. 3. Before the Correlation Market
  4. 4. The Gaussian Copula Model <ul><li>The Gaussian copula model </li></ul><ul><ul><li>Construction of default times consistent with marginal credit curves </li></ul></ul><ul><ul><li>Typically via a single correlation parameter (1F) </li></ul></ul><ul><ul><li>Fast semi-analytical formulas for pricing and greeks </li></ul></ul><ul><li>Typical trade, long mezzanine protection, delta hedged </li></ul><ul><ul><li>Positive carry trade </li></ul></ul><ul><ul><li>Short Idiosyncratic default risk </li></ul></ul><ul><ul><li>Gamma </li></ul></ul><ul><ul><ul><li>Short idiosyncratic gamma </li></ul></ul></ul><ul><ul><ul><li>Long parallel gamma </li></ul></ul></ul>Manifestation of correlation risk Parallel gamma Idiosyncratic gamma Default
  5. 5. Gaussian Copula Model in Action Many sudden credit events occurring early Several credit events Few credit events Hedging simulation of [3-6%] long protection position, delta hedged only
  6. 6. Model Risk : Choice of Copula <ul><li>From first to last to default swap premiums (bp pa) </li></ul>36 0.06 0.06 0.04 0.04 10 36 0.39 0.35 0.25 0.28 9 36 1.5 1.3 1.1 1.2 8 36 4.3 4.0 3.5 3.6 7 36 11 10 10 11 6 36 25 25 24 24 5 37 56 55 55 55 4 53 123 122 122 122 3 160 274 276 278 277 2 723 723 723 723 723 1 Marshall-Olkin Copula Clayton Copula Student-t copula (12 dof) Student-t copula (6 dof) Gaussian Copula Rank 10 names, spreads from 60 bps to 150 bps, recovery = 40%, maturity = 5 years, Gaussian correlation = 30%
  7. 7. Black Scholes compared to GCM <ul><li>Black-Scholes </li></ul><ul><li>Dynamic Model describing evolution of underlyings </li></ul><ul><li>Gaussian Copula Model </li></ul><ul><li>Static representation of default times </li></ul><ul><li>Price defined by unique replicating portfolio </li></ul><ul><li>Delivered volatility  Price </li></ul><ul><li>Replicating portfolio more complicated and not tradeable </li></ul><ul><li>Obvious economic intuition </li></ul><ul><li>Economics not obvious (tenuous intepretation via Merton model) </li></ul><ul><li>Delivered correlation is a complex function of greeks (gamma, realised defaults) </li></ul><ul><li>Natural extensions (e.g. stochastic volatility) linked to observation of market implied skew </li></ul><ul><li>Not so obvious how to extend and overcoming shortcomings </li></ul>
  8. 8. The Correlation Skew
  9. 9. <ul><li>Tranches of the Dow Jones CDX and iTraxx portfolios are now traded as liquid products to allow investors to express views on credit spread and default risk. </li></ul>Standard Index Tranches The growth of the index market has led the development of liquid tranched credit markets DJ iTraxx Europe Super Senior 22-100% Equity 0-3% 3-6% 6-9% 125 equally weighted names 12-22% 9-12% Tranched DJ iTraxx Europe
  10. 10. A Traded Correlation Market Market GCM 24.00% 19.3% 82.5 234.7 26.5 82.0 14.0 32.9 8.75 6.99 3.53 0.05 <ul><li>Dependency is defined by a single correlation parameter </li></ul><ul><ul><li>No concept of idiosyncratic default </li></ul></ul><ul><ul><li>No concept of systemic default </li></ul></ul>Super Senior 22-100% Equity 0-3% 3-6% 6-9% 12-22% 9-12%
  11. 11. Base Correlation
  12. 12. Base Correlation – Interpolation and Extrapolation [16-17%] tranchelet Base Correlation Curve “ Tranchelet” Premiums
  13. 13. Arbitrage-free Loss Interpolation <ul><li>Build base tranche expected loss curve as attachment point increases </li></ul><ul><li>Restrictions to be arbitrage-free </li></ul><ul><ul><li>Must be increasing (tranche expected tranche losses cannot be negative) </li></ul></ul><ul><ul><li>Must be concave (a more senior tranche cannot be more risky) </li></ul></ul><ul><ul><li>Must eventually hit index level (before 100%) </li></ul></ul>
  14. 14. Tranchelet Pricing – Some Extremes Maximum concavity, maximum dispersion, idiosyncratic risk 3 6 3 6 3 6 Tranche notional [0-3%] equity tranche  [0-1%], [1-2%] and [2-3%] tranchelets Minimum concavity, systemic risk effect 1,201 2-3% 1,201 1-2% 1,201 0-1% 61 2-3% 1,214 1-2% 3,090 0-1%
  15. 15. Pricing Tranchelets We know for example [0-3%] and [3-6%] Where would we price [0-1%], [1-2%], [2-3%], [3-4%], [4-5%] and [5-6%] ? <ul><li>All fit the 2 market prices </li></ul><ul><li>- All are arbitrage-free </li></ul>
  16. 16. CDO Models
  17. 17. CDO Models <ul><li>Many Examples </li></ul><ul><li>Extensions to Gaussian copula model </li></ul><ul><ul><li>Random factor loadings / local correlation </li></ul></ul><ul><ul><li>Stochastic correlation </li></ul></ul><ul><ul><li>Double-t / Double-NIG </li></ul></ul><ul><ul><li>Levy process / intensity gamma </li></ul></ul><ul><li>Dynamic models </li></ul><ul><ul><li>Stochastic intensity models </li></ul></ul><ul><ul><li>Dynamic loss models </li></ul></ul><ul><li>Typically quite hard to fit the market </li></ul>Implied copula
  18. 18. Difficulty in fitting the market Implied Compound Correlation
  19. 19. The Toothpaste Tube Analogy Super Senior 22-100% Equity 0-3% 3-6% 6-9% Index [0-100%] 12-22% 9-12% Index = Sum of tranches [22-100%] = [0-100%] – [0-3%] – [3-6%] – [6-9%] – [9-12%] – [12-22%]
  20. 20. The Toothpaste Tube Analogy (II) <ul><li>Not really correct </li></ul><ul><ul><li>Small changes in equity default timing assumptions can change the size of the tube…. </li></ul></ul>[22-100%] > [12-22%] [22-100%] = 0
  21. 21. Fitting the Market - Summary <ul><li>For hedging purposes need to fit tranches and index </li></ul><ul><li>Super senior risk causes real problems </li></ul><ul><ul><li>[22-100%] tranche can withstand 45 credit events at 40% recovery – very out of the money </li></ul></ul><ul><ul><li>Must have flexibility over timing of credit events </li></ul></ul><ul><ul><li>Shouldn’t try and boostrap the market </li></ul></ul><ul><ul><ul><li>example : 7Y equity gives information about 5Y super senior </li></ul></ul></ul><ul><ul><ul><li>example : 5Y equity tranchelets give information about 10Y equity </li></ul></ul></ul><ul><li>Very technical market </li></ul><ul><ul><li>Leveraged super senior issuance can move equity premiums </li></ul></ul><ul><ul><li>Dislocation between maturities </li></ul></ul><ul><li>Greeks </li></ul><ul><ul><li>If we don’t fit precisely how can we characterise / calculate greeks? </li></ul></ul>
  22. 22. Bespoke Tranches – Normalisation Methods If the portfolio is more risky then an equivalent tranche is more risky How to we adjust the correlation curve we use to account for this? Index portfolio Bespoke portfolio Expected loss Tranche
  23. 23. Bespoke Tranches – Normalisation Methods (II)
  24. 24. Structural Models Lead only one way <ul><li>Implied Copula Approach (Hull and White) </li></ul><ul><ul><li>Can fit index tranche market perfectly </li></ul></ul><ul><ul><li>Bespoke prices are not uniquely defined </li></ul></ul>
  25. 25. The Future
  26. 26. Index Correlation – off the run tranches Index rolls give us more maturity information detach maturity correlation CDX.5 CDX.6 CDX.7 CDX.8 CDX.5 CDX.6 CDX.7 CDX.8 CDX.5 CDX.6 CDX.7 CDX.8 5Y 7Y 10Y “ Base Correlation” Surface CDX.4 CDX.4 CDX.4
  27. 27. Index Correlation – HY/IG Different indices may provide complimentary information [15-30%] [25-35%] [10-15%] [15-25%] [7-10%] [10-15%] [3-7%] [0-10%] [0-3%] CDX HY CDX IG
  28. 28. Index Correlation – HY/IG (II) Test out your pricing method 3% 7% 10% 15% 30% IG HY 3% 35% 25% 15% 10% IG HY
  29. 29. Index Correlation – HY/IG (III) Obvious implications for Barbell portfolios [15-30%] [25-35%] [10-15%] [15-25%] [7-10%] [10-15%] [3-7%] [0-10%] [0-3%] CDX HY CDX IG
  30. 30. Bespoke CDO Pricing <ul><li>Many possible mapping techniques / models to go from index to bespoke </li></ul><ul><li>Shouldn’t really be expecting a unique solution </li></ul><ul><ul><li>Bespoke portfolio may not overlap / share characteristics with index from which it is valued </li></ul></ul><ul><li>Better approach to look at the whole picture </li></ul><ul><ul><ul><li>IG / HY / XO tranches </li></ul></ul></ul><ul><ul><ul><li>On-the-run and off-the-run tranches </li></ul></ul></ul><ul><ul><ul><li>Different regions </li></ul></ul></ul>iTraxx.6 5Y iTraxx.6 7Y iTraxx.6 10Y iTraxx.5 5Y iTraxx.5 7Y iTraxx.5 10Y HY tranches XO tranches Bespokes spread Maturity MODEL
  31. 31. Product Development <ul><li>Exotic Payoffs </li></ul><ul><ul><li>Cross-region, cross-asset </li></ul></ul><ul><ul><li>Long/short </li></ul></ul><ul><ul><li>IO/PO structures </li></ul></ul><ul><ul><li>CDO^2 </li></ul></ul><ul><li>Forward correlation </li></ul><ul><ul><li>Forward starting CDO </li></ul></ul><ul><ul><li>Amortising CDO </li></ul></ul><ul><li>Options </li></ul><ul><ul><li>Tranche options </li></ul></ul><ul><ul><li>Leveraged super senior tranches </li></ul></ul>Payoff sensitive to credit spread distributions aswell as default times Payoffs only depend on default times Large area of interest tackling these exotic CDOs Now we need a model based approach that can characterise maturity term structure Stochastic Intensity and Dynamic Loss Models Base correlation, implied copula approach
  32. 32. The Challenges and Solutions Tranchelet pricing Bespoke Pricing Forward Starting Loss Surface Construction Tranche Options Enhanced Base Correlation Methods Implied Copula Approach Stochastic Intensity Models Dynamic Loss Models There is no one to one mapping in the above  Tranche options pricing may be very sensitive to tranchelet pricing “ Exotic” CDOs
  33. 33. Disclaimer <ul><li>This presentation has been prepared by Barclays Capital - the investment banking division of Barclays Bank PLC and its affiliates worldwide (‘Barclays Capital’). This publication is provided to you for information purposes, any pricing in this report is indicative and is not intended as an offer or solicitation for the purchase or sale of any financial instrument. The information contained herein has been obtained from sources believed to be reliable but Barclays Capital does not represent or warrant that it is accurate and complete. The views reflected herein are those of Barclays Capital and are subject to change without notice. Barclays Capital and its respective officers, directors, partners and employees, including persons involved in the preparation or issuance of this document, may from time to time act as manager, co-manager or underwriter of a public offering or otherwise deal in, hold or act as market-makers or advisors, brokers or commercial and/or investment bankers in relation to the securities or related derivatives which are the subject of this report. </li></ul><ul><li>Neither Barclays Capital, nor any officer or employee thereof accepts any liability whatsoever for any direct or consequential loss arising from any use of this publication or its contents. Any securities recommendations made herein may not be suitable for all investors. Past performance is no guarantee of future returns. Any modeling or backtesting data contained in this document is not intended to be a statement as to future performance. </li></ul><ul><li>Investors should seek their own advice as to the suitability of any investments described herein for their own financial or tax circumstances. </li></ul><ul><li>This communication is being made available in the UK and Europe to persons who are investment professionals as that term is defined in Article 19 of the Financial Services and Markets Act 2000 (Financial Promotion Order) 2001. It is directed at persons who have professional experience in matters relating to investments. The investments to which is relates are available only to such persons and will be entered into only with such persons. </li></ul><ul><li>Barclays Capital - the investment banking division of Barclays Bank PLC, authorised and regulated by the Financial Services Authority (‘FSA’) and member of the London Stock Exchange. </li></ul><ul><li>Copyright in this report is owned by Barclays Capital (© Barclays Bank PLC, 2004) - no part of this report may be reproduced in any manner without the prior written permission of Barclays Capital. Barclays Bank PLC is registered in England No. 1026167. Registered office 54 Lombard Street, London EC3P 3AH. EUxxx </li></ul>trant: Note to Production Team Don’t forget to enter EU number EUXXX