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Increment Variance Reduction Techniques with an Application to Multi-name Credit Derivatives

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Abstract

Increment variance reduction techniques are add-ons to Monte Carlo (MC) simulations. They make MC simulations converging faster by repeating the number of simulations with an incremental rate derived from mathematical functions. Besides speeding up MC simulations, the major advantage of increment techniques is their ability to handle large numbers of simulations avoiding memory saturation and overflow which occur when a plain MC simulation is involved in the pricing of multi-name credit derivatives. A trend among authors pricing financial securities with MC simulation has been to choose Quasi-Monte Carlo (QMC) methods using deterministic sequences instead of MC methods involving pseudorandom generators such as congruential generator and Mersenne twister. The Increment family models circumvent the constraint of identifying the optimal QMC sequence to price a given security by using a common generator such as Matlab-LCG-Xor RNG and determining the optimal mathematical function of incrementation of MC simulations that will make the pricing of the security adequate. Market participants in need of selecting a reliable numerical method for pricing complex financial securities such as multi-name credit derivatives will find our paper appealing.

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References

  • Andersen, L., & Sidenius, J. (2005). Extensions to the Gaussian copula: Random recovery and random factor loadings. Journal of Credit Risk,1, 29–70.

    Google Scholar 

  • Andersen, L., Sidenius, J., & Basu, S. (2003). All your hedges in one basket. Risk,16, 67–72.

    Google Scholar 

  • Berry, A. C. (1941). The accuracy of the Gaussian approximation to the sum of independent variates. Transactions of the American Mathematical Society,49, 122–136.

    Google Scholar 

  • Biancardi, M., & Villani, G. (2017). Robust Monte Carlo method for R&D real options valuation. Computational Economics,49(3), 481–498. https://doi.org/10.1007/s10614-016-9578-z.

    Google Scholar 

  • Black, F., & Cox, J. (1976). Valuing corporate securities—Some effects of bond indenture provisions. Journal of Finance,31, 351–367.

    Google Scholar 

  • Brent, R. P. (2004). Note on Marsaglia’s xorshift random number generators. Journal of Statistical Software,11, 1–4.

    Google Scholar 

  • Brigo, D. & Errais, E. (2005). A correlation bridge between structural models and reduced form models for multiname credit derivatives. www.defaultrisk.com. Accessed March 3, 2018.

  • Burtschell, X., Gregory, J., & Laurent, J.-P. (2005). A comparative analysis of CDO pricing models. Working paper, BNP Paribas.

  • Burtschell, X., Gregory, J., & Laurent, J.-P. (2007). Beyond the Gaussian copula: Stochastic and local correlation. Journal of Credit Risk,3, 31–62.

    Google Scholar 

  • Burtschell, X., Gregory, J., & Laurent, J.-P. (2009). A comparative analysis of CDO pricing models under the factor Copula framework. Journal of Derivatives,16, 9–37.

    Google Scholar 

  • Clewlow, L. J., & Carverhill, A. (1994). On the simulation of contingent claims. Journal of Derivatives,2, 66–74.

    Google Scholar 

  • Conghui, H., Xun, Z., & Qiuming, G. (2015). Synthetic CDO pricing: the perspective of risk integration. Applied Economics,47, 1574–1587.

    Google Scholar 

  • Demarta, S., & McNeil, A. J. (2005). The t copula and related copulas. International Statistical Review,73, 111–129.

    Google Scholar 

  • Duffie, D. (2004). Time to adapt copula methods for modelling credit risk correlation. Risk, 77.

  • Duffie, D., & Gârleanu, N. (2001). Risk and the valuation of collateralized debt obligations. Financial Analyst Journal,57, 41–59.

    Google Scholar 

  • Duffie, D., & Singleton, K.J. (1998). Simulating correlated defaults. Working paper, Stanford University.

  • Duffie, D., & Singleton, K. J. (1999). Modeling term structures of defaultable bonds. Review of Financial Studies,12, 687–720.

    Google Scholar 

  • Eberlein, E., Frey, R. & Von Hammerstein, E. (2008). Advanced credit portfolio modeling and CDO pricing. Mathematics-Key Technology for the Future. http://link.springer.com/chapter/10.1007/978-3-540-77203-3_17 Accessed March 3, 2018.

  • Elouerkhaoui, Y. (2003a). Credit derivatives: Basket asymptotics. Working paper, UBS Warburg.

  • Elouerkhaoui, Y. (2003b). Credit risk: Correlation with a difference. Working paper, UBS Warburg.

  • Embrechts, P., Lindskog, F., & McNeil, A. (2003). Modelling dependence with copulas and applications to risk management. In S. Rachev (Ed.), Handbook of heavy tailed distributions in finance. Amsterdam: Elsevier.

    Google Scholar 

  • Esseen, C.-G. (1942). On the Liapunoff limit of error in the theory of probability. Arkiv för matematik, astronomi och fysik, A28, 1–19. ISSN 0365-4133.

  • Esseen, C.-G. (1956). A moment inequality with an application to the central limit theorem. Skand, Aktuarietidskr,39, 160–170.

    Google Scholar 

  • Feldhütter, P. (2008). An empirical investigation of an intensity-based model for pricing CDO tranches. Working paper http://www.feldhutter.com/CDOpaper.pdf. Accessed March 3, 2018.

  • Finger, C. (2004). Issues in the pricing of synthetic CDOs. Journal of Credit Risk,1, 113–124.

    Google Scholar 

  • Frey, R., & McNeil, A. (2003). Dependent defaults in models of portfolio credit risk. Journal of Risk,6, 59–92.

    Google Scholar 

  • Friend, A., & Rogge, E. (2004). Correlation at first sight. Economic notes: Review of banking. Finance and Monetary Economics,34, 155–183.

    Google Scholar 

  • Giesecke, K. (2003). A simple exponential model for dependent defaults. Journal of Fixed Income,13, 74–83.

    Google Scholar 

  • Glasserman, P., & Suchintabandid, S. (2007). Correlation expansions for CDO pricing. Journal of Banking & Finance,31, 1375–1398.

    Google Scholar 

  • Glynn, P. W., & Iglehart, D. L. (1989). Importance sampling for stochastic simulations. Management Science,35, 1367–1393.

    Google Scholar 

  • Goodman, L. S. (2002). Synthetic CDOs: An introduction. Journal of Derivatives, Spring,9, 60–72.

    Google Scholar 

  • Greenberg, A., Mashal, R., Naldi, M., & Schlogl, L. (2004). Tuning correlation and tail risk to the market prices of liquid tranches. Lehman Brothers: Quantitative Research Quarterly.

    Google Scholar 

  • Gregory, J. and Laurent, J-P. (2003) I will survive. Risk, June, 103-107.

  • Hager, S. (2008). Pricing portfolio credit derivatives by means of evolutionary algorithms. Deutsche National bibliografie, Dissertation Universität Tübingen, Gabler Edition Wissenschaft.

  • Hammersley, J. M., & Morton, K. W. (1956). A new Monte Carlo technique: Antithetic variates. Mathematical Proceedings of the Cambridge Philosophical Society,52(3), 449–475. https://doi.org/10.1017/S0305004100031455.

    Google Scholar 

  • Hull, J., Predescu, M., & White, A. (2005). The valuation of correlation dependent credit derivatives using a structural model. www.defaultrisk.com. Accessed March 3, 2018.

  • Hull, J., & White, A. (2004). Valuation of a CDO and an nth to default CDS without Monte Carlo simulation. Journal of Derivatives,12, 8–23.

    Google Scholar 

  • Hull, J., & White, A. (2006). Valuing credit derivatives using an implied copula approach. Journal of Derivatives,14, 8–28.

    Google Scholar 

  • Jäckel, P. (2002). Monte Carlo methods in finance. Chichester: Wiley.

    Google Scholar 

  • Joy, C., Boyle, P. P., & Tan, K. S. (1996). Quasi-Monte Carlo methods in numerical finance. Management Science,42, 926–938.

    Google Scholar 

  • Kalemanova, A., Schmid, B., & Werner, R. (2007). The normal inverse Gaussian distribution for synthetic CDO pricing. The Journal of Derivatives,14, 80–94.

    Google Scholar 

  • Kolman, M. (2013). Selected one-factor methods to price synthetic CDOs. Working paper, University of Economics, Prague.

  • Lamb, R., Perraudin, W., & Van Landschoot, A. (2008). Dynamic Pricing of synthetic collateralized debt obligations. Imperial College. http://econpapers.repec.org/paper/ecbecbwps/20080910.htm. Accessed March 3, 2018.

  • Lando, D. (1994). Three essays on contingent claims pricing. Ph. D. thesis, Cornell University.

  • Laurent, J.-P., & Gregory, J. (2005). Basket default swaps, CDOs and factor copulas. Journal of Risk,7, 103–122.

    Google Scholar 

  • L’Ecuyer, P., & Simard R. (2007). TestU01: A C library for empirical testing of random number generators. ACM Transactions on Mathematical Software, 33, article 22.

  • Li, D. X. (2000). On default correlation: A copula function approach. The Journal of Fixed Income,9, 43–54.

    Google Scholar 

  • Lindskog, F., & McNeil, A. (2003). Common Poisson shock models: Applications to insurance and credit risk modeling. ASTIN Bulletin,33, 209–238.

    Google Scholar 

  • Madan, D. B., Konikov, & Marinescu, M. (2004). Credit and basket default swaps. Journal of Credit Risk, 2.

  • Marsaglia, G. (2003). Xorshift RNGs. Journal of Statistical Software,8, 1–6.

    Google Scholar 

  • Mashal, R., Naldi, M. & Zeevi, A. (2003). On the dependence of equity and asset returns Risk, October, 83–87.

  • Mashal, R., & Zeevi, A. (2003). Inferring the dependence structure of financial assets: empirical evidence and implications. Working paper, Columbia Business School.

  • Matsumoto, M., & Kurita, Y. (1992). Twisted GFSR Generators II. Working paper, Research Institute for Mathematical Sciences, National Research Laboratory of Metrology, Tsukuba 305, Kyoto University, Kyoto 606, Japan.

  • Matsumoto, M., & Nishimura, T. (1998). Mersenne twister: A 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Transactions on Modeling and Computer Simulation,8, 3–30.

    Google Scholar 

  • Merton, R. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance,29, 449–470.

    Google Scholar 

  • Michello, F. A., & Deme, M. (2012). Communication failures, synthetic CDOs, and the 2008 financial crisis. Academy of Accounting & Financial Studies Journal,16, 105–121.

    Google Scholar 

  • Moosbrucker, T. (2006). Pricing CDOs with correlated variance gamma distributions. Journal of Fixed Income,12, 1–30.

    Google Scholar 

  • Mortensen, A. (2006). Semi-analytic valuation of basket credit derivatives in intensity-based models. Journal of Derivatives,13, 8–26.

    Google Scholar 

  • Mounfield, C. C. (2008). Synthetic CDOs: Modelling, valuation and risk management (mathematics, finance and risk). Cambridge: Cambridge University Press.

    Google Scholar 

  • O’Kane, D. (2009). Modelling single-name and multi-name credit derivatives. Chichester: Wiley.

    Google Scholar 

  • Panneton, F., & L’Ecuyer, P. (2005). On the xorshift random number generators. ACM Transactions on Modeling and Computer Simulation,15, 346–361.

    Google Scholar 

  • Rogge, E., & Schonbucher, P. J. (2003). Modelling dynamic portfolio credit risk. Working paper, Imperial College London and ETH Zurich.

  • Roncoroni, A., & Fusai, G. (2008). Implementing models in quantitative finance: Methods and cases. Berlin: Springer.

    Google Scholar 

  • Rostan, P., & Rostan, A. (2013). Testing quasi-random versus pseudorandom numbers on bond option pricing. The IEB International Journal of Finance,6, 96–115.

    Google Scholar 

  • Rostan, P., Rostan, A., & Racicot, F.-E. (2015). Pricing discrete double barrier options with a numerical method. The Journal of Asset Management,16, 243–271. https://doi.org/10.1057/jam2015.6.

    Google Scholar 

  • Särndal, C.-E., Swensson, B., & Wretman, J. (2003). Stratified sampling. Model assisted survey sampling. New York: Springer.

    Google Scholar 

  • Schloegl, L., & O’Kane, D. (2005). A note on the large homogeneous portfolio approximation with the student-t copula. Finance and Stochastics,9, 577–584.

    Google Scholar 

  • Schonbucher, P. J. (2002). Taken to the limit: Simple and not-so-simple loan loss distributions. Working paper, University of Bonn.

  • Schönbucher, P., & Schubert, D. (2001). Copula-dependent default risk in intensity models. TIK-Report No. 103. http://mx.nthu.edu.tw/~jtyang/Teaching/Risk_management/Papers/Correlations/Copula-Dependent%20Default%20Risk%20in%20Intensity%20Models.pdf. Accessed March 3, 2018.

  • Schruben, L. W. (2011). Common random numbers. Chichester: Wiley Encyclopedia of Operations Research and Management Science, Wiley. https://doi.org/10.1002/9780470400531.eorms0166.

    Google Scholar 

  • Sowers, R. (2010). Exact pricing asymptotics of investment-grade tranches of synthetic CDO’s: A large homogeneous pool. International Journal of Theoretical and Applied Finance,13, 367–401.

    Google Scholar 

  • Traub, J. F., & Paskov, S. (1995). Faster evaluation of financial derivatives. Journal of Portfolio Management,22, 113–120.

    Google Scholar 

  • Veremyev, A., Tsyurmasto, P., Uryasev, S., & Rockafellar, R. T. (2012). Convex–concave–convex distributions in application to CDO pricing. Working paper, Risk Management and Financial Engineering Lab, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL.

  • Wang, D., Rachev, S. T., & Fabozzi, F. J. (2009). Pricing of credit default index swap tranches with one-factor heavy-tailed copula models. Journal of Empirical Finance,16, 201–215.

    Google Scholar 

  • Whetten, M., & Adelson, M. (2004). The Bespoke [bispóuk] A Guide to Single-Tranche Synthetic CDOs. Nomura Fixed Income Research, http://www.derivativeslawyer.com/doctemplates/1000065.pdf. Accessed March 3, 2018.

  • Wong, D. (2000). Copula from the limit of a multivariate binary model. Working paper, Bank of America Corporation.

  • Wu, J.-L., & Yang, W. (2010). Pricing CDO tranches in an intensity based model with the mean reversion approach. Mathematical and Computer Modelling,52, 814–825.

    Google Scholar 

  • Xiao, Y., & Wang, X. (2017). Enhancing Quasi-Monte Carlo simulation by minimizing effective dimension for derivative pricing. Computational Economics. https://doi.org/10.1007/s10614-017-9732-2.

    Google Scholar 

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Authors and Affiliations

  1. American University of Paris, 102, rue Saint Dominique, 75007, Paris, France

    Pierre Rostan & Alexandra Rostan

  2. Telfer School of Management, University of Ottawa, 55 Laurier Avenue East, Ottawa, ON, K1N 6N5, Canada

    François-Éric Racicot

  3. Affiliate Research Fellow, IPAG Business School, Paris, France

    François-Éric Racicot

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  1. Pierre Rostan
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Correspondence to Pierre Rostan.

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Rostan, P., Rostan, A. & Racicot, FÉ. Increment Variance Reduction Techniques with an Application to Multi-name Credit Derivatives. Comput Econ 55, 1–35 (2020). https://doi.org/10.1007/s10614-018-9828-3

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