Abstract
The computational effort of pricing an m-th to default swap depends highly on the size n of the underlying basket. Usually, n different default times are modeled, but in fact the valuation only depends on the m-th smallest default time of this tuple. In this paper we attain an analytical formula for the distribution of this m-th default time. With the help of this distribution we simplify the valuation problem from an n-dimensional quadrature to a one-dimensional quadrature and break the curse of dimensionality. Applications of this modification are efficient pricing of m-th to default swaps, estimation of sensitivities and pricing of European max/min options.
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References
Duffie, D., Singleton, K.J.: Credit Risk: Pricing, Measurement and Management. Princeton University Press, Princeton (2003)
Li, D.X.: On default correlation: a copula function approach. J. Fixed Income 9(4), 43–54 (2000)
Embrechts, P., Lindskog, F., McNeil, A.: Modelling dependence with copulas and applications to risk management. In: Rachev, S.T. (ed.) Handbook of Heavy Tailed Distributions in Finance. Elsevier, Amsterdam (2003)
Marshall, A.W., Olkin, I.: Families of multivariate distributions. J. Am. Stat. Assoc. 83(403), 834–841 (1988)
Hofert, M.: Sampling archimedean copulas. Comput. Stat. Data Anal. 52(12), 5163–5174 (2008)
Joshi, M.S., Kainth, D.: Rapid and accurate development of prices and Greeks for th to default credit swaps in the Li model. Quant. Finance 4(3), 266–275 (2004)
Chen, Z., Glasserman, P.: Fast pricing of basket default swaps. Oper. Res. 56(2), 286–303 (2008)
Schröter, A., Heider, P.: Numerical methods to quantify the models risk of basket default swaps. Preprint. http://www.mi.uni-koeln.de/~aschroet/preprint.pdf (2010)
Schönbucher, P.J.: Credit Derivatives Pricing Models: Models, Pricing and Implementation. Wiley, New York (2003)
Sklar, A.: Random variables, joint distribution functions and copulas. Kybernetika 9(6), 449–460 (1973)
Nelsen, R.B.: An Introduction to Copulas. Springer, Berlin (2006)
Takacs, L.: On the method of inclusion and exclusion. J. Am. Stat. Assoc. 62(317), 102–113 (1967)
Gallier, J.: Discrete Mathematics. Springer, Berlin (2011)
Cheney, W., Kincaid, D.: Numerical Mathematics and Computing. Brooks/Cole, Pacific Grove (2007)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964)
Genz, A., Bretz, F.: Computation of Multivariate Normal and t Probabilities. Springer, Berlin (2009)
Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, Berlin (2002)
Bodie, Z., Kane, A., Marcus, A.J.: Investments, 8th edn. McGraw-Hill, New York (2009)
Shampine, L.F.: Matlab program for quadrature in 2d. Appl. Math. Comput. 202(1), 266–274 (2008)
Shampine, L.F.: Vectorized adaptive quadrature in Matlab. J. Comput. Appl. Math. 211(2), 131–140 (2008)
Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, Berlin (2004)
Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)
Seydel, R.U.: Tools for Computational Finance. Springer, Berlin (2009)
Haug, E.G.: The Complete Guide to Option Pricing Formulas. McGraw-Hill, New York (2006)
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Schröter, A., Heider, P. An analytical formula for pricing m-th to default swaps. J. Appl. Math. Comput. 41, 229–255 (2013). https://doi.org/10.1007/s12190-012-0589-1
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DOI: https://doi.org/10.1007/s12190-012-0589-1
Keywords
- m-th to default swaps
- Principle of inclusion and exclusion
- Copula models
- Credit derivatives sensitives
- European max/min options