Binomial approximation to locally dependent collateralized debt obligations. (English) Zbl 1523.62047
Summary: In this paper, we develop Stein’s method for binomial approximation using the stop-loss metric that allows one to obtain a bound on the error term between the expectation of call functions. We obtain the results for a locally dependent collateralized debt obligation (CDO), under certain conditions on moments. The results are also exemplified for an independent CDO. Finally, it is shown that our bounds are sharper than the existing bounds.
MSC:
| 62E17 | Approximations to statistical distributions (nonasymptotic) |
| 60E05 | Probability distributions: general theory |
| 60F05 | Central limit and other weak theorems |
| 62E20 | Asymptotic distribution theory in statistics |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
References:
| [1] | Barbour, AD; Holst, L.; Janson, S., Poisson Approximation (1992), Oxford University Press · Zbl 0746.60002 |
| [2] | Barbour, AD; Čekanavičius, V., Total variation asymptotes for sum of independent integer random variables, Ann Prob, 30, 509-545 (2002) · Zbl 1018.60049 · doi:10.1214/aop/1023481001 |
| [3] | Barbour, AD; Xia, A., Poisson perturbation, ESAIM Probab Statist, 3, 131-150 (1999) · Zbl 0949.62015 · doi:10.1051/ps:1999106 |
| [4] | Boutsikas, MV; Vaggelatou, E., On the distance between convex-ordered random variables, with applications, Adv Appl Probab, 34, 349-374 (2002) · Zbl 1005.60036 · doi:10.1239/aap/1025131222 |
| [5] | Brown, TC; Phillips, MJ, Negative binomial approximation with Stein’s method, Methodol Comput Appl Probab, 1, 407-421 (1999) · Zbl 0992.62015 · doi:10.1023/A:1010094221471 |
| [6] | Brown, TC; Xia, A., Stein’s method and birth-death processes, Ann Probab, 29, 1373-1403 (2001) · Zbl 1019.60019 · doi:10.1214/aop/1015345606 |
| [7] | Čekanavičius, V.; Vellaisamy, P., Discrete approximations for sums of \(m\)-dependent random variables, ALEA Lat Am J Probab Math Stat, 12, 765-792 (2015) · Zbl 1331.60045 |
| [8] | Čekanavičius, V.; Vellaisamy, P., Compound Poisson approximations in \(\ell_p\)-norm for sums of weakly dependent vectors, J Theoret Probab, 34, 2241-2264 (2021) · Zbl 1478.62052 · doi:10.1007/s10959-020-01042-9 |
| [9] | Chen, LHY, Poisson approximation for dependent trials, Ann Probability, 3, 534-545 (1975) · Zbl 0335.60016 · doi:10.1214/aop/1176996359 |
| [10] | Eichelsbacher, P.; Reinert, G., Stein’s method for discrete Gibbs measures, Ann Appl Probab, 18, 1588-1618 (2008) · Zbl 1146.60011 · doi:10.1214/07-AAP0498 |
| [11] | El Karoui, N.; Jiao, Y., Stein’s method and zero bias transformation for CDO tranche pricing, Finance Stoch, 13, 151-180 (2009) · Zbl 1199.91063 · doi:10.1007/s00780-008-0084-6 |
| [12] | El Karoui, N.; Jiao, Y.; Kurtz, D., Gaussian and Poisson approximation: applications to CDOs tranche pricing, J Comput Finance, 12, 31-58 (2008) · Zbl 1175.91177 · doi:10.21314/JCF.2008.180 |
| [13] | Hull, JC; White, AD, Valuation of a CDO and an \(n^{\text{th}}\) to default CDS without Monte Carlo simulation, J Deriv, 12, 8-23 (2004) · doi:10.3905/jod.2004.450964 |
| [14] | Kumar, AN, Approximations to weighted sums of random variables, Bull Malays Math Sci Soc, 44, 2447-2464 (2021) · Zbl 1472.62030 · doi:10.1007/s40840-021-01077-z |
| [15] | Kumar AN (2022) Bounds on negative binomial approximation to call function. To appear in Revstat Stat J |
| [16] | Kumar, AN; Upadhye, NS; Vellaisamy, P., Approximations related to the sums of \(m\)-dependent random variables, Braz J Probab Stat, 36, 349-368 (2022) · Zbl 1487.60092 · doi:10.1214/21-BJPS529 |
| [17] | Kumar, AN; Vellaisamy, P.; Viens, F., Poisson approximation to the convolution of power series distributions, Probab Math Statist, 42, 63-80 (2022) · Zbl 07596636 · doi:10.37190/0208-4147.00056 |
| [18] | Mattner, L.; Roos, B., A shorter proof of Kanter’s Bessel function concentration bound, Probab Theory Relat Fields, 139, 191-205 (2007) · Zbl 1127.60018 · doi:10.1007/s00440-006-0043-0 |
| [19] | Neammanee, K.; Yonghint, N., Poisson approximation for call function via Stein-Chen method, Bull Malays Math Sci Soc, 43, 1135-1152 (2020) · Zbl 1434.60074 · doi:10.1007/s40840-019-00729-5 |
| [20] | Röllin, A., Symmetric and centered binomial approximation of sums of locally dependent random variables, Electron J Probab, 13, 756-776 (2008) · Zbl 1189.60053 |
| [21] | Stein C (1972) A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proc. Sixth Berkeley Symp. Math. Statist. Probab. II. Probability Theory, Univ. California Press, Berkeley, Calif., pp 583-602 · Zbl 0278.60026 |
| [22] | Vellaisamy, P.; Punnen, AP, On the nature of the binomial distribution, J Appl Probab, 38, 36-44 (2001) · Zbl 0987.60023 · doi:10.1239/jap/996986641 |
| [23] | Vellaisamy, P.; Upadhye, NS; Čekanavičius, V., On negative binomial approximation, Theory Probab Appl, 57, 97-109 (2013) · Zbl 1268.60017 · doi:10.1137/S0040585X97985819 |
| [24] | Yonghint, N.; Neammanee, K.; Chaidee, N., Poisson approximation for locally dependent CDO, Comm Statist Theory Methods, 51, 2073-2081 (2022) · Zbl 07897022 · doi:10.1080/03610926.2020.1759638 |
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