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The \(\beta \)-variance gamma model. (English) Zbl 1232.91713

Summary: A. Kuznetsov [Ann. Appl. Probab. 20, No. 5, 1801–1830 (2010; Zbl 1222.60038)] introduces a 10-parameter family of Lévy processes for which the Wiener-Hopf factors and the distribution of the running supremum (infimum) can be determined semi-analytically. In this text we will examine the numerical performance of this so-called \(\beta \)-family, both in the equity world and in the field of credit risk. In order to do this, we will calibrate a particular member of this family to a vanilla option surface (by means of the fast Fourier transform-technique due to P. Carr and D. P. Madan [“Option valuation using the fast Fourier transform”, J. Comput. Finance 2, No. 4, 61–73 (1999)] and use the resulting parameters to determine the prices of a digital down-and-out barrier (DDOB) option, written on the same underlying. In a second experiment, we will try and calibrate the model to some real-life credit default swap (CDS) term structures. The parameters of the model under investigation are chosen such that its Lévy density is approximately equal to that of the famous variance gamma (VG) process, which will serve as a benchmark. Hence, the former will be referred to as the \(\beta \)-VG model. The option prices will be determined both semi-analytically (using the formulas derived by Kuznetsov [loc.cit.]) and through a Monte-Carlo simulation. However, the CDS spreads will only be determined semi-analytically, due to the very close relation between pricing DDOB options and determining the par spread of a CDS. Furthermore, in both cases, the results will be compared with the ones obtained using the VG model; see [W. Schoutens, Lévy processes in finance: pricing financial derivatives. Chichester: Wiley (2003); W. Schoutens and J. Cariboni, Lévy processes in credit risk. Chichester: Wiley (2009; Zbl 1192.91008)]. It will turn out that, w.r.t. vanilla option prices, the \(\beta \)-VG model performs almost identically as the VG model, whereas the semi-analytical expressions by Kuznetsov [loc.cit.] lead to a (fast and) accurate pricing of DDOB options and CDSs.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
91G20 Derivative securities (option pricing, hedging, etc.)
91G40 Credit risk

Software:

Algorithm 368
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Full Text: DOI

References:

[1] Abate J., Whitt W. (1995) Numerical inversion of laplace transforms of probability distributions. ORSA Journal on Computing 7: 36–43 · Zbl 0821.65085 · doi:10.1287/ijoc.7.1.36
[2] Abate J., Whitt W. (1992) The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10: 5–88 · Zbl 0749.60013 · doi:10.1007/BF01158520
[3] Asmussen S., Rosiński J. (2001) Approximations of small jumps of Lévy processes with a view towards simulation. Journal of Applied Probability 38: 482–493 · Zbl 0989.60047 · doi:10.1239/jap/996986757
[4] Cariboni J., Schoutens W. (2009) Levy processes in credit risk. Wiley, Chichester · Zbl 1192.91008
[5] Carr P., Chang E. C., Madan D. B. (1998) The variance gamma process and option pricing. European Finance Review 2: 79–105 · Zbl 0937.91052 · doi:10.1023/A:1009703431535
[6] Carr P., Madan D. B. (1999) Option valuation using the fast Fourier transform. Journal of Computational Finance 2(4): 61–73
[7] Chen N., Kou S. G. (2009) Credit spreads, optimal capital structure, and implied volatility with endogenous default and jump risk. Mathematical Finance 19(3): 343–378 · Zbl 1168.91379 · doi:10.1111/j.1467-9965.2009.00375.x
[8] Kuznetsov, A. (2009). Wiener-Hopf factorization and distribution of extrema for a family of Lévy processes. Annals of Applied Probability (to appear).
[9] Kuznetsov, A. (2010). Wiener-Hopf factorization for a family of Lévy processes related to theta functions. (preprint). · Zbl 1223.60029
[10] Kuznetsov, A., Kyprianou, A. E., Pardo, J. C., & van Schaik, K. (2010). A Wiener-Hopf Monte-Carlo simulation technique for Levy processes (preprint). · Zbl 1245.65005
[11] Merton R. C. (1974) On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance 2: 449–470
[12] Rosiński J. (2008) Simulation of Levy processes, encyclopedia of statistics in quality and reliability: Computationally intensive methods and simulation. Wiley, New York
[13] Rydberg T. H. (1997) The normal inverse Gaussian Lévy process: Simulation and approximation. Stochastic Models 13(4): 887–910 · Zbl 0899.60036 · doi:10.1080/15326349708807456
[14] Schoutens W. (2003) Lévy processes in finance: Pricing financial derivatives. Wiley, Chichester
[15] Stehfest H. (1970) Algorithm 368 numerical inversion of Laplace transforms. Communications of the ACM 13: 479–490 (erratum 13, 624)
[16] Talbot A. (1979) The accurate numerical inversion of Laplace transforms. Journal of Applied Mathematics 23(1): 97–120 · Zbl 0406.65054
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