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On Asian option pricing for NIG Lévy processes. (English) Zbl 1107.91042

Summary: We derive approximations and bounds for the Esscher price of European-style arithmetic and geometric average options. The asset price process is assumed to be of exponential Lévy type with normal inverse Gaussian (NIG) distributed log-returns. Numerical illustrations of the accuracy of these bounds as well as approximations and comparisons of the NIG average option prices with the corresponding Black–Scholes prices are given.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
60G99 Stochastic processes
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[1] Abramowitz, M.; Stegun, I., Handbook of mathematical functions, (1968), Dover Publications New York
[2] Barndorff-Nielsen, O.E., Exponentially decreasing distributions for the logarithm of particle size, Proc. roy. soc. London A, 353, 401-419, (1977)
[3] O.E. Barndorff-Nielsen, Normal inverse Gaussian processes and the modelling of stock returns, Research Report 300, Department Theoretical Statistics, Aarhus University, 1995.
[4] Barndorff-Nielsen, O.E., Normal inverse Gaussian distributions and stochastic volatility modelling, Scand. J. statist., 24, 1-13, (1997) · Zbl 0934.62109
[5] Barndorff-Nielsen, O.E., Processes of normal inverse Gaussian type, Finance stoch., 2, 1, 41-68, (1998) · Zbl 0894.90011
[6] Bühlmann, H.; Delbaen, F.; Embrechts, P.; Shiryaev, A.N., No-arbitrage, change of measure and conditional esscher transforms, CWI quart., 9, 291-317, (1996) · Zbl 0943.91037
[7] A. Cherny, No-arbitrage and completeness for the linear and exponential models based on Lévy processes, Research Report 2001-33, MaPhySto, University of Aarhus, 2001.
[8] Dhaene, J.; Denuit, M.; Goovaerts, M.J.; Kaas, R.; Vyncke, D., The concept of comonotonicity in actuarial science and financetheory, Insur. math. econom., 31, 3-33, (2001) · Zbl 1051.62107
[9] Eberlein, E., Application of generalized hyperbolic Lévy motions to finance, (), 319-337 · Zbl 0982.60045
[10] Eberlein, E.; Keller, U., Hyperbolic distributions in finance, Bernoulli, 1, 281-299, (1995) · Zbl 0836.62107
[11] Eberlein, E.; Prause, K., The generalized hyperbolic modelfinancial derivatives and risk measures, ()
[12] Geman, H., Pure jump Lévy processes for asset price modelling, J. bank. finance, 26, 1297-1316, (2002)
[13] Gerber, H.U.; Shiu, E.S.W., Option pricing by esscher transforms, Trans. soc. actuaries, XLVI, 99-140, (1994)
[14] Gerber, H.U.; Shiu, E.S.W., Author’s reply to the discussions on the paper, “option pricing by esscher transforms”, Trans. soc. actuaries, XLVI, 175-177, (1994)
[15] Grandits, P., The p-optimal martingale measure and its asymptotic relation with the minimal-entropy martingale measure, Bernoulli, 5, 225-247, (1999) · Zbl 0923.60045
[16] J. Hartinger, M. Predota, Pricing Asian options in the hyperbolic model: a fast Quasi-Monte Carlo approach, Grazer Math. Ber. 345 (2002) 1-33. · Zbl 1053.91059
[17] Jarrow, R.; Rudd, A., Approximate option valuation for arbitrary stochastic processes, J. financial econom., 10, 347-369, (1982)
[18] Kemna, A.G.Z.; Vorst, A.C.F., A pricing method for options based on average asset values, J. bank. finance, 14, 113-129, (1990) · Zbl 0638.90013
[19] Levy, E., Pricing European average rate currency options, J. internat. money finance, 11, 474-491, (1992)
[20] Madan, D.B.; Milne, F., Option pricing with V.G. martingale components, Math. finance, 1, 39-55, (1991) · Zbl 0900.90105
[21] K. Prause, The generalized hyperbolic model: estimation, financial derivatives and risk measures, Ph.D. Thesis, University of Freiburg, 1999. · Zbl 0944.91026
[22] S. Raible, Lévy processes in Finance: Theory, Numerics and Empirical Facts, Ph.D. Thesis, University of Freiburg, 2000. · Zbl 0966.60044
[23] Rydberg, T., The normal inverse Gaussian Lévy processsimulation and approximation, Comm. statist. stoch. models, 13, 887-910, (1997) · Zbl 0899.60036
[24] Rydberg, T., Generalized hyperbolic diffusion processes with applications in finance, Math. finance, 9, 183-201, (1999) · Zbl 0980.91039
[25] Simon, S.; Goovaerts, M.J.; Dhaene, J., An easy computable upper bound for the price of an arithmetic Asian option, Insur. math. econom., 26, 175-183, (2000) · Zbl 0964.91021
[26] Turnbull, S.; Wakeman, L., A quick algorithm for pricing European average options, J. financial quant. anal., 26, 377-389, (1991)
[27] Vorst, T.C.F., Prices and hedge ratios of average exchange rate options, Internat. rev. financial anal., 1, 3, 179-193, (1992)
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