An efficient transform method for Asian option pricing.

*(English)*Zbl 1357.91053##### MSC:

91G60 | Numerical methods (including Monte Carlo methods) |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

65T50 | Numerical methods for discrete and fast Fourier transforms |

65D07 | Numerical computation using splines |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

60E10 | Characteristic functions; other transforms |

91G20 | Derivative securities (option pricing, hedging, etc.) |

##### Keywords:

arithmetic Asian options; fast Fourier transform; Lévy processes; basis; characteristic function; Carverhill-Clewlow factorization; PROJ; COS; FFT; frame projection; option pricing; B-spline; exotic options
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##### References:

[1] | H. Albrecher, P. Mayer, and W. Schoutens, General lower bounds for arithmetic Asian option prices, Appl. Math. Finance, 15 (2008), pp. 123–149. · Zbl 1134.91394 |

[2] | H. Albrecher and M. Predota, Bounds and approximations for discrete Asian options in a variance-gamma model, Grazer Math. Ber., 345 (2002), pp. 35–57. · Zbl 1053.91056 |

[3] | B. Alziary, J. P. Dechamps, and P. F. Koehl, PDE approach to Asian options: Analytical numerical evidence, J. Banking Finance, 21 (1997), pp. 613–640. |

[4] | J. Andreasen, The pricing of discretely sampled Asian and lookback options: A change of numeraire approach, J. Comput. Finance, 1 (1998), pp. 15–36. |

[5] | O. Barndorff-Nielsen, Processes of normal inverse Gaussian type, Finance Stoch., 2 (1997), pp. 41–68. · Zbl 0894.90011 |

[6] | E. Benhamou, Fast Fourier transform for discrete Asian options, J. Comput. Finance, 6 (2002), pp. 49–61. |

[7] | S. Boyarchenko and S. Levendorskii, Non-Gaussian Merton-Black-Scholes Theory, Adv. Ser. Stat. Sci. Appl. Probab. 9, World Scientific, River Edge, NJ, 2002. |

[8] | S. Boyarchenko and S. Levendorskii, Efficient versions of the Fourier transform in applications to option pricing, J. Comp. Finance, 18 (2014). |

[9] | S. Boyarchenko and S. Levendorskii, Option pricing for truncated Levy processes, Int. J. Theor. App. Finance, 3 (2000), pp. 549–552. · Zbl 0973.91037 |

[10] | P. Carr, H. Geman, D. B. Madan, and M. Yor, The fine structure of asset returns: An empirical investigation, J. Business, 75 (2002), pp. 305–332. |

[11] | A. Carverhill and L. Clewlow, Flexible convolution: Valuing average rate (Asian) options, Risk, 3 (1990), pp. 25–29. |

[12] | A. Cerny and I. Kyriakou, An improved convolution algorithm for discretely sampled Asian options, Quant. Finance, 11 (2011), pp. 381–389. · Zbl 1232.91653 |

[13] | O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003. · Zbl 1017.42022 |

[14] | T. Dai and Y. Lyuu, Accurate and efficient lattice algorithms for American-style Asian options with range bounds, Appl. Math. Comput., 209 (2009), pp. 238–253. · Zbl 1156.91365 |

[15] | M. De Innocentis and S. Levendorskii, Pricing discrete barrier options and credit default swaps under Levy processes, Quant. Finance, 14 (2014), pp. 1337–1365. · Zbl 1402.91825 |

[16] | P. Den Iseger and E. Oldenkamp, Pricing guaranteed return rate products and discretely sampled Asian options, J. Comput. Finance, 9 (2006), pp. 1–39. |

[17] | J. Dewynne and P. Wilmott, Asian options as linear complementary problems, Adv. Futures Options Res., 8 (1995), pp. 145–173. |

[18] | E. Eberlein and A. Papapantoleon, Equivalence of floating and fixed strike Asian and lookback options, Stochastic Process. Appl., 115 (2005), pp. 31–40. · Zbl 1114.91049 |

[19] | F. Fang and C. W. Oosterlee, A novel pricing method for European options based on Fourier-cosine series expansions, SIAM J. Sci. Comput., 31 (2009), pp. 826–848. · Zbl 1186.91214 |

[20] | L. Feng and X. Lin, Inverting analytic characteristic functions and financial applications, SIAM J. Finan. Math., 4 (2013), pp. 372–398. · Zbl 1282.60021 |

[21] | L. Feng and V. Linetsky, Pricing discretely monitored barrier options and defaultable bonds in Levy process models: A fast Hilbert transform approach, Math. Finance, 18 (2008), pp. 337–384. · Zbl 1141.91438 |

[22] | G. Fusai, D. Marazzina, and M. Marena, Pricing discretely monitored Asian options by maturity randomization, SIAM J. Finance Math., 2 (2011), pp. 383–403. · Zbl 1215.91079 |

[23] | G. Fusai and A. Meucci, Pricing discretely monitored Asian options under Levy processes, J. Banking Finance, 32 (2008), pp. 2076–2088. |

[24] | H. Geman and D. B. Madan, Risks in returns: A pure jump perspective, in Exotic Option Pricing and Advanced Levy Models, A. Kyprianou, W. Schountens and P. Wilmott, eds., Wiley, Chichester, 2005, pp. 51–66. |

[25] | C. Heil, A Basis Theory Primer. Birkhauser, New York, 2011. · Zbl 1227.46001 |

[26] | N. Ju, Pricing Asian and basket options via Taylor expansion, J. Comput. Finance, 5 (2002), pp. 79–103. |

[27] | A. Kemna and A. Vorst, A pricing method for options based on average asset values, J. Banking and Finance, 14 (1990), pp. 113–129. |

[28] | J. L. Kirkby, Efficient option pricing by frame duality with the fast Fourier transform, SIAM J. Finan. Math., 6 (2015), pp. 713–747. · Zbl 1320.91155 |

[29] | J. L. Kirkby, Robust option pricing with characteristic functions and the B-spline order of density projection, J. Comput. Finance, to appear. |

[30] | J. L. Kirkby and S. Deng, Static hedging and pricing of exotic options with payoff frames, Math. Finance., under review. · Zbl 1411.91567 |

[31] | D. Lemmens, L. Z. J. Liang, J. Tempere, and A. De Schepper, Pricing bounds for discrete arithmetic Asian options under Levy models, Phys. A, 389 (2010), pp. 5193–5207. |

[32] | S.Z. Levendorskii and J. Xie, Pricing Discretely sampled Asian Options Under Levy Processes, , 2012. |

[33] | D. Madan and E. Seneta, The variance gamma (v.g.) model for share market returns, J. Business, 63 (1990), pp. 511–524. |

[34] | J. Nielsen and K. Sandmann, Pricing bounds on Asian options, J. Finance Quant. Anal., 38 (2003), pp. 449–473. |

[35] | P. Sabino, Monte Carlo methods and path-generation techniques for pricing multi-asset path- dependent options, preprint, , 2007. |

[36] | K-I. Sato, Levy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999. · Zbl 0973.60001 |

[37] | F. Stenger, Numerical Methods based on Sinc and Analytic Functions, Springer-Verlag, New York, 1993. · Zbl 0803.65141 |

[38] | M. Unser and I. Daubechies, On the approximation power of convolution-based least squares versus interpolation, IEEE Trans. Signal Process., 45 (1997), pp. 1697–1711. · Zbl 0879.94005 |

[39] | M. Unser, Vanishing moments and the approximation power of wavelet expansions, In Proceedings of the 1996 IEEE International Conference on Image Processing, IEEE, Piscataway, NJ, 1996, pp. 629–632. |

[40] | J. Vecer, A new PDE approach for pricing arithmetic average Asian options, J. Comput. Finance, 4 (2001), pp. 105–115. |

[41] | B. Zhang and C. W. Oosterlee, Efficient pricing of European-style Asian options under exponential Lévy processes based on Fourier cosine expansions, SIAM J. Finan. Math., 4 (2013), pp. 399–426. · Zbl 1282.65023 |

[42] | J. E. Zhang, Pricing continuously sampled Asian options with perturbation method, J. Futures Markets, 23 (2003), pp. 535–560. |

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