# zbMATH — the first resource for mathematics

Efficient pricing of European options on two underlying assets by frame duality. (English) Zbl 1465.91120
The authors consider the option pricing method of density projection (onto B-splines) by frame duality, previously applied to pricing European options on one underlying asset, and extend it to higher dimensions, especially two-dimensions in which some exotic options can be priced. The technique does not require an a-priori truncation of the integration range, and exhibits excellent performance compared with other state-of-the-art methods, particularly for fatter-tailed short maturity models. Numerical results on implementation of this method to price for popular two-assets options, under both the geometric Brownian motion and variance-gamma dynamics, demonstrate remarkable accuracy and robustness.

##### MSC:
 91G20 Derivative securities (option pricing, hedging, etc.) 60G51 Processes with independent increments; Lévy processes 91G60 Numerical methods (including Monte Carlo methods)
Full Text:
##### References:
 [1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55 (1965), Courier Corporation [2] Borovkova, S.; Permana, F.; Van Der Weide, J., American basket and spread option pricing by a simple binomial tree, J. Deriv., 19, 4, 29-38 (2012) [3] Carr, P.; Madan, D., Option valuation using the fast Fourier transform, J. Comput. Finance, 2, 4, 61-73 (1999) [4] Christensen, O., An Introduction to Frames and Riesz Bases (2016), Birkhauser Boston · Zbl 1348.42033 [5] Chui, C., An Introduction to Wavelets (1992), Academic Press · Zbl 0925.42016 [6] Chui, C. K., Wavelets: A Mathematical Tool for Signal Analysis, vol. 1 (1997), SIAM · Zbl 0903.94007 [7] Colldeforns-Papiol, G.; Ortiz-Gracia, L.; Oosterlee, C. W., Two-dimensional Shannon wavelet inverse Fourier technique for pricing European options, Appl. Numer. Math., 117, 115-138 (2017) · Zbl 1414.91409 [8] Cui, Z.; Kirkby, J. L.; Nguyen, D., Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps, Insur. Math. Econ., 74, 46-62 (2017) · Zbl 1394.91206 [9] Cui, Z.; Kirkby, J. L.; Nguyen, D., Nonparametric density estimation by b-spline duality, Econom. Theory, 1-42 (2019) [10] De Boor, C.; DeVore, R. A.; Ron, A., Approximation from shift-invariant subspaces of $$l_2( \mathbf{R}^d)$$, Trans. Am. Math. Soc., 341, 2, 787-806 (1994) · Zbl 0790.41012 [11] Fang, F.; Oosterlee, C. W., A novel pricing method for European options based on Fourier-cosine series expansions, SIAM J. Sci. Comput., 31, 2, 826-848 (2008) · Zbl 1186.91214 [12] Fang, F.; Oosterlee, C. W., Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions, Numer. Math., 114, 1, 27 (2009) · Zbl 1185.91176 [13] Fang, F.; Oosterlee, C. W., A Fourier-based valuation method for Bermudan and barrier options under Heston’s model, SIAM J. Financ. Math., 2, 1, 439-463 (2011) · Zbl 1236.65163 [14] Feng, L.; Lin, X., Inverting analytic characteristic functions and financial applications, SIAM J. Financ. Math., 4, 1, 372-398 (2013) · Zbl 1282.60021 [15] Hurd, T. R.; Zhou, Z., A Fourier transform method for spread option pricing, SIAM J. Financ. Math., 1, 1, 142-157 (2010) · Zbl 1188.91218 [16] Kirkby, J. L., Efficient option pricing by frame duality with the fast Fourier transform, SIAM J. Financ. Math., 6, 1, 713-747 (2015) · Zbl 1320.91155 [17] Kirkby, J. L., An efficient transform method for Asian option pricing, SIAM J. Financ. Math., 7, 1, 845-892 (2016) · Zbl 1357.91053 [18] Kirkby, J. L., Robust option pricing with characteristic functions and the b-spline order of density projection, J. Comput. Finance, 21, 2, 101-127 (2017) [19] Kirkby, J. L., Robust barrier option pricing by frame projection under exponential Lévy dynamics, Appl. Math. Finance, 24, 4, 337-386 (2017) · Zbl 1398.91672 [20] Kirkby, J. L.; Deng, S., Static hedging and pricing of exotic options with payoff frames, Math. Finance, 29, 2, 612-658 (2019) · Zbl 1411.91567 [21] Kirkby, J. L.; Deng, S.-J., Swing option pricing by dynamic programming with B-spline density projection, Int. J. Theor. Appl. Finance, Article 1950038 pp. (2019) · Zbl 1430.91113 [22] Kirkby, J. L.; Nguyen, D.; Cui, Z., A unified approach to Bermudan and barrier options under stochastic volatility models with jumps, J. Econ. Dyn. Control, 80, 75-100 (2017) · Zbl 1401.91533 [23] Leentvaar, C.; Oosterlee, C. W., Multi-asset option pricing using a parallel Fourier-based technique, J. Comput. Finance, 12, 1-26 (2008) [24] Ortiz-Gracia, L.; Oosterlee, C. W., Robust pricing of European options with wavelets and the characteristic function, SIAM J. Sci. Comput., 35, 5, B1055-B1084 (2013) · Zbl 1281.62227 [25] Ortiz-Gracia, L.; Oosterlee, C. W., A highly efficient Shannon wavelet inverse Fourier technique for pricing European options, SIAM J. Sci. Comput., 38, 1, B118-B143 (2016) · Zbl 1330.91184 [26] Ruijter, M. J.; Oosterlee, C. W., Two-dimensional Fourier cosine series expansion method for pricing financial options, SIAM J. Sci. Comput., 34, 5, B642-B671 (2012) · Zbl 1258.91222 [27] Semeraro, P., A multivariate variance gamma model for financial applications, Int. J. Theor. Appl. Finance, 11, 01, 1-18 (2008) · Zbl 1152.91548 [28] Tavella, D.; Randall, C., Pricing Financial Instruments: The Finite Difference Method, vol. 13 (2000), John Wiley & Sons [29] Trefethen, L. N.; Weideman, J., The exponentially convergent trapezoidal rule, SIAM Rev., 56, 3, 385-458 (2014) · Zbl 1307.65031 [30] Zhang, B.; Oosterlee, C. W., Efficient pricing of European-style Asian options under exponential Lévy processes based on Fourier cosine expansions, SIAM J. Financ. Math., 4, 1, 399-426 (2013) · Zbl 1282.65023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.