zbMATH — the first resource for mathematics

Numerical valuation of options with jumps in the underlying. (English) Zbl 1117.91028
Summary: A jump-diffusion model for a single-asset market is considered. Under this assumption the value of a European contingency claim satisfies a general partial integro-differential equation (PIDE). The equation is localized and discretized in space using finite differences and finite elements and in time by the second order backward differentiation formula (BDF2). The resulting system is solved by an iterative method based on a simple splitting of the matrix. Using the fast Fourier transform, the amount of work per iteration may be reduced to $$O(n\log_2n)$$ and only $$O(n)$$ entries need to be stored for each time level. Numerical results showing the quadratic convergence of the methods are given for Merton’s model and Kou’s model.

MSC:
 91B28 Finance etc. (MSC2000)
Full Text:
References:
 [1] Andersen, L.; Andreasen, J., Jump-diffusion processes: volatility smile Fitting and numerical methods for option pricing, Rev. derivatives res., 4, 231-262, (2000) · Zbl 1274.91398 [2] Barndorff, O.E., Processes of normal inverse Gaussian type, Finance stochast., 2, 41-68, (1998) · Zbl 0894.90011 [3] Bertoin, J., Lévy processes, Cambridge tracts in mathematics, vol. 121, (1996), Cambridge University Press Cambridge · Zbl 0861.60003 [4] Black, F.; Scholes, M.S., The pricing of options and corporate liabilities, J. political economy, 7, 637-654, (1973) · Zbl 1092.91524 [5] Boyarchenko, S.I.; Levendorskiı̆, S.Z., Non-Gaussian merton – black – scholes theory, Advanced series on statistical science & applied probability, vol. 9, (2002), World Scientific River Edge, NJ · Zbl 0997.91031 [6] M. Briani, C. La Chioma, R. Natalini, Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory, Numer. Mat., in press · Zbl 1065.65145 [7] Carr, P.P.; Geman, H.; Madan, D.B., The fine structure of asset returns: an empirical investigation, J. business, (2002) [8] Carr, P.P.; Madan, D.B., Option valuation using the fast Fourier transform, J. comput. finance, 2, 61-73, (1999) [9] Coleman, T.F.; Li, Y.; Verma, A., Reconstructing the unknown local volatility function, J. comput. finance, 2, 77-100, (1999) [10] Das, S.R.; Foresi, S., Exact solutions for bond and option prices with systematic jump risk, Rev. derivatives res., 1, 7-24, (1996) · Zbl 1274.91448 [11] Delbaen, F.; Schachermayer, W., A general version of the fundamental theorem of asset pricing, Math. ann., 300, 463-520, (1994) · Zbl 0865.90014 [12] d’Halluin, Y.; Forsyth, P.A.; Labahn, G., A penalty method for American options with jump diffusion processes, Numer. math., 97, 321-352, (2004) · Zbl 1126.91036 [13] Dupire, B., Pricing with a smile, RISK magazine, 1, 18-20, (1999) [14] Eberlein, E., Application of generalized hyperbolic Lévy motions to finance, (), 319-336 · Zbl 0982.60045 [15] Gerber, H.U.; Shiu, E.S.W., Option pricing by esscher transforms, Trans. soc. actuaries, 46, 99-140, (1995) [16] Harrison, J.M.; Kreps, D.M., Martingales and arbitrage in multiperiod securities markets, J. econom. theory, 20, 381-408, (1979) · Zbl 0431.90019 [17] Harrison, J.M.; Pliska, S.R., Martingales and stochastic integrals in the theory of continuous trading, Stochastic process. appl., 11, 215-260, (1981) · Zbl 0482.60097 [18] Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. financ. stud., 6, 327-343, (1993) · Zbl 1384.35131 [19] Horn, R.A.; Johnson, C.R., Topics in matrix analysis, (1994), Cambridge University Press Cambridge · Zbl 0801.15001 [20] Hull, J.; White, A., The pricing of options with stochastic volatilities, J. finance, 42, 281-300, (1987) [21] Karatzas, I.; Shreve, S.E., Methods of mathematical finance, Applications of mathematics, vol. 39, (1998), Springer New York · Zbl 0941.91032 [22] Kou, S.G., A jump diffusion model for option pricing, Management sci., 48, 1086-1101, (2002) · Zbl 1216.91039 [23] S.G. Kou, H. Wang, Option pricing under a double exponential jump diffusion model, Working paper, Columbia University, 2001 [24] A.L. Lewis, A simple option formula for general jump-diffusion and other exponential Lévy processes, in: 8th. Annual CAP Workshop on Derivative Securities and Risk Management, November 2001 [25] A.M. Matache, T. von Petersdorff, C. Schwab, Fast deterministic pricing of options on Lévy driven assets, Working paper, ETH, Zürich, 2002 · Zbl 1072.60052 [26] Merton, R.C., Option pricing when the underlying stocks are discontinuous, J. financ. econ., 5, 125-144, (1976) · Zbl 1131.91344 [27] Meyer, G.H., The numerical valuation of options with underlying jumps, Acta math. univ. Comenian, 67, 69-82, (1998) · Zbl 0936.91019 [28] S. Raible, Lévy Processes in Finance: Theory, Numerics, and Empirical Facts, Ph.D. Thesis, Inst. für Mathematische Stochastik, Albert-Ludwigs-Universität Freiburg, Freiburg, Germany, 2000 · Zbl 0966.60044 [29] Samuelson, P.A., Rational theory of warrant pricing, Indust. management rev., 6, 13-32, (1965) [30] Sato, K.-i., Basic results on Lévy processes, (), 3-37 · Zbl 0974.60036 [31] Van Loan, C., Computational frameworks for the fast Fourier transform, Frontiers in applied mathematics, vol. 10, (1992), SIAM Philadelphia, PA · Zbl 0757.65154 [32] Young, D.M., Iterative solution of large linear systems, (1971), Academic Press New York · Zbl 0204.48102
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.