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The Jacobi stochastic volatility model. (English) Zbl 1402.91746
Summary: We introduce a novel stochastic volatility model where the squared volatility of the asset return follows a Jacobi process. It contains the Heston model as a limit case. We show that the joint density of any finite sequence of log-returns admits a Gram-Charlier A expansion with closed-form coefficients. We derive closed-form series representations for option prices whose discounted payoffs are functions of the asset price trajectory at finitely many time points. This includes European call, put and digital options, forward start options, and can be applied to discretely monitored Asian options. In a numerical study, we show that option prices can be accurately and efficiently approximated by truncating their series representations.

MSC:
91G20 Derivative securities (option pricing, hedging, etc.)
91G60 Numerical methods (including Monte Carlo methods)
60H30 Applications of stochastic analysis (to PDEs, etc.)
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[1] Abken, P.A., Madan, D.B., Ramamurtie, B.S.: Estimation of risk-neutral and statistical densities by Hermite polynomial approximation: with an application to Eurodollar futures options. Working Paper 96-5, Federal Reserve Bank of Atlanta (1996). Available online at https://www.researchgate.net/profile/Dilip_Madan · Zbl 0236.60037
[2] Ackerer, D., Filipović, D.: Option pricing with orthogonal polynomial expansions. Swiss Finance Institute Research Paper No. 17-41, 2017. Available online at https://ssrn.com/abstract=3076519 · Zbl 1298.91171
[3] Ahdida, A.; Alfonsi, A., A mean-reverting SDE on correlation matrices, Stoch. Process. Appl., 123, 1472-1520, (2013) · Zbl 1271.65014
[4] Ait-Sahalia, Y., Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach, Econometrica, 70, 223-262, (2002) · Zbl 1104.62323
[5] Al-Mohy, A.H.; Higham, N.J., Computing the action of the matrix exponential, with an application to exponential integrators, SIAM J. Sci. Comput., 33, 488-511, (2011) · Zbl 1234.65028
[6] Albrecher, H.; Mayer, P.; Schoutens, Wim; Tistaert, Jurgen, The little Heston trap, Wilmott Mag., January, 83-92, (2007)
[7] Andersen, L.B.G.; Piterbarg, V.V., Moment explosions in stochastic volatility models, Finance Stoch., 11, 29-50, (2007) · Zbl 1142.65004
[8] Backus, D.K., Foresi, S., Wu, L.: Accounting for biases in Black-Scholes, Working Paper (2004). Available online at https://ssrn.com/abstract=585623
[9] Bakshi, G.; Madan, D.B., Spanning and derivative-security valuation, J. Financ. Econ., 55, 205-238, (2000)
[10] Bernis, G.; Scotti, S., Alternative to beta coefficients in the context of diffusions, Quant. Finance, 17, 275-288, (2017) · Zbl 1402.91976
[11] Billingsley, P.: Probability and Measure, 3rd edn. Wiley Series in Probability and Statistics. Wiley, New York (1995) · Zbl 0822.60002
[12] Black, F.; Scholes, M.S., The pricing of options and corporate liabilities, J. Polit. Econ., 81, 637-654, (1973) · Zbl 1092.91524
[13] Brenner, M., Eom, Y.H.: No-arbitrage option pricing: New evidence on the validity of the martingale property. NYU Working Paper No. FIN-98-009 (1997). Available online at https://ssrn.com/abstract=1296404 · Zbl 1092.91026
[14] Broadie, M.; Kaya, Ö., Exact simulation of stochastic volatility and other affine jump diffusion processes, Oper. Res., 54, 217-231, (2006) · Zbl 1167.91363
[15] Carr, P.; Madan, D.B., Option valuation using the fast Fourier transform, J. Comput. Finance, 2, 61-73, (1999)
[16] Chen, H.; Joslin, S., Generalized transform analysis of affine processes and applications in finance, Rev. Financ. Stud., 25, 2225-2256, (2012)
[17] Corrado, C.J.; Su, T., Skewness and kurtosis in S&P 500 index returns implied by option prices, J. Financ. Res., 19, 175-192, (1996)
[18] Corrado, C.J.; Su, T., Implied volatility skews and stock index skewness and kurtosis implied by S&P 500 index option prices, J. Deriv., 4, 8-19, (1997)
[19] Cuchiero, C.; Keller-Ressel, M.; Teichmann, J., Polynomial processes and their applications to mathematical finance, Finance Stoch., 16, 711-740, (2012) · Zbl 1270.60079
[20] Delbaen, F.; Shirakawa, H., An interest rate model with upper and lower bounds, Asia-Pac. Financ. Mark., 9, 191-209, (2002) · Zbl 1071.91020
[21] Demni, N.; Zani, M., Large deviations for statistics of the Jacobi process, Stoch. Process. Appl., 119, 518-533, (2009) · Zbl 1158.60008
[22] Drimus, G.G., Necula, C., Farkas, W.: Closed form option pricing under generalized Hermite expansions. Working Paper (2013). Available online at https://ssrn.com/abstract=2349868 · Zbl 1384.35131
[23] Duffie, D.; Filipović, D.; Schachermayer, W., Affine processes and applications in finance, Ann. Appl. Probab., 13, 984-1053, (2003) · Zbl 1048.60059
[24] Dufresne, D.: The integrated square-root process. Working Paper No. 90, Centre for Actuarial Studies, University of Melbourne (2001). Available online at https://hdl.handle.net/11343/33693 · Zbl 0884.90029
[25] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. Vols. I, II. McGraw-Hill, New York (1953) · Zbl 0052.29502
[26] Eriksson, B.; Pistorius, M., Method of moments approach to pricing double barrier contracts in polynomial jump-diffusion models, Int. J. Theor. Appl. Finance, 14, 1139-1158, (2011) · Zbl 1229.91304
[27] Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence, 2nd edn. Wiley, New York (1986) · Zbl 0592.60049
[28] Fang, F.; Oosterlee, K., A novel pricing method for European options based on Fourier-cosine series expansions, SIAM J. Sci. Comput., 31, 826-848, (2009) · Zbl 1186.91214
[29] Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1. Wiley, New York (1960) · Zbl 0158.34902
[30] Filipović, D.; Larsson, M., Polynomial diffusions and applications in finance, Finance Stoch., 20, 931-972, (2016) · Zbl 1386.60237
[31] Filipović, D.; Mayerhofer, E.; Schneider, P., Density approximations for multivariate affine jump-diffusion processes, J. Econom., 176, 93-111, (2013) · Zbl 1284.62110
[32] Geman, H.; Yor, M., Bessel processes, Asian options, and perpetuities, Math. Finance, 3, 349-375, (1993) · Zbl 0884.90029
[33] Gouriéroux, C.; Jasiak, J., Multivariate Jacobi process with application to smooth transitions, J. Econom., 131, 475-505, (2006) · Zbl 1337.62365
[34] Heston, S.L., 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
[35] Heston, S.L.; Rossi, A.G., A spanning series approach to options, Rev. Asset Pricing Stud., 7, 2-42, (2016)
[36] Hochbruck, M.; Lubich, C., On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34, 1911-1925, (1997) · Zbl 0888.65032
[37] Jäckel, P.: A note on multivariate Gauss-Hermite quadrature. Technical report (2005). Available online at http://www.jaeckel.org · Zbl 1270.60079
[38] Jacquier, A.; Roome, P., Asymptotics of forward implied volatility, SIAM J. Financ. Math., 6, 307-351, (2015) · Zbl 1339.60021
[39] Jarrow, R.; Rudd, A., Approximate option valuation for arbitrary stochastic processes, J. Financ. Econ., 10, 347-369, (1982)
[40] Kahl, C.; Jäckel, P., Not-so-complex logarithms in the Heston model, Wilmott Mag., September, 94-103, (2005)
[41] Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991) · Zbl 0734.60060
[42] Karlin, S., Taylor, H.M.: A Second Course in Stochastic Processes. Academic Press, San Diego (1981) · Zbl 0469.60001
[43] Kruse, S.; Nögel, U., On the pricing of forward starting options in heston’s model on stochastic volatility, Finance Stoch., 9, 233-250, (2005) · Zbl 1092.91026
[44] Larsson, M.; Pulido, S., Polynomial preserving diffusions on compact quadric sets, Stoch. Process. Appl., 127, 901-926, (2017) · Zbl 1355.60107
[45] Li, H.; Melnikov, A.; Fung, W.K. (ed.); etal., On polynomial-normal model and option pricing, 285-302, (2012), Singapore · Zbl 1277.91172
[46] Longstaff, F., Option pricing and the martingale restriction, Rev. Financ. Stud., 8, 1091-1124, (1995)
[47] Madan, D.B.; Milne, F., Contingent claims valued and hedged by pricing and investing in a basis, Math. Finance, 4, 223-245, (1994) · Zbl 0884.90042
[48] Mazet, O.; Azéma, J. (ed.); etal., Classification des semi-groupes de diffusion sur ℝ associés à une famille de polynômes orthogonaux, No. 1655, 40-53, (1997), Berlin · Zbl 0883.60072
[49] Necula, C., Drimus, G., Farkas, W.: A general closed form option pricing formula. Swiss Finance Institute Research Paper No. 15-53 (2015). Available online at https://ssrn.com/abstract=2210359
[50] Pagès, G.; Printems, J., Optimal quadratic quantization for numerics: the Gaussian case, Monte Carlo Methods Appl., 9, 135-165, (2003) · Zbl 1029.65012
[51] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999) · Zbl 0917.60006
[52] Rogers, L.C.G.; Shi, Z., The value of an Asian option, J. Appl. Probab., 32, 1077-1088, (1995) · Zbl 0839.90013
[53] Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales: Volume 1, Foundations, 2nd edn. Cambridge University Press, Cambridge (2000) · Zbl 0949.60003
[54] Dacheng, X., Hermite polynomial based expansion of European option prices, J. Econom., 179, 158-177, (2014) · Zbl 1298.91171
[55] Yamada, T.; Watanabe, S., On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11, 155-167, (1971) · Zbl 0236.60037
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