×

Correlators of polynomial processes. (English) Zbl 1479.91391

Summary: In the setting of one-dimensional polynomial jump-diffusion dynamics, we provide an explicit formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula appears as a linear combination of exponentials of the generator matrix, extending the well-known moment formula for polynomial processes. The developed framework can, for example, be applied in financial pricing, such as for path-dependent options and in a stochastic volatility models context. In applications to options, having closed and compact formulations is attractive for sensitivity analysis and risk management, since Greeks can be derived explicitly.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
60J74 Jump processes on discrete state spaces
60J60 Diffusion processes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] D. Ackerer and D. Filipovic, Linear credit risk models, Finance Stoch., 24 (2020), pp. 169-214. · Zbl 1445.91066
[2] D. Ackerer, D. Filipovic, and S. Pulido, The Jacobi stochastic volatility model, Finance Stoch., 22 (2018), pp. 667-700. · Zbl 1402.91746
[3] D. Ackerer and D. Filipovic, Option pricing with orthogonal polynomial expansions, Math. Finance, 30 (2020), pp. 47-84. · Zbl 1508.91546
[4] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed., Cambridge University Press, Cambridge, UK, 2009. · Zbl 1200.60001
[5] O. E. Barndorff-Nielsen, F. E. Benth, and A. E. D. Veraart, Ambit Stochastics, Springer, Cham, 2018. · Zbl 1472.60002
[6] F. E. Benth, J. Kallsen, and T. Meyer-Brandis, A non-Gaussian Ornstein-Uhlenbeck process for electricity spot price modeling and derivatives pricing, Appl. Math. Finance, 14 (2007), pp. 153-169. · Zbl 1160.91337
[7] F. E. Benth, M. Groth, and R. Kufakunesu, Valuing volatility and variance swaps for a non-Gaussian Ornstein-Uhlenbeck stochastic volatility model, Appl. Math. Finance, 14 (2007), pp. 347-363. · Zbl 1141.91015
[8] P. Carr and D. B. Madan, Option valuation using the fast Fourier transform, J. Comput. Finance, 2 (1999), pp. 61-73.
[9] C. Cuchiero, Affine and Polynomial Processes, Ph.D. thesis, ETH Zürich, Zürich, Switzerland, 2011.
[10] C. Cuchiero, M. Keller-Ressel, and J. Teichmann, Polynomial processes and their applications to mathematical finance, Finance Stoch., 16 (2012), pp. 711-740. · Zbl 1270.60079
[11] C. Cuchiero, Polynomial processes in stochastic portfolio theory, Stochastic Process. Appl., 129 (2018), pp. 1829-1872. · Zbl 1426.91243
[12] C. Cuchiero and S. Svaluto-Ferro, Infinite-dimensional polynomial processes, Finance Stoch., 25 (2021), pp. 383-426. · Zbl 1461.91310
[13] F. Delbaen and H. Shirakawa, An interest rate model with upper and lower bounds, Asia-Pacific Financial Markets, 9 (2002), pp. 191-209. · Zbl 1071.91020
[14] D. Fasino, Spectral properties of Hankel matrices and numerical solutions of finite moment problems, J. Comput. Appl. Math., 65 (1995), pp. 145-155. · Zbl 0855.65042
[15] D. Filipović, M. Larsson, and A. B. Trolle, Linear-rational term structure models, J. Finance, 72 (2016), pp. 655-704.
[16] D. Filipović and M. Larsson, Polynomial diffusions and applications in finance, Finance Stoch., 20 (2016), pp. 931-972. · Zbl 1386.60237
[17] M. Fiedler, Polynomials and Hankel matrices, Linear Algebra Appl., 66 (1985), pp. 235-248. · Zbl 0569.15005
[18] D. Filipović and M. Larsson, Polynomial jump-diffusion models, Stoch. Syst., 10 (2020), pp. 71-97. · Zbl 1450.60038
[19] N. Golyandina, V. Nekrutkin, and A. A. Zhigljavsky, Analysis of Time Series Structure: SSA and Related Techniques, CRC Press, Boca Raton, FL, 2001. · Zbl 0978.62073
[20] H. Hassani and D. Thomakos, A review on singular spectrum analysis for economic and financial time series, Stat. Interface, 3 (2010), pp. 377-397. · Zbl 1245.91078
[21] N. J. Higham, The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 1179-1193, https://doi.org/10.1137/04061101X. · Zbl 1081.65037
[22] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0729.15001
[23] P. Jain and R. B. Pachori, Event-based method for instantaneous fundamental frequency estimation from voiced speech based on eigenvalue decomposition of the Hankel matrix, IEEE/ACM Trans. Audio Speech Language Process., 22 (2014), pp. 1467-1482.
[24] P. Jain and R. B. Pachori, An iterative approach for decomposition of multi-component non-stationary signals based on eigenvalue decomposition of the Hankel matrix, J. Franklin Institute, 352 (2015), pp. 4017-4044. · Zbl 1395.94122
[25] A. G. Z. Kemna and T. A. C. F. Vorst, A pricing method for options based on average asset values, J. Banking Finance, 14 (1990), pp. 113-129.
[26] X. Kleisinger-Yu, V. Komaric, M. Larsson, and M. Regez, A multifactor polynomial framework for long-term electricity forwards with delivery period, SIAM J. Financial Math., 11 (2020), pp. 928-957, https://doi.org/10.1137/19M1283264. · Zbl 1452.91310
[27] D. Kressner, R. Luce, and F. Statti, Incremental computation of block triangular matrix exponentials with application to option pricing, Electron. Trans. Numer. Anal., 47 (2017), pp. 57-72. · Zbl 1371.15007
[28] S. Lavagnini, Pricing Asian Options with Correlators, preprint, https://arxiv.org/abs/2104.11684, 2021.
[29] J. R. Magnus and H. Neudecker, The elimination matrix: Some lemmas and applications, SIAM J. Algebraic Discrete Methods, 1 (1980), pp. 422-449, https://doi.org/10.1137/0601049. · Zbl 0497.15014
[30] J. Munkhammar, L. Mattsson, and J. Rydén, Polynomial probability distribution estimation using the method of moments, PLOS ONE, 12 (2017), e0174573.
[31] V. Peller, Hankel Operators and Their Applications, Springer, New York, 2003. · Zbl 1030.47002
[32] A. Townsend, M. Webb, and S. Olver, Fast polynomial transforms based on Toeplitz and Hankel matrices, Math. Comp., 87 (2018), pp. 1913-1934. · Zbl 1478.65147
[33] T. Ware, Polynomial processes for power prices, Appl. Math. Finance, 26 (2019), pp. 453-474. · Zbl 1433.91104
[34] R. Weron, Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach, Wiley Finance Ser. 403, John Wiley & Sons, New York, 2007.
[35] S. Willems, Asian option pricing with orthogonal polynomials, Quant. Finance, 19 (2019), pp. 605-618. · Zbl 1420.91481
[36] H. Zhou, Itô conditional moment generator and the estimation of short-rate processes, J. Financial Econometrics, 1 (2003), pp. 250-271.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.