-
Previous Article
Law of large numbers and central limit theorem under nonlinear expectations
- PUQR Home
- This Issue
-
Next Article
Correction to: “Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting”
Affine processes under parameter uncertainty
1. Department of Mathematical Stochastics, University of Freiburg, Ernst-Zermelo Str. 1, 79104 Freiburg, Germany; |
2. Nanyang Technological University, Division of Mathematical Sciences, Singapore, Singapore; |
3. Freiburg Institute of Advanced Studies(FRIAS), Freiburg im Breisgau, Germany; |
4. University of Strasbourg Institute for Advanced Study(USIAS), Strasbourg, France |
We then develop an appropriate Itô formula, the respective term-structure equations, and study the nonlinear versions of the Vasiček and the Cox-Ingersoll-Ross (CIR) model. Thereafter, we introduce the nonlinear Vasiček-CIR model. This model is particularly suitable for modelling interest rates when one does not want to restrict the state space a priori and hence this approach solves the modelling issue arising with negative interest rates.
References:
[1] |
Acciaio, B, Beiglböck, M, Penkner, F, Schachermayer, W:A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance. 26(2), 233-251(2016) |
[2] |
Amadori, AL:Uniqueness and comparison properties of the viscosity solution to some singular HJB, equations. Nonlinear Differ. Equ. Appl. NoDEA. 14(3-4), 391-409(2007) |
[3] |
Avellaneda, M, Levy, A, Parás, A:Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance. 2(2), 73-88(1995) |
[4] |
Bannör, KF, Scherer, M:Capturing parameter risk with convex risk measures. Eur. Actuar. J. 3(1), 97-132(2013) |
[5] |
Barrieu, P, Scandolo, G:Assessing financial model risk. Eur. J. Oper. Res. 242(2), 546-556(2015) |
[6] |
Bergenthum, J, Rüschendorf, L:Comparison of semimartingales and Lévy processes. Ann. Probab. 35(1), 228-254(2007) |
[7] |
Biagini, S, Bouchard, B, Kardaras, C, Nutz, M:Robust fundamental theorem for continuous processes. Math. Finance. 27(4), 963-987(2017) |
[8] |
Bielecki, TR, Cialenco, I, Rutkowski, M:Arbitrage-free pricing of derivatives in nonlinear market models. Probab. Uncertain. Quant. Risk. 3(1), 2(2018) |
[9] |
Bouchard, B, Touzi, N:Weak dynamic programming principle for viscosity solutions. SIAM J. Control. Optim. 49(3), 948-962(2011) |
[10] |
Breuer, T, Csiszár, I:Measuring distribution model risk. Math. Financ. 26(2), 395-411(2016) |
[11] |
Carver, L:Negative rates:Dealers struggle to price 0% floors. Risk Mag. (2012) |
[12] |
Cont, R:Model uncertainty and its impact on the pricing of derivative instruments. Math. Financ. 16, 519-542(2006) |
[13] |
Costantini, C, Papi, M, D'Ippoliti, F:Singular risk-neutral valuation equations. Financ. Stochast. 16(2), 249-274(2012) |
[14] |
Crandall, MG, Ishii, H, Lions, P-L:User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27(1), 1-67(1992) |
[15] |
da Fonseca, J, Grasselli, M:Riding on the smiles. Quant. Financ. 11(11), 1609-1632(2011) |
[16] |
Denis, L, Martini, C:A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16(2), 827-852(2006) |
[17] |
Denk, R, Kupper, M, Nendel, M:A semigroup approach to nonlinear Lévy processes (2017). arXiv:1710.08130v1 |
[18] |
Duffie, D, Filipović D, Schachermayer, W:Affine processes and applications in finance. Ann. Appl. Probab. 13, 984-1053(2003) |
[19] |
Eberlein, E, Madan, DB, Pistorius, M, Yor, M:Bid and ask prices as non-linear continuous time G-expectations based on distortions. Math. Financ. Econ. 8(3), 265-289(2014) |
[20] |
El Karoui N, Tan, X:Capacities, measurable selection and dynamic programming part I:Abstract framework (2013a). arXiv:1310.3363v1 |
[21] |
El Karoui, N, Tan, X:Capacities, measurable selection and dynamic programming part II:Application in stochastic control problems (2013b). arXiv:1310.3363v1 |
[22] |
Epstein, LG, Ji, S:Ambiguous volatility and asset pricing in continuous time. Rev. Financ. Stud. 26(7), 1740-1786(2013) |
[23] |
Evans, LC:Partial differential equations. Grad. Stud. Math. Am. Math. Soc. 19(2012) |
[24] |
Feller, W:Two singular diffusion problems. Ann. Math. 54, 173-182(1951) |
[25] |
Filipović, D:Term Structure Models:A Graduate Course. Springer Verlag, Berlin Heidelberg New York(2009) |
[26] |
Fleming, WH, Soner, HM:Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York (2006) |
[27] |
Fouque, J-P, Ren, B:Approximation for option prices under uncertain volatility. SIAM J. Financ. Math. 5(1), 360-383(2014) |
[28] |
Gikhman, I:A short remark on Fellerś square root condition (2011). Available on SSRN |
[29] |
Guillaume, F, Schoutens, W:Calibration risk:Illustrating the impact of calibration risk under the Heston model. Rev. Deriv. Res. 15(1), 57-79(2012) |
[30] |
Guo, G, Tan, X, Touzi, N:Tightness and duality of martingale transport on the Skorokhod space. Stochast. Process. Appl. 127(3), 927-956(2017) |
[31] |
Guyon, J, Henry-Labordère, P:Nonlinear option pricing. Chapman and Hall/CRC Financial Mathematics Series (2013) |
[32] |
Heider, P:Numerical methods for non-linear Black-Scholes equations. Appl. Math. Financ. 17(1), 59-81(2010) |
[33] |
Heston, S:A closed-form solution for options with stochastic volatility and applications to bond and currency options. Rev. Financ. Stud. 6, 327-343(1993) |
[34] |
Jacod, J, Protter, P:Probability essentials. Springer Verlag Berlin Heidelberg GmbH, Heidelberg (2004) |
[35] |
Kallenberg, O:Foundations of modern probability, Probability and its Applications (New York), second edn. Springer-Verlag, New York (2002) |
[36] |
Karatzas, I, Shreve, SE:Brownian Motion and Stochastic Calculus. Springer Verlag, Berlin Heidelberg New York (1988) |
[37] |
Kijima, M:Monotonicity and convexity of option prices revisited. Math. Financ. 12(4), 411-425(2002) |
[38] |
Madan, DB:Benchmarking in two price financial markets. Ann. Financ. 12(2), 201-219(2016) |
[39] |
Muhle-Karbe, J, Nutz, M:A risk-neutral equilibrium leading to uncertain volatility pricing. Financ. Stochast. 22(2), 281-295(2018) |
[40] |
Neufeld, A, Nutz, M:Measurability of semimartingale characteristics with respect to the probability law. Stochast. Process. Appl. 124(11), 3819-3845(2014) |
[41] |
Neufeld, A, Nutz, M:Nonlinear Lévy processes and their characteristics. Trans. Am. Math. Soc. 369(1), 69-95(2017) |
[42] |
Nutz, M, van Handel, R:Constructing sublinear expectations on path space. Stochas. Process. Appl. 123(8), 3100-3121(2013) |
[43] |
Patel, J, Russo, V, Fabozzi, FJ:Using the right implied volatility quotes in times of low interest rates:An empirical analysis across different currencies. Financ. Res. Lett. 25, 196-201(2018) |
[44] |
Peng, S:Backward SDE and related g-expectation. Backward stochastic differential equations, Vol. 364 of Pitman Res. Notes Math. Ser, pp. 141-159. Longman Scientific & Technical (1997) |
[45] |
Peng, S:G-Brownian motion and dynamic risk measure under volatility uncertainty. Lect. Notes (2007a) |
[46] |
Peng, S:G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stochast. Anal. Appl. 2, 541-567(2007b) |
[47] |
Revuz, D, Yor, M:Continuous martingales and Brownian motion. Springer Verlag, Berlin (1999) |
[48] |
Russo, V, Fabozzi, FJ:Calibrating short interest rate models in negative rate environments. J. Deriv. 24(4), 80-92(2017) |
[49] |
Vorbrink, J:Financial markets with volatility uncertainty. J. Math. Econ. 53, 64-78(2014) |
[50] |
Wilmott, P, Oztukel, A:Uncertain parameters, an empirical stochastic volatility model and confidence limits. Int. J. Theor. Appl. Financ. 1(1), 175-189(1998) |
show all references
References:
[1] |
Acciaio, B, Beiglböck, M, Penkner, F, Schachermayer, W:A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance. 26(2), 233-251(2016) |
[2] |
Amadori, AL:Uniqueness and comparison properties of the viscosity solution to some singular HJB, equations. Nonlinear Differ. Equ. Appl. NoDEA. 14(3-4), 391-409(2007) |
[3] |
Avellaneda, M, Levy, A, Parás, A:Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance. 2(2), 73-88(1995) |
[4] |
Bannör, KF, Scherer, M:Capturing parameter risk with convex risk measures. Eur. Actuar. J. 3(1), 97-132(2013) |
[5] |
Barrieu, P, Scandolo, G:Assessing financial model risk. Eur. J. Oper. Res. 242(2), 546-556(2015) |
[6] |
Bergenthum, J, Rüschendorf, L:Comparison of semimartingales and Lévy processes. Ann. Probab. 35(1), 228-254(2007) |
[7] |
Biagini, S, Bouchard, B, Kardaras, C, Nutz, M:Robust fundamental theorem for continuous processes. Math. Finance. 27(4), 963-987(2017) |
[8] |
Bielecki, TR, Cialenco, I, Rutkowski, M:Arbitrage-free pricing of derivatives in nonlinear market models. Probab. Uncertain. Quant. Risk. 3(1), 2(2018) |
[9] |
Bouchard, B, Touzi, N:Weak dynamic programming principle for viscosity solutions. SIAM J. Control. Optim. 49(3), 948-962(2011) |
[10] |
Breuer, T, Csiszár, I:Measuring distribution model risk. Math. Financ. 26(2), 395-411(2016) |
[11] |
Carver, L:Negative rates:Dealers struggle to price 0% floors. Risk Mag. (2012) |
[12] |
Cont, R:Model uncertainty and its impact on the pricing of derivative instruments. Math. Financ. 16, 519-542(2006) |
[13] |
Costantini, C, Papi, M, D'Ippoliti, F:Singular risk-neutral valuation equations. Financ. Stochast. 16(2), 249-274(2012) |
[14] |
Crandall, MG, Ishii, H, Lions, P-L:User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27(1), 1-67(1992) |
[15] |
da Fonseca, J, Grasselli, M:Riding on the smiles. Quant. Financ. 11(11), 1609-1632(2011) |
[16] |
Denis, L, Martini, C:A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16(2), 827-852(2006) |
[17] |
Denk, R, Kupper, M, Nendel, M:A semigroup approach to nonlinear Lévy processes (2017). arXiv:1710.08130v1 |
[18] |
Duffie, D, Filipović D, Schachermayer, W:Affine processes and applications in finance. Ann. Appl. Probab. 13, 984-1053(2003) |
[19] |
Eberlein, E, Madan, DB, Pistorius, M, Yor, M:Bid and ask prices as non-linear continuous time G-expectations based on distortions. Math. Financ. Econ. 8(3), 265-289(2014) |
[20] |
El Karoui N, Tan, X:Capacities, measurable selection and dynamic programming part I:Abstract framework (2013a). arXiv:1310.3363v1 |
[21] |
El Karoui, N, Tan, X:Capacities, measurable selection and dynamic programming part II:Application in stochastic control problems (2013b). arXiv:1310.3363v1 |
[22] |
Epstein, LG, Ji, S:Ambiguous volatility and asset pricing in continuous time. Rev. Financ. Stud. 26(7), 1740-1786(2013) |
[23] |
Evans, LC:Partial differential equations. Grad. Stud. Math. Am. Math. Soc. 19(2012) |
[24] |
Feller, W:Two singular diffusion problems. Ann. Math. 54, 173-182(1951) |
[25] |
Filipović, D:Term Structure Models:A Graduate Course. Springer Verlag, Berlin Heidelberg New York(2009) |
[26] |
Fleming, WH, Soner, HM:Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York (2006) |
[27] |
Fouque, J-P, Ren, B:Approximation for option prices under uncertain volatility. SIAM J. Financ. Math. 5(1), 360-383(2014) |
[28] |
Gikhman, I:A short remark on Fellerś square root condition (2011). Available on SSRN |
[29] |
Guillaume, F, Schoutens, W:Calibration risk:Illustrating the impact of calibration risk under the Heston model. Rev. Deriv. Res. 15(1), 57-79(2012) |
[30] |
Guo, G, Tan, X, Touzi, N:Tightness and duality of martingale transport on the Skorokhod space. Stochast. Process. Appl. 127(3), 927-956(2017) |
[31] |
Guyon, J, Henry-Labordère, P:Nonlinear option pricing. Chapman and Hall/CRC Financial Mathematics Series (2013) |
[32] |
Heider, P:Numerical methods for non-linear Black-Scholes equations. Appl. Math. Financ. 17(1), 59-81(2010) |
[33] |
Heston, S:A closed-form solution for options with stochastic volatility and applications to bond and currency options. Rev. Financ. Stud. 6, 327-343(1993) |
[34] |
Jacod, J, Protter, P:Probability essentials. Springer Verlag Berlin Heidelberg GmbH, Heidelberg (2004) |
[35] |
Kallenberg, O:Foundations of modern probability, Probability and its Applications (New York), second edn. Springer-Verlag, New York (2002) |
[36] |
Karatzas, I, Shreve, SE:Brownian Motion and Stochastic Calculus. Springer Verlag, Berlin Heidelberg New York (1988) |
[37] |
Kijima, M:Monotonicity and convexity of option prices revisited. Math. Financ. 12(4), 411-425(2002) |
[38] |
Madan, DB:Benchmarking in two price financial markets. Ann. Financ. 12(2), 201-219(2016) |
[39] |
Muhle-Karbe, J, Nutz, M:A risk-neutral equilibrium leading to uncertain volatility pricing. Financ. Stochast. 22(2), 281-295(2018) |
[40] |
Neufeld, A, Nutz, M:Measurability of semimartingale characteristics with respect to the probability law. Stochast. Process. Appl. 124(11), 3819-3845(2014) |
[41] |
Neufeld, A, Nutz, M:Nonlinear Lévy processes and their characteristics. Trans. Am. Math. Soc. 369(1), 69-95(2017) |
[42] |
Nutz, M, van Handel, R:Constructing sublinear expectations on path space. Stochas. Process. Appl. 123(8), 3100-3121(2013) |
[43] |
Patel, J, Russo, V, Fabozzi, FJ:Using the right implied volatility quotes in times of low interest rates:An empirical analysis across different currencies. Financ. Res. Lett. 25, 196-201(2018) |
[44] |
Peng, S:Backward SDE and related g-expectation. Backward stochastic differential equations, Vol. 364 of Pitman Res. Notes Math. Ser, pp. 141-159. Longman Scientific & Technical (1997) |
[45] |
Peng, S:G-Brownian motion and dynamic risk measure under volatility uncertainty. Lect. Notes (2007a) |
[46] |
Peng, S:G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stochast. Anal. Appl. 2, 541-567(2007b) |
[47] |
Revuz, D, Yor, M:Continuous martingales and Brownian motion. Springer Verlag, Berlin (1999) |
[48] |
Russo, V, Fabozzi, FJ:Calibrating short interest rate models in negative rate environments. J. Deriv. 24(4), 80-92(2017) |
[49] |
Vorbrink, J:Financial markets with volatility uncertainty. J. Math. Econ. 53, 64-78(2014) |
[50] |
Wilmott, P, Oztukel, A:Uncertain parameters, an empirical stochastic volatility model and confidence limits. Int. J. Theor. Appl. Financ. 1(1), 175-189(1998) |
[1] |
Giséle Ruiz Goldstein, Jerome A. Goldstein, Rosa Maria Mininni, Silvia Romanelli. A generalized Cox-Ingersoll-Ross equation with growing initial conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1513-1528. doi: 10.3934/dcdss.2020085 |
[2] |
Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi, Gregory I. Sivashinsky. A fully nonlinear equation for the flame front in a quasi-steady combustion model. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1415-1446. doi: 10.3934/dcds.2010.27.1415 |
[3] |
Zhenzhong Zhang, Enhua Zhang, Jinying Tong. Necessary and sufficient conditions for ergodicity of CIR model driven by stable processes with Markov switching. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2433-2455. doi: 10.3934/dcdsb.2018053 |
[4] |
Daozhou Gao, Yijun Lou, Shigui Ruan. A periodic Ross-Macdonald model in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3133-3145. doi: 10.3934/dcdsb.2014.19.3133 |
[5] |
Alexandre Montaru. Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 231-256. doi: 10.3934/dcdsb.2014.19.231 |
[6] |
Eleonora Messina. Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation. Conference Publications, 2015, 2015 (special) : 826-834. doi: 10.3934/proc.2015.0826 |
[7] |
Philippe Michel, Bhargav Kumar Kakumani. GRE methods for nonlinear model of evolution equation and limited ressource environment. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6653-6673. doi: 10.3934/dcdsb.2019161 |
[8] |
Andrei Cozma, Christoph Reisinger. Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3359-3377. doi: 10.3934/dcdsb.2016101 |
[9] |
Shair Ahmad, Alan C. Lazer. On a property of a generalized Kolmogorov population model. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 1-6. doi: 10.3934/dcds.2013.33.1 |
[10] |
Francesca Biagini, Katharina Oberpriller. Reduced-form setting under model uncertainty with non-linear affine intensities. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 159-188. doi: 10.3934/puqr.2021008 |
[11] |
Pavol Kútik, Karol Mikula. Diamond--cell finite volume scheme for the Heston model. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 913-931. doi: 10.3934/dcdss.2015.8.913 |
[12] |
Zhibo Cheng, Xiaoxiao Cui. Positive periodic solution for generalized Basener-Ross model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4361-4382. doi: 10.3934/dcdsb.2020101 |
[13] |
Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 |
[14] |
Youshan Tao, J. Ignacio Tello. Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 193-207. doi: 10.3934/mbe.2016.13.193 |
[15] |
Belinda A. Batten, Hesam Shoori, John R. Singler, Madhuka H. Weerasinghe. Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 83-107. doi: 10.3934/dcdsb.2018162 |
[16] |
Chonghu Guan, Fahuai Yi, Xiaoshan Chen. A fully nonlinear free boundary problem arising from optimal dividend and risk control model. Mathematical Control and Related Fields, 2019, 9 (3) : 425-452. doi: 10.3934/mcrf.2019020 |
[17] |
Joachim Escher, Boris Kolev, Marcus Wunsch. The geometry of a vorticity model equation. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1407-1419. doi: 10.3934/cpaa.2012.11.1407 |
[18] |
Evelyne Miot, Mario Pulvirenti, Chiara Saffirio. On the Kac model for the Landau equation. Kinetic and Related Models, 2011, 4 (1) : 333-344. doi: 10.3934/krm.2011.4.333 |
[19] |
Dong Li, Xiaoyi Zhang. On a nonlocal aggregation model with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 301-323. doi: 10.3934/dcds.2010.27.301 |
[20] |
Fadia Bekkal-Brikci, Khalid Boushaba, Ovide Arino. Nonlinear age structured model with cannibalism. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 201-218. doi: 10.3934/dcdsb.2007.7.201 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]