# American Institute of Mathematical Sciences

• 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”
January  2019, 4: 5 doi: 10.1186/s41546-019-0039-1

## 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

Received  June 20, 2018 Revised  April 23, 2019 Published  May 2019

We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call nonlinear affine processes. This is done as follows: given a set Θ of parameters for the process, we construct a corresponding nonlinear expectation on the path space of continuous processes. By a general dynamic programming principle, we link this nonlinear expectation to a variational form of the Kolmogorov equation, where the generator of a single affine process is replaced by the supremum over all corresponding generators of affine processes with parameters in Θ. This nonlinear affine process yields a tractable model for Knightian uncertainty, especially for modelling interest rates under ambiguity.
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.
Citation: Tolulope Fadina, Ariel Neufeld, Thorsten Schmidt. Affine processes under parameter uncertainty. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 5-. doi: 10.1186/s41546-019-0039-1
##### 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: