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

Affine processes under parameter uncertainty

Tolulope Fadina 1,, , Ariel Neufeld 2, and  Thorsten Schmidt 3,4,

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

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.
Keywords: Affine processes, Knightian uncertainty, Riccati equation, Vasiček model, Cox-Ingersoll-Ross model, Nonlinear Vasiček/CIR model, Heston model, Itô formula, Kolmogorov equation, Fully nonlinear PDE, Semimartingale.
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),

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