Reduced-form setting under model uncertainty with non-linear affine intensities

Francesca Biagini; Katharina Oberpriller

doi:10.3934/puqr.2021008

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September 2021, 6(3): 159-188. doi: 10.3934/puqr.2021008

Reduced-form setting under model uncertainty with non-linear affine intensities

Francesca Biagini ^1,, and Katharina Oberpriller ^2,

1.	Department of Mathematics, Workgroup Financial and Insurance Mathematics, University of Munich (LMU), Theresienstraße 39, 80333 Munich, Germany
2.	Department of Mathematics of Natural, Social and Life Sciences, Gran Sasso Science Institute (GSSI), Viale F. Crispi 7, 67100 L’Aquila, Italy

Email: francesca.biagini@math.lmu.de

Received December 23, 2020 Accepted July 06, 2021 Published September 2021

In this paper we extend the reduced-form setting under model uncertainty introduced in [5] to include intensities following an affine process under parameter uncertainty, as defined in [15]. This framework allows us to introduce a longevity bond under model uncertainty in a way consistent with the classical case under one prior and to compute its valuation numerically. Moreover, we price a contingent claim with the sublinear conditional operator such that the extended market is still arbitrage-free in the sense of “no arbitrage of the first kind” as in [6].

Keywords: Sublinear expectation, Reduced-form framework, Non-linear affine processes, Arbitrage-free pricing.

Mathematics Subject Classification: 60G65, 91B70, 91G15, 91G40.

Citation: 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

References:

[1]	Beatrice Acciaio, Mathias Beiglböck, Friedrich Penkner and Walter Schachermayer, A modelfree version of the fundamental theorem of asset pricing and the super-replication theorem, Mathematical Finance, 2021, 26(2): 233-251.
[2]	Bahar Akhtari, Francesca Biagini, Andrea Mazzon, and Katharina Oberpriller, Generalized Feynman-Kac Formula under volatility uncertainty, arXiv: 2012.08163, 2012.
[3]	Anna Aksamit and Monique Jeanblanc, Enlargement of Filtration with Finance in View, Springer, 2017.
[4]	Erhan Bayraktar, Yuchong Zhang, and Zhou Zhou, A note on the fundamental theorem of asset pricing under model uncertainty, Risks, 2014, 2(4): 425-433.
[5]	Francesca Biagini and Yinglin Zhang, Reduced-form framework under model uncertainty, The Annals of Applied Probability, 2019, 29(4): 2481-2522.
[6]	Sara Biagini, Bruno Bouchard, Constantinos Kardaras, and Marcel Nutz, Robust fundamental theorem for continuous processes, Mathematical Finance, 2017, 27(4): 963-987.
[7]	Thomasz R. Bielecki and Marek Rutkowski, Credit Risk: Modeling, Valuation and Hedging, Springer, 2004.
[8]	Enrico Biffis, Affine processes for dynamic mortality and actuarial valuations, SSRN Electronic Journal, 2004, https://dx.doi.org/10.2139/ssrn.647421.
[9]	Bruno Bouchard and Marcel Nutz, Arbitrage and duality in nondominated discrete-time models, The Annals of Applied Probability, 2015, 25(2): 823-859.
[10]	Andrew Cairns, David Blake, and Kevin Dowd, Pricing death: Frameworks for the valuation and securitization of mortality risk, ASTIN Bulletin, 2006, 36(1): 79-120.
[11]	Giorgia Callegaro, Monique Jeanblanc, and Behnaz Zargari, Carthaginian enlargement of filtrations, arXiv: 1111.3073v1, 2018.
[12]	Mikkel Dahl, Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts, Insurance: Mathematics and Economics, 2004, 35(1): 113-136. doi: 10.1016/j.insmatheco.2004.05.003.
[13]	Nicole El Karoui and Xiaolu Tan, Capacities, measurable selection & dynamic programming, Part I: Abstract framework, arXiv: 1310.3363v1, 2013.
[14]	Nicole El Karoui and Xiaolu Tan, Capacities, measurable selection & dynamic programming, Part Ⅱ: Application in stochastic control problems, arXiv: 1310.3364v2, 2015.
[15]	Tolulope Fadina, Ariel Neufeld, and Thorsten Schmidt, Affine processes under parameter uncertainty, Probability, Uncertainty and Quantitative Risk, 2019, 4: 5. doi: 10.1186/s41546-019-0039-1.
[16]	Tolulope Fadina and Thorsten Schmidt, Default ambiguity, Risks, 2019, 7(2): 64.
[17]	Damir Filipovic, Term-Structure Models: A Graduate Course, Springer, 2009.
[18]	Hans Föllmer and Philip Protter, Local martingales and filtration shrinkage, ESAIM: Probability and Statistics, 2011, 15: S25-S28.
[19]	Hans Föllmer and Alexander Schied, Stochastic Finance: An Introduction in Discrete Time, De Gruyter, 2016.
[20]	Djibril Gueye, Monique Jeanblanc, and Libo Li, Models of default times and Cox model revisited, Freiburg FRIAS: Finance and Insurance, November 2019.
[21]	Daniel Hollender, Lèvy-type processes under uncertainty and related nonlocal equations, PhD Thesis, TU Dresden, 2016.
[22]	Julian Hölzermann, The Hull-White model under Knightian uncertainty about the volatility, arXiv: 1808.03463v2, 2019.
[23]	Julian Hölzermann, Pricing interest rate derivatives under volatility uncertainty, arXiv: 2003.04606v1, 2020.
[24]	Julian Hölzermann and Lian Quian, Term structure modeling under volatility uncertainty, arXiv: 1904.02930, 2020.
[25]	Jean Jacod and Albert N. Shiryaev, Limit Theorems for Stochastic Processes, Springer, 2013.
[26]	Yuri Kabanov, Constantinos Kardaras, and Shiqi Song, No arbitrage of the first kind and local martingale numèraires, Finance and Stochastics, 2016, 20(4): 1097-1108. doi: 10.1007/s00780-016-0310-6.
[27]	Constantinos Kardaras, Finitely additive probabilities and the fundamental theorem of asset pricing, In: Contemporary Quantitative Finance, Springer, 2010: 19-34.
[28]	Dimitri O. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Probability Theory and Related Fields, 1996, 105(4): 459-479. doi: 10.1007/BF01191909.
[29]	David Lando, On Cox processes and credit risky securities, Review of Derivatives Research, 1998, 2(2-3): 99-120.
[30]	Elisa Luciano and Elena Vigna, Non-mean reverting affine processes for stochastic mortality, SSRN Electronic Journal, 2005, https://ssrn.com/abstract=724706.
[31]	Ariel Neufeld and Marcel Nutz, Superreplication under volatility uncertainty for measurable claims, Electronic Journal of Probability, 2013, 18(48): 1-14.
[32]	Ariel Neufeld and Marcel Nutz, Measurability of semimartingale characteristics with respect to the probability law, Stochastic Processes and their Applications, 2014, 124(11): 3819-3845. doi: 10.1016/j.spa.2014.07.006.
[33]	Ariel Neufeld and Marcel Nutz, Nonlinear Lèvy processes and their characteristics, Transactions of the American Mathematical Society, 2017, 369(1): 69-95. doi: 10.1090/tran/6656.
[34]	Ariel Neufeld and Marcel Nutz, Robust utility maximization with Lévy processes, Mathematical Finance, 2018, 28(1): 82-105.
[35]	Marcel Nutz, Random G-expectations, The Annals of Applied Probability, 2013, 23(5): 1755-1777.
[36]	Marcel Nutz, Superreplication under model uncertainty in discrete time, Finance and Stochastics, 2014, 18(4): 791-803. doi: 10.1007/s00780-014-0238-7.
[37]	Marcel Nutz, Robust superhedging with jumps and diffusion, Stochastic Processes and their Applications, 2015, 125(12): 4543-4555. doi: 10.1016/j.spa.2015.07.008.
[38]	Marcel Nutz and Roman Van Handel, Constructing sublinear expectations on path space, Stochastic Processes and their Applications, 2013, 123(8): 3100-3121. doi: 10.1016/j.spa.2013.03.022.
[39]	Marcel Nutz and Mete Soner, Superhedging and dynamic risk measures under volatility uncertainty, SIAM Journal on Control and Optimization (SICON), 2012, 50(4): 2065-2089. doi: 10.1137/100814925.
[40]	Shige Peng, Nonlinear expectations and stochastic calculus under uncertainty, arXiv: 1002.456v1, 2010.
[41]	Philip E, Protter, Stochastic Integration and Differential Equations, Springer, 2005.
[42]	Daniel Revuz and Marc Yor, Continuous Martingales and Brownian Motion, Springer, 2005.
[43]	Thorsten Rheinländer and Jenny Sexton, Hedging Derivatives, World Scientific, 2011.
[44]	David F. Schrager, Affine stochastic mortality, Insurance: Mathematics and Economics, 2006, 38(1): 81-97. doi: 10.1016/j.insmatheco.2005.06.013.
[45]	Jörg Vorbrink, Financial markets with volatility uncertainty, Journal of Mathematics Economics, 2014, 53: 64-78. doi: 10.1016/j.jmateco.2014.05.008.
[46]	Yinglin Zhang, Insurance modeling in continuous time, PhD Thesis, Ludwig-Maximilians University Munich, 2018.

show all references

1Galmarino’s Test [42, Exercise 4.21]: Let $ \Omega = C({\mathbb{R}}_+, {\mathbb{R}}) $ , $ {\cal{F}} $ the Borel $ \sigma $ -algebra with respect to the topology of locally uniform convergence, and $ {\mathbb{F}} $ be the raw filtration generated by the canonical process $ B $ on $ \Omega $ . Then, an $ {\cal{F}} $ -measurable function $ \tau: \Omega \to {\mathbb{R}}_+ $ is an $ {\mathbb{F}} $ -stopping time if and only if $ \tau(\omega) \leq t $ and $ \omega\vert_{[0, t]} = \omega'\vert_{[0, t]} $ imply $ \tau(\omega) = \tau(\omega') $ . Furthermore, given an $ {\mathbb{F}} $ -stopping time $ \tau $ , an $ {\cal{F}} $ -measurable function $ f $ is $ {\cal{F}}_{\tau} $ -measurable if and only if $ f = f \circ \iota_{\tau} $ , where $ \iota_{\tau}: \Omega \to \Omega $ is the stopping map $ (\iota_{\tau}(\omega))_t = \omega_{t \wedge \tau(\omega)} $ .

2By the same arguments regarding the filtration as in Remark 5.2, $ S $ is also a $ (\tilde{P}, {\mathbb{G}}^{*, {\tilde{\cal{P}}}}_+) $ -semimartingale for all $ \tilde{P} \in \tilde{\cal{P}} $ .

3The sigma-martingale property holds with respect to the filtration $ {\mathbb{G}}^{*, {\tilde{\cal{P}}}}_+ $ for all $ \tilde{P} \in \tilde{\cal{P}} $ .

4Note, the assumption $ \sup_{P \in {{\cal{Z}}}} E^{{P}}[e^{-\int_0^T B_s^{\mu} {\rm{d}}s }] < \infty $ is always satisfied for $ B^{\mu} > 0 $ which is the case for a mortality intensity.

References:

[1]	Beatrice Acciaio, Mathias Beiglböck, Friedrich Penkner and Walter Schachermayer, A modelfree version of the fundamental theorem of asset pricing and the super-replication theorem, Mathematical Finance, 2021, 26(2): 233-251.
[2]	Bahar Akhtari, Francesca Biagini, Andrea Mazzon, and Katharina Oberpriller, Generalized Feynman-Kac Formula under volatility uncertainty, arXiv: 2012.08163, 2012.
[3]	Anna Aksamit and Monique Jeanblanc, Enlargement of Filtration with Finance in View, Springer, 2017.
[4]	Erhan Bayraktar, Yuchong Zhang, and Zhou Zhou, A note on the fundamental theorem of asset pricing under model uncertainty, Risks, 2014, 2(4): 425-433.
[5]	Francesca Biagini and Yinglin Zhang, Reduced-form framework under model uncertainty, The Annals of Applied Probability, 2019, 29(4): 2481-2522.
[6]	Sara Biagini, Bruno Bouchard, Constantinos Kardaras, and Marcel Nutz, Robust fundamental theorem for continuous processes, Mathematical Finance, 2017, 27(4): 963-987.
[7]	Thomasz R. Bielecki and Marek Rutkowski, Credit Risk: Modeling, Valuation and Hedging, Springer, 2004.
[8]	Enrico Biffis, Affine processes for dynamic mortality and actuarial valuations, SSRN Electronic Journal, 2004, https://dx.doi.org/10.2139/ssrn.647421.
[9]	Bruno Bouchard and Marcel Nutz, Arbitrage and duality in nondominated discrete-time models, The Annals of Applied Probability, 2015, 25(2): 823-859.
[10]	Andrew Cairns, David Blake, and Kevin Dowd, Pricing death: Frameworks for the valuation and securitization of mortality risk, ASTIN Bulletin, 2006, 36(1): 79-120.
[11]	Giorgia Callegaro, Monique Jeanblanc, and Behnaz Zargari, Carthaginian enlargement of filtrations, arXiv: 1111.3073v1, 2018.
[12]	Mikkel Dahl, Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts, Insurance: Mathematics and Economics, 2004, 35(1): 113-136. doi: 10.1016/j.insmatheco.2004.05.003.
[13]	Nicole El Karoui and Xiaolu Tan, Capacities, measurable selection & dynamic programming, Part I: Abstract framework, arXiv: 1310.3363v1, 2013.
[14]	Nicole El Karoui and Xiaolu Tan, Capacities, measurable selection & dynamic programming, Part Ⅱ: Application in stochastic control problems, arXiv: 1310.3364v2, 2015.
[15]	Tolulope Fadina, Ariel Neufeld, and Thorsten Schmidt, Affine processes under parameter uncertainty, Probability, Uncertainty and Quantitative Risk, 2019, 4: 5. doi: 10.1186/s41546-019-0039-1.
[16]	Tolulope Fadina and Thorsten Schmidt, Default ambiguity, Risks, 2019, 7(2): 64.
[17]	Damir Filipovic, Term-Structure Models: A Graduate Course, Springer, 2009.
[18]	Hans Föllmer and Philip Protter, Local martingales and filtration shrinkage, ESAIM: Probability and Statistics, 2011, 15: S25-S28.
[19]	Hans Föllmer and Alexander Schied, Stochastic Finance: An Introduction in Discrete Time, De Gruyter, 2016.
[20]	Djibril Gueye, Monique Jeanblanc, and Libo Li, Models of default times and Cox model revisited, Freiburg FRIAS: Finance and Insurance, November 2019.
[21]	Daniel Hollender, Lèvy-type processes under uncertainty and related nonlocal equations, PhD Thesis, TU Dresden, 2016.
[22]	Julian Hölzermann, The Hull-White model under Knightian uncertainty about the volatility, arXiv: 1808.03463v2, 2019.
[23]	Julian Hölzermann, Pricing interest rate derivatives under volatility uncertainty, arXiv: 2003.04606v1, 2020.
[24]	Julian Hölzermann and Lian Quian, Term structure modeling under volatility uncertainty, arXiv: 1904.02930, 2020.
[25]	Jean Jacod and Albert N. Shiryaev, Limit Theorems for Stochastic Processes, Springer, 2013.
[26]	Yuri Kabanov, Constantinos Kardaras, and Shiqi Song, No arbitrage of the first kind and local martingale numèraires, Finance and Stochastics, 2016, 20(4): 1097-1108. doi: 10.1007/s00780-016-0310-6.
[27]	Constantinos Kardaras, Finitely additive probabilities and the fundamental theorem of asset pricing, In: Contemporary Quantitative Finance, Springer, 2010: 19-34.
[28]	Dimitri O. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Probability Theory and Related Fields, 1996, 105(4): 459-479. doi: 10.1007/BF01191909.
[29]	David Lando, On Cox processes and credit risky securities, Review of Derivatives Research, 1998, 2(2-3): 99-120.
[30]	Elisa Luciano and Elena Vigna, Non-mean reverting affine processes for stochastic mortality, SSRN Electronic Journal, 2005, https://ssrn.com/abstract=724706.
[31]	Ariel Neufeld and Marcel Nutz, Superreplication under volatility uncertainty for measurable claims, Electronic Journal of Probability, 2013, 18(48): 1-14.
[32]	Ariel Neufeld and Marcel Nutz, Measurability of semimartingale characteristics with respect to the probability law, Stochastic Processes and their Applications, 2014, 124(11): 3819-3845. doi: 10.1016/j.spa.2014.07.006.
[33]	Ariel Neufeld and Marcel Nutz, Nonlinear Lèvy processes and their characteristics, Transactions of the American Mathematical Society, 2017, 369(1): 69-95. doi: 10.1090/tran/6656.
[34]	Ariel Neufeld and Marcel Nutz, Robust utility maximization with Lévy processes, Mathematical Finance, 2018, 28(1): 82-105.
[35]	Marcel Nutz, Random G-expectations, The Annals of Applied Probability, 2013, 23(5): 1755-1777.
[36]	Marcel Nutz, Superreplication under model uncertainty in discrete time, Finance and Stochastics, 2014, 18(4): 791-803. doi: 10.1007/s00780-014-0238-7.
[37]	Marcel Nutz, Robust superhedging with jumps and diffusion, Stochastic Processes and their Applications, 2015, 125(12): 4543-4555. doi: 10.1016/j.spa.2015.07.008.
[38]	Marcel Nutz and Roman Van Handel, Constructing sublinear expectations on path space, Stochastic Processes and their Applications, 2013, 123(8): 3100-3121. doi: 10.1016/j.spa.2013.03.022.
[39]	Marcel Nutz and Mete Soner, Superhedging and dynamic risk measures under volatility uncertainty, SIAM Journal on Control and Optimization (SICON), 2012, 50(4): 2065-2089. doi: 10.1137/100814925.
[40]	Shige Peng, Nonlinear expectations and stochastic calculus under uncertainty, arXiv: 1002.456v1, 2010.
[41]	Philip E, Protter, Stochastic Integration and Differential Equations, Springer, 2005.
[42]	Daniel Revuz and Marc Yor, Continuous Martingales and Brownian Motion, Springer, 2005.
[43]	Thorsten Rheinländer and Jenny Sexton, Hedging Derivatives, World Scientific, 2011.
[44]	David F. Schrager, Affine stochastic mortality, Insurance: Mathematics and Economics, 2006, 38(1): 81-97. doi: 10.1016/j.insmatheco.2005.06.013.
[45]	Jörg Vorbrink, Financial markets with volatility uncertainty, Journal of Mathematics Economics, 2014, 53: 64-78. doi: 10.1016/j.jmateco.2014.05.008.
[46]	Yinglin Zhang, Insurance modeling in continuous time, PhD Thesis, Ludwig-Maximilians University Munich, 2018.

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Reduced-form setting under model uncertainty with non-linear affine intensities

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