2000 character limit reached
Real-world models for multiple term structures: a unifying HJM semimartingale framework (2411.01983v2)
Published 4 Nov 2024 in q-fin.MF and math.PR
Abstract: We develop a unified framework for modeling multiple term structures arising in financial, insurance, and energy markets, adopting an extended Heath-Jarrow-Morton (HJM) approach under the real-world probability. We study market viability and characterize the set of local martingale deflators. We conduct an analysis of the associated stochastic partial differential equation (SPDE), addressing existence and uniqueness of solutions, invariance properties and existence of affine realizations.
- K. Amin and R. Jarrow. Pricing foreign currency options under stochastic interest rates. Journal of International Money and Finance, 10:310–329, 1991.
- D. A. Bálint and M. Schweizer. Large financial markets, discounting, and no asymptotic arbitrage. Theory of Probability and its Applications, 65(2):191–223, 2020.
- J. Barbarin. Heath–Jarrow–Morton modelling of longevity bonds and the risk minimization of life insurance portfolios. Insurance: Mathematics and Economics, 43:41–55, 2008.
- F. E. Benth and S. Koekebakker. Stochastic modeling of financial electricity contracts. Energy Economics, 30:1116–1157, 2008.
- F. E. Benth and P. Krühner. Representation of infinite-dimensional forward price models in commodity markets. Communications in Mathematics and Statistics, 2(1):47–106, 2014.
- Mean-reverting additive energy forward curves in a Heath-Jarrow-Morton framework. Mathematics and Financial Economics, 13:543–577, 2019.
- T. Bielecki and M. Rutkowski. Credit Risk: Modeling, Valuation and Hedging. Springer, Berlin - Heidelberg, 2002.
- Towards a general theory of bond markets. Finance and Stochastics, 1(2):141–174, 1997.
- Bond market structure in the presence of marked point processes. Mathematical Finance, 7(2):211–239, 1997.
- T. Björk and L. Svensson. On the existence of finite dimensional realizations for nonlinear forward rate models. Mathematical Finance, 11(2):205–243, 2001.
- Real-world jump-diffusion term structure models. Quantitative Finance, 10(1):23–37, 2010.
- Optimal investment with intermediate consumption under no unbounded profit with bounded risk. Journal of Applied Probability, 54:710–719, 2017.
- T. Choulli and M. Schweizer. Locally ϕitalic-ϕ\phiitalic_ϕ-integrable σ𝜎\sigmaitalic_σ-martingale densities for general semimartingales. Stochastics, 88(2):191–266, 2016.
- A general HJM framework for multiple yield curve modeling. Finance and Stochastics, 20(2):267–320, 2016.
- Affine multiple yield curve models. Mathematical Finance, 29(2):568–611, 2019.
- A new perspective on the fundamental theorem of asset pricing for large financial markets. Theory of Probability and its Applications, 60(4):561–579, 2016.
- M. De Donno and A. Pratelli. A theory of stochastic integration for bond markets. Annals of Applied Probability, 15(4):2773–2791, 2005.
- M. De Donno and A. Pratelli. Stochastic integration with respect to a sequence of semimartingales. In M. Émery and M. Yor, editors, In Memoriam Paul-André Meyer - Séminaire de Probabilités XXXIX, pages 119 –135. Springer, Berlin - Heidelberg, 2006.
- D. Filipović. Consistency Problems for Heath-Jarrow-Morton Interest Rate Models, volume 1760 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2001.
- D. Filipović. Term-Structure Models. A Graduate Course. Springer, Berlin - Heidelberg, 2009.
- Jump-diffusions in Hilbert spaces: existence, stability and numerics. Stochastics, 82(5):475–820, 2010.
- Term structure models driven by Wiener processes and Poisson measures: existence and positivity. SIAM Journal on Financial Mathematics, 1(1):523–554, 2010.
- D. Filipović and J. Teichmann. Existence of invariant manifolds for stochastic equations in infinite dimension. Journal of Functional Analysis, 197(2):398–432, 2003.
- C. Fontana. Weak and strong no-arbitrage conditions for continuous financial markets. International Journal of Theoretical and Applied Finance, 18(1):1550005, 2015.
- Term structure modeling for multiple curves with stochastic discontinuities. Finance and Stochastics, 24:465–511, 2020.
- Z. Grbac and W. J. Runggaldier. Interest Rate Modeling: Post-Crisis Challenges and Approaches. Springer, 2015.
- Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica, 60(1):77–105, 1992.
- J. Jacod. Un théorème de représentation pour les martingales discontinues. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 34:225–244, 1976.
- J. Jacod and A. Shiryaev. Limit Theorems for Stochastic Processes. Springer, Berlin - Heidelberg - New York, second edition, 2003.
- R. Jarrow and S. Turnbull. Pricing derivatives on financial securities subject to credit risk. Journal of Finance, 50(1):53–85, 1995.
- R. Jarrow and S. Turnbull. A unified approach for pricing contingent claims on multiple term structures. Review of Quantitative Finance and Accounting, 10:5–19, 1998.
- I. Karatzas and C. Kardaras. The numéraire portfolio in semimartingale financial models. Finance and Stochastics, 11(4):447–493, 2007.
- C. Kardaras. On the closure in the Emery topology of semimartingale wealth-process sets. Annals of Applied Probability, 23(4):1355–1376, 2013.
- C. Kardaras. A time before which insiders would not undertake risk. In Y. Kabanov, M. Rutkowski, and T. Zariphopoulou, editors, Inspired by Finance - The Musiela Festschrift, pages 349–362. Springer, Cham, 2014.
- C. Kardaras. Stochastic integration with respect to arbitrary collections of continuous semimartingales and applications to mathematical finance. Annals of Applied Probability, 34(3):2566–2599, 2024.
- N. Koval. Time-inhomogeneous Lévy Processes in Cross-Currency Market Models. PhD thesis, University of Freiburg, 2005.
- M. Métivier. Semimartingales: a Course on Stochastic Processes. de Gruyter Studies in Mathematics, Berlin - New York, 1982.
- S. Miller and E. Platen. A two-factor model for low interest rate regimes. Asia-Pacific Financial Markets, 11(1):107–133, 2004.
- A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1983.
- E. Platen and D. Heath. A Benchmark Approach to Quantitative Finance. Springer, Berlin - Heidelberg, 2006.
- E. Platen and S. Tappe. Real-world forward rate dynamics with affine realizations. Stochastic Analysis and Applications, 33(4):573–608, 2015.
- S. Tappe. An alternative approach on the existence of affine realizations for HJM term structure models. Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, 466(2122):3033–3060, 2010.
- S. Tappe. Existence of affine realizations for Lévy term structure models. Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, 468(2147):3685–3704, 2012.
- S. Tappe. The Yamada-Watanabe theorem for mild solutions to stochastic partial differential equations. Electronic Communications in Probability, 18(24):1–13, 2013.
- S. Tappe. Invariance of closed convex cones for stochastic partial differential equations. Journal of Mathematical Analysis and Applications, 451(2):1077–1122, 2017.
- S. Tappe. Invariant cones for jump-diffusions in infinite dimensions. Nonlinear Differential Equations and Applications, 31:107, 2024.
- S. Tappe and S. Weber. Stochastic mortality models: an infinite-dimensional approach. Finance and Stochastics, 18(1):209–248, 2014.
- M. Veraar. The stochastic Fubini theorem revisited. Stochastics, 84(4):543–551, 2012.
