Markovian short rates in multidimensional term structure Lévy models. (English) Zbl 1460.91279
Jakubowski, Jacek (ed.) et al., Stochastic modeling and control. Based on the Simons semester, Warsaw, Poland, January 2 – March 31, 2019. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 122, 93-106 (2020).
Summary: We study a bond market model and the related term structure of interest rates in which the prices of zero coupon bonds are driven by a multidimensional Lévy process. We show that the short rate forms a Markov process if and only if the deterministic forward rate volatility coefficients are decomposed into products of two factors where the factor depending on the maturity time is the same for all components. The proof is based on the analysis of sample path properties of the underlying multidimensional process.
For the entire collection see [Zbl 1460.93006].
For the entire collection see [Zbl 1460.93006].
MSC:
| 91G30 | Interest rates, asset pricing, etc. (stochastic models) |
| 60G51 | Processes with independent increments; Lévy processes |
| 60J60 | Diffusion processes |
| 60J25 | Continuous-time Markov processes on general state spaces |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
References:
| [1] | T. Björk, Yu. M. Kabanov, W. Runggaldier,Bond market structure in the presence of marked point processes, Math. Finance 7 (1997), 211-239. · Zbl 0884.90014 |
| [2] | T. Björk, G. di Masi, Yu. M. Kabanov, W. Runggaldier,Towards a general theory of bond markets, Finance Stoch. 1 (1997), 141-174. · Zbl 0889.90019 |
| [3] | A. Carverhill,When is the short rate Markovian?Math. Finance 4 (1994), 305-312. · Zbl 0884.90017 |
| [4] | C. Chiarella, O. K. Kwon,Forward rate dependent Markovian transformations of the Heath-Jarrow-Morton term structure model, Finance Stoch. 5 (1997), 237-258. · Zbl 0972.60058 |
| [5] | J. C. Cox, J. E. Ingersoll, S. A. Ross,A theory of the term structure of interest rates, Econometrica 53 (1985), 385-407. · Zbl 1274.91447 |
| [6] | E. Eberlein,Applications of generalized hyperbolic Lévy motions to finance, in: Lévy Processes — Theory and Applications, Birkhäuser, Boston, 2001, 319-336. · Zbl 0982.60045 |
| [7] | E. Eberlein, S. Raible,Term structure models driven by general Lévy processes, Math. Finance 9 (1999), 31-53. · Zbl 0980.91020 |
| [8] | P. V. Gapeev,On arbitrage and Markovian short rates for fractional bond markets, Statist. Probab. Lett. 70 (2004), 211-222. · Zbl 1060.60083 |
| [9] | P. V. Gapeev, U. Küchler,On Markovian short rates in term structure models driven by jump-diffusion processes, Statist. Decisions 24 (2006), 255-271. · Zbl 1152.60065 |
| [10] | D. Heath, R. Jarrow, A. Morton,Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation, Econometrica 60 (1992), 77-105. · Zbl 0751.90009 |
| [11] | J. Jacod, A. N. Shiryaev,Limit Theorems for Stochastic Processes, Grundlehren Math. Wiss. 288, Springer, Berlin, 1987. · Zbl 0635.60021 |
| [12] | J. Jakubowski, J. Zabczyk,Exponential moments for HJM models with jumps, Finance Stoch. 11 (2007), 429-445. · Zbl 1164.60038 |
| [13] | U. Küchler,On integrals with respect to Lévy processes, Statist. Probab. Lett. 66 (2004), 145-151. · Zbl 1103.60050 |
| [14] | U. Küchler, E. Naumann,Markovian short rates in a forward rate model with a general class of Lévy processes, Discussion Paper 6 of Sonderforschungsbereich 373, Humboldt University, Berlin, 2003. |
| [15] | U. Küchler, S. Tappe,Bilateral gamma distributions and processes in financial mathematics, Stochastic Process. Applications 118 (2008), 261-283. · Zbl 1133.62089 |
| [16] | U. Küchler, S. Tappe,On the shapes of bilateral Gamma densities, Statist. Probab. Lett. 78 (2008), 2478-2484. · Zbl 1146.62309 |
| [17] | D. B. Madan,Purely discontinuous asset price processes, in: Option Pricing, Interest Rates and Risk Management, Handb. Math. Finance, Cambrige Univ. Press, Cambridge, 2001, 105-153. · Zbl 1005.91047 |
| [18] | D. B. Madan, E. Seneta,The variance gamma (VG) model for share market returns, J. of Business 63 (1990), 511-524. |
| [19] | H. Shirakawa,Interest rate option pricing with Poisson-Gaussian forward rate curve processes, Math. Finance 1 (1991), no. 4, 77-94. · Zbl 0900.90107 |
| [20] | A. N. Shiryaev,Essentials of Stochastic Finance, Adv. Ser. Stat. Sci. Appl. Probab. 3, World Scientific, River Edge, NJ, 1999. · Zbl 0926.62100 |
| [21] | O. A. Vasiček,An equilibrium characterization of the term structure, J. Financ. Econ. 5 (1977), 177-188. · Zbl 1372.91113 |
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