Abstract
In affine term structure models (ATSM) the stochastic Jacobian under the forward measure plays a crucial role for pricing, as discussed in Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). Their approach leads to a deterministic integro-differential equation which, apparently, has the advantage of by-passing the solution to the Riccati ODE obtained by the standard Feynman-Kac argument. In the generic multi-dimensional case, we find a procedure to reduce such integro-differential equation to a non linear matrix ODE. We prove that its solution does necessarily require the solution of the vector Riccati ODE. This result is obtained proving an extension of the celebrated Radon Lemma, which allows us to highlight a deep relation between the geometry of the Riccati flow and the stochastic calculus of variations for an ATSM.
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References
Brown R. and Schaefer S. (1994). Interest rate volatility and the shape of the term structure. Philos. Trans. R. Soc. Phys. Sci. Eng. 347: 449–598
Cox J.C., Ingersoll J.E. and Ross S.A. (1985). A theory of the term structure of interest rates. Econometrica 53: 385–408
Dai Q. and Singleton K. (2000). Specification analysis of affine term structure models. J. Finance 55: 1943–1978
Duffie D. and Kan R. (1996). A yield-factor model of interest rates. Math. Finance 6(4): 379–406
Duffie D., Pan J. and Singleton K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68: 1343–1376
Duffie D., Filipović D. and Schachermayer W. (2003). Affine Processes and Applications in Finance. Ann. Appl. Probab 13(3): 984–1053
Elliott R.J. and van der Hoek J. (2001). Stochastic flows and the forward measure. Finance Stoch. 5: 511–525
Freiling G. (2002). A survey of nonsymmetric Riccati equations. Linear Algebra Appl. 351–352: 243–270
Grasselli M. and Tebaldi C. (2004). Bond price and impulse response function for the Balduzzi, Das, Foresi and Sundaram (1996) model. Econ. Notes 33(3): 359–374
Grasselli, M., Tebaldi, C.: Solvable affine term structure models. Math. Finance (to appear) (2004)
Hamilton J.D. (1994). Time Series Analysis. Princeton university press, Princeton
Piazzesi M. (2005). Bond yields and federal reserve. J. Polit. Econ. 113(21): 311–344
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We are grateful to two anonymous referees for their careful reading of the paper.
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Grasselli, M., Tebaldi, C. Stochastic Jacobian and Riccati ODE in affine term structure models. Decisions Econ Finan 30, 95–108 (2007). https://doi.org/10.1007/s10203-007-0071-y
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DOI: https://doi.org/10.1007/s10203-007-0071-y