Stochastic Jacobian and Riccati ODE in affine term structure models

Grasselli, Martino; Tebaldi, Claudio

doi:10.1007/s10203-007-0071-y

Cited by 5 publications

(12 citation statements)

References 13 publications

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“…More generally, mathematical finance and biology provide examples of elliptic differential operators which become degenerate along the boundary of a 'quadrant', R d−m × R m + , and R d−m ×R m + is a state space for the corresponding Markov process. Examples primarily motivated by mathematical finance include affine processes [1,14,17,22,23,24,40,41], which may be viewed as extensions of geometric Brownian motion (see, for example, [71]), the Heston stochastic volatility process [51], and the Wishart process [14,42,45,46,47]. Examples of this kind which arise in mathematical biology include the multi-dimensional Kimura diffusions and their local model processes [27,Equations (1.5) and (1.20)].…”

Section: 2mentioning

confidence: 99%

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

Feehan

2013

Communications in Partial Differential Equations

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Abstract. We prove weak and strong maximum principles, including a Hopf lemma, for C 2 subsolutions to equations defined by second-order, linear elliptic partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the C 2 subsolutions along this boundary vanishing locus ensures that these maximum principles hold irrespective of the sign of the Fichera function. Boundary conditions need only be prescribed on the complement in the domain boundary of the principal symbol's vanishing locus. We obtain uniqueness and a priori maximum principle estimates for C 2 solutions to boundary value and obstacle problems defined by these boundary-degenerate elliptic operators with partial Dirichlet or Neumann boundary conditions. We also prove weak maximum principles and uniqueness for W 1,2 solutions to the corresponding variational equations and inequalities defined with the aide of weighted Sobolev spaces. The domain is allowed to be unbounded when the operator coefficients and solutions obey certain growth conditions. Contents List of

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Section: 2mentioning

confidence: 99%

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

Feehan

2013

Communications in Partial Differential Equations

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“…Grasselli and Tebaldi [16,17] study the relationship between the flows approach, Riccati equations, and interest rate risk-management by algebraic methods. The flows approach must necessarily be equivalent to solving the Riccati equation in cases where the ATSM is well-posed in the sense of admissibility defined by Dai and Singleton [8] and consistency defined by Levendorskiȋ [27,28].…”

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confidence: 99%

“…The flows approach must necessarily be equivalent to solving the Riccati equation in cases where the ATSM is well-posed in the sense of admissibility defined by Dai and Singleton [8] and consistency defined by Levendorskiȋ [27,28]. Grasselli and Tebaldi [16] take as a starting point for certain calculations that the conditional expectation of the Jacobian of the stochastic flow is deterministic as claimed Elliott and van der Hoek [14]. However, Grasselli and Tebaldi [16] assume the admissibility conditions of Dai and Singleton [8] and, as we shall show, these conditions are sufficient to ensure that the conditional expectation of the Jacobian of the stochastic flow is deterministic.…”

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confidence: 99%

A forward–backward SDE approach to affine models

Hyndman

2009

Math Finan Econ

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We consider factor models for interest rates and asset prices where the riskneutral dynamics of the factors process is modelled by an affine diffusion. We characterize the factors process and bond price in terms of forward-backward stochastic differential equations (FBSDEs), prove an existence and uniqueness theorem which gives the solution explicitly, and characterize the bond price as an exponential affine function of the factors in a new way. Our approach unifies the results, based on stochastic flows, of Elliott and van der Hoek (Finance Stoch 5: [511][512][513][514][515][516][517][518][519][520][521][522][523][524][525] 2001) with the approach, based on the Feynman-Kac formula, of Duffie and Kan (Math Finance 6(4): 1996), and addresses a mistake in the approach of Elliott and van der Hoek (Finance Stoch 5:511-525, 2001). We extend our results on the bond price to consider the futures and forward price of a risky asset or commodity.Keywords Affine models · Forward-backward stochastic differential equations · Stochastic flows · Bond price · Futures price · Forward price JEL Classification E43 · G12 · G13 Mathematics Subject Classification (2000) 60G35 · 60H20 · 60H30 · 91B28 · 91B70

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“…487-488). In higher dimensional cases the stochastic flow method requires the addition of some parametric restrictions to the model similar to the ATSM case which was studied by Grasselli and Tebaldi (2007). Nevertheless, the example considered in this section illustrates the similarities of the stochastic flow method in the one-dimensional QTSM with Gaussian factor process to the one-dimensional ATSM model and provides further motivation for the necessity of the change of measure.…”

Section: Where A(τ) B(τ) and C(τ) Is Given By (77) (78) And (85) Rmentioning

confidence: 99%

“…The second approach, which we briefly consider, is based on the stochastic flows method studied by Elliott and van der Hoek (2001), Grasselli and Tebaldi (2007), and Hyndman (2007bHyndman ( , 2009. This method gives a closed-form solution to the pricing problems for certain ATSMs.…”

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Explicit Solutions of Quadratic FBSDEs Arising From Quadratic Term Structure Models

Hyndman

Zhou

2015

Stochastic Analysis and Applications

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We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to similar results on affine term structure models of Hyndman (Math. Financ. Econ. 2(2):107-128, 2009) due to the relationship between quadratic functionals of Gaussian processes and linear functionals of affine processes. Similar to the affine case a sufficient condition for the explicit solutions to hold is the solvability in a fixed interval of Riccati-type ordinary differential equations. However, in contrast to the affine case, these Riccati equations are easily associated with those occurring in linear-quadratic control problems. We also consider quadratic models for a risky asset price and characterize the futures price and forward price of the asset in terms of similar FBSDEs. An example is considered, using an approach based on stochastic flows that is related to the FBSDE approach, to further emphasize the parallels between the affine and quadratic models. An appendix discusses solvability and explicit solutions of the Riccati equations.

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Stochastic Jacobian and Riccati ODE in affine term structure models

Cited by 5 publications

References 13 publications

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

A forward–backward SDE approach to affine models

Explicit Solutions of Quadratic FBSDEs Arising From Quadratic Term Structure Models

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