Stochastic flows and the forward measure

Elliott, Robert J.; Hoek, John van der

doi:10.1007/s007800000039

Cited by 28 publications

(89 citation statements)

References 5 publications

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“…Further, these results suggest a correspondence between our results and the results of Gombani and Runggaldier (2013) In the next section we briefly consider an example of the application to QTSMs of the method of stochastic flows due to Elliott and van der Hoek (2001) which was originally developed in the context of ATSMs. A general formulation of the method of stochastic flows for QTSMs similar to that presented for ATSMs in Section 4 of Hyndman (2009) (2009) for APMs.…”

Section: Remark 49 Comparing Theorems 33 43 and 47 We See That mentioning

confidence: 80%

“…A general formulation of the method of stochastic flows for QTSMs similar to that presented for ATSMs in Section 4 of Hyndman (2009) (2009) for APMs. However, the main motivation for our consideration of the flows method in the context of quadratic models driven by Gaussian factors, in contrast to the case for affine models driven by Gaussian factors which was considered in Elliott and van der Hoek (2001) and Hyndman (2007b), is to show that a measure change is necessary for the method to be effective. This objective can easily be accomplished by consideration of an example in the one-dimensional case.…”

Section: Remark 49 Comparing Theorems 33 43 and 47 We See That mentioning

confidence: 99%

“…By taking the derivative of P(t, T, x) with respect to the initial condition, and then using properties of stochastic flows and their Jacobians, it is possible to express P(t, T, x) as an ordinary differential equation (ODE) which illuminates the nature of the functional dependence of the bond price on the factor process. In Elliott and van der Hoek (2001) when the factors process is Gaussian and the interest rate is an affine function of the factors process the fact that the derivative of the factor process with respect to x is deterministic leads to a linear ODE for P (t, T, x). In this case it is then immediate that the bond price is an exponential affine function of the factor process.…”

Section: Stochastic Flowsmentioning

confidence: 99%

“…The FBSDE approach to characterizing the bond price in ATSM introduced in Hyndman (2009), and extended to QTSMs in this paper, was motivated by the stochastic flow approach introduced by Elliott and van der Hoek (2001). The stochastic flow approach expresses the price at time t of the T -maturity zero coupon bond as a function of the value x at time t of the factor process, and this dependence is denoted P (t, T, x).…”

Section: Stochastic Flowsmentioning

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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confidence: 99%

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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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Section: Remark 49 Comparing Theorems 33 43 and 47 We See That mentioning

confidence: 80%

Section: Remark 49 Comparing Theorems 33 43 and 47 We See That mentioning

confidence: 99%

Section: Stochastic Flowsmentioning

confidence: 99%

Section: Stochastic Flowsmentioning

confidence: 99%

mentioning

confidence: 99%

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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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The entropic measure transform

2020

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We introduce the entropic measure transform (EMT) problem for a general process and prove the existence of a unique optimal measure characterizing the solution. The density process of the optimal measure is characterized using a semimartingale BSDE under general conditions. The EMT is used to reinterpret the conditional entropic risk-measure and to obtain a convenient formula for the conditional expectation of a process which admits an affine representation under a related measure. The entropic measure transform is then used provide a new characterization of defaultable bond prices, forward prices, and futures prices when the asset is driven by a jump diffusion. The characterization of these pricing problems in terms of the EMT provides economic interpretations as a maximization of returns subject to a penalty for removing financial risk as expressed through the aggregate relative entropy. The EMT is shown to extend the optimal stochastic control characterization of default-free bond prices of Gombani and Runggaldier (Math. Financ. 23(4):659-686, 2013). These methods are illustrated numerically with an example in the defaultable bond setting.

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Stochastic Jacobian and Riccati ODE in affine term structure models

Grasselli

Tebaldi

2007

Decisions Econ Finan

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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.We are grateful to two anonymous referees for their careful reading of the paper.

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Stochastic flows and the forward measure

Cited by 28 publications

References 5 publications

Explicit Solutions of Quadratic FBSDEs Arising From Quadratic Term Structure Models

Explicit Solutions of Quadratic FBSDEs Arising From Quadratic Term Structure Models

The entropic measure transform

Stochastic Jacobian and Riccati ODE in affine term structure models

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