A forward–backward SDE approach to affine models

Hyndman, Cody B.

doi:10.1007/s11579-009-0016-z

Cited by 10 publications

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References 42 publications

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“…Assumption 2.1 and equation (1) may now be used to extend the forward-backward stochastic differential equation (FBSDE) approach to term structure modelling for ATSMs of Hyndman (2009) to QTSMs. The main difference from Hyndman (2009) is that in this paper the dynamics of the factor process are given by a Gaussian (rather than affine) process and the riskless interest rate is a quadratic (rather than affine) functional of the factors process.…”

Section: Assumption 21 the Instantaneous Riskless Interest Rate Procmentioning

confidence: 99%

“…The first approach, and our main focus, is based on forward-backward stochastic differential equations (FBSDEs), which we henceforth refer to the FBSDE approach and was previously introduced in the context of ATSMs in Hyndman (2005Hyndman ( , 2007aHyndman ( , 2009. By first characterizing the factor process and bond price in terms of the solution of coupled nonlinear FBSDEs and then demonstrating existence, uniqueness, and explicit solutions of the the FBSDEs the pricing problem is solved.…”

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confidence: 99%

“…The key result of the FBSDE approach is the extension of a technique due to Yong (1999) of a method for proving the existence, uniqueness, and explicit solution of certain coupled linear FBSDEs to the nonlinear FBSDEs which characterize the bond pricing problem in ATSMs. The same techniques were employed to characterize futures prices and forward prices in affine price models (APMs) in Hyndman (2009). In this paper we extend the FBSDE approach to the bond pricing problem in the context of QTSMs and futures and forward prices of quadratic price models (QPMs) for a risky asset.…”

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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%

“…Section 2 briefly introduces the modelling framework and notation. Section 3 reviews the FBSDE approach for the zero-coupon bond price and extends the results of Hyndman (2009) from ATSMs to QTSMs. Section 4 considers models where a risky asset price is an exponential quadratic function of the factor process (QPM), which includes the zero-coupon bond price, and applies the FBSDE approach to the futures price and forward price.…”

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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: Assumption 21 the Instantaneous Riskless Interest Rate Procmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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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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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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A Convolution Method for Numerical Solution of Backward Stochastic Differential Equations

Hyndman

Ngou

2015

Methodol Comput Appl Probab

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We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast Fourier transform (FFT). The problem of error control is addressed and a local error analysis is provided. We consider the extension of the method to forward-backward stochastic differential equations (FBSDEs) and reflected FBSDEs. Numerical examples are considered from finance demonstrating the performance of the method. comments of the meeting participants are gratefully acknowledged. 1 C. B. HYNDMAN AND P. OYONO NGOU Convolution method for BSDEs has a unique solution u ∈ C 1,2 then the solution (Y, Z) for the one-dimensional BSDE (1.1) with terminal condition ξ = g(W T ) admits the representation). (1.4) Conversely, the solution of the PDE (1.2) can be interpreted in terms of the solution of the BSDE (1.1). General formulations of the nonlinear Feynman-Kac formula for FBSDEs, quasilinear parabolic PDEs, and viscosity solutions have been studied extensively. Deriving an explicit solution to a nontrivial (F)BSDE is possible only in very few situations, such as [40], [23] and [37]. Thus, numerical methods for BSDEs have been studied extensively. Numerical methods for (F)BSDEs can be classified into three main groups: PDE based methods, spatial discretization based methods, and Monte-Carlo based methods. PDE based methods, which started with the finite difference approach of [16], consider a numerical resolution of the nonlinear parabolic PDE related to the (F)BSDE. The two other methods rely on a time discretization of the (F)BSDE. Spatial discretization based methods (see [12], [2], [15], [13], [38] or [35] among others) use a deterministic space grid. On the other hand, the space discretization is random in Monte-Carlo based methods (for instance, [9], [22], and [3]).In this paper, we propose an alternative spatial discretization method for BSDEs and illustrate its implementation in the one-dimensional case. To the best of our knowledge, the most efficient approach, in terms of speed and accuracy, in this simple case is the binomial method of [35] which has connections with the theoretical work of [10] and [29]. However, our method avoids a notable drawback of the binomial method: the contraction of the space grid leading to the approximation of the Wiener process by means of scaled random walks. Indeed, we use a fixed equidistant space grid, thus allowing an exact simulation of the Wiener process at time nodes. The FFT algorithm, which plays a key role in our method, helps in producing an efficient algorithm. As in [11] and [28] in the context of option pricing under Lévy processes, we employ the FFT algorithm to compute quadratures. The presence of dynamic programming through the Euler scheme is a major similarity between our method and [28]. The method presented in [38] is somewhat similar in t...

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A forward–backward SDE approach to affine models

Cited by 10 publications

References 42 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

A Convolution Method for Numerical Solution of Backward Stochastic Differential Equations

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