## Affine processes and applications in finance (2003)

Venue: | Annals of Applied Probability |

Citations: | 38 - 5 self |

### BibTeX

@ARTICLE{Duffie03affineprocesses,

author = {D. Duffie and D. Filipović and W. Schachermayer},

title = {Affine processes and applications in finance},

journal = {Annals of Applied Probability},

year = {2003},

volume = {13},

pages = {984--1053}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuousstate branching processes with immigration and Ornstein-Uhlenbeck type processes. We show, and provide foundations for, a wide range of financial applications for regular affine processes.

### Citations

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Citation Context ...ton [56] approach to option pricing, building on earlier work of Stein and Stein [88] that did not exploit the properties of affine processes. Heston’s objective was to extend the Black-Scholes model =-=[15]-=-, for which the underlying price process is a geometric Brownian motion, to allow “stochastic volatility.” In [56], the underlying asset price is eZt ,where(Y,Z) is the affine process (m = n =1) defin... |

1211 |
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Citation Context ...) Early prominent models of interest-rate behavior were based on such simple models of the short rate L(X) as the Vasicek (Gaussian Ornstein-Uhlenbeck) process [90], or the Cox-Ingersoll-Ross process =-=[30]-=-, which is the continuous branching diffusion of Feller [43]. Both of these short-rate processes are of course themselves affine (L(x) =x), as are many variants [20, 23, 30, 49, 58, 59, 72, 90, 74]. I... |

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(Show Context)
Citation Context ...lay between the existence of moments of a bounded measure on RN and the regularity of its characteristic function. 1.1. Basic Notation. For the stochastic background and notation we refer to [57] and =-=[76]-=-. Let k ∈ N. Wewrite R k + = {x ∈ R k | xi ≥ 0, ∀i}, R k ++ = {x ∈ R k | xi > 0, ∀i}, C k + = {z ∈ C k | Re z ∈ R k +}, C k ++ = {z ∈ C k | Re z ∈ R k ++},sAFFINE PROCESSES 5 and analogously R k − , R... |

671 |
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Citation Context ...he corresponding ODEs. Regular affine processes include continuous-state branching processes with immigration (CBI) (for example, [62]) and processes of the Ornstein-Uhlenbeck (OU) type (for example, =-=[79]-=-). Roughly speaking, the regular affine processes with state space R m + are CBI, and those with state space Rn are of OU type. For any regular affine process X =(Y,Z) inR m + × R n , we show that the... |

603 | An equilibrium characterization of the term structure
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Citation Context ...the State Space. The preceding approach requires nonnegativity of L. But there is a large literature on affine term structures for which the short rate is not necessarily nonnegative. See for example =-=[90]-=- and [31]. We shall provide a different approach using the martingale argument from Theorem 2.12. Let L be as at the beginning of Section 11. For r ∈ R write R r t := r + � t 0 L(Xs) ds. It can be sho... |

532 |
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Citation Context ...tion 11, on a shortrate process L(X). We shall view Qtf(Xs) as the price at time s of a financial asset paying the amount f(Xs+t) attimes + t. This implies a particular “risk-neutral” interpretation (=-=[52, 32]-=-) of the semi-group (Pt) that we shall not detail here. We emphasize, however, that statistical analysis of time series of X, ormeasurement of the risk of changes in market values of financial assets,... |

426 | Modelling term structures of defaultable bonds
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Citation Context ...13.1), although with a different effective discount rate. The popularity of affine models of interest rates has thus led to the common application of affine processes to default modeling, as in [35], =-=[41]-=-, and [63]. A defaultable bond with maturity t is a financial asset paying 1 {τ>t} at t. Applying the doubly-stochastic property, Lando [63] showed that the defaultable bond has a price of Ex � e − � ... |

385 | Transform analysis and asset pricing for affine jump-diffusions
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Citation Context ...ξ)+···+ymµm(dξ), minus the killing rate C(x) =c + 〈γ,y〉. An informal definition of an affine process could consist of the requirement that A(x), B(x), C(x) andM(x, dξ) haveaffine dependence on x, see =-=[40]-=-. The particular kind of this affine dependence in the present setup is implied by the geometry of the state space D. First, we notice that A(x) ∈ Sem d , C(x) ≥ 0andM(x, D) ≥ 0, for all x ∈ D. Whence... |

382 | A yield-factor model of interest rates
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Citation Context ...e (11.10)). Indeed, given the desire to model interest rates with ever increasing realism, various higher-dimensional (d >1) variants have appeared [5, 6, 11, 14, 23, 30, 31, 54, 64, 69], and efforts =-=[13, 18, 31, 39, 46, 45]-=-, includingsAFFINE PROCESSES 51 this paper, have been directed to the classification and unification of affine termstructure models. Beyond the scope of our analysis here, there are in fact “infinited... |

371 |
Empirical Performance of Alternative Option Pricing Models
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Citation Context ...faultable option may be likewise priced by replacing L(Xt) withL(Xt) + Λ(Xt), where {Λ(Xt−) : t ≥ 0} determines the default intensity, as for defaultable bond pricing. Numerous affine generalizations =-=[3, 4, 7, 8, 9, 24, 26, 40, 83, 84]-=- of the Heston model have been directed toward more realistic stochastic volatility and jump behavior. Pan [73] conducted a time-series analysis of the S-and-P 500 index data, both the underlying retu... |

336 | Specification analysis of affine term structure models
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(Show Context)
Citation Context ..., for any conservative regular affine process X =(Y,Z) inRm + × Rn , the sharp-brackets and jump characteristics of X depend only on the CBI component Y . This completes and extends the discussion in =-=[31]-=-, where they provide sufficient conditions for affine diffusion (and hence continuous) Markov processes to be well defined and classify them by the number m of Yis that can enter the conditional varia... |

278 |
Jumps and Stochastic Volatility: Exchange Rate
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(Show Context)
Citation Context ...faultable option may be likewise priced by replacing L(Xt) withL(Xt) + Λ(Xt), where {Λ(Xt−) : t ≥ 0} determines the default intensity, as for defaultable bond pricing. Numerous affine generalizations =-=[3, 4, 7, 8, 9, 24, 26, 40, 83, 84]-=- of the Heston model have been directed toward more realistic stochastic volatility and jump behavior. Pan [73] conducted a time-series analysis of the S-and-P 500 index data, both the underlying retu... |

267 | Foundations of Modern Analysis - Dieudonne - 1969 |

236 |
On cox processes and credit risky securities
- Lando
- 1998
(Show Context)
Citation Context ...hough with a different effective discount rate. The popularity of affine models of interest rates has thus led to the common application of affine processes to default modeling, as in [35], [41], and =-=[63]-=-. A defaultable bond with maturity t is a financial asset paying 1 {τ>t} at t. Applying the doubly-stochastic property, Lando [63] showed that the defaultable bond has a price of Ex � e − � t 0 L(Xs) ... |

234 | Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics
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(Show Context)
Citation Context ...faultable option may be likewise priced by replacing L(Xt) withL(Xt) + Λ(Xt), where {Λ(Xt−) : t ≥ 0} determines the default intensity, as for defaultable bond pricing. Numerous affine generalizations =-=[3, 4, 7, 8, 9, 24, 26, 40, 83, 84]-=- of the Heston model have been directed toward more realistic stochastic volatility and jump behavior. Pan [73] conducted a time-series analysis of the S-and-P 500 index data, both the underlying retu... |

232 |
Martingales and stochastic integrals in the theory of continuous trading
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(Show Context)
Citation Context ... process. We also show that a regular affine process X is (up to its lifetime) a semimartingale with respect to every Px, a crucial property in most financial applications because the standard model (=-=[53]-=-) of the financial gain generated by trading a security is a stochastic integral with respect to the underlying price process. We provide a one-to-one relationship between the coefficients of the char... |

213 | The jump-risk premia implicit in options: evidence from an integrateed time-series study
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- 2002
(Show Context)
Citation Context ...defaultable bond pricing. Numerous affine generalizations [3, 4, 7, 8, 9, 24, 26, 40, 83, 84] of the Heston model have been directed toward more realistic stochastic volatility and jump behavior. Pan =-=[73]-=- conducted a time-series analysis of the S-and-P 500 index data, both the underlying returns as well as option prices, based on an affine jumpdiffusion model of returns. Special numerical methods for ... |

190 |
Stock Price Distribution with Stochastic Volatility: An Analytic Approach,” Review of Financial Studies
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(Show Context)
Citation Context ...give the solution under non-negativity of L(X), or under conditions described at the end of Section 11. This is the Heston [56] approach to option pricing, building on earlier work of Stein and Stein =-=[88]-=- that did not exploit the properties of affine processes. Heston’s objective was to extend the Black-Scholes model [15], for which the underlying price process is a geometric Brownian motion, to allow... |

173 |
Maximum Likelihood Estimation for a Multi-factor Equilibrium Model of the Term Structure of Interest Rates
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(Show Context)
Citation Context ...aultable option may be likewise priced by replacing L(Xt) with L(Xt) + Λ(Xt), where {Λ(Xt−) : t ≥ 0} determines the default intensity, as for defaultable bond pricing. Numerous affine generalizations =-=[3, 4, 7, 8, 9, 24, 27, 40, 79, 80]-=- of the Heston model have been directed toward more realistic stochastic volatility and jump behavior. Pan [72] conducted a time-series analysis of the S-and-P 500 index data, both the underlying retu... |

169 |
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Citation Context ...f work, summarized in Section 13, on pricing and measuring default risk exploits the properties of a doubly-stochastic counting process N driven by an affine process X. The stochastic intensity of N (=-=[16]-=-) is assumed to be of the form {Λ(Xt−) : t ≥ 0}, for some affine x ↦→ Λ(x). The time of default of a financial counterparty, such as a borrower or option writer, �s4 D. DUFFIE, D. FILIPOVIĆ, AND W. SC... |

149 |
Interest rate volatility and the term structure: A two-factor general equilibriurn model
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(Show Context)
Citation Context ... equations for a broad range of affine processes (see (11.10)). Indeed, given the desire to model interest rates with ever increasing realism, various higher-dimensional (d >1) variants have appeared =-=[5, 6, 11, 14, 23, 30, 31, 54, 64, 69]-=-, and efforts [13, 18, 31, 39, 46, 45], includingsAFFINE PROCESSES 51 this paper, have been directed to the classification and unification of affine termstructure models. Beyond the scope of our analy... |

147 | G.: Markov processes - Ethier, Kurtz - 1986 |

121 | Pricing the Risks of Default
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(Show Context)
Citation Context ...� t 0 (L(Xs)+Λ(Xs)) ds� . Because x ↦→ L(x) +Λ(x) is affine, the defaultable bond price is again of the tractable form of the default-free bond price (13.1), with new coefficients. Various approaches =-=[63, 60, 41, 71]-=- to modeling non-zero recovery at default have been adopted. For a model of the default times τ1,... ,τk of k>1 different financial contracts, an approach is to suppose that τi is the first jump time ... |

108 |
An Exact Bond Option Formula
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(Show Context)
Citation Context ... or the Cox-Ingersoll-Ross process [30], which is the continuous branching diffusion of Feller [43]. Both of these short-rate processes are of course themselves affine (L(x) =x), as are many variants =-=[20, 23, 30, 49, 58, 59, 72, 90, 74]-=-. In general, because 1 = e 〈0,x〉 , the bond price Qt1(x) =e A(t)+〈B(t),x〉 (13.1) is easily calculated from the generalized Riccati equations for a broad range of affine processes (see (11.10)). Indee... |

105 | Characteristic Functions - Lukacs - 1970 |

102 | The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes
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(Show Context)
Citation Context ...tion 11, on a shortrate process L(X). We shall view Qtf(Xs) as the price at time s of a financial asset paying the amount f(Xs+t) attimes + t. This implies a particular “risk-neutral” interpretation (=-=[52, 32]-=-) of the semi-group (Pt) that we shall not detail here. We emphasize, however, that statistical analysis of time series of X, ormeasurement of the risk of changes in market values of financial assets,... |

97 |
A Markov model for the Term Structure of credit spreads
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(Show Context)
Citation Context ...� t 0 (L(Xs)+Λ(Xs)) ds� . Because x ↦→ L(x) +Λ(x) is affine, the defaultable bond price is again of the tractable form of the default-free bond price (13.1), with new coefficients. Various approaches =-=[63, 60, 41, 71]-=- to modeling non-zero recovery at default have been adopted. For a model of the default times τ1,... ,τk of k>1 different financial contracts, an approach is to suppose that τi is the first jump time ... |

90 |
Two singular diffusion problems
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(Show Context)
Citation Context ...d on such simple models of the short rate L(X) as the Vasicek (Gaussian Ornstein-Uhlenbeck) process [90], or the Cox-Ingersoll-Ross process [30], which is the continuous branching diffusion of Feller =-=[43]-=-. Both of these short-rate processes are of course themselves affine (L(x) =x), as are many variants [20, 23, 30, 49, 58, 59, 72, 90, 74]. In general, because 1 = e 〈0,x〉 , the bond price Qt1(x) =e A(... |

84 | Methoden der Mathematischen Physik - Courant, Hilbert - 1931 |

81 |
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Citation Context ...en based on approximation of the likelihood function [68, 40], on generalized method of moments [48] or on spectral properties, making use of the easily calculated complex moments of affine processes =-=[61, 21, 86]-=-. 13.2. Default Risk. In order to model the timing of default of financial contracts, we suppose that N is a non-explosive counting process [16] (defined on an enlarged probability space) that is doub... |

76 | Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets
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Citation Context ...ts, such as jumps and stochastic volatility in various forms. In addition to applications summarized below regarding the valuation of financial assets in settings of affine processes, recent progress =-=[17, 22, 66, 67, 80, 81, 82, 93]-=- in the modeling of optimal dynamic portfolio and consumption choice has exploited the special structure of controlled affine state-process models. We fix a conservative regular affine process X with ... |

66 | Dynamic asset allocation with event risk
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(Show Context)
Citation Context ...ts, such as jumps and stochastic volatility in various forms. In addition to applications summarized below regarding the valuation of financial assets in settings of affine processes, recent progress =-=[17, 22, 66, 67, 80, 81, 82, 93]-=- in the modeling of optimal dynamic portfolio and consumption choice has exploited the special structure of controlled affine state-process models. We fix a conservative regular affine process X with ... |

59 | Optimal consumption and portfolio selection with stochastic differential utility
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(Show Context)
Citation Context ...ts, such as jumps and stochastic volatility in various forms. In addition to applications summarized below regarding the valuation of financial assets in settings of affine processes, recent progress =-=[17, 22, 66, 67, 80, 81, 82, 93]-=- in the modeling of optimal dynamic portfolio and consumption choice has exploited the special structure of controlled affine state-process models. We fix a conservative regular affine process X with ... |

51 |
The Central Tendency: A Second Factor in Bond Yields
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(Show Context)
Citation Context ...equations for a broad range of affine processes (see (11.10)). Indeed, given the desire to model interest rates with ever increasing realism, various higher-dimensional (d > 1) variants have appeared =-=[5, 6, 11, 14, 23, 30, 31, 63, 68]-=-, and efforts [13, 18, 31, 39, 46, 45], includingAFFINE PROCESSES AND APPLICATIONS IN FINANCE 51 this paper, have been directed to the classification and unification of affine termstructure models. B... |

50 |
Affine term structure models and the forward premium anomaly
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Citation Context ...7]. Empirical analyses of interest-rate behavior based on the properties of affine models include [19, 25, 31, 34, 36, 50, 55, 65, 74, 75, 91], with a related analysis of foreign-currency forwards in =-=[2]-=-. Statistical methods developed specifically for the analysis of time-series data from affine models have been based on approximation of the likelihood function [68, 40], on generalized method of mome... |

46 |
A Multivariate Model of the Term Structure
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Citation Context ... equations for a broad range of affine processes (see (11.10)). Indeed, given the desire to model interest rates with ever increasing realism, various higher-dimensional (d >1) variants have appeared =-=[5, 6, 11, 14, 23, 30, 31, 54, 64, 69]-=-, and efforts [13, 18, 31, 39, 46, 45], includingsAFFINE PROCESSES 51 this paper, have been directed to the classification and unification of affine termstructure models. Beyond the scope of our analy... |

40 |
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Citation Context ... of our analysis here, there are in fact “infinitedimensional affine term-structure models” [51, 28, 27]. Empirical analyses of interest-rate behavior based on the properties of affine models include =-=[19, 25, 31, 34, 36, 50, 55, 65, 74, 75, 91]-=-, with a related analysis of foreign-currency forwards in [2]. Statistical methods developed specifically for the analysis of time-series data from affine models have been based on approximation of th... |

39 |
Branching Processes with Immigration and Related Limit Theorems. Theory Probab
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Citation Context ...ciated with the generator are in a one-to-one relation with those of the corresponding ODEs. Regular affine processes include continuous-state branching processes with immigration (CBI) (for example, =-=[62]-=-) and processes of the Ornstein-Uhlenbeck (OU) type (for example, [79]). Roughly speaking, the regular affine processes with state space R m + are CBI, and those with state space Rn are of OU type. Fo... |

38 |
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Citation Context ... equations for a broad range of affine processes (see (11.10)). Indeed, given the desire to model interest rates with ever increasing realism, various higher-dimensional (d >1) variants have appeared =-=[5, 6, 11, 14, 23, 30, 31, 54, 64, 69]-=-, and efforts [13, 18, 31, 39, 46, 45], includingsAFFINE PROCESSES 51 this paper, have been directed to the classification and unification of affine termstructure models. Beyond the scope of our analy... |

38 | Do bonds span the fixed income markets? Theory and evidence for unspanned stochastic volatility
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Citation Context ...r, have been directed to the classification and unification of affine termstructure models. Beyond the scope of our analysis here, there are in fact “infinitedimensional affine term-structure models” =-=[51, 26, 28]-=-. Empirical analyses of interest-rate behavior based on the properties of affine models include [19, 25, 31, 34, 36, 50, 54, 64, 73, 87], with a related analysis of foreign-currency forwards in [2]. S... |

36 |
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Citation Context ...cess. We provide a one-to-one relationship between the coefficients of the characteristic function of a conservative regular affine process X and (up to a version) its semimartingale characteristics (=-=[57]-=-) (B,C,ν) (after fixing a truncation of jumps), of which B is the predictable component of the canonical decomposition of X, C is the “sharpbrackets” process, and ν is the compensator of the random ju... |

36 |
Integrated Time-Series Analysis of Spot and Option Prices, working paper
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Citation Context ...defaultable bond pricing. Numerous affine generalizations [3, 4, 7, 8, 9, 24, 27, 40, 79, 80] of the Heston model have been directed toward more realistic stochastic volatility and jump behavior. Pan =-=[72]-=- conducted a time-series analysis of the S-and-P 500 index data, both the underlying returns as well as option prices, based on an affine jumpdiffusion model of returns. Special numerical methods for ... |

34 |
The Term Structure of Interest Rates as a Random Field
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(Show Context)
Citation Context ...r, have been directed to the classification and unification of affine termstructure models. Beyond the scope of our analysis here, there are in fact “infinitedimensional affine term-structure models” =-=[51, 28, 27]-=-. Empirical analyses of interest-rate behavior based on the properties of affine models include [19, 25, 31, 34, 36, 50, 55, 65, 74, 75, 91], with a related analysis of foreign-currency forwards in [2... |

31 |
market structure in the presence of marked point processes
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Citation Context ...e (11.10)). Indeed, given the desire to model interest rates with ever increasing realism, various higher-dimensional (d >1) variants have appeared [5, 6, 11, 14, 23, 30, 31, 54, 64, 69], and efforts =-=[13, 18, 31, 39, 46, 45]-=-, includingsAFFINE PROCESSES 51 this paper, have been directed to the classification and unification of affine termstructure models. Beyond the scope of our analysis here, there are in fact “infinited... |

30 |
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Citation Context ...v process with state space Rm + , independently of z ∈ Rn . Theorem 2.7 generalizes and unifies two classical types of stochastic processes. For the notion of a CBI process we refer to [94], [62] and =-=[85]-=-. For the notion of an OU type process see [79, Definition 17.2]. Corollary 2.10. Let X =(Y,Z) be regular affine. Then (Y,(P (y,z))y∈R m + ) is a CBI process, for every z ∈ R n .Ifm =0then X is an OU ... |

27 |
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Citation Context |

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Consistency Problems for Heath-Jarrow-Morton Interest Rate Models, Lecture Notes in Mathematics 1760
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(Show Context)
Citation Context ...e (11.10)). Indeed, given the desire to model interest rates with ever increasing realism, various higher-dimensional (d >1) variants have appeared [5, 6, 11, 14, 23, 30, 31, 54, 64, 69], and efforts =-=[13, 18, 31, 39, 46, 45]-=-, includingsAFFINE PROCESSES 51 this paper, have been directed to the classification and unification of affine termstructure models. Beyond the scope of our analysis here, there are in fact “infinited... |

27 |
Affine term structure models. In
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(Show Context)
Citation Context ... of our analysis here, there are in fact “infinitedimensional affine term-structure models” [51, 28, 27]. Empirical analyses of interest-rate behavior based on the properties of affine models include =-=[19, 25, 31, 34, 36, 50, 55, 65, 74, 75, 91]-=-, with a related analysis of foreign-currency forwards in [2]. Statistical methods developed specifically for the analysis of time-series data from affine models have been based on approximation of th... |

21 |
Stochastic Mean and Stochastic Volatility – A Three-Factor Model of the Term Structure of Interest Rates and its Applications to the Pricing of Interest Rate Derivatives
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(Show Context)
Citation Context ... or the Cox-Ingersoll-Ross process [30], which is the continuous branching diffusion of Feller [43]. Both of these short-rate processes are of course themselves affine (L(x) =x), as are many variants =-=[20, 23, 30, 49, 58, 59, 72, 90, 74]-=-. In general, because 1 = e 〈0,x〉 , the bond price Qt1(x) =e A(t)+〈B(t),x〉 (13.1) is easily calculated from the generalized Riccati equations for a broad range of affine processes (see (11.10)). Indee... |

17 |
Stochastic interest rates and the bond-stock mix. The European Finance Review 4
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(Show Context)
Citation Context |