Estimation in Financial Models - RiskLab

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Introduction Over the last few years various new derivative instruments have emerged in nancial markets leading to a demand for versatile estimation methods for relevant model parameters. Typical examples include volatility, covariances and correlations. In this paper we give a survey on statistical estimation methods for both discrete as well as continuous time stochastic models. The text is organized as follows: in Chapter 1 we rst motivate a model in which volatility ofa price process is assumed to follow astochastic process. Out of the variety ofcontinuous time stochastic volatility models introduced in the literature wechoose two empirically relevant ones, that is an arithmetic Ornstein-Uhlenbeck process and a square root diusion model. Those two models serve as reference models in some of the later chapters. As for discrete time stochastic volatility models, we concentrate on log-AR(p) processes, ARCH(q) and GARCH(p; q) processes all of which will be discussed in detail in Section 3.2. Approximations of diusion models by discrete time models and vice versa are described in Chapter 2. The convergence result on which these approximations are based is stated. As applications the diusion limits of GARCH(1,1)- type processes and of AR(1) E-ARCH processes are derived. In addition, we present strategies for approximating diusions and briey compare dierent discretizations of a diusion model. In Section 3.1 estimation of an unknown parameter in a general diusion process is discussed. For continuously observed processes we develop the classical theory of maximum likelihood estimation including properties like consistency and asymptotic normality. In the case of discrete observations the maximum likelihood estimator retains all the 'good' properties if the transition densities of the process are known. However, in most cases relevant for nance, we do not have explicit expressions for the underlying transition densities and the use of approximate likelihood functions leads to inconsistent estimators when the time between observations is bounded away from zero. We describe alternative estimation methods, as there are martingale estimat- 3
ing functions or methods based on approximating the transition densities. As a result, we are able to obtain consistent and asymptotically normal estimators. The methods introduced will be tested on some examples. In Section 3.2 discrete AR, ARCH and GARCH models and their asymptotic properties are discussed. Finally, a brief description of Bayesian estimation is given. Concerning nonparametric estimation in diusions, in Section 4.1 we discuss estimation of a probability density and estimation of an unknown signal. Section 4.2 presents two dierent nonparametric techniques for discrete models, namely kernel estimators and Fourier type estimators. In Chapter 5 we show how to solve linear stochastic dierential equations explicitly and give some classes of nonlinear stochastic dierential equations that can be reduced to linear ones. Some necessary background material is briey introduced in the appendices. An extensive bibliography guides the interested reader to further published material. Acknowledgements: In working out the material presented, I was fortunate to have been able to discuss with various people aspects of the text. I am especially grateful to Prof. P. Embrechts for his invaluable advice and constant encouragement. I also take great pleasure in thanking A.N. Shiryaev, A. Dassios, G. Parker and H. Schurz for their much appreciated help. I thank M. Kafetzaki Boulamatsis very much for all the helpful discussions and her support during the preparation of the manuscript. The fruitful discussions within Risklab also were a constant source of inspiration. 4
Page 1 and 2: Estimation in Financial Models RISK
Page 3: 5 Some Diusion Models with Explicit
Page 7 and 8: wonder why people still use the Bla
Page 9 and 10: Chapter 2 Approximations 2.1 Diusio
Page 11 and 12: 1) fX kh g: the family of discrete
Page 13 and 14: By means of Theorem 1we now obtain
Page 15 and 16: 2.2 Discrete time models by diusion
Page 17 and 18: Chapter 3 Parameter Estimation 3.1
Page 19 and 20: which in the case (3.6) has the for
Page 21 and 22: (see [49], p.249f). Asymptotic norm
Page 23 and 24: a convergence in probability result
Page 25 and 26: time between observations is bounde
Page 27 and 28: For xed 0 s < t, x 2 IR d , 2 an
Page 29 and 30: After these theoretical considerati
Page 31 and 32: Considering the system (3.26) we di
Page 33 and 34: An Application As an application of
Page 35 and 36: If depends on we use the same est
Page 37 and 38: The asymptotic properties of the es
Page 39 and 40: Corollary 2 For we have F (X (i,1)
Page 41 and 42: disadvantage of ~G n and G n motiv
Page 43 and 44: 3.2 Discrete models In section 1.2
Page 45 and 46: The sequence hMi n is called the sq
Page 47 and 48: of time varying variances and covar
Page 49 and 50: The Hessian is @ 2 l t @@ 0 = , 1 @
Page 51 and 52: 3.2.3 The Bayesian estimation metho
Page 53 and 54: f() with respect to (jx). If 1 ; 2
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p( 1 j 2 ) and we have the a priori
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Chapter 4 Nonparametric Estimation
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for model (4.2) and the estimator ^
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Chapter 5 Some Diusion Models with
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and thus X t = t;t0 X t0 + Z t t 0
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If we do not specialize at this poi
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where we assume the choice of b 1 b
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where h is given by (5.42). Again w
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Appendix A The Kalman-Bucy Filter A
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A.2 The discrete case For reasons o
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we can write the Euler scheme (B.1)
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Appendix C The It^o Formula C.1 The
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Appendix D The Radon-Nikodym Theore
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[10] Billingsley,P., 1968, \Converg
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[38] Ibragimov, I.A. and R.Z. Has'm
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[63] Pedersen, A.R., 1994, \Statist
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