Estimation in Financial Models - RiskLab

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1.2 Discrete time In the previous section we dealt with continuous time stochastic volatility models based on systems of stochastic dierential equations. In the discrete time approach we consider time series models which are systems of stochastic dierence equations. In analogy to the continuous time approach we assume the stock price S n , S n > 0, to follow the process S n = S n ( n + n z n ); (1.6) with and dependent on time and z n N (0; 1). We consider the logarithm of the stock price H n ln S n . With the notation we derive the process H n = y 1 + :::+ y n ; H n = y n =( n , 2 n 2 )+ nz n : (1.7) We will concentrate on three dierent discrete stochastic volatility models. First we consider a rather simple model in which the conditional variance of the time series fh n g follows a logarithmic AutoRegressive (log-AR) process (see Jacquier, Polson and Rossi [42]). I. log-AR(p)/stochastic volatility: v n = 0 + 1 v n,1 + :::+ p v n,p + ~z n ; (1.8) with v n =ln 2 n and (z n ; ~z n ) independent N (0; 1). The famous AutoRegressive Conditional Heteroskedastic (ARCH) model proposed by Engle [22] will be the second model to look at. The key insight offered by the ARCH model lies in the distinction between the conditional and the unconditional second order moments. While the unconditional covariance matrix for the variables of interest may be time invariant, the conditional variances and covariances often depend non-trivially on the past. II. ARCH(q): 2 n = 0 + 1 y 2 n,1 + :::+ qy 2 n,q : (1.9) A long lag length q and a large number of parameters are often needed in empirical applications of ARCH(q). To avoid this problem Bollerslev introduced the Generalized ARCH (GARCH) model [12]. III. GARCH(p; q): 2 n = 0 + 1 y 2 n,1 + :::+ q y 2 n,q + 1 2 n,1 + :::+ p 2 n,p: (1.10) For a discussion of AR, ARCH and GARCH models see section 3.2. 7
Chapter 2 Approximations 2.1 Diusion models by discrete time models The theory of nance is mainly treated in terms of stochastic dierential equations, whereas in practice observations can be made only at discrete time intervals. We want to bridge this gap by approximating diusion models by discrete time models and vice versa. In this section we will deal with discrete time models as diusion approximations. The advantage of approximating a diusion model by a sequence of discrete time models mainly lies in estimating and forecasting. Whereas the likelihood function of a discrete time model is easy to compute and to maximize, there may arise problems in deriving the likelihood of a discretely observed nonlinear stochastic dierential equation system, e.g. when the diffusion coecient depends on an unknown parameter or when there are unobservable state variables like conditional variance (see [54], [14]). In the case of a diusion model with continuous observations, see 3.1.1 for the denition of the maximum likelihood estimator, its properties and the mentioned problems like parameter dependence. When the diusion model is observed at discrete time points and the transition densities are unknown, discretizing the continuous time log-likelihood leads to the problem of inconsistent estimators. This problem is described in 3.1.2 where in addition three dierent approaches are given to overcome this problem. The latter also in the case of incomplete observations. In summary rather than estimating and forecasting with a diusion model observed at discrete time points it may bemuch easier to use a discrete time model directly, e.g. an ARCH model, as basic approximation. 8
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Page 5 and 6: ing functions or methods based on a
Page 7: wonder why people still use the Bla
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Page 15 and 16: 2.2 Discrete time models by diusion
Page 17 and 18: Chapter 3 Parameter Estimation 3.1
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Page 21 and 22: (see [49], p.249f). Asymptotic norm
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Page 31 and 32: Considering the system (3.26) we di
Page 33 and 34: An Application As an application of
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Page 39 and 40: Corollary 2 For we have F (X (i,1)
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Page 43 and 44: 3.2 Discrete models In section 1.2
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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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