Estimation in Financial Models - RiskLab

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When the transition densities of X are unknown, the usual alternative estimator is dened by approximating the log-likelihood function for based on continuous observations of X. For this log-likelihood function to be dened, the diusion coecient (t; x; ) =(t; x) has to be known (see Section 3.1.1, p.17). Under some assumptions (see [51] x7) the log-likelihood function for based on continuous observations of X in [0;t n ] can be written in terms of integrals l c t n () = Z tn 0 , 1 2 b (s; X s ; ) T (s; X s )(s; X s ) T ,1 dXs Z tn 0 b (s; X s ; ) T (s; X s )(s; X s ) T ,1 b (s; Xs ; ) ds; (3.16) and the usual approximation of these integrals leads to the approximate loglikelihood function for based on discrete observations of X ~ ln () = nX i=1 , 1 2 with the notation b (t i,1 ;X ti,1 ; ) T i,1 (X ti , X ti,1 ) nX i=1 b (t i,1 ;X ti,1 ; ) T i,1 b (t i,1 ;X ti,1 ; )(t i , t i,1 ); (3.17) i,1 (t i,1 ;X ti,1 ) (t i,1 ;X ti,1 ) T ,1 : When the diusion coecient also depends on an unknown parameter, the question arises how the parameter is to be estimated. If divides into two parts =( 1 ; 2 ), such that b(; ; ) =b(; ; 1 ) and (; ; ) is known up to the scalar factor 2 , that means (; ; ) = 2 ~(; ), we may avoid the problem of parameter dependence. In this case 2 2 can be estimated in advance by a quadratic variance-like formula (see Florens-Zmirou [25]), and by inserting this estimate in (; ; ) = 2 ~(; ), the diusion term can be assumed "known". Then the estimate of the approximate log-likelihood function ~ l n can be used to estimate 1 . In the case where the diusion coecient (; ; ) depends on more generally, Hutton and Nelson [37] show that the discretized score function corresponding to lt c n can under certain regularity conditions still be used to estimate . That means in this case we are able to estimate based on discrete observations of X as well. However, estimation methods for discrete observations that arise from the theory of continuous observations have the undesirable property that the estimators are strongly biased unless max 1in jt i , t i,1 j is "small". If the 23
time between observations is bounded away from zero, especially in the case of time-equidistant observations with xed, Florens-Zmirou [25] shows that the estimator ~ n of obtained by maximizing the approximate log-likelihood function ~ l n () is inconsistent. To overcome the diculties regarding parameter dependence and the dependence of ~ l n ()onmax 1in jt i ,t i,1 j,we will propose three dierent estimation approaches. The basic idea of the rst two approaches is to nd good approximations to the transition densities. In the third approach we construct martingale estimating functions. Approximation to the transition densities of X by a sequence of transition densities of approximating Markov processes The following approach is proposed by Pedersen [58, 60]. We shall derive a sequence (l n;N ()) 1 N=1 of approximations to l n (), that builds a connection between ~ l n () (see (3.17)) and l n () in the following sense: the approximation l n;1 () is a generalization of ~ l n () with no restrictions on (; ; ) regarding parameter dependence, each l n;N () for N 2 is an improvement of l n;1 () and as N tends to innity l n;N () converges for each in probability tol n (). The crucial point of the approach is to approximate the transition densities p(s; x; t; y; ) of X by a sequence of transition densities (p N (s; x; t; y; )) 1 N=1 of approximating Markov processes which converges to p(s; x; t; y; ) as N tends to innity, and then to dene the approximate log-likelihood functions l n;N () = nX i=1 log (p N (t i,1 ;X ti,1 ;t i ;X ti ; )): (3.18) In the following we will derive the approximating densities p N (s; x; t; y; ). We remark that we may relax the Lipschitz assumption made in (3.1) for b and a little bit, namely we only assume b and to be locally Lipschitz continuous as dened below. Under the following conditions (A1), (A2) and (A3) which must hold for all 2 , the stochastic dierential equation (3.1) has a weak solution for all x 0 and and has the pathwise-uniqueness property which implies the uniqueness in law (see [65], p.132 and p.151; in this context see also [73], x5-x8). (A1) b and are continuous in t for all x 2 IR d . 24
Page 1 and 2: Estimation in Financial Models RISK
Page 3 and 4: 5 Some Diusion Models with Explicit
Page 5 and 6: ing functions or methods based on a
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: a convergence in probability result
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
Page 55 and 56: p( 1 j 2 ) and we have the a priori
Page 57 and 58: Chapter 4 Nonparametric Estimation
Page 59 and 60: for model (4.2) and the estimator ^
Page 61 and 62: Chapter 5 Some Diusion Models with
Page 63 and 64: and thus X t = t;t0 X t0 + Z t t 0
Page 65 and 66: If we do not specialize at this poi
Page 67 and 68: where we assume the choice of b 1 b
Page 69 and 70: where h is given by (5.42). Again w
Page 71 and 72: Appendix A The Kalman-Bucy Filter A
Page 73 and 74: A.2 The discrete case For reasons o
Page 75 and 76:
we can write the Euler scheme (B.1)
Page 77 and 78:
Appendix C The It^o Formula C.1 The
Page 79 and 80:
Appendix D The Radon-Nikodym Theore
Page 81 and 82:
[10] Billingsley,P., 1968, \Converg
Page 83 and 84:
[38] Ibragimov, I.A. and R.Z. Has'm
Page 85:
[63] Pedersen, A.R., 1994, \Statist
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