Estimation in Financial Models - RiskLab

More documents

Recommendations

Info

Appendix B Numerical Methods In our representation we follow [45], x9 and x10. B.1 The Euler Scheme The Euler approximation is a basic discrete time method to approximate an Itô process. Consider an Itô process X = fX(t);t 0 t T g following the scalar stochastic dierential equation dX t = a(t; X t )dt + b(t; X t )dW t ; with t 0 t T and the initial condition X t0 = X 0 . A discretization t 0 = t 0
we can write the Euler scheme (B.1) in the impressive form Y n+1 = Y n + a n + bW n (B.2) for n =0; 1;:::;N , 1. In order to compute the sequence fY n ;n = 0; 1;:::;N , 1g of values of the Euler approximation we have to generate the random increments W n for n = 0; 1;:::;N , 1 of the Wiener process W = fW t ;t 0g. These increments are independent Gaussian random variables with E(W n ) = 0 and Var(W n )= n and can be generated by a random number generator (see e.g. [45], x1.3). For the multi-dimensional case of the Euler scheme see e.g. [45], x10.2. Note that when the diusion coecient b is identically zero the stochastic iterative scheme (B.2) reduces to the well-known deterministic Euler scheme for the ordinary dierential equation x 0 = a(t; x). We introduce the notion of strong convergence. Denition 1 A time discrete approximation Y with maximum step size converges strongly to X at time T if lim E(jX T , Y (T )j) =0: #0 The rate of strong convergence is crucial if we want to compare dierent time discrete approximation methods. Denition 2 A time discrete approximation Y converges strongly with order > 0 at time T , if there exists a constant C > 0, independent of , and a 0 > 0 such that for each 2 (0; 0 ). E(jX T , Y (T )j) C Under some regularity assumptions the Euler scheme converges strongly with order =0:5 : Theorem 8 Suppose E(jX 0 j 2 ) < 1; E jX 0 + Y 0 j 2 1 2 C 1 1 2 ; ja(t; x) , a(t; y)j + jb(t; x) , b(t; y)j C 2 jx , yj; ja(t; x)j + jb(t; x)j C 3 (1 + jxj) 74
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 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
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: A.2 The discrete case For reasons o
Page 77 and 78: Appendix C The Itô 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
show all

Estimation in Financial Models - RiskLab

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?