Estimation in Financial Models - RiskLab

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Chapter 1 Stochastic Volatility Models 1.1 Continuous time The value of a stock price S is supposed to follow the process dS t = S t ( t dt + t dW t ); (1.1) where W t is a Wiener process and and are functions of t. Considering the logarithm of the stock price H ln S and using It^o's formula we derive the process followed by H dH t =( t , 2 t 2 ) dt + t dW t : (1.2) In the following consider the process H t instead of the equivalent process S t . For many purposes this leads to a more tractable dierential equation. Also from a statistical point of view do the increments of H t (i.e. the so-called log-returns) behave in a 'nicer' way as the increments of S t . In the case t and t , S t follows geometric Brownian motion. This is assumed in the Black-Scholes option pricing model which has been used as an eective tool for the pricing of options for more than two decades. When comparing the calculated option values using the Black-Scholes model with the option prices there is usually a dierence. Among these biases in model prices the well-known \smile"-eect is important: the Black-Scholes option pricing formula tends to underprice out-of-the-money-options and to overprice at-the-money-options, that means implied volatility changes with the striking price (see Ball [1]). This eect arises from the assumption in the Black-Scholes model that volatility is a known constant (\I sometimes 5
wonder why people still use the Black-Scholes formula, since it is based on such simple assumptions { unrealistically simple assumptions". Black [11]). In real life volatility is not constant at all. It is non-uniform, that means on days with major economic events volatility is usually higher than on other days (especially on non-trading days). In addition, sometimes stock prices jump which can be thought of as a sudden large increase in the stock's volatility. Furthermore many economic time series have a mean-reversion tendency: when the value of a random variable reaches a very high level then it will rather go down than up and vice versa. Put dierently, it tends away from extremely high or low values and reverts to some long-term mean. All these observations lead to the assumption that volatility is a random variable and a lot of stochastic volatility models have been recently introduced in the literature. Now the question arises in which way stochastic volatility 2 t = 2 (t; !) can be modeled. The standard framework assumes the volatility specication dv = m(v) dt + k(v) d ~W t ; (1.3) with v = 2 t or v = ln 2 t and ~ W t a Wiener process. Out of the variety of stochastic volatility models we will consider the following two: I: dH t = ( t , 2 t 2 ) dt + t dW t dv t = ( , v t ) dt + d~W t ; with v t =ln 2 t (1.4) II: dH t = ( t , 2 t 2 ) dt + t dW t dv t = b (a , v t ) dt + c p v t d ~W t ; with v t = 2 t (1.5) where ; ; ; a; b and c are xed constants, W t and W ~ t are independent Wiener processes. In the model (1.4) volatility, ormore exactly ln 2 , is governed by an arithmetic Ornstein-Uhlenbeck (or rst order autoregressive) process with a meanreverting tendency. A large and growing literature treats this case as an empirically relevant one (see Bollerslev, Engle and Nelson [14], Stein and Stein [72], Wiggins [75]). In (1.5) a square root diusion model for stochastic volatility is suggested. For this model see Ball [1] and literature given there. One further reason why we have restricted our attention to the models (1.4) and (1.5) above is to be able to show explicitly how certain statistical estimation procedures are to be implemented and also compared and contrasted in a specic nance context. Most of the techniques introduced do apply to much more general set-ups. 6
Page 1 and 2: Estimation in Financial Models RISK
Page 3 and 4: 5 Some Diusion Models with Explicit
Page 5: ing functions or methods based on a
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
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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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