Quantitative Finance Conditional Heteroskedastic Models

Transcription

1 Quantitative Finance Conditional Heteroskedastic Models Miloslav S. Vosvrda Dept of Econometrics ÚTIA AV ČR

2 MV1 Robert Engle Professor of Finance Michael Armellino Professorship in the Management of Financial Services Joined Stern 000 Phone: (1) Fax : (1) rengle@stern.nyu.edu Office: KMEC West Fourth Street Suite 9-6 New York, NY

3 Snímek MV1 Prof Engle je nositelem Nobelovy ceny za ekonomii pro rok 003 Miloslav Vosvrda;

4 Recent Awards The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel 003 for methods of analyzing economic time series with time-varying volatility (ARCH)

5 The Pivotal Interest in almost all financial applications is The Predictability of Price Changes

6 The Pivotal Requirement in almost all financial applications is Any Volatility Model Must Be Capable to Forecast the Volatility

7 Financial Time Series The volatile behavior in financial markets is usually referred to as the volatility. Volatility has become a very important concept in different areas in financial theory and practice. Volatility is usually measured by variance, or standard deviation. The financial markets are sometimes more volatile, and sometimes less active. Therefore a conditional heteroskedasticity and stylized facts of the volatility behavior observed in financial time series are tuypical features for each financial market.

8 ARCH, GARCH This one is a class of (generalized) autoregressive conditional heteroskedasticity models which are capable of modeling of time varying volatility and capturing many of the stylized facts of the volatility behavior observed in financial time series.

9 Stylized Facts 1) Non-Gaussian, heavy-tailed, and skewed distributions. The empirical density function of returns has a higher peak around its mean, but fatter tails than that of the corresponding normal distribution. The empirical density function is tailer, skinnier, but with a wider support than the corresponding normal density.figure ) Volatility clustering (ARCH-effects).Figure 3) Temporal dependence of the tail behavior. NEXT

10 dnormx (, 0, 0.7) dt( x, 3) x Back

11 Back Series and Conditional SD 0.1 Original Series Values Conditional SD Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q

12 4) Short- and long-range dependence. 5) Asymmetry-Leverage effects. There is evidence that the distribution of stock returns is slightly negatively skewed. Agents react more strongly to negative information than to positive information.figure Next

13 EGARCH(1,1) 40 returns r n n time Back

14 Capital Market Price Models The Random Walk Model (RWM) Martingale (MGL) Conditional Heteroscedastic Model (CHM) Models of the first category ARCH GARCH IGARCH EGARCH CHARMA ARCH-M GARCH-M Models of the second category Stochastic volatility model Next

15 St The Random Walk Model (RWM) the price observed at the beginning of time ε an error term with E ε =0 and Var ε = σ ( ) ( ) t t t values of the independent of each other S = S + ε t t 1 t 1 S S = ε t t t t ε t S = ε t j j = 1 RWM was first hypothesis about how financial prices move. Fama (1965) compiled EMH.

16 A Prediction by the RWM The 1-step ahead forecast at the forecast origin h is ˆ Sh = E Sh+ 1 Sh Sh 1 = S h ( 1 ) (,,...) The -step ahead forecast at the forecast origin h is Sˆ = E S S, S,... = ( ) ( ) h h+ h h 1 = E( S + ε S, S,...) = Sˆ () 1 = S For any l > 0 forecast horizon we have ˆh ( ) S l = S h+ 1 h+ h h 1 h h h Therefore, the RWM is not mean-reverting

17 Forecast Error The l-step ahead forecast error is So that () = e l ε ε h h l h ( ) Var eh l = lσ ε which diverges to infinity as l The RWM is not predictable.

18 RWM with drift RWM with drift for the price S where S = µ + S + ε t t 1 t ( ) µ = E S S t t 1 + Then S = S + t t 0 and Var t i= 1 ε i = tσ ε t t is µ ε j= 1 Therefore the conditional standard deviation grows slower than the conditional expectation j tσ ε

19 Le Roy s critique of RWM The critique of RWM by Le Roy (1989) led to some serious questions about RWM as the theoretical model of financial markets. The assumption that price changes are independent was found to be too restrictive. Therefore was, after broad discussion, suggested the following model r t = S S + D + t 1 t t S t where r is a return D is a dividend

20 ( ) We assume that E r I = r is a constant t Taking expectations at time t of the both sides r t = S S D + + t t 1 t t S t and we get S t = ( D ) E S t 1 t t + + I 1+ r We assume that reinvesting of dividends is x = h S t t t and ( ) h S = h S + D t+ 1 t+ 1 t t+ 1 t Back

21 Thus ( ) ( ) ( ) E x I = E h S I = h S + D = t+ 1 t t+ 1 t+ 1 t t t+ 1 t ( 1 r) h S ( 1 ) = + = + r x t t t x. For, is a submartingale, because i.e. that is a martingale t r > 0 x r < 0 t ( ) E x x t For, is a supermartingale, because x t 1 t t + I ( ) E x x t 1 t t + I

22 Very Important Distinction A stochastic process following a RWM is more restrictive than the stochastic process that follows a martingale. A financial series is known to go through protracted both quiet periods and periods of turbulence. This type of behavior could be modeled by a process with conditional variances. Such a specification is consistent with a martingale, but not with RWM.

23 Back Martingale processes lead to non-linear stochastic processes that are capable of modeling higher conditional moments. Such models are called Conditional Heteroskedastic Models. These ones are very important for a modeling of the volatility. The volatility is an very important factor in options trading. The volatility means the conditional variance of underlying asset return ( ) ( ) ( ),where ( ) σ = Var r I = E r µ I µ = E r I t t t 1 t t t 1 t t t 1

24 Conditional Heteroskedastic r = + t t t µ µ ε ( ) = E r I 1 t t t p Model (CHM) Let r t is a stationary ARMA(p,q) model, i.e., = + r + µ µ φ θ ε t 0 i t i i t i i= 1 i= 1 ( ) ( ) ( ) ( ) σ = Var r I = E r µ I = E ε I q t t t 1 t t t 1 t t 1

25 We distinguish two categories of CHM The first category: An exact function to a governing of the evolution of the s t is used The second category: A stochastic equation to a describing of the s t is used The models coming under the first category are models of type ARCH or GARCH. These models may catch three out of the five stylized features, namely 1), ), and 4). The models coming under the second category are models of type Stochastic Volatility Model. Back

26 ARCH model This type of model was introduced by Engle in the paper "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation," Econometrica 50 (198), pp , for modeling the predictive variance for U.K. inflation rates.

27 ARCH model The basic idea r = µ + ε t t t ε = µ t rt t t t = σ η innovations = t 0 1 t 1 m t m σ α αε α ε t ( ) Var ( ) η η η, E = 0, = t > 0, 0,, 0 α α α t ( 0,1) or t( ) η N η t m t

28 ε = σ η α t t t > 0, α > ARCH(1) η t N ( 0,1) so that = + t 0 1 t 1 σ α αε This process is stationary if, and only if, σ ε Unconditional variance is equal The fourth moment is finite if 1 α 3α < 1. α 1 < 1. /1. ( α ) 0 1 ( ) ( ) The kurtosis is so given by α1 α1 31 /1 3.

29 10 ARCH(1) 5 returns r n 0 µ n := β0 + β1 n n time α0 := 0.1 α1 := 0.1 β0 := 0.0 β1 := ( ) r := µ n n + ε n α0 + α1 ε n 1

30 ARCH(1) 0 returns r n µ n n time 500

31 ARCH(1) Volatility volatility α0+ α1 ε n 1 σε ( ) n time 110 3

32 1 1 Autocorrelation Function A i AA i σε 0.5 σε r-mi (r-mi)^ +Stadard deviation -Stadard deviation i 6

33 1 1 Partial Autocorrelation Function P i PP i 0.5 σε σε r-mi (r-mi)^ +Stadard deviation -Stadard deviation i 6

34 Simulated ARCH(1) errors Simulated ARCH(1) volatility e(t) sigma(t)

35 Sample Quantiles: min 1Q median 3Q max Sample Moments: mean std skewness kurtosis Number of Observations: 500

36 The fourth moment ( rr, 0, 4) = 6.14 The fourth moment ( rrr, 0, 4) = The third moment ( rr, 0, 3) = 1.85 The third moment ( rrr, 0, 3) =

37 Daily Stock Returns of FORD Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q

38 Series : ford.s Lag Series : ford.s^ Lag ACF ACF

39 The kurtosis exceeds 3, so that the unconditional distribution of ε is fatter tailed than the normal. Testing for ARCH effects. This test is constructed on a simple Lagrange Multiplier (LM) test. The null hyphothesis is that: There are no ARCH effects,i.e., α = = α = 0 1 m The test statistic is LM = N R ~ χ ( p)

40 Test for ARCH Effects: LM Test Null Hypothesis: no ARCH effects Test Statistics: FORD Test Stat p.value Dist. under Null: chi-square with 1 degrees of freedom Total Observ.: 000

41 A practical problem with this model is that, with m increasing, the estimation of coefficients often leads to the violation of the non-negativity of the a s coefficients that are needed to ensure that conditional variance s t is always positive. A natural way to achieve positiveness of the conditional variance is to rewrite an ARCH (m) model as σ = α +Ε ΩΕ, Ε = ε,, ε ( ) T t 0 m, t 1 m, t 1 m, t 1 t 1 t m T

42 Forecasting Forecasts of the ARCH model can be obtained recursively. Consider an ARCH(m) model. At forecast origin h, the 1-step ahead forecast of σ is h + 1 ( ) σ 1 = α 0 + α 1ε + + α ε + 1 The -step ahead forecast of σ h + is ( ) ( ) σ = α + ασ 1 + α ε + + α ε + where h h m h m h 0 1 h h m h m The l-step ahead forecast of h l is m h 0 i h i= 1 σ l = α + ασ l i () ( ) ( ) σh ε + l i = if l i 0 h l i σ +

43 15 ARCH(1) Forecasting r n 10 returns r nn rr nn rrr nn nnn, time

44 Weakness of ARCH Models The model assumes that positive and negative shocks have the same effects on volatility because it depends on the square of the previous shocks. The ARCH model is a little bit restrictive because for a finite fourth moment is necessary to be α 1 [ 0,1/ 3 ). This constraint becomes complicated for higher order of the ARCH. The ARCH model provides only way to describe the behavior of the conditional variance. It gives no any new insight for understanding the source of variations of a financial time series. ARCH models are likely to overpredict the volatility because they respond slowly to large isolated shocks.

45 Although the ARCH model is simple, it often requires many parameters to adequately describe the volatility process. To obtain more flexibility, the ARCH model was generalised to GARCH model. This model works with the conditional variance function. Back

46 0 GARCH model r = µ + ε, ε = σ η t t t t t t m s = + + t 0 i t i j t j i= 1 j= 1 σ α α ε β σ ( ms) ( ) Var( ) η η η, E = 0, = 1 t t t α α β max, i= 1 > 0, 0, 0, i j ( α + β ) < 1, and α = 0 for i > m, i i i β j = 0 for j > s

47 A Connection to the ARMA process For better underestanding the GARCH model is useful to use the following transform: t= t t so that t = t t, and t i = t i t i Substitute these expressins into GARCH equation and we get as follows ξ ε σ σ ε ξ σ ε ξ ( ms) max, s = + ( + ) + t 0 i i t i t j t j i= 1 j= 1 ε α α β ε ξ β ξ The process { ξ } is a martingale difference series, i.e., t ( ) Eξt = 0, cov ξt, ξt j = 0, for j 1.

48 GARCH(max(m,s),s) The process is an application of the ARMA idea to the squared series. Thus ε t ( ms) max, s = + ( + ) + t 0 i i t i t j t j i= 1 j= 1 ε α α β ε ξ β ξ ( ms) max, = / 1 + t 0 i i i= 1 ( ) Eε α α β

49 The model provides a simple parametric function for describing the volatility evolution. or ε σ t 1 t 1 ε t 1 1 ε α + β > 0 GARCH(1,1) = + + t 0 1 t 1 1 t 1 σ α α ε β σ α > 0,1 > α 0,1 > β 0, α + β < 1 A large gives rise to a large. A large tends to be followed by large ε t generating the well-known behavior of volatility clustering in financial time series. If then kurtosis ( ) t t ( ( α β ) ) ε ( α β ) ( ) /1 + > 3 σ t

50 GARCH(1,1) 5 returns r n n time 110 3

51 GARCH(1,1) Volatility.5 volatility δ n n time 110 3

52 σ h Forecasting by GARCH(1,1) We assume that the forecast origin is h. For 1-step ahead forecast For -step ahead forecast ( 1) α 0 βσ 1 h 1 h α ε = + + σ For l-step ahead forecast ( ( ) l 1 α 1 α + β ) () h l = α β h ( ) ( α + β ) σ ( 1) l h 1 1 σ α β σ Therefore σ () l = α + 1 ( ) ( ) α 0 h l 1 α1 1 β 0 h

53 The multistep ahead volatility forecasts of GARCH(1,1) model converge to the unconditional variance of εt as the forecast horizon increases to infinity provided that Var(ε t ) exists.

54 GARCH(1,1) Forecasting r n 10 r nn returns rr nn rrr nn n, nn original realization forecasting +standard deviation -standard deviation time

55 Mean Equation: ford.s ~ 1 Conditional Variance Equation: ~ garch(1, 1) Coefficients: C 7.708e-004 A 6.534e-006 ARCH(1) 7.454e-00 GARCH(1) 9.10e-001

56 Series and Conditional SD Original Series Conditional SD Values Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q1 Q Q3 Q4 Q

57 Series with Conditional SD Superimposed ford.mod11 Values Q Q4 Q Q4 Q Q4 Q Q4 Q Q4 Q Q4 Q Q4 Q Q

58 ACF of Observations 1.0 ACF Lags

59 ACF of Squared Observations 1.0 ACF Lags

60 Cross Correlation CCF Lags

61 GARCH Model Residuals residuals Residuals Q Q4 Q Q4 Q Q4 Q Q4 Q Q4 Q Q4 Q Q4 Q Q

62 GARCH Volatility Conditional SD volatility Q Q4 Q Q4 Q Q4 Q Q4 Q Q4 Q Q4 Q Q4 Q Q

63 GARCH Standardized Residuals residuals Standardized Residuals Q Q4 Q Q4 Q Q4 Q Q4 Q Q4 Q Q4 Q Q4 Q Q

64 ACF of Std. Residuals 1.0 ACF Lags

65 ACF of Squared Std. Residuals 1.0 ACF Lags

66 QQ-Plot of Standardized Residuals QQ-Plot 1/3/ Standardized Residuals /13/ /19/ Quantiles of gaussian distribution

67 garch(formula.mean = ford.s ~ 1, formula.var = ~ garch(1,1)) Mean Equation: ford.s ~ 1 Conditional Variance Equation: ~ garch(1, 1) Conditional Distribution: gaussian Estimated Coefficients: Value Std.Error t value Pr(> t ) C 7.708e e A 6.534e e ARCH(1) 7.454e e GARCH(1) 9.10e e

68 AIC(4) = BIC(4) = Normality Test: Jarque-Bera P-value Shapiro-Wilk P-value

69 Ljung-Box test for standardized residuals: Statistic P-value Chi^-d.f Ljung-Box test for squared standardized residuals: Statistic P-value Chi^-d.f

70 Lagrange multiplier test: Lag 1 Lag Lag 3 Lag 4 Lag 5 Lag 6 Lag Lag 8 Lag 9 Lag 10 Lag 11 Lag 1 C TR^ P-value F-stat P-value

71 Test for Residual Autocorrelation Test for Autocorrelation: Ljung-Box Null Hypothesis: no autocorrelation Test Statistics: Test Stat p.value 0.516

72 Test for Residual^ Autocorrelation Test for Autocorrelation: Ljung-Box Null Hypothesis: no autocorrelation Test Statistics: Test Stat p.value 0.984

73 GARCH(1,1) 0. returns r n n time 110 3

74 GARCH(1,1) Volatility volatility δ nk nk 500 time

75 GARCH(1,1) Forecasting 0.4 r n r nn 0. returns rr nn rrr nn n, nn original realization forecasting +standard deviation -standard deviation time

76 Weakness of GARCH Models This model encounters the same weakness as the ARCH model. In addition, recent empirical studies of high-frequency financial time series indicate that the tail behavior of GARCH models remains too short even with standardised Student t-innovations. Back

77 ARCH-M model The return of a security may depend on its volatility. If we take conditional deviation as a measure for risk, it is possible to use risk as a regressor in returns modeling. To model such a phenomenon, one may consider the ARCH-M model, where M means in mean. This type model was introduced by Engle in the paper "Estimation of Time Varying Risk Premia in the Term Structure:the ARCH-M Model," (with David Lilien and Russell Robins), Econometrica 55 (1987):

78 A simple ARCH-M(1) model r = µ + cσ η t t t t = t 0 1 t 1 m t m σ α αε α ε The parameter c is called the risk premium parameter. A positive c indicates that the return is positively related to its past volatility.

79 ARCH-M(1) 6 returns r n n time Back

80 GARCH-M Model A simple GARCH(1,1)-M model r = µ + c σ + ε t t t t ε = σ η t t t = + + t 0 1 t 1 1 t 1 σ α α ε β σ The parameter c is called the risk premium parameter. A positive c indicates that return is positively related to its past volatility. By formulation we can see that there are serial correlations in the return series r t.

81 GARCH-M(1,1) 5 returns r n n time Back

82 IGARCH IGARCH models are unit-root (intergrated) GARCH models. This type model was introduced by Engle and Bollerslev in the paper Modelling the Persistence of Conditional Variances," Econometric Reviews,5, (1986), pp A key feature of IGARCH models is that the impact of past squared shocks ζ = ε σ for i > 0 on is persistent. t i t i t i ε t

83 IGARCH(1,1) An IGARCH(1,1) model can be written as r µ ε ε ση = +, =, t t t t t t = + + t 0 1 t 1 1 t 1 σ α βσ αε where ε 1 { } t is defined as before and 1> β > 0.

84 IGARCH(1,1) 10 returns r n n time 110 3

85 Forecasting by IGARCH(1,1) When α + β = repeated substitutions in give ( l) = + ( + ) ( l ) 1 h h σ α α β σ ( ) ( ) ( ) h h 0 ( ) σ σ α where h is the forecast origin. The effect of on future volatilities is also persistent, and the volatility forecasts form a straight line with slope α 0. The case ofα 0 = 0 is of particular interest in studying the IGARCH(1,1) model. The volatility forecasts are simply = 1 + 1, 1, σ h 1 ( ) 1 σ h for all forecast horizons.

86 IGARCH(1,1) Forecasting 30 r n r nn 0 returns rr nn rrr nn nnn, time Back

87 FARIMA General IGARCH(p,q) process is called FARIMA(p,d,q): r µ ε ε ση = +, =, t t t t t t σ α β σ ε α ε α β ε t 0 1 t 1 t 1 1 t t = + ( ) + (1 + ) + ( + )

88 EGARCH Model This model allows to consider asymmetric effects between positive and negative asset returns through the weighted innovation g ( η ) θη γ η E( η ) = + t t t t where θ and γ are real constant.

89 Leverage Effects Negative shocks tend to have a larger impact on volatility than positive shocks. Negative shocks tend to drive down the shock price, thus increasing the leverage ( i.e. the debt-equity ratio) of the shock and causing the shock to be more volatile. The asymmetric news impact is usually refferred to as the leverage effect.

90 Because η t E E g( η ) = 0. ( η ) t is iid sequence then t So EGARCH(m,s) can be written as r µ ε, ε = t t t t t t = + σ η 1+ βl+ + β L s = exp 1 s g t + 0 t 1 1 L L m 1 m σ α η α α ( )

91 r = µ + ε, ε = σ η t t t t t t EGARCH(1,0) 1 σ exp ( ) t = α0 + g ηt 1 1 α1l The asymmetry of can easily be seen as g ( η ) t g η ( ) t θ + γ ηt γ ηt if ηt 0 = ( θ γ) η ( ) t γe ηt if ηt < 0 ( ) E ( )

92 and ε ( θ γ) ε σt = σt 1exp α0( 1 α1) γ / π ε ( θ γ) ε < t 1 exp + if t 1 0 σ t 1 t 1 exp if t 1 0 σ t 1

93 EGARCH(1,1) 40 returns r n n time Back

94 garch(formula.mean = hp.s ~ 1, formula.var = ~ egarch(1,1), leverage = T, trace = F) Mean Equation: hp.s ~ 1 Conditional Variance Equation: ~ egarch(1, 1) Coefficients: C A ARCH(1) GARCH(1) LEV(1)

95 CHARMA This model uses random coefficients to produce a conditional heteroscedasticity. The CHARMA model has the following form where r = µ + ε t t t t 1t t 1 t t mt t m t η t, ε δε δε δ ε η = { } 1,..., T ( 0, σ ) η, { δ } = ( δ δ ) t t mt is a sequence of iid random vectors with mean zero and non-negative definite covariance matrix Ώ and { δ } t is independent of { η } t

96 CHARMA() r = µ + ε, ε = δ ε + δ ε + η t t t t 1t t 1 t t t The CHARMA model can easily be generalized so that the volatility of ε t may depend on some m explanatory variables. Let { x it} t= 1 be m explanatory variables available at time t. Consider the model r = µ + ε, ε = δ x + η t t t t it i, t 1 t i= 1 { } where { } (,..., ) T δ and { η } t are sequences of t = δ1 t δmt random vectors and random variables. m

97 CHARMA() 10 returns r n n time 110 3

98 Then the conditional variance of ε t ( x,..., x ) ( x,..., x ) σ = σ + Ω t η 1, t 1 m, t 1 1, t 1 m, t 1 where Ω is a diagonal matrix is T Back

99 Stochastic Volatility Model r = + + t 0 1 t 1 t ( ) = µ + ε ε = σ η, g, t t t t t t σ α ασ υ where { } iid ( 0,1) ηt { υ } iid ( 0, σ ) t υ { η } { υ } t t Back

100 The Routes Ahead High frequency volatility High dimension correlation Derivative pricing Modeling non-negative processes Analysing conditional simulations by Least Squares Monte Carlo

101 Thank you very much for your attention