18.S096 Problem Set 5 Fall 2013 Volatility Modeling Due Date: 10/29/2013
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1 18.S096 Problem Set 5 Fall 2013 Volatility Modelig Due Date: 10/29/ Sample Estimators of Diffusio Process Volatility ad Drift Let {X t } be the price of a fiacial security that follows a geometric Browia motio process: where dx(t) X(t) = µ dt + σdw (t), σ > 0, is the volatility parameter µ (, ), is the drift parameter dx(t) is the ifiitesimal icremet i price. dw (t) is the icremet of a stadard Wieer Process, i.e, ifiitesimal icremets W (t+dt) W (t) are i.i.d. Normal radom variables with zero mea ad variace equal to dt. Cosider samplig values of the price process over a fixed time period t [0, T ], at equal time icremets h = T/. Defie t i = i h, i = 0, 1,..., X i = X(t i ), i = 0, 1,..., Y i = log(x i /X i Accept as give that: 1 ), i = 1, 2,..., Y i are i.i.d. N(µ h, σ 2 h) radom variables, (this is prove with the theory of diffusio processes/stochastic differetial equatios, with µ = µ σ 2 1 ). 2 1(a) Prove that the Maximum-Likelihood Estimates: µˆ ad σˆ for a sample: y 1, y 2,..., y, are give by µˆ = 1 i=1 Y i σˆ2 = 1 i=1 (Y i µˆ) 2 1(b) Derive the distributio of µˆ ; give specific formulas for the expectatio ad variace of µˆ. 1
2 1(c) Derive the distributio of σˆ2 ; give specific formulas for the expectatio ad variace of σˆ2. 1(d) Cosider icreasig the umber of icremets o the fixed time period [0, T ], ad let µˆ ad σˆ2 be the correspodig MLEs of the parameters. Determie the limitig distributios of µˆ ad σˆ2. 1(e) A sequece of estimators θˆ for a parameter θ, is weakly cosistet if lim P r( ˆθ θ ) = 0. For each of µˆ ad σˆ2, determie whether the sequece of estimators is weakly cosistet. 2. Cosider the same process as i problem 1, but ow, for fixed values of µ ad σ, cosider samplig values of the price process over a fixed time perio d t [0, T ], at variable icremets h i > 0, i = 1, 2,...,, such that i=1 h i = T. Defie i t i = j=1 h j, i = 0, 1,..., X i = X(t i ), i = 0, 1,..., Y i = log(x i /X i Accept as give that: 1 ), i = 1, 2,..., Y i are i.i.d. N(µ h i, σ 2 h i ) radom variables, (this is prove with the theory of diffusio processes/stochastic differetial equatios). 2(a) Derive the MLE for µ ad its distributio for a fixed set of samplig icremets {h i } : i=1 h i = T. 2(b) Derive the MLE for σ 2 ad its distributio for a fixed set of samplig icremets {h i } : i=1 h i = T. 2(c) If limited to samplig + 1 price poits of {X t }, (icludig X 0 ad X T ) prove that For estimatig σ 2, samplig, the ML estimators vary with the icremet spacig, but the variace of these estimators are all equal, regardless of the icremet spacig. For estimatig µ, all ML estimators are the same ad have the same variace, regardless of the icremet spacig. 2
3 3. ARCH(1) Model Properties Let y t = log(s t /S t 1 ) be the discrete returs of the price of a security/portfolio {S t, t = 1, 2,...}, ad supppose that y t ARCH(1), i.e. y t = µ t + ɛ t, where µ t is the mea retur, coditioal o F t 1, the iformatio available up to time (t 1) ad ɛ t = Z t σ t, where Z t iid with E[Z t ] = 0, ad var[z t ] = 1, ad σ 2 2 t = α 0 + α 1 ɛ t 1. Additioally, suppose that E[Z 3 t ] = 0, ad E[Z 4 t ] = κ. (The parameter κ is the Kurtosis of the Z t distributio with uit variace; if Z t is Gaussia/ormal, the κ = 3. Prove that: 3(a) E[ɛ 2 t ] = α 0 /(1 α 1 ) 3(b) E[ɛ 3 t ] = 0 κα 3(c) E[ɛ 4 ] = 0(1+α 2 1 ) t (1 α 1)(1 κα1 2) 3(d) What costraits o α 0, α 1 must be made i (c), to maitai 4-th order statioarity (bouded). 3(e) The kurtosis of ɛ t is κ ɛ = E[ɛ 4 t ]/(E[ɛ 2 t ]) 2. (The fourth momet is ormalized to be scale-free). If the distributio Z t is Gaussia/ormal (i.e., the scaled, coditioal error distributio of ɛ t ), does the ucoditioal distributio of ɛ t, have a higher tha that of the Gaussia distributio, ( i.e., heavier tails)? 3
4 4. Usig Daily Ope/High/Low/Close Data o the S&P500 Idex from , aual sample variaces were computed of chages i the log idex value of the daily Close. The followig table gives the aual sample variaces, day couts, ad aualized volatilities Aual Sample Variaces of Logarithmic Returs: daily.variace days volatility e e e e e e e The differeces i the sample variaces ad volatilities appears quite large for some years. Are the year-by-year differeces sigificat? To address this questio, cosider modelig the returs for ay give year as a simple radom sample from a Gaussia distributio: {y 1, y 2,..., y 2 } : y i i.i.d. N(µ, σ ). The table gives values of σˆ2,, ad σˆ, where σˆ2 = 1 1 i=1 (y i µˆ) 2, with µˆ = 1 i=1 y i. 4(a) Uder the Gaussia model, give, µ, σ 2, prove that the distributio of σˆ2 is σˆ2 σ2 1 χ2 1. that is, a scaled Chi-square distributio with degrees of freedom equal to ( 1), ad scale factor equal to σ2 1 4(b) Statistical methodology defies cofidece itervals for ukow parameters by computig a likely iterval for the parameter estimate give the ukow parameter, ad the ivertig the iterval to correspod to the parameter istead of the estimate. For a 95% cofidece iterval (two-sided), the developmet is as follows: 4
5 With 95% probability, the χ 2 1 radom variable will fall withi the iterval from the percetile, to the percetile, i.e., P r(q < χ 2 1 q ) = = 0.95 where P r(χ 2 1 q 0.025) = P r(χ 2 1 q 0.975) = Replacig the radom variable χ with ( )σˆ2 gives: σ 2 P r(q < ( 1 )σˆ2 q σ ) = 0.95 which ca be iverted to ( 1) ( 1) P r(σˆ2 σ 2 σˆ2 q q ) = 0.95 The followig table gives the percetiles of the Chi-square distributios for degrees of freedom ragig from 249 to 252 (oe less tha the aual day couts). df q0.025 q0.975 ll.factor ul.factor The last two colums are ll.factor = 1 ad ul.factor = 1 q q which whe multiplied by the ubiased sample estimate σˆ2, defie the cofidece iterval for σ 2. Usig data for 2008, compute the two-sided 95% cofidece iterval for σ 2, based o daily log returs. Express the iterval i terms of the aualized volatility ( 253σ). Does the sample aual volatility for ay other year fall i the cofidece iterval for 2008? 4(c) The retur variace / volatility varies cosiderably from year to year. To evaluate the statistical sigificace of the differece i values for ay two years, we ca use the F -Distributio. Cosider 2007 ad Uder the assumptio (i.e., a ull hypothesis H 0 ) of Gaussia/ormal daily returs ad that the variaces of the returs are costat/ the same for all days i the two years it follows from 4(a) that: X = ( ) 2 )σˆ22007 χ2 df, where df X = ( ) σ 1 Y = ( ) )σˆ22008 σ 2 χ2 df, where df Y = ( ) 5
6 ad X ad Y are idepedet radom variables. The statistic Y/df Y σˆ2 S = = X/df X ( 2008 ) σˆ22007 has the F -Distributio with degrees of freedom df Y for the umerator ad df X for the deomiator. (Verify by lookig up the defiitio of the F -distributio.) Uder the ull hypothesis, the umerator ad deomiator of S are estimates of the same retur variace. Their ratio varies about 1 due to the idepedet variatio i the umerator ad deomiator of scaled Chi-squared radom variables. The methodology of hypothesis testig i statistics uses the fact that the test statistic has a kow distributio uder the ull hypothesis. The ull hypothesis is accepted / rejected so log as the test statistic is ot extreme. We choose a test α-level, the probability of (falsely) rejectig the ull hypothesis if true, say α = From this, extreme rages of the test statistic are defied that occur with probability α whe the ull hypothesis is true. For α = 0.05, a two sided alterative is cosidered usig q ad q 0.975, the percetiles of the F distributio give by: P r(f dfy,df X < q ) = P r(f dfy,df X < q ) = The ull hypothesis is accepted if q < S < q From the package R, we provide the percetiles of the F -distributio whe df 1 = , ad df 2 = : > qf(0.025, df1=252, df2=250) [1] > qf(0.975, df1=252, df2=250) [1] so the ull hypothesis is accepted if q = < S < = q Compute the test statistic S = S 0 for testig the daily retur variace for 2008 is equal to the daily retur variace for Give the value of the test statistic S 0, determie the α level at which the ull hypothesis is o the boudary of beig just accepted/rejected. 6
7 (This level is called the P -value of the test statistic. Reportig a test statistic s P -value provides evidece cocerig for/agaist the test ull hypothesis which ca be provided without havig to specify a α-level.) Repeat the previous two questios, for testig the equality of the retur variace for 2008 to that for (Note: the degrees of freedom for 2006 are the same as those for 2007 so the same F distributio is applicable) 7
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