Economics 883: The Basic Diffusive Model, Jumps, Variance Measures. George Tauchen. Economics 883FS Spring 2015

Transcription

1 Economics 883: The Basic Diffusive Model, Jumps, Variance Measures George Tauchen Economics 883FS Spring 2015

2 Main Points 1. The Continuous Time Model, Theory and Simulation 2. Observed Data, Plotting and Pitfalls 3. Empirical Variance Measures 4. Options and Implied Volatility

3 1. The Continuous Time Model Dynamics of the log price (p) process: Continuous dp(t) = µ(t)dt + σ(t)dw(t). With jumps dp(t) = µ(t)dt + σ(t)dw(t) + dj(t), where J(t) J(s) = s<τ t κ(τ).

4 Basic Diffusion x 10 4

5 Simulation Generated by Euler Scheme Generate Gaussian (normal) random variables: Z j N(0, 1), j = 1, 2,..., J M steps per day: M = 80 = 1/80 Standard deviation σ = percent per day T days:

6 Euler Scheme (continued) Wiener increments dw j = Z j j = 1, 2,..., J Initialize p 0 = log(75) ($75 per share) Iterate: p j = p j 1 + µ + σ dw j, j = 1, 2,..., J Convert to levels: {P j } J j=1, P j = e p j j = 1, 2,,..., J(= M T ),

7 Euler Scheme: Time Varying Variance Wiener increments dw j = Z j j = 1, 2,..., J Initialize p 0 = log(75) ($75 per share) Iterate: p j = p j 1 + µ + σ j 1 dw j, j = 1, 2,..., J Convert to levels: {P j } J j=1, P j = e p j j = 1, 2,,..., J(= M T ), and make a nice plot.

8 Simulation of Jump Diffusion by an Euler Scheme Sample the number of jumps N Poisson, then draw N jumps κ k N(0, σ jmp ), k = 1,..., N, and scatter the jumps randomly.

9 Jump Diffusion With jumps dp(t) = µ(t)dt + σ(t)dw(t) + dj(t),

10 2. Observed Data: XOM September-October XOM: September October Sep 02, 2008 Sep 16, 2008 Oct 01, 2008 Oct 16, 2008 Oct 31, 2008

11 Within-day geometric returns X = 100 p (log-price) r t,i n i X

12 XOM Returns, September-October, XOM, Return: September October Sep 02, 2008 Sep 16, 2008 Oct 01, 2008 Oct 16, 2008 Oct 31, 2008

13 Default Plots from Graphics are not Acceptable Poorly labeled axis, scales, and colors:

14 Default gives very poor graphic: 6 Default no good

15 Care with Data: e.g., Stock Splits Consider Apple (APPL): AAPL Price Jan 02, 2002 Apr 06, 2004 Jul 05, 2006 Oct 03, 2008 Dec 30, 2010

16 Clearly, something is wrong, or needs attention: AAPL Returns Jan 02, 2002 Apr 06, 2004 Jul 05, 2006 Oct 03, 2008 Dec 30, 2010

17 Culprit Data recording error or maybe a stock split? AAPL (Yahoo Finance, Charts, click max) Splits: Jun 16, 1987 [2:1], Jun 21, 2000 [2:1], Feb 28, 2005 [2:1] Need to adjust the data for any stock splits. The convention is to adjust backwards. Watch it, sometimes there can be a reverse stock split such as 1:3.

18 3. Variance Measures The Realized Variance For integer t RV t = n n i X 2 i=1 As 0, n, then from advanced probability theory: RV t t s=t 1 c s ds + t 1<s t X s RV t converges to the integrated variance plus sum of all withinday jumps squared.

19 Bipower Variation The Bipower variation of Barndorff-Nielsen and Shephard (2003) BV t = µ 1 1 n n 1 M n i X n i X j=2 As 0, M, and BV t t s=t 1 σ 2 (s) ds BV t converges to the integrated variance. It is jump robust. In the absence of jumps RV t BV t 0.

20 Continuous and Jump Variation Model: dx t = µ t dt + c t dw t + J t, Split the quadratic variation into two pieces: RV t QV t = t s=t 1 σ 2 (s) ds + t 1<s t κ 2 s QV t = IV t + T JV t IV t = t s=t 1 σ 2 (s) ds, T JV t = t 1<s t κ 2 s

21 Decompose the Realized Variance RV t = CV t + JV t CV t = RV t I(z t c) + BV t I(z t > c) JV t = (RV t BV t ) I(z t > c)

22 Realized Variance (Annualized) XOM Annualized Realized Volatility Jan 03, 2007 Oct 19, 2007 Aug 11, 2008 Jun 01, 2009 Mar 17, 2010 Dec 30, 2010

23 Bipower Variation (Annualized) XOM Annualized Bipower Variation Jan 03, 2007 Oct 19, 2007 Aug 11, 2008 Jun 01, 2009 Mar 17, 2010 Dec 30, 2010

24 Relative Contribution of Jumps XOM A poorly scaled graphic: 0.5 XOM: Relative Contribution of Jumps Jan 03, 2007 Oct 19, 2007 Aug 11, 2008 Jun 01, 2009 Mar 17, 2010 Dec 30, 2010

25 Relative Contribution of Jumps XOM A reasonable well-scaled graphic: XOM: Relative Contribution of Jumps Jan 03, 2007 Oct 19, 2007 Aug 11, 2008 Jun 01, 2009 Mar 17, 2010 Dec 30, 2010

26 Truncated Variance (TV) as Estimator of IV t Threshold Estimator(Mancini, 2009) Truncated Variance: T V t = n n i X 2 I [ n i X cutoff t ] i=1 where I[ ] is the 0-1 indicator function (1 = true, 0 = false).

27 Value of cutoff for Threshold Variation What is the cutoff? Something like 4 (or 2, 3) standard deviations would be about right, but what standard deviation? Papers by Ait-Sahalia, Todorov, and others provide some guidance. A reasonable choice might be be cutoff t = 3.5 BV t 1 ( n ) ω, 0 < ω < 1 2 where BV t 1 is the bipower variation of the preceding day. But this does not allow for the deterministic intraday diurnal pattern.

28 Diurnal Adjustment Think of the diffusive part X c of the process. With constant daily variance n i Xc = σ n Z i, Z i N(0, 1) The fluctuations are on the order σ n. Suppose we had deterministic factors τ i that re-distribute the daily σ 2. Then n i Xc = τ i σ n Z i, Z i N(0, 1) We need n n 1 τ i = 1 to preserve the interpretation of σ 2 as the total daily variance.

29 Estimate Jump Robust Diurnal Adjustment Let v i = π 1 2 T T t=1 n (t 1)n+i X n (t 1)n+i 1X, i = 2,..., n be the jump robust estimate of variance for interval i. v 1 = v 2 (harmless). Then Set τ i = v i (1/n) n 1 v j We can think of the returns as n i X = n τi σ 2 Z i

30 Example, Microsoft 5-min data, τ i versus i, i = 1,..., Time of Day

31 4. Options and Implied Volatility Later in other courses