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This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers
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1. 1.       Advanced  Financial  Models     under  construc2on
2. 2. Learning  Objec-ves     ¨  Lognormal  Distribu-ons     ¨  Rela-ons  between     ¤  Normal  &  lognormal   n  Pdfs   n  Sta-s-cs   ¤  Simple  and  natural  log  rates     2
3. 3. Hypotheses  and  Models     ¨  Explana-ons  of  phenomenon   ¤  Hypothesis   n  A  proposed  explana-on  for  a   phenomena   ¤  Law   n  Statement  of  a  cause  and  eﬀect     without  explana-on   n  Newton’s  law  of  gravity     ¤  Theory   n  A  well-­‐established  explana-on  for   a  phenomenon   n  Einstein’s  theory  of  gravity   ¨  A  model  is  a  mathema-cal  or   physical  representa-on  of  a   phenomenon   ¤  The  “Bohr  atomic  model”     ¤  Newton’s  inverse  square  law  of   gravity       ¤  Einstein’s  Theory  of  General   Rela-vity         3 2 21 r mm GF ⋅ ⋅=
4. 4. SPX  Daily  Ln  Rate  Histogram:  Zoom   4
5. 5. SPX  Daily  Ln  Rate  Histogram:    More  Zoom   5 Again this histogram includes daily return rates from 1950 <-4.5% should happen less than once in a thousand years, but there have been 31 such days since 1950 or about once every two years -22.9% day should not have happened (Oct 19, 1987)
6. 6. SPX  Daily  Ln  Rate:  August  –  December  2008   6
7. 7. SPX  Daily  Ln  Rate:  Mean   7 -­‐350% -­‐250% -­‐150% -­‐50% 50% 150% 250% 1/5/51 11/9/57 9/13/64 7/19/71 5/23/78 3/27/85 1/30/92 12/4/98 10/8/05 Annualized  mean   22  day  annualized  tailing  mean   252  day  annualized  tailing  mean   Long  term  annualized  tailing  mean
8. 8. SPX  Daily  Ln  Rate:  Mean   8 -­‐80% -­‐60% -­‐40% -­‐20% 0% 20% 40% 60% 80% 1/2/90 3/12/92 5/21/94 7/29/96 10/7/98 12/15/00 2/23/03 5/3/05 7/12/07 9/19/09 Zoom  in  on  Annualized  mean   252  day  annualized  tailing  mean   Long  term  annualized  tailing  mean
9. 9. 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 1/3/1950 3/22/1958 6/8/1966 8/25/1974 11/11/1982 1/28/1991 4/16/1999 7/3/2007 SPX  Daily  Ln  Rate:  Standard  Devia-on     9 Annualized  standard  devia-ons  (‘vola-lity’)   22  day  annualized  trailing  vola-lity   252  day  annualized  trailing  vola-lity   Long  term  annualized  trailing  vola-lity
10. 10. 0% 5% 10% 15% 20% 25% 30% 1/2/1990 9/28/1992 6/25/1995 3/21/1998 12/15/2000 9/11/2003 6/7/2006 SPX  Daily  Ln  Rate:  Standard  Devia-on     10 Zoom  in  on  Annualized  standard  devia-ons  (‘vola-lity’)   252  day  annualized  trailing  vola-lity   Long  term  annualized  trailing  vola-lity
11. 11. SPX  Daily  Ln  Rate:  Autocorrela-on  Cluster   11
12. 12. SPX  Daily  Ln  Rate:  Autocorrelogram     12 Natural  log  daily  return  rates  for  SPX,  v     1950  –  2011   15471  days     Rates  do  look  rather  uncorrelated
13. 13. SPX  Daily  Ln  Rate:  Autocorrelogram     13 Natural  log  daily  de-­‐trended  squares  of  return  rates   (variance)    for  SPX,  (v-­‐u)2   1950  –  2011    15471  days     There  is  some  posi-ve  autocorrela-on  (persistence)     Might  even  be  greater  persistence  over  shorter  periods
14. 14. SPX  Daily  Ln  Rate:  Histogram  of  Annualized   Daily  Variance   14 Histogram  of  Annualized  Daily  Variance
15. 15. SPX:  Annual  Accumula-on  of  Daily  Returns   15 10,000  annual  sums  of  252  day   (1  year)  con-guous  return  rates   randomly  selected  from  1950  to   2011     This  histogram  doesn’t  look   normal  at  all  as  the  addi-ve  CLT   would  indicate     So  the  rates  are  not  IID/  FV
16. 16. SPX:  Ln  Rate  Q-­‐Q  Plot   16 A  Q-­‐Q  plot  compares  the   measured  rates  to  ideal  normal   rates  from  measured  mean  and   variance
17. 17. Natural  Log  Rate  –  More  Tests   ¨  Jarque  Bera  normality  test     ¤  JB  is  a  Chi  Squared  sta-s-c  with  2  dof     ¤  Normality  via  chi  squared  considera-on  of  s   skew,  S,  and  kurtosis,  K,  the  3rd  and  4th     moments  of  distribu-on  which  measure     asymmetry     17 440,657           4 3)(29.0600 1.05621 6 15471           4 3)(K S 6 n JB 2 2 2 2 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − +−= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − += JB                       (χ2   statistic)   If  normality  is   rejected,  what  is   the  probability  of  a   rejection  error   0.0000 100.00% 4.6051 10.00% 5.9914 5.00% 9.2103 1.00% 10.0000 0.67% 15.0000 0.06% 20.0000 0.00% 25.0000 0.00% 30.0000 0.00% 35.0000 0.00% 40.0000 0.00% 45.0000 0.00% 50.0000 0.00% So  there  is  ~0%  probability  of  incorrectly  rejec-ng  the  normal  hypothesis
18. 18. Natural  Log  Rate  –  Tests  For  Normality     18
19. 19. Stock  Return  Rate  Summary   ¨  Historical  stock  return  rates,  r  and  v,  are  characterized  by     ¤  Leptokurtosis   n  Fat  or  heavy  tails:  more  extreme  events  than  ‘normal’   n  More  return  rates  near  the  mean  than  ‘normal’   ¤  Nega-ve  skew   n  More  extreme  downside  events  than  upside     ¨  Dependence  in  return  rate  vola-lity     ¤  Rate  vola-lity  clustering,  short  term  persistence  then  reversion  to  mean   ¨  Less  frequent  sampling  e.g.,  weekly  and  monthly  would  show  some   smoothing,  but  s-ll  not  normal     ¤  However,  quarterly  or  annual  sampling  would  ignore  important  rate  of   return  informa-on     19
20. 20. Lognormal  Pdf   20 The  lognormal  pdf  is   asymmetric,  is  not  nega-ve,   over  -me  the  mean,  mode,   and  median  drii  further   apart,  and  the  distribu-on   skews  more  posi-vely.
21. 21. Lognormal  Pdf   21   0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Mode       Median       Mean  (expected)
22. 22. 22 ( ) ( ) ∞>> ⋅⋅ = − ⋅ − −    x      0 e π2σx 1 σ  μ,  |x  f 1s,uNL~r                       σ2 μ)(lnx x 2 2 2 ( ) ( ) ∞>>∞ ⋅ = ⋅ − − x    -­‐ e π2σ 1 σ  μ,  |x  f s,uN~v                       σ2 μ)(x x 2 2 2 u  is  mean,  median,  and  mode     The  parameters  is  the  normal  pdf   above  are  also  the  sta-s-cs  –  the   mean  and    variance   The  mean,  mode,  and  median  are   all  diﬀerent       The  parameters  is  the  lognormal   pdf  are  the  same  as  for  the  normal   pdf,  but  they  are  not  the  sta-s-cs,   not  the  mean  or  variance
23. 23.     ¨  Why  simple  returns  can’t  really  be  normal   ¤  Simple  returns  are  compounded  over  -me  increments,  but   normal  random  variables  are  mul-plied   ¤  (1+r)n   ¤  u·∙n   23 ( )r1lnv +=  ... 3 r 2 r r)r1ln(v )1x(                                                ... 3 x 2 x x)x1ln( 32 32 −+−=+= −≠−+−=+
24. 24. Variance  of  Simple  and  Log  Returns   24 [ ] [ ] [ ] [ ] ( ) ( ) ( ) [ ]( ) ( )                                                                                                                                           1er1E                       1ee                         1ee                           1ee                         ee                         ee                       xExE                       dr1VarrVar 2 2 2 2 2 22 22 222 s2 s 2 2 s u s2 s u2 ssu2 s2us2u2 2 2 s u 2 s2 u2 22 2 −⋅+= −⋅ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ = −⋅= −= −= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −= −= =+= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +⋅ +⋅ +⋅+⋅ + ⋅ +⋅ [ ]( ) ( ) ( ) ( ) ( ) ( ) ( ) 1)s      1,(a                                  sd 1ed d1e d1lns 1)(a                          1    a1 a1 d 1lns 1ea1               1er1E    d   22 s2 2s 22 2 2 2 2 s2 s22 2 2 2 2 <<<<≈≈ −≈ +≈ +≈ <<≈+ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + += −⋅+= −⋅+= ( ) [ ] [ ] [ ] [ ] 2 2 22 s2u22 2 s u 2 sk uk k 2 v e        xE e            xE                  e    xE   su,NL~                      er1X ⋅+⋅ + ⋅ +⋅ = = = =+=
25. 25. Variance  of  Simple  and  Log  Returns   25 Future  Value  Factor:  1+r  =  ev [ ] ( )  1eer1var 22 ssu2 −⋅=+ +⋅ [ ] * 2 u2 s u eer1E ≡=+ + [ ] u er1M =+
26. 26. 26 ( ) ( ) [ ] [ ] [ ] [ ] ( )[ ] [ ] ondistributi  normal  log  for  Median                            er1MM[x]          ondistributi  lognormal  for  moment  2        er1E  xE ondistributi  normal  log  for  (mean)  moment  1                          er1E  xE ondistributi  normal  log  for  moment  k                                            exE          su,NL~        er1 u nds2u222 st2 s u th2 sk uk k 2v 2 2 22 =+= =+= =+= = =+ ⋅+⋅ + ⋅ +⋅
27. 27. Central  Limit  Theorem     27 ( ) ( )2n n n 1i i n 1i i0n s,uN~ n y u n y n v vSln)Sln()Sln( →= =Δ=− ∑ ∑ = = ( ) 1)x,x(NL~r g1f)r(1 fr1 S S n 1 n n 1 n 1i i n n 1i i 0 n − +→=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ≡+= ∏ ∏ = = Assume  that  n  is  large  and  r  and  v  are  IID/FV
28. 28. 28   ( ) ( ) ( ) ∑∏ ∑∏ ∑∏ == == == =⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +=⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + n 1i i n 1i v n 1i i n 1i v n 1i i n 1i i veln r1lneln r1lnr1ln i i n21 i vvv 0 n 1i v 0n n210 n 1i i0n e  ...e  e  S               eS    S )r(1....)r(1)r(1  S             )r(1S    S ⋅⋅⋅⋅= ⋅= +⋅⋅+⋅+⋅= +⋅= ∏ ∏ = = ( ) ( ) ( ) ( ) ( ) ( ) n210 n 1i i0n n210 n 1i i0n v...vvSln                     vSln    Sln )rln(1....)rln(1)rln(1Sln                       )rln(1Sln    Sln ++++= += +++++++= ++= ∑ ∑ = =
29. 29. Mean  Natural  Log  Return  Rates   29 Example:  v  is  distributed   uniformly  from  -­‐10%  to   +20%     Average  of  sums  of  vi  are   normal  (sum  of  n  rates)         ( )2 n 1i i s,uN~ n v∑=
30. 30. Simple  Future  Value  Factors     30 Example:  r  is  distributed   uniformly  from  -­‐10%  to  +20%   )x,x(NL~)r(1f n 1 n 1i i n 1 n ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ += ∏=
31. 31. Simple  Future  Value  Factors     31 ( ) )x,x(NL~r1f n 1i in ∏= += Example:  r  is  distributed  uniformly  from  -­‐10%  to  +20%    f  is  distributed  lognormal
32. 32. We  did  plot  a  histogram  of  natural  return  rates,  v,  for  the  SPX.    It  did  have  the   general  appearance  of  normality.    But  a  Levy  stable  seems  like  a  bemer  ﬁt,  but   has  disadvantages.      However,  the  typical  assump-on  in  ﬁnance  is  that  v  is   normally  distributed  which  has  a  number  of  advantages.         One  advantage  is  that  the  loca-on  sta-s-cs  are  iden-cal  –  mode,  median,  and   mean  –  it’s  a  symmetric     32  v)r1(ln                      e)r1(  v)r1(ln                    e)r1( v ii v i i =+=+ =+=+ For  stocks  or  other  ﬁnancial  assets,  so  far  there  has  been  no  assump-on  on   the  distribu-ons  of  v  and  r  other  than  being  IID/FV     But  the  rela-onship  between  r  and  v  has  been  deﬁned  as     ( ) ( )2 s,uN~r1lnv +=
33. 33. 33 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1s,uNL1e~r s,uNLe~r1 s,uN~r1lnv 2s,uN 2s,uN 2 2 2 −≡− ≡+ += Another  advantage  is  the  normal  distribu-on  scale  linearly  in  -me.    The  mean   driis  to  the  right  while  the  variance  increases.     ( )2 sn,unN~vn ⋅⋅⋅ Another  advantage  is  the  normal  distribu-on  scale  linearly  in  -me.    The  mean   driis  to  the  right  while  the  variance  increases.      Yet  another  advantage  is  the   rela-on  between  the  normal  and  lognormal  distribu-ons  is  similar  to  the   rela-on  between  the  na-ral  log  rate  and  simple  rate     Therefore  the  simple  rate,  r,  is  lognormal  under  assump-on  that  the  natural   log  rate  is  lognormal
34. 34. 34
35. 35. 35
36. 36. Natural  Log  Rate  Autocorrelogram     36 Natural  log  daily  absolute  return  rates  for  SPX,  |v|   Daily  range       1950  –  2011   15471  days
37. 37. Common  PDFs  in  Finance     ¨  Gaussian  /  Normal   ¤  IID  /  FV,  two  parameters     ¤  CLT  for  sums  of  IID/FV  random   variables   ¤  Special  case  Levy  stable  and  ellip-c   distribu-ons       ¨  Ellip-c     ¤  IID  /  FV,  two  parameters     ¤  unimodal,  no  skew,  no  kurtosis  other   than  Gaussian  case   ¤  Linear  correla-on  deﬁnes  linear   dependence   ¤  Used  in  MPT  and  CAPM   ¤  Includes    Gauss,  Cauchy,  t-­‐distr,   Laplace,  symmetric  Levy  Stable     ¨  Lognormal   ¤  IID  /  FV   ¤  CLT  for  products  of  IID/FV  random   variables   ¤  Posi-ve     ¤  Mode,  median,  mean  non-­‐coincident     ¨  Levy  stable   ¤  IID,  not  generally  FV,  4  parameters     ¤  Unimodal,  skew,  kurtosis  other  than   Gaussian  case   ¤  Central  limit  theorem  for  IID  and  stable   but  not  FV  random  variables  converges   to  a  Levy  stable  distribu-on     ¤  Includes  Gaussian,  Cauchy,  Levy   37
38. 38. More  on  Covar  &  Corre   38 [ ] [ ] [ ] [ ]                                                     yExEyxEy,xCov ⋅−⋅=
39. 39. Monthly  Idealized  PDFs  From  SPX  History     39 ln(1+r)  =  v   Normally   distributed     N(u,s2)         u  and  s2  are   normal  pdf   parameters  and   sta-s-cs  -­‐  mean   and  variance     (1+r)=  ev    Lognormally   distributed      NL(u,s2)       Same  pdf  parameters,   but  diﬀerent  mean  and   variance
40. 40. Monthly  Idealized  PDFs  From  SPX  History     40 Future  value   factor,  (1+r)  =  ev   lei  shiied  by  -­‐1,  r
41. 41. Return  Rate  PDFs:    Sta-s-cs  Increase  With  Time   41   -­‐75% -­‐50% -­‐25% 0% 25% 50% 75% 100% 125% 150% 175% 200% 225% 250% 275% 300% Natural  log  rates   (v)  are  assumed   normal.    The  mean   and  variance  of  a   normal  distribu-on   scale  linear  in  -me     The  future  value  factors  (1+r)  are   assumed  log  normally  distributed.     The  mean  and  variance  do  not  scale   linearly  in  -me.
42. 42. 42   Future  Value  Factor:  1+r  =  ev [ ] ( )  1eer1var 22 ssu2 −⋅=+ +⋅ [ ] * 2 u2 s u eer1E ≡=+ + [ ] u er1M =+
43. 43. 43   [ ] [ ] [ ] [ ] ( ) ( ) ( ) [ ]( ) ( )                                                                                                                                             1er1E                       1ee                         1ee                           ondistributi  normal  logof    Variance        1ee                         ee                         ee                       xExEr1VarrVar 2 2 2 2 2 22 22 222 s2 s 2 2 s u s2 s u2 ssu2 s2us2u2 2 2 s u 2 s2 u2 22 −⋅+= −⋅ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ = −⋅= −= −= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −= −=+= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +⋅ +⋅ +⋅+⋅ + ⋅ +⋅
44. 44. 44
45. 45. -­‐1.25 -­‐1.00 -­‐0.75 -­‐0.50 -­‐0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 Idealized  PDFs  for  36  Months     45   Forecast  36  month   natural  log  rate  of   return     normally  distributed     N(36·∙u,  36·∙s2)       Forecast  36   month  future   value  factor     lognormally   distributed     Forecast  36  month  simple  rate  of  return       lognormally  distributed
46. 46. Lognormal  Distribu-on   46 ( ) ( ) [ ] [ ] [ ] [ ] ( )[ ] [ ] ondistributi  normal  log  for  Median                            er1MM[x]          ondistributi  lognormal  for  moment  2        er1E  xE ondistributi  normal  log  for  (mean)  moment  1                          er1E  xE ondistributi  normal  log  for  moment  k                                            exE          su,NL~        er1 u nds2u222 st2 s u th2 sk uk k 2v 2 2 22 =+= =+= =+= = =+ ⋅+⋅ + ⋅ +⋅ ( ) ( ) ( ) ( ) ( ) ( ) n210 n 1i i0n n210 n 1i i0n v...vvSln                     vSln    Sln )rln(1....)rln(1)rln(1Sln                       )rln(1Sln    Sln ++++= += +++++++= ++= ∑ ∑ = =     S SS r )r1(SS rate  return  Simple 1i 1ii i i1ii − − − − = +⋅= ( ) ( ) )r1ln(           SlnSln S S lnv eSS rate  return  log  Natural i 1ii 1i i i v 1ii i += −=⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ⋅= − − −
47. 47. Lognormal  Distribu-on   47 ( ) ( ) [ ] [ ] [ ] [ ] ( )[ ] [ ] ondistributi  normal  log  for  Median                            er1MM[x]          ondistributi  lognormal  for  moment  2        er1E  xE ondistributi  normal  log  for  (mean)  moment  1                          er1E  xE ondistributi  normal  log  for  moment  k                                            exE          su,NL~        er1 u nds2u222 st2 s u th2 sk uk k 2v 2 2 22 =+= =+= =+= = =+ ⋅+⋅ + ⋅ +⋅
48. 48. GARCH  Time  Series     ¨  Similar  to  historic  vola-lity     ¤  Simple  condi-onal  dependence  in  the  second  moment  (vola-lity)     n  Vola-lity  clustering    or  persistence     ¨  The  GARCH  vola-lity  has  three  contribu-ons   ¤  Long  term  average  vola-lity,  s2,  so  there’s  a  reversion  of  the  mean   ¤  Short  term  dependence  on  recent  square  of  return  rate,  v2       ¤  Short  term  dependence  on  recent  Garch  vola-lity,  h   ¨  To  Do   n  Is  there  a  probability  distribu-on?  Maybe  not       n  Plot  the  resul-ng  rates  and  look  for  fat  tails     n  So  it  looks  good  historically,  but  how  can  it  be  used  in  decision  making  ?     48
49. 49. GARCH  Time  Series   ¨  The  GARCH(1,1)  vola-lity  model  with  the  natural  log  rate  process   model  vola-lity  has  three  contribu-ons   49 ( ) 0βλ,α, 1βαγ β  ,α  ,γ      :weights hβvαsβα1         hβvαsγh zh  uv   1i 2 1i 2 1i 2 1i 2 i iii > =++ ⋅+⋅+⋅−−= ⋅+⋅+⋅= ⋅+= −− −− The    Gaussian  rate  process  is     vi  =  u  +  s  ·∙zi     s  is  the  (tradi-onal)  long  term   average  standard  devia-on   z  is  the  standard  normal  random   variable   h  is  the  Garch  variance     v  is  the  nat  log  return  rates       Example:   α = .85 , β = .1 , γ = .05
50. 50. GARCH  Time  Series   50 Single  simulated  GARCH(1,1)  vola-lity   for  15,461  days
51. 51. GARCH  Time  Series   51
52. 52. GARCH  Time  Series   52
53. 53. Adendum:  Nat  Log  &  Exp   53   ( ) ( ) ( ) y+xyxyxyx 32 32 x )xln( e  =  e  e                    )(e  =  e    ... 3 1x 2 1x 1x)xln( )1x(                                                ... 3 x 2 x x)x1ln( x 1 )xln( dx d )yln()xln( y x ln )yln()xln()yxln( x)eln( )0x(                                                                                                              xe ⋅ − − + − −−= −≠−+−=+ = −=⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +=⋅ = >= ⋅
54. 54. Addendum   54   Rate Periodic   mean   Annual   mean Periodic   standard   deviation Annual   standard   deviation a α g γ vi u µ s σ d  =  Var(r)  =  Var(1+r) ri d δ
55. 55. Addendum   55          dwσdt 2 σ μ              dln(S) 2 * ⋅+⋅⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −=( ) ( ) Tσ Tσ.5r K S ln d Tσ Tσ.5r K S ln d 2*0 2 2*0 1 ⋅ ⋅⋅−+⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⋅ ⋅⋅++⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = [ ] ( )  1eer1var 22 ssu22 −⋅=+=δ +⋅ [ ] * 2 u2 s u eer1E ≡=+ + ( ) ( ) [ ] [ ] tsz 1ii s,0N 1i i v 1ii 2 i i1i-­‐i eSS e~ S S eSS s,0N~v  vSln    Sln 2 i Δ⋅⋅ − − − ⋅= ⋅= += tsz i i tsz i1ii i )rln(1v ii ii i ii er )r(1e )r(1SS )r(1ee tszv )rln(1        v   Δ⋅⋅ Δ⋅⋅ − + = += +⋅= +== Δ⋅⋅= +=
56. 56. Addendum     56   ( ) [ ] [ ] ( ) ( ) ( ) −+−=+ + Δ⋅⋅ + Δ⋅⋅ +Δ⋅⋅+== −+−=+ ++++= =⋅δ⋅+ =+≡δ ⋅δ⋅+⋅= ⋅δ⋅== Δ⋅⋅ Δ⋅⋅ 3 r 2 r r)r1ln( ... 6 tsz 2 tsz tsz1ee 3 x 2 x x)x1ln( 6 x 2 x x1e e  ΔtZ1 rSDr)(1SD  ΔtZ1SS ΔtZ?r 3 i 2 i ii 3 i 2 i i tszv 32 32 x tsz i itt ii ii 1-­‐ii Not  yet  ready  to  related  normal  and   lognormal  distribu-ons.    Need   lognormal  sta-s-cs  and  Ito’s  Lemma   Normal    Natural  log  rates    Natural  log  prices   Lognormal    Simple  rates    Future  value  factors    Prices
57. 57. Levy  Stable  Distribu-on   ¤  Bemer  ﬁts  historical  rates  of  return   n  Can  model  Leptokurtosis  and  skew   n  Constant  parameters   n  Generalized  Central  Limit  Theorem     n  Normal  distribu-on  is  a  special  case     n  Problems  included   n  Inﬁnite  variance     n  Variance  cant  be  used  as  a  measure  of  risk  or  vola-lity     n  CAPM,  MPT,  B-­‐S     n  PDF  models  not  applicable     n  Generally  no  analy-c  representa-on     ¤  To  Do     n  Fit  data  to  a  distribu-on  and  graph     n  Why  does  FMH  without  IID  invoke  this  model     n  How  does  it  relate  to  power  law  model  (Has  an  α  >  2  ?)     57
58. 58. Levy  Stable  Distribu-ons     58 [ ] ( ) ( ) parameter  location  ,μ parameter  scale  0,c parameter  skewness  1,1β parameter  stability  (0,2]α Parameters ∞∞−∈ ∞∈ −∈ ∈ undefined  otherwise2,α  when  0  :kurtosis  excess undefined  otherwise2,α  when  0:skew infinite  otherwise2,α  when  c2    :variance undefined  otherwise  1,α  whenμ      :mean 2 = = =⋅ >
59. 59. Levy  Stable  Distribu-ons     ¨  DJIA:  α  =  1.5958  β  =  -­‐.0995  µ  =  .0002  σ  =  .0056     ¤  (5/26/1896  –  1/16/2004  daily)   ¨  SPX  =  α  =  1.6735  β  =  .1064  µ  =  -­‐.0002  σ  =  .0049       ¤  (3/1/1950  –  5/27.2005  daily)             ¨  Only  three  sets  of  parameters  result  in  closed  form       ¤  Gaussian   n  Actually  two  of  the  four  parameters  are  zero  (?)  or  reduced  to  diﬀerent  2  ?     n  Finite  variance     ¤  Levy   ¤  Cauchy   59 [ ] ( ) ( ) parameter  location  ,μ parameter  scale  0,c parameter  skewness  1,1β parameter  stability  (0,2]α Parameters ∞∞−∈ ∞∈ −∈ ∈
60. 60. Power  Law     60
61. 61. Power  Law  Method   ¨  Coopera-on,  herding,  cri-cality     ¤  How  Nature  Works  –  Bak   ¤  Ubiquity  –  Buchanan     ¨  The  ubiquity  of  scale-­‐free  behavior   and  self-­‐organiza-on  in  Nature  led   Bak,  Tang  and  Wiesenfeld  (BTW)  to   coin  the  term  Self-­‐Organized   Cri-cality  (SOC)  to  explain  the   emergence  of  complexity  in   dynamical  systems  with  many   interac-ng  degrees  of  freedom   without  the  presence  of  any  external   agent  ;  SOC  was  devised  to  be  a  sort   of  supergeneral  theory  of  complexity.   61
62. 62. Power  Law     ¨  Confusion  based  on  Fractal  Market  Hypothesis:    Is  it  stable  or  power  law??   ¨  Hurst  soiware  shows  a  random  series  to  be  persistent  ??     ¨  Hurst  exponent   ¤  0.5  is  Brownian    t1/2        √t   ¤  0  <  H  <  0.5  :  an--­‐persistent,  mean  rever-ng     ¤  .5  <  H  ≤  1.0    :  persistent     ¨  Stability  parameter   ¤  α  =  1  /  H,    example  Gaussian:  α =  2,  H  =  .5   ¨  Correla-on  (?)    C  =  22H-­‐1  –  1     ¨  Example   ¤  SPX:  3/1/1950  –  5/27/2005  daily   n     α  =  1.6735  β  =  .1064  µ  =  -­‐.0002  σ  =  .0049     n   H  =  .5976        C=  .1448   ¤  SPX:    1/3/1950  –  6/24/2011    daily     n  H=  .562    α=  1.779        C=  .090   62
63. 63. Reference:  Nat  Log  &  Exp   63   )1x(  ... 3 x 2 x x)x1ln( x 1 )xln( dx d )yln()xln( y x ln )yln()xln()yxln( x)eln( )0x(                                      xe 32 x )xln( −≠−+−=+ = −=⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +=⋅ = >= ( ) ( ) ( ) ++++= ⋅ − − + − −−= ⋅ 6 x 2 x x1e e  =  e  e                    )(e  =  e    ... 3 1x 2 1x 1x)xln( 32 x y+xyxyxyx 32 Rate Periodic   mean   Annual   mean Annual   standard   deviation Period   standard   deviation Rate   pdf a α g γ vi u µ s σ Normal d  =  SD(r)  =  SD(1+r) ri d δ Log   normal
64. 64. Related  Concepts     ¨  Expected  Rate  of  Return  On  Equity   ¤  CAPM  requires  that  the  return  rate  is  normally  distributed  with  a  trend     ¤  Ordinary  least  squares           ¨  Theore-cal  basis  for  r  being  an  independent  random  variable   ¤  Eﬃcient  Market  Hypothesis   ¨  Theore-cal  basis  for  r  being  an  independent  random  variable  with  a  trend   ¤  Ra-onal  Market  Hypothesis         64 ( ) ( ) ( ) ( )FMFEE iE1ii E1ii rrβr  k    r    zsr1SS                  r1SSE −⋅+== ⋅++⋅= +⋅= − −
65. 65. Geometric  Brownian  Mo-on   65 ( ) ( ) [ ] [ ] tsz 1ii s,0N 1i i v 1ii 2 i i1i-­‐i eSS e~ S S eSS s,0N~v  vSln    Sln 2 i Δ⋅⋅ − − − ⋅= ⋅= += tsz i i tsz i1ii i )rln(1v ii ii i ii er )r(1e )r(1SS )r(1ee tszv )rln(1        v   Δ⋅⋅ Δ⋅⋅ − + = += +⋅= +== Δ⋅⋅= +=        dwσdt 2 σ μ              dln(S) 2 * ⋅+⋅⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −=( ) ( ) Tσ Tσ.5r K S ln d Tσ Tσ.5r K S ln d 2*0 2 2*0 1 ⋅ ⋅⋅−+⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⋅ ⋅⋅++⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = [ ] ( )  1eer1var 22 ssu22 −⋅=+=δ +⋅ [ ] * 2 u2 s u eer1E ≡=+ +
66. 66. Geometric  Brownian  Mo-on   66 ( ) [ ] [ ] ( ) ( ) ( ) ... 6 tsz 2 tsz tsz1ee 3 x 2 x x)x1ln( 6 x 2 x x1e e  ΔtZ1 rSDr)(1SD  ΔtZ1SS ΔtZ?r 3 i 2 i i tszv 32 32 x tsz i itt ii ii 1-­‐ii + Δ⋅⋅ + Δ⋅⋅ +Δ⋅⋅+== −+−=+ ++++= =⋅δ⋅+ =+≡δ ⋅δ⋅+⋅= ⋅δ⋅== Δ⋅⋅ Δ⋅⋅ Not  yet  ready  to  related  normal  and   lognormal  distribu-ons.    Need   lognormal  sta-s-cs  and  Ito’s  Lemma   Normal    Natural  log  rates    Natural  log  prices   Lognormal    Simple  rates    Future  value  factors    Prices   u,  s        µ, σ r,  d        α, δ g,              γ,
67. 67. Alterna-ves     ¨  Fat  Tail  Models     ¤  Power  law  not  exponen-al  tails     ¤  Leptokurtosis,  ﬁnite  variance  ?     ¤  Examples     n  Student  t  –  no  skew     n  Levy  stable  –  skew     ¨  Non  IID  Models  –  non-­‐sta-onary  process     ¤  Correla-on  in  rate  vola-lity,  but  not  in  rate,  so  s-ll  ‘unpredictable’     ARCH  models     ¤  Used  with  normal  or  other  distribu-on   67
68. 68. Addendum   68 ( ) ( ) ( ) ( ) ( )SlnSlnSln-­‐Sln eSS Δtσz        Δt 2 σ μ        1SS Δtσz        ΔtμSlnSln 1-­‐ii it 2 ii i1-­‐ii i1-­‐ii tt Δtσz    Δt 2 σ μ 1tt t 2 tt ttt Δ≠Δ= = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⋅⋅+⋅⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ++⋅= ⋅⋅+⋅+= ⋅⋅+⋅⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⋅− ( ) ( ) nszunSlnSln 1i-­‐1-­‐ni ⋅⋅+⋅+=+ ( ) ( ) [ ] [ ] tsztu 1ii s,uN 1i i v 1ii 2 i i1i-­‐i eSS e~ S S eSS s,uN~v  vSln    Sln 2 i Δ⋅⋅+⋅ − − − ⋅= ⋅= +=
69. 69. 69 [ ] [ ] [ ] [ ]2 εii1i-­‐i 2 εii1i-­‐i 1i-­‐i 2 εii1i-­‐i s0,N~ε            εSS s0,IID~ε            εSS SSE                                                     s0,~ε            εSS += += = += ( ) ( ) [ ] ( ) ( ) ( )[ ] ( ) Δtσz   tt tt 1i-­‐1-­‐ni i 1ii 1ii SlnSlnE tszSlnSln 1,0N~z                                                       nszSlnSln ⋅⋅ + = Δ⋅⋅+= ⋅⋅+= − − Generally Rate Periodic   mean   Annual   mean Annual   standard   deviation Period   standard   deviation Rate   pdf a α g γ v u µ s σ Normal d  =  SD(r)  =  SD(1+r) r d δ Log   normal
70. 70. 70 [ ] [ ] tΔBzSS  tΔB    ,0N~ε                                                           tttΔ                            tΔ    ,0N~ε            εSS i1-­‐ii i ii1-­‐ii ttt 2 t 1i-­‐itttt ⋅⋅+= ⋅ −=+= [ ] [ ] mszum 1i1mi sm,umN m 1i v 2 m 1i i i 2 i eSS e~e sm,umN~v    increment  d,multiperio ⋅⋅+⋅ −−+ ⋅⋅ = = ⋅= ⋅⋅ ∏ ∑ [ ] [ ] it 1i i 1ii eS           eSS tzt ,N~sm,umN~ t tzt tt tt 22 t µ Δ⋅σ⋅+Δ⋅µ ⋅= ⋅= Δ⋅σ⋅+Δ⋅µ=µ σµ⋅⋅µ − − ( ) ΔtσzΔtμ S S ΔtσzΔtμ1SS t * i * tt 1-­‐ii ⋅⋅+⋅= Δ ⋅⋅+⋅+⋅= ΔtzΔw tttΔ SSSΔ 1i-­‐i tt 1-­‐ii ⋅= −= −=
71. 71. 71 ( ) 1)r(1g 1fg )r(1f f S S )r(1     S S n 1 n 1i in n 1 nn n 1i in n 0 n n 1i i 0 n −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ += −= += = += ∏ ∏ ∏ = = = n s u )r(1lns v S S ln n n n 1i in n 1i i 0 n = += =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∑ ∑ = =
72. 72. Levy  Stable  Distribu-ons     72
73. 73. Levy  Stable  Distribu-ons     73
74. 74. 74
75. 75. Power  Law     ¨  Power  law  with  rescaled  range     ¨  Many  natural  phenomena  modeled     with  power  law   ¨  Nonlinear  feedback     ¨  Hurst  exponent  is  the  slope     ¨  Fractal  and  self  similar     ¨  Complexity     ¨  How  can  it  be  used  in  decision     making?     ¨  The  rescaled  range  follows  a     power  law   75
76. 76. 76 [ ] [ ] tΔBzSS  tΔB    ,0N~ε                                                           tttΔ                            tΔ    ,0N~ε            εSS i1-­‐ii i ii1-­‐ii ttt 2 t 1i-­‐itttt ⋅⋅+= ⋅ −=+= Rate based process is Geometric Brownian Motion (GBM) [ ] [ ] mszum 1i1mi sm,umN m 1i v 2 m 1i i i 2 i eSS e~e sm,umN~v    increment  d,multiperio ⋅⋅+⋅ −−+ ⋅⋅ = = ⋅= ⋅⋅ ∏ ∑[ ] [ ] it 1i i 1ii eS           eSS tzt ,N~sm,umN~ t tzt tt tt 22 t µ Δ⋅σ⋅+Δ⋅µ ⋅= ⋅= Δ⋅σ⋅+Δ⋅µ=µ σµ⋅⋅µ − − ( ) ΔtσzΔtμ S S ΔtσzΔtμ1SS t * i * tt 1-­‐ii ⋅⋅+⋅= Δ ⋅⋅+⋅+⋅= ΔtzΔw tttΔ SSSΔ 1i-­‐i tt 1-­‐ii ⋅= −= −=
77. 77. Appendix:  Exponen-als  and  Natural  Logs     77 ( ) ( ) dx dy y 1 dx ln(y)d e dx dy dx ed y y ⋅= ⋅= +++++= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ += ∞→ !4 x !3 x !2 x x1e n 1 1lime 432 x n n xlndx X 1 e a 1 dxe xaxa = ⋅= ∫ ∫ ⋅⋅
78. 78. Appendix:  Exponen-als  and  Natural  Logs     78
79. 79. Price  as  a  Stochas-c  Diﬀ  Eqn     79 ( ) ( )1eSSd 1eSS eSSS e S SS e S S eSS dwtd tzt t tzt tt tzt t t tzt t t tzt tt i 1i i 1i1i i 1i 1i i 1i i i 1ii −⋅= −⋅=Δ ⋅=+Δ = +Δ = ⋅= ⋅σ+⋅µ Δ⋅σ⋅+Δ⋅µ Δ⋅σ⋅+Δ⋅µ Δ⋅σ⋅+Δ⋅µ Δ⋅σ⋅+Δ⋅µ Δ⋅σ⋅+Δ⋅µ − −− − − − − ( )SfF =
80. 80. 80 ( )[ ] ( )[ ] ...  1eS S F 2 1 1eS S F dt t F dF ...  dS S F 2 1 dS S F dt t F dF 2dwtd 2 2 dwtd 2 2 2 +−⋅ ∂ ∂ ⋅+−⋅ ∂ ∂ + ∂ ∂ = + ∂ ∂ ⋅+ ∂ ∂ + ∂ ∂ = ⋅σ+⋅µ⋅σ+⋅µ ( ) ( ) ( ) dx dy y 1 dx dy y 1 dx ln(y)d e dx yd   dx ed e dx dy dx ed 2 y 2 2 2 y2 y y ⋅=⋅= ⋅= ⋅= dx dS S 1 dx dS S 1 dx  d dx dS S 1 dx ln(S)  d 2 ⋅−=⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ ⋅= ( ) n 0 nu 0 tμ 0t * * n* nnu n 0 nu 0 )a1(SeSeS]E[S )a1ln(u )a1ln(n n 1 u )a1(lnnu )a1(e )a1(SeS ** * * +⋅=⋅=⋅= += +⋅⋅= +=⋅ += +⋅=⋅ ⋅⋅ ⋅ ⋅
81. 81. ¨  Actually,  they  [power  laws]  aren’t  special  at  all.  They  can  arise  as  natural  consequences  of   aggrega-on  of  high  variance  data.  You  know  from  sta-s-cs  that  the  Central  Limit  Theorem   says  distribu-ons  of  data  with  limited  variability  tend  to  follow  the  Normal  (bell-­‐shaped,  or   Gaussian)  curve.  There  is  a  less  well-­‐known  version  of  the  theorem  that  shows  aggrega-on   of  high  (or  inﬁnite)  variance  data  leads  to  power  laws.  Thus,  the  bell  curve  is  normal  for  low-­‐ variance  data  and  the  power  law  curve  is  normal  for  high-­‐variance  data.  In  many  cases,  I   don’t  think  anything  deeper  than  that  is  going  on.   81