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by: Kwan

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# Managing Financial Risk Week VI Notes BU.230.730.53.SP16

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Kwan
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Options Management
COURSE
Managing Financial Risk
PROF.
Nicola Fusari
TYPE
Class Notes
PAGES
9
WORDS
KARMA
25 ?

## Popular in Finance

This 9 page Class Notes was uploaded by Kwan on Thursday April 28, 2016. The Class Notes belongs to BU.230.730.53.SP16 at Johns Hopkins University taught by Nicola Fusari in Spring 2016. Since its upload, it has received 37 views. For similar materials see Managing Financial Risk in Finance at Johns Hopkins University.

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Date Created: 04/28/16
Risk  VI Tuesday,  April  26,  2016 13:37 •1.  Assignment <-­‐-­‐ How  to  prepare  Final &  Formulas:  cheat  sheet  (posted) A  conference:  The  role  of  derivatives  in  asset  pricing:  June  4,  2016  (Baltimore,   MD) A  dinner Audio  recording  started:  13:40  Wednesday,  April  27,  2016 Assignment 1) Rt  ~  N  (miu,  sigma)  -­‐-­‐ average  Rt;  stdev  Rt Unconditional  (the  same  sigma):   (Rt-­‐miu)/sigma  ~N  (0,1) Sorted  return:  =small Normal  Quantile:  =quantile  (array,  1/3520) Compare:  plot  (tails  are  not  normal  distribution) Conditional Rt/sigma-­‐t    =  zt Rt  ~  N  (0,  sigma-­‐t) Is  z  normal  distribution Sigma-­‐t:  ghost  (RM  <  GARCH  <  RV:  normal) If  RM/GARCH  ,  compensate:  zt  -­‐-­‐ fat  tail,  t-­‐student 2) d? Method  of  Moments (4th  moment) If  RM/GARCH  ,  compensate:  zt  -­‐-­‐ fat  tail,  t-­‐student 2) d? Method  of  Moments (4th  moment) Data                                                                  Model Ex.  Kurt  (zs)                                        Ex.  Kurt=6/(d -­‐4) Ri/sigma-­‐i  =  zi =kurt(array) Compare:  if  kurt  =  0,  normal  (d  -­‐-­‐>  infinite) T.INV  (p,  d)                                           TINV  (p,  d)   t-­‐student:  one  tail                  Two  tails Negative                                                    Positive                                                                                     =-­‐TINV(2*p,  d) FHS:  (only  t+1) -­‐2:  possibility  1/4 VaR=  -­‐2*sigma_t+1 -­‐2  =  percentile  (zs,  0.25)  or  =  percentile  (array,  0.01) 4) Which  is  better? HS,  Normal  Distribution,  t-­‐student,  FHS How  many  times  actual  loss  larger  than  VaR  (violations): =if  [It  =  1  if  loss  >=  VaR,  else    0:  ∏(1-­‐tao)^(1-­‐It)*tau^It] L(^tau)=(1-­‐^tao)^T0*^tao^T1 -­‐2  =  percentile  (zs,  0.25)  or  =  percentile  (array,  0.01) 4) Which  is  better? HS,  Normal  Distribution,  t-­‐student,  FHS How  many  times  actual  loss  larger  than  VaR  (violations): =if  [It  =  1  if  loss  >=  VaR,  else    0:  ∏(1-­‐tao)^(1-­‐It)*tau^It] L(^tau)=(1-­‐^tao)^T0*^tao^T1 L(p)=(1-­‐p)^T0*^p^T1 L(p)=<  L(^tau);  as  ^tao  the  maximum 2 -­‐2*ln(L(p)/L(^tau))>=0  ~  ???? P=0.01,  total=754 tau  compared  with  p 5) Monte  Carlo  Simulation: 1st  Matrix:  random  numbers  from  normal  distribution Multiply Compute  average -­‐-­‐>  Crystal  ball 1st  Matrix:  random  numbers  from  normal  distribution Multiply Compute  average -­‐-­‐>  Crystal  ball FHS:  only  z  differs Only  1st  Matrix  changes:  =  index(array  of  past  zs,  Ceil(Rand()*500):  #  1~ 500) •2.  Options Risk  Management Lec 6 1) Black-­‐Scholes :  50  years Assumptions:  ……sigma  as  constant…. Can  replicate  any  portfolio  in  the  future:  market  is  complete miu  =  Rf e Rt+T=e ln(St+T/St) E(X):  expected  value  (continuous,  like  ∑) If  Rt+1  ~  t-­‐student Sigma_t+1   -­‐-­‐>  GARCH -­‐-­‐>  Use  Monte  Carlo  Simulation 2) Implied  volatility: C  =  BS  (St,  K,  T,  rf,  d,  sigma)  &  put-­‐call  parity  for  Put <-­‐-­‐ But  market:  "-­‐-­‐>"  as  Demand  &  supply  \$;  thus  sigma Eg.  25=BS(1000,1000,1m,0.05,2%,  ?);  sigma  =  0.15  -­‐-­‐>  cheap,  sigma  =  0.7  -­‐-­‐>   expensive If  BS  true  -­‐-­‐>  Implied  Volatility  Surface Market  changed: Oct26,  1987:  S&P  sharp  down,  no  longer  a  level  plane  -­‐-­‐>  put  option,  higher   price,  more  expensive More  options Possible  model  extensions t  (today):  buy  call  oppion  Ct  =(St,  K,  t,  sigma..)  -­‐-­‐>  t+1  Ct+1  =(St+1,  K,  T,  sigma) Risk:  Ct+1  -­‐ Ct   t+1‐-(percentile):  unknown Rt+1~  N(0,  sigma):  known Ct+1=f(Rt+1)  ? t  (today):  buy  call  option  Ct  =(St,  K,  t,  sigma..)  -­‐-­‐>  t+1  Ct+1  =(St+1,  K,  T,  sigma) p Risk:  Ct+1  -­‐ Ct   t+1‐-(percentile):  unknown Rt+1~  N(0,  sigma):  known Ct+1=f(Rt+1)  ? 3) Greeks Delta: c(St+1)  =  c(St)  +  δ8 \*(St+1  -­‐St)  *St/St (St+1  -­‐ St)  /St  =  rt+1  (simple  return)   ≈ Rt Ct+1  -­‐Ct  ≈ δ*Rt+1*St NB.  X~N(0,δ)  -­‐-­‐>  aX  ~  N(0,  a δ ) VaR:  approximation  (not  good)  :  where  differences  are  large  is  used Gamma:  Taylor  expansion Better  approximation With  uncertain  chi  distributio-­‐-­‐  > 4) Full  Valuation  (Monte  Carlo  Simulation): BS:  easy  cuz Ct  is  easy t  -­‐-­‐>  t+T  (sigma  constant) Ct  is  easy t  -­‐-­‐>  t+T  (sigma  constant) (Final  -­‐-­‐easier) 5) Excel: Options:  365 Volatility:  252 -­‐-­‐>  change  sigma  to  calendar  day Delta=N(d1) Delta-­‐Approx:   Full  Valuation  (Monte  Carlo): Visual  Basic:  to  define  BS  formula -­‐-­‐Trajectories -­‐-­‐Percentile [book]

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