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Quantitative Financial Analysis Week III Notes

by: Kwan

Quantitative Financial Analysis Week III Notes BU.230.710.52.SP16

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Monte Carlo & Options Pricing
Quantitative Financial Analysis
Stuart Urban
Class Notes
25 ?




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This 5 page Class Notes was uploaded by Kwan on Friday April 1, 2016. The Class Notes belongs to BU.230.710.52.SP16 at Johns Hopkins University taught by Stuart Urban in Spring 2016. Since its upload, it has received 68 views. For similar materials see Quantitative Financial Analysis in Finance at Johns Hopkins University.


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Date Created: 04/01/16
Quant  III Friday,  April  1,  2016 09:03 1.HW2 Right  click  the  function,  open  the  function  file .*:  in  case  of  vectors Syntax:  [B,D]=… More  flexible:  more  variables Use  a  single  variable  to  store  the  repeated  calculation A  =  xlsread('xxx','Sheet1'):  read  all  data ID  =  A(:,1) Max  -­‐-­‐>  get  vectors 2.Monte  Carlo  Simulation   A  series  of  simulations Central  limit  theorem:  handy  -­‐-­‐>  what  distribution  (what  result)  not  im--t    can   calculate  confidence… B-­‐S  Model:  constant  interest  rate  and  constant  volatility  [not  true  in  reality] -­‐-­‐>  Monte  Carlo   Eg.  Estimating  Pi -­‐-­‐>as  a  function  file:  no  inputs  and  outputs  (shootingDart:  No  functions  in  Script) Formatting:   fprintf:  format  string  (&)  s:  strings;  d:  integer;  %10d\n:  new  line;  matrix:   transposed Eg.  Integration X  =  rand(N,1); I=  mean(cos(2*pi*X)); Shift  +  enter:  change  all  names Function  inside  a  script:  cannot  be  used  in  other  scripts MC_estimatePi  -­‐-­‐>  MC_estimateInteger  (not  a  function,  change  only  a  few  lines) Convergence Standard  error:  SEM Norinv(.975)  [5% -­‐-­‐>  2.5%] file:  MC_estimatePi_withError.m   3.BUILDING  A  STOCK  PRICE  MOVEMENT  PROCESS   Noise/random  shock/random  price  shock:   ???? ????:  only  variable,  ~N 252:  days  in  US  stock  market  per  year S(0)  →  index  1  in  matlab:  s(1)  =  40; Closse  all:  close  all pups-­‐‑ Matlab:  2  lines  (Monte  Carlo)-­‐‑  -­‐‑ milion Don't  use  MC_stockPrice.m  -­‐‑-­‐‑>Ito  (log  normal) Closse  all:  close  all pups-­‐‑ Matlab:  2  lines  (Monte  Carlo)-­‐‑  -­‐‑ milion Don't  use  MC_stockPrice.m  -­‐‑-­‐‑>Ito  (log  normal) Motivation:  our  stock  prices  don't  go  below  zero… 4.OPTIONS  PRICING   Risk-­‐‑Newtral  Valuation: No  arbitrage  oportuinies;  Miu=r MC_pricingEUCall.m   Drift  &  diffusion HW3: Don't  overthink:  change  to  PUT  /butterfly  function  (payoff)  &  change  nam-s>    Done! Cf.  Black-­‐‑Scholes  Pricing  Formula  (pricing) Normcdf:  d1,  d2 -­‐‑-­‐‑ Delta:  slope  of  Call  Price  vs  S0 Gamma:  slope  of  delta Graph  -­‐‑→  magnifying  around  42,  the  price  is  4.76 5.Variance  Reduction  Techniques Multiple  dimensions Random  -­‐‑→  Quasi-­‐‑random Skewed  -­‐‑→  even  out  (mean  is  0) 5.Variance  Reduction  Techniques Multiple  dimensions Random  -­‐‑→  Quasi-­‐‑random Skewed  -­‐‑→  even  out  (mean  is  0) 6.Asion  Options  &  tips Path/trajectory Mean  -­‐‑-­‐‑>  around  T=20,  40,  60,  80,  100 How  good  the  estimate  is--­‐>  standard  error  (4*std  error  =  range)  [1.96] Simulation:  don't  interact  with  each  other Min? Path-­‐dependent  option  or  not? More  steps  -­‐-­‐>  price:  random  walk  (expect  to  go  up) Payoff:  Lower  bound,  infinity;  With  K,  no  cancel  out  below  0 4  steps  per  day  (in  real  world) Increment  matrix: Adding  from  250  -­‐-­‐>251  (logS0+delta  s1  +  delta  s2:  the  first  column) LogPaths  =  cumsum([log(S0)*ones(NbTraj,1)  ,  Increments]  ,  2);  :-­‐ d  own,  2  -­‐ cross Cumulative  sum [Conclusion:  don't  touch  the  GenerePaths function]


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