Monte Carlo Simulation

1
Monte Carlo Simulation

• ??????? ?????????????
• ?????????????????????????? ?????????????????????
• 10 ??????? 2550

2
?????????

• ??????????????????????????????????????????????????
  ?????????????? ?????????????????????????????
• ??????????????????????????? closed-form formula

3
????????? (2)

• Monte Carlo Simulation ?????????????????????????
  ?????????
• ????????????????? ????????????????????????????????
  ?????????
• ??????????????????????????????????????????????????
  ?
• ???????????????????????????????????

4
Monte Carlo Simulation

• ??????????????????????????????? ?? real world ???
• ???????????????????????????
• ???????????????????????????????
  ????????????????????

5
Random Walk Process

• ?????????? stochastic process
• ???????????????????????????????????????????????
  (random variable) ????????????????????????????????
  ????? ???? ????
• ??????????????????????????????????????????????????
  ??????? ?????????? ??????????????????????????????
  ????????????????????????????

6
??????????????????????????????? x

• ????????????????? Probability Density Function f
  x ? ?????? T ??????????????????????????????????
  ??????????????? x
• ?????????????? x ??????????????????????? f x ?
  ???????
• ??????????????????????????????????????????????????
  ??????????????????????????????????????

7
?????????????????????????? x

• ???????? standard deviation ?????????????????
  ??????????? error ????????????????????????????
  ???? d
• ???????????????????????? ?????? S ??? ???
  standard deviation ???????????????????????????????
  ? N ??? ?????????????????
• ??????????????????????????????????????????????????
  ????? mean ???????????????????????????????????????
  ?????

8
Law of Large Number

• ??? x1, x2,, xN ????????????????????????? i.i.d.
  ????? mean ??????? ? ?????? variance ??????? s2
  ????????? e ??????????????? 0

9
Central Limit Theorem

• ??? x1, x2,, xN ????????????????????????? i.i.d.
  ????? mean ??????? ? ?????? variance ??????? s2
  ????? N ??????????????????
• ?????? ??? cumulative distribution
  function ??? standard normal distribution ? ??? z

10
????????

• ???????? ?????????????????? x ?????? Pb measure
  P

N x
1 -0.30
2 -1.28
3 0.24
4 1.28
5 1.20
11
????????

• ????????
• ???????????? g(x) ?????? Pb measure P

N x g(x)
1 -0.30 0.00
2 -1.28 0.00
3 0.24 0.00
4 1.28 0.28
5 1.20 0.20
12
????????????????????? Monte Carlo Simulation
???????????????????????????

• ??????????????????? AT ?????????????? A ????
  ???????? ?????????????? ST
• (1) ???????????????? f ST T ?????? Pb
  Measures Q
• (2) ?????????????????????????????????????????????
  ?? AT ??????? ??????? ST
• (3) ????????????????? ST,n ??? n 1 2 N
• ??????????????????? N ? 8
• (4) ???????????? AT,n g(ST,n) ??? n 1 2 N
• (5) ?? ????????? ??? g (ST) ????? AT EQg
  (ST) ???? ERNg (ST)

13
Real World VS Risk Neutral World

• Real world (Pb measure P)
• ??????????????????????????????????????????????????
  ???????????
• Risk neutral world (Martingale Pb measure Q)
• ??????????????????????????????????????????????????
  ???? risk preference ??????????
• ???????????????????????????????????
• ????? ?????? Pb measure P ???? Pb measure Q
  ??????????

14
??????? Pb Measure

• ??????????????????????????? ST ?? Real World ???
  Pb Measure P
• ??????? Geometric Brownian Motion ????????
• dSt S µ dt s S dzP
• ????????????????????????????????????????????? r
  ?????????? dt
• ???????????????????? Martingale Pb Measure Q
  ??????????????????????????????? ST ???????
  ??????????????????? PB Measure Q
• ?????? The Girsanov Theorem
• dSt S r dt s S dzQ

15
???????????????? f ST TQ

• ???????????????? ST ?? continuous time dSt S
  r dt s S dzQ dzQ Normal (0.00, dt) ??????
  dt ? 0.00
• ??? Itos Lemma Theory d ln St (r - s2/2) dt
  s dzQ

16
?????????????? Euler Method

• Discrete Time Approximation
• ???????? dt 1 ??? dzQ e Normal
  (0.00,1.00)

17
??????????????????????????

• 1. ?????????????????? S(T) ????? n N ?????
• ???????? n 1
• ? ???????????? t 0 ?????????????????????? S(0)
• dt 1 ??? ??????????????? T ???
• ???? v ???????? 1 ??????????????????????????????
  N(0.00, 1.00)
• S1(T) S(0) exp(r - s2/2) T s sqrt(T) v1
• ???????? n N
• SN(T) S(0) exp(r - s2/2) T s sqrt(T) vN

18
?????????????????????????? (2)

• 2. ???????????????????????? ? ????????????????????
  ????????????????? N ????? ?????????????? ?
  ????????
• AT,n g (ST,n)
• A0,n g(ST,n) exp-rT
• 3. ????????????????????????????

19
?????????????????????????????????????????????

• ???????????????????????? Monte Carlo Simulation
  ??????????????????????????? 3 ???
• ??????????????????????? S0 ? ???????????
• ????????????????????????????????? r
• ?????????? s ??????? Standard Deviation ??? rS
  ?????? Measure Q
• ???????????????????????????????????????? ????
  ??????????????????????????? ThaiBMA

20
????????????????????? s

• ?????????? s ??? Standard Deviation ??? rS ??????
  Measure Q
• ?????????? s ??? rS ?????? Martingale Measure Q
• ??? Real-World, Objective Measure P ???????????
• ?????????????????????? s ??? rS
• ??? ???????????????? ??? rS ?????????????????????
  ????????????Standard Deviation
• ?????????????????????.xls

21
????????

• European Call Option ?? SET50 Index ?????? T 1
  ??
• SET500 100.00 ??? ????????? Selling at the
  Money ????? X 100 ???
• ?????????? r 0.25 ????????
• ????????????? s 6.14 ????????
• ???????????? SET50T,n ??? CT,n ??????????????????
  n ???

22
???????? (2)

• ??????????????? CT,n ????????????????????? ???????

• ??????? ???????? C0,n ?????? n 1 2 N ??? N
  Large ??????????
• ?????????????? Monte Carlo Price

23
???????????????? SET50T,n
24
???????????? CT
Distribution of Possible Call Value
25
?????????

• ???????????? ST ??????????????? Geometric
  Brownian Motion
• dSt St r dt s St dzQ
• ???????????????????????? Black-Scholes Option
  Pricing Model
• ??????? ????????????? ????????????? Black-Scholes
  Price
• ??????? ??? Monte Carlo Method ???????????????????
  ???????
• Monte Carlo Price Black-Scholes Price

26
??????????????

• ??????????????? ???????????????????????????????
  Monte Carlo Method.xls

27
??????????? Black-Scholes Model

• BSOPM ????????????? ST ??????????????? Geometric
  Brownian Motion
• dSt St r dt s St dzQ
• ??????????????????? ?????? r s ??????????
• ????? ????????? 2550 ?????????????????????????????
  ??????????????????????????????????????????????????
  ??? Geometric Brownian Motion

28
GARCH(1,1)

• Duan(1995) ???????????????????????????????????????
  ????? GARCH (1,1) ????????????????????????????????
  ????
• ?????? b0 gt 0 ???? b1 0 ??? b2 0 ???l
  ??????????????????????

29
GARCH(1,1) ?? Risk Neutral World

• ??????????????????????????????????????????????????
  ?????????? ???? Duan ?????????????? Locally
  risk-neutral world ??????????????????????????????
  ??????????
• ?????? ?????????????????????? ??????????????

30
????????????????????????????????????????????????

• ??????????????????? AT ?????????????? A ????
  ???????? ?????????????? ST
• (1) ???????????????? f ST T ?????? Pb
  Measures Q
• (2) ?????????????????????????????????????????????
  ?? AT ??????? ??????? ST
• (3) ????????????????? ST,n ??? n 1 2 N
• ??????????????????? N ? 8
• (4) ???????????? AT,n g(ST,n) ??? n 1 2 N
• (5) ?? ????????? ??? g (ST) ????? AT EQg
  (ST) ???? ERNg (ST)

31
????????????????????? GARCH(1,1) ??? Monte Carlo
Simulation ??????????????????????????

• 1. ??????????????????? ?t-1 ??? ht-1
• 2. ?????????????????? S(T) ????? n N ?????
• ???????? n 1
• ? ???????????? t 0 ?????????????????????? S(0)
• ?t 1 ??? ??????????????? T ???
• ???? ? ???????? 1 ??????????????????????????????
  N(0.00, ht)
• r1(T) rf 0.5hT ?1
• S1(T) S(0) expr1(T)
• ???????? n N
• SN(T) S(0) exprN(T)

32
????????????????????? GARCH(1,1) ??? Monte Carlo
Simulation ?????????????????????????? (2)

• 3. ???????????????????????? ? ????????????????????
  ????????????????? N ????? ?????????????? ?
  ????????
• AT,n g (ST,n)
• A0,n g(ST,n) exp-rT
• 4. ????????????????????????????

33
????????

• European Call Option ?? SET50 Index ?????? T 1
  ??
• SET500 100.00 ??? ????????? Selling at the
  Money ????? X 100 ???
• ?????????? r 0.25 ????????
• ???????????? SET50T,n ?????????????????? n ???

Sn(T) S(0) exprn (T)
34
???????? (2)

• ???????????? ??? CT,n ?????????????????? n ???
• ??????????????? CT,n ????????????????????? ???????

• ??????? ???????? C0,n ?????? n 1 2 N ??? N
  Large ??????????
• ?????????????? Monte Carlo Price

35
???????????????????? SET50T,n
36
???????????? CT
37
????????????
38
??????? ?n ??? ?n ??? SN (0.00,1.00)

• ???????????????????????????????? ?n ??? ?1 ??? SN
  (0.00,1.00)
• ???????????????????????????
• ????????????????? 3 ???
• ?????? function RAND( ) ??? NORMSINV( )?? EXCEL
• ?????? Random Number Generation ?? EXCEL
• ?????????????????????? _at_RISK

39
?????? function RAND( ) ??? NORMSINV( )?? EXCEL
N RAND( ) NORMSINV( )
1 0.350951 -0.38276
2 0.054902 -1.59907
3 0.404053 -0.24287
4 0.41467 -0.21555
5 0.681525 0.471969
6 0.471857 -0.0706
7 0.314412 -0.48338
8 0.055441 -1.59424
9 0.696701 0.514936
10 0.512691 0.031817
40
?????? Random Number Generation ?? EXCEL (1)
41
?????? Random Number Generation ?? EXCEL (2)
42
?????????????????? _at_RISK

• _at_RISK ?????????????????? Add-Ins
• ???????????????????????? Monte Carlo Analyses
  ????????????
• ????????????????? ???? ???? Download Trial
  Version ?????? FREE Trial Versions, Risk
  Analysis, Decision Analysis Software - Palisade
  Asia-Pacific

43
????????? _at_RISK (1)

• ???????????????? RISKNORMAL(0.00,1.00)
• ???????????????? ?n ??? SN (0.00,1.00)

44
????????? _at_RISK (2)

• ???????? ??? (Number of N Simulations) ??????????
  ??? ??????????????

45
??????????????????? Monte Carlo Simulation

• ????????????????????? ????????????????????????????
  ??????????????????????
• Small bias ??????????????? ???????????????????????
  ???????????????

46
References

• ????? ?????????. 2547. ???????????????????????????
  ???????? ??????????. ???????????????????????????.
• Duan, J.C.. 1995. The Garch Option Pricing Model.
  Mathematica Finance 5. pp. 13-32.
• Chan, N.G. and H.Y. Wong. 2006. Simulation
  Techniques in Financial Risk Management. Wiley.
  New Jersey.