Changes in Partnership Interest

Changes in Partnership Interest

Monte Carlo Simulation 10 2550 1 closed-form formula 2 (2) Monte Carlo Simulation

3 Monte Carlo Simulation real world 4 Random Walk Process stochastic process (random variable) random variable)

5 x Probability Density Function f {x| } x x f {x| } 2 1 x f x | , Exp 0.5 2

6 x standard deviation error d S d N S standard deviation N mean

7 Law of Large Number x1, x2,, xN i.i.d. mean variance Pb x N2 0 as N 0 8 Central Limit Theorem x1, x2,, xN i.i.d. mean variance 2 N ( xN ) Pb N z ( z )

(z ) cumulative distribution function standard normal distribution z 9 N 1 x 0. 3 0 1. 2 x

Pb measure 1 NP P E x Lim xn N N n 1 1 E x 0.3 1.28 0.24 1.28 1.20 0.23 5 P 10 N g(x x )

1 0 . 3 0.0 0 0 1 g ( x) max x 1,0 g(random variable) x) 1 Pb measure P E P g ( x) 0 0 0 0.28 0.2 0.10 5 11 Monte Carlo Simulation

AT A ST (random variable) 1) f {ST |} Pb Measures Q (random variable) 2) AT ST (random variable) 3) ST,n n = 1 2 N N (random variable) 4) AT,n = g(random variable) ST,n) n = 1 2 N (random variable) 5) g (random variable) ST) AT = EQ[g (random variable) ST)] ERN[g (random variable) ST)] 12 Real World VS Risk Neutral World Real world (random variable) Pb measure P) Risk neutral world (random variable) Martingale Pb measure Q) risk preference

Pb measure P Pb measure Q 13 Pb Measure ST Real World Pb Measure P Geometric Brownian Motion dSt = S dt + S dzP r dt Martingale Pb Measure Q ST PB Measure Q The Girsanov Theorem dSt = S r dt + S dzQ 14 f {ST |}Q ST

continuous time dSt = S r dt + S dzQ dzQ ~ Normal (random variable) 0.00, dt) dt 0.00 1 2Theory ~r Q T ~z Q Itos Lemma r T S T 2

d ln St = (random variable) r - 2/2) dt + dzQ T ~z Q d ~z Q ~ Normal 0.00, Tdt T T 0 15 Euler Method Discrete Time Approximation dt = 1 dzQ ~ Normal (random variable) 0.00,1.00) 2 T ~

Q ~ 1 u ST S0 exp rS T S0 exp r 2 u 1 2 T ~ T u ST S0 exp r 2 u 1 2 ~ ~ T T v ST S0 exp r 2

16 1. S(random variable) T) n = N n = 1 t = 0 S(random variable) 0) dt = 1 T v 1 N(random variable) 0.00, 1.00) S1(random variable) T) = S(random variable) 0) * exp{(random variable) r - 2/2) T + sqrt(random variable) T) v1} n = N SN(random variable) T) = S(random variable) 0) * exp{(random variable) r - 2/2) T + sqrt(random variable) T) vN} 17

(2) 2. N AT,n = g (random variable) ST,n) A0,n =N g(random variable) ST,n) * exp{-rT} A N1 An 3. n 1 18 ~ ~ T T v S S exp r 2

T 0 2 Monte Carlo Simulation 3 S0 r Standard Deviation rS Measure Q ThaiBMA 19

Standard Deviation rS Measure Q rS Martingale Measure Q Real-World, Objective Measure P rS rS Standard Deviation K 1 2 .xls r r SD K 1 S, k S

k 1 20 European Call Option SET50 Index T = 1 SET500 = 100.00 Selling at the Money X = 100 r = 0.25% = 6.14% 0 . 0614 C 12T,n S 100 exp 0.0025 - SET50

0.0614 12T,n v 2 n 2 T, n n C T, n Max ST, n 100,0.00 21 (2) CT,n C 0,n Exp - 0.0025 12 C12,n C0,n n = 1 2 N N = Large Monte Carlo Price

1 N Monte Carlo Pr ice C 0 C 0,n N n 1 22 SET50T,n Distribution for Possible S(T) under Measure Q/B17 0.020 Mean=103.0448 0.018 230 0.016 210 190 0.013 170 0.011

150 130 0.009 110 90 0.007 70 50 0.004 0.002 0.000 40 90 5%

140 90% 70.9911 190 240 5% 142.9254 23 CT Distribution of Possible Call Value Distribution for ST T ... 0.120 Mean=10.20538 0.100

0.080 0.060 0.040 0.020 0.000 0 35 90% 0 70 105 140 5%

42.9254 24 ST Geometric Brownian Motion dSt = St r dt + St dzQ Black-Scholes Option Pricing Model Black-Scholes Price Monte Carlo Method Monte Carlo Price = Black-Scholes Price 25 Monte Carlo Price Monte Carlo Method.xls

@RISK (1) Martingale Pricing Monte Carlo Method (2) Monte Carlo Method Errors . S0 100.0000 X 100.0000 T 12.0000 r 0.2500% STD = SMA 6.1400% z ~ SN (0.00,1.00) z ~ SN (0.00,1.00) n 0.0000 n T rS 0.7380% ST T n

100.7408 ST T n 0.7408 C0 n 0.7189 9.9038 Monte Carlo Call-Option Price 9.9045 Black-Scholes Call-Option Price -0.0072% 26 BlackScholes Model BSOPM ST Geometric Brownian Motion dSt = St r dt + St dzQ r 2550 Geometric Brownian Motion

27 GARCH(1,1) Duan(random variable) 1995) GARCH (random variable) 1,1) 1 ~ r h h t t t t 2 ht

0 1 2t -1 2 h t -1 0 > 0 1 0 2 0 28 GARCH(1,1) Risk Neutral World Duan Locally risk-neutral world 1 rt rf h t ~t 2 ht 0 1 t 1 h t 1

2 2 h t -1 ~t ~ N 0.00, ht 29 A A T ST (random variable) 1) f {ST |} Pb Measures Q (random variable) 2) AT ST (random variable) 3) ST,n n = 1 2 N

N (random variable) 4) AT,n = g(random variable) ST,n) n = 1 2 N (random variable) 5) g (random variable) ST) AT = EQ[g (random variable) ST)] ERN[g (random variable) ST)] 30 Monte Carlo Simulation 1. t-1 ht-1 2. S(random variable) T) n = N n = 1 t = 0 S(random variable) 0) t = 1 T 1 N(random variable) 0.00, ht) r1(random variable) T) = rf 0.5hT + 1 S1(random variable) T) = S(random variable) 0) * exp{r1(random variable) T)} n = N SN(random variable) T) = S(random variable) 0) * exp{rN(random variable) T)} 31

Monte Carlo Simulation (2) 3. N AT,n = g (random variable) ST,n) A0,n =N g(random variable) ST,n) * exp{-rT} A N1 An 4. n 1 32 European Call Option SET50 Index T = 1 SET500 = 100.00 Selling at the Money X = 100 r =

1 0.25% ~ r (T) r h 2 SET50T,n 2 0.00357 0 . 0338 0 . 1568 h T n T 1 0.000h T-1 1 n

hT f T T Sn(T) = S(0) * exp{rn (T)} 33 (2) CT,n ST, C T, n Max n n 100,0.00 C C Exp - 0.0025C 12 T,n 0, n

12,n C0,n n = 1 2 N N = Large Monte Carlo Price 1 N Monte Carlo Pr ice C 0 C 0,n N n 1 34 SET50T,n Distribution for Possible SET50 Index Value/F24 0.020 Mean=102.9994 0.018 180 160 0.016

140 0.013 120 0.011 100 80 0.009 60 0.007 40 0.004 0.002 0.000 40

100 5% 71.3532 160 90% 220 5% 143.2377 35 CT Distribution for Possible Call Option Value/J24 0.120 0.100 Mean=9.675842 0.080

0.060 0.040 0.020 0.000 0 40 90% 0 80 120 5% 41.9598 36 GARCH (1,1)

0.250000% 0.157005 Lambda 0.003570 a0 0.033738 a1 0.000030 b1 SET 50 ( t = 0 ) 100.00 100.00 ( ) 1 2 3 4 5 6 7

8 9 10 11 12 . SET 50 GARCH (1,1) Duan (1995) add-in @RISK Website http://www.palisade.com.au/risk/ 1 Simulation @RISK B E GARCH (1,1) Option Pricing

2.510973 3.632061 4.523951 5.294084 5.988380 6.596101 7.228683 7.795588 8.345522 8.810840 9.239307 9.675843 0.528550 0.544274 0.555895 0.560836 0.561968 0.575067 0.576387

0.580210 0.584455 0.585419 0.590018 0.593432 2.261416 3.135030 3.775333 4.297729 4.748265 5.117125 5.498843 5.821002 6.116532 6.348011 6.561133 6.765105 -0.471450 -0.455726 -0.444105 -0.439164 -0.438032 -0.424933

-0.423613 -0.419790 -0.415545 -0.414581 -0.409982 -0.406568 37 n n SN (0.00,1.00) n 1 SN (random variable) 0.00,1.00) 3 function RAND( ) NORMSINV( ) EXCEL Random Number Generation EXCEL @RISKRISK 38 function RAND( ) NORMSINV( ) EXCEL N

1 2 3 4 5 6 7 8 9 10 RAND( ) 0.350951 0.054902 0.404053 0.41467 0.681525 0.471857 0.314412 0.055441 0.696701 0.512691 NORMSINV( )

-0.38276 -1.59907 -0.24287 -0.21555 0.471969 -0.0706 -0.48338 -1.59424 0.514936 0.031817 39 Random Number Generation EXCEL (1) 40 Random Number Generation EXCEL (2) 41 @RISKRISK

@RISK Add-Ins Monte Carlo Analyses Download Trial Version FREE Trial Versions, Risk Analysis, Decision Analysis Software - Palisade Asia-Pacific 42 @RISKRISK (1) = RISKNORMAL(random variable) 0.00,1.00) n SN (random variable) 0.00,1.00) 43 @RISKRISK (2) (random variable) Number of N Simulations) 44 Monte Carlo

Simulation Small bias 45 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. 46

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