8 4 2005 8 JOURNAL OF MANAGEMENT SCIENCES IN CHINA Vol 8 No 4 Aug 2005 1 2 3 (1 110004 ; 2 300222 ; 3 310012) : ; : ; ; ; ; : C931 2 : A : 1007-9807(2005) 04-0068 - 06 0 ; [3 4 ] ; ; [5 ] [1 2 ] ; : 2003-06 - 03 ; : 2005-06 - 06 : (1964 )
4 : 69 1 1 1 1 2 g n [8 ] n h ( z) F( z) D n D H ( z) E ( ) h ( z) F( z) z D D [6 ] : g F( z) : D D sup H ( z) < 1 E[ G( Z) I D ] = E [ G( Z) e - z + (1/ 2) z D ID ] (1) D Z N ( 0 I) I n 0 D n Z h ( z) n + 1 = g F ( n ) ( Z + ) g = h 0 ( z) E [ G( Z) e - z + (1/ 2) ID ] = E[ G( Z + ) e - Z + (1/ 2) ID ( Z + ) ] (2) (2) G( Z + ) e - Z + (1/ 2) ID ( Z + ) min E [ G( Z) 2 e - 2 Z + ID ] F ( z) = ln G ( z) gg( ) / G( ) = [7 ] i G( ) G( ) = G( i ) + g G( i ) ( - i ) min max [2 F( z) - z + 1 Z D 2 1-2 z z ] (3) g G( ) g G( i ) D Z g G( i ) i + 1 = (5) G( G ( z) D i ) + g G( i ) ( i + 1 - i ) = ( D i + 1 ) = 1/ [ G ( i ) + g G( D 0 i ) ( - i ) ] i + 1 = g G( i ) 2 g F( z) - - z = 0 (5) g F( ) = g G( i ) 2 2 + ( G( i ) - g G( i ) i ) - 1 = 0 Taloy F( ) + g F( e ) Z + O ( Z Z) - ID = e F( ) - (1/ 2) i i + 1 e O ( Z Z) I D (4) F ( z)
70 2005 8 U0 i [0 1 ] 2 u F ( z) Z i = X i + C Y i Y i N ( 0 I n ) X i C C C = I n - C u I n - F ( z) 2 1 2 U = ( 1 2 k ) i n Z 1 k n U U = I k F ( z) [1 ] U Z N ( 0 I k ) ( Z U Z = a) N ( U a I - UU ) k 2 1 1 2 k Taylor N = m k k [0 1) m ID = e F( ) - (1/ 2) [0 1) K N F( + Z) - Z - (1/ 2) e (1/ 2) Z e I D (6) H( ) Duffie [9 ] ( Z Z = a ) N ( a I n - ) ( i - 1) / N P( Z B i ) i/ N V i [ ( i - 1) / N i/ N ] X i = - 1 ( V i ) X i Z B i - 1 F( z) - 1 ( ) 2 1 1 n n Z n m k Z Z N R N (Latin Hypercube Sampling) k B 1 B 2 B N X i Z B i Z 2 2 Z N C = E( f ( Z) ) X i i = 1 2 N E(f 2 ( Z) ) < Z Z 1 Z 2 Z n Z < [10] X i i = 1 2 N min var ( f ( Z) Z = x) <( x) d x (8) = 1 B i ( i - 1) / N i/ N F( z) V i = i - 1 f ( Z) = exp 1 N + Ui 0 2 Z H ( ) Z (9) ( i = 1 2 N ) (7) N H( ) f ( Z) Z =
4 : 71 = kb F ( Z) bb gf ( b ) (1/ 2) Z min E[Var (e H ( ) Z Z) ] (10) = 1 f ( b Z) ( b R n ) [10 ] : v H( ) n 1 2 v n v = / 1/ 2 v i i 3 j H ( ) = v 3 j F( Z) u = v 3 j F( Z) ( 1/ 2) Forunie [11 ] CIR b F ( Z) = 3 2 3 MC F ( Z) IMC H ( ) SIMC H ( ) v 1 EIMC SIMC SIMC2CV 3 1 F( Z) = f ( b Z) ( b R n ) Benedecte A Jean2Paul [12 ] G( z) = e - rt max( S - K0) r e - rt Taylor S 0 S 1 S 2 S N S i F ( Z) i S i t t = T/ N : g F( Z) = b gf ( b Z) gf ( ) S S = (1/ N) N S i i = 1 b gf ( b ) = = ( z1 3 z2 3
72 2005 8 z 3 N ) H ( ) t 2 3 S i z 3 i = j j = j = 1 2 N (11) NG( ) z 3 j + 1 = z 3 j - j (12) [13 NG( z) 3 2 z1 3 = S 0 = 60 G( ) T = 1 r = 0 05 ; 0 15 ( r - (1/ 2) s j = S j - 1 e ) t + tz 3 j j = 1 2 N - 1 0 25 ; K 40 55 60 ; N = 16 m = 50 M = 1 000 F ( z) = ( z1 3 100 z2 3 z 3 N ) H ( ) 1 F ( z) = ln (max ( S - K 1 a b 1 0) ) S > K z 000 SIMC2CV EIMC 100 1 Table 1 The ratios of estimated standard deviation for the simulated model a b IMC SIMC SIMC2CV EIMC 0 15 0 25 45 55 60 45 55 60 6 000 1 902 0 215 7 101 4 093 2 054 5 998 1 911 0 215 7 100 4 091 2 057 3 2 1 2 5 4 13 2 0 2 5 7 CV 3 2 2 SIMC EIMC SIMC EIMC SIMC EIMC S 0 = 60 K = 55 = 0 25 T = 1 0 05 n = 16 1 5 SIMC2CV EIMC = (0 22 0 207 0 194 0 182 0 169 SIMC2CV EIMC = (0 23 0 217 0 204 0 191 0 178 6 7 23 71 12 15 20 7 3 30 7 98 0 14 7 19 2 25 7 7 8 34 102 17 6 21 4 27 0 SIMC2 0 027 0 205)
4 : 73 0 036 0 013) ; EIMC SIMC SIMC SIMC2 CV EIMC 4 (2) (3) H ( ) 1 : (1) : [1 ]Broadie P Glasserman P Monte Carlo methods for securities pricing[j ] Journal of Economic Dynamic and Control 1997 21 : 1263 1321 [2 ]Bratley P A Guide to Simulation[M] Berlin :Springer 1983 438 471 [3 ]Broadie M Detemple Recent advances in numerical methods for pricing derivative securities[ A ] Numerical Methods in Finance [M] Cambridge : Cambridge University Press 1997 170 229 [4 ]Boyle P A Monte Carlo approach[j ] Journal of Financial Economics 1977 4 : 328 338 [5 ]Owen A Safe and Effective Importance Sampling[ R] Technique Report Standford : Standford University Press 1998 210 242 [6 ]Veach E Guiba P Optimally Combining Sampling Technique for Monte Carlo Rendering[ C] Los Angeles : In SIGGRAPH 1995 Conference Proceeding 1995 172 210 [7 ] [M] : 2002 147 149 Ma Junhai Numerical Analysis Methods for Pricing Financial Derivative Securities[M] Hangzhou : People Press of Zhejiang 2002 147 149 (in Chinese) [8 ]Aiworth P Broadie M Glasserman P Monte Carlo and Quasi2Monte Carlo Methods for Scientific Computing [ M] New York : Springer2Verlag 1997 211 240 [9 ]Duffie D Efficient Monte Carlo simulation of security prices[j ] Annals of Applied Probability 1995 5 : 897 905 [ 10 ] Glasserman P Gaussian P Importance Sampling and Stratification : Computation Issuer[ C] Proceeding of the 1998 Winter Simu2 lation Conference New York : IEEE Press 1998 685 693 [11 ]Fournie E Lasry J Numerical Methods in Finance[M] Cambridge : Cambridge University Press 1999 321 370 [ 12 ]Benedecte A Jean2Paul D A PDE approach to Asian options : Analytical and numerical evidence[j ] Journal of Banking & Fi2 nance 1997 21 : 613 640 [13 ] Hull J White A The use of the control variate technique in option pricing[j ] Journal of Financial and Quantitative Analysis 1988 23(4) : 237 251 ( 79 )
4 : 79 [21 ]Schwartz A Bankruptcy workouts and debt contracts[j ] Journal of Law and Economics 1993 36 : 595 632 [22 ]Moody Default and recovery rates of corporate bond issuers : A statistical review of Moodyπs Ratings performance 1970 2001 [ ED/ OL ] http :/ / www moodys com/ 2002 [23 ]Lando D On Cox processes and credit2risky securities[j ] Review of Derivatives Research 1998 2 : 99 120 [24 ]Vasicek O An equilibrium characterization of the term structure[j ] Journal of Financial Economics 1977 5(2) : 177 188 [25 ]Cox J C Ingersoll J E Ross S A An intertemporal general equilibrium model of asset prices[j ] Econometrica 1985 53 : 363 384 [26 ]Cox J C Ingersoll J E Ross S A A theory of the term structure of interest rates[j ] Econometrica 1985 53 : 385 407 Study of credit risk term structure with stochastic default intensity LIANG Shi2dong GUO Bing FANG Zhao2ben Department of Stat & Finance University of Science & Technology of China Hefei 230026 China Abstract : The credit risk pricing model constructed in this paper belongs to the Intensity Model Category We con2 structed the frame model for term structure model of credit risk discussed the example of two-factors models and got the closed2form solution to the price of default2able bonds Finally we analyzed the pricing problem of credit risk derivatives Key words : credit risk ; term structure ; pricing ; intensity model ( 73 ) Comprehensive variance reduction techniques of Monte Carlo simulation methods for pricing options MA J un2hai ZHANG Wei LIU Feng2qin 1 Post2Doctor Work Station of Computer Science and Technology Northeastern University Shenyang 110004 China ; 2 Tianjin University of Finance and Economics Tianjin 300222 China ; 3 Zhejiang University of Finance and Economics Hangzhou 310012 China Abstract : In this paper combining the stronger ability to deal with some special financial derivative securities given by importance sampling technique and the characters of simple and flexible of mulit2control variable technique and optimum stratified sampling technique we will put forward some more effective comprehensive variance reduction techniques on Monte Carlo simulation method for pricing financial derivative securities by introducing control variable technique and optimum stratified sampling technique into the analysis framework of importance sampling technique At last we make some practical analysis by using an arithmetic Asian option Key words : options ; Monte Carlo simulation ; variance reduction technique ; importance sampling technique ; optimum stratified sampling technique