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Ὑπατος Κορομηλάς
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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) : ; : ;

1 JOURNAL OF MANAGEMENT SCIENCES IN CHINA Vol 8 No 4 Aug ( ; ; ) : ; : ; ; ; ; : C931 2 : A : (2005) ; [3 4 ] ; ; [5 ] [1 2 ] ; : ; : : (1964 )

2 4 : 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 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 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 ) ( 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)

3 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) 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 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 ( ) 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 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) = 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

5 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 ; K ; N = 16 m = 50 M = F ( z) = ( z z2 3 z 3 N ) H ( ) 1 F ( z) = ln (max ( S - K 1 a b 1 0) ) S > K z 000 SIMC2CV EIMC Table 1 The ratios of estimated standard deviation for the simulated model a b IMC SIMC SIMC2CV EIMC CV SIMC EIMC SIMC EIMC SIMC EIMC S 0 = 60 K = 55 = 0 25 T = n = SIMC2CV EIMC = ( SIMC2CV EIMC = ( SIMC )

6 4 : ) ; 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 : [2 ]Bratley P A Guide to Simulation[M] Berlin :Springer [3 ]Broadie M Detemple Recent advances in numerical methods for pricing derivative securities[ A ] Numerical Methods in Finance [M] Cambridge : Cambridge University Press [4 ]Boyle P A Monte Carlo approach[j ] Journal of Financial Economics : [5 ]Owen A Safe and Effective Importance Sampling[ R] Technique Report Standford : Standford University Press [6 ]Veach E Guiba P Optimally Combining Sampling Technique for Monte Carlo Rendering[ C] Los Angeles : In SIGGRAPH 1995 Conference Proceeding [7 ] [M] : Ma Junhai Numerical Analysis Methods for Pricing Financial Derivative Securities[M] Hangzhou : People Press of Zhejiang (in Chinese) [8 ]Aiworth P Broadie M Glasserman P Monte Carlo and Quasi2Monte Carlo Methods for Scientific Computing [ M] New York : Springer2Verlag [9 ]Duffie D Efficient Monte Carlo simulation of security prices[j ] Annals of Applied Probability : [ 10 ] Glasserman P Gaussian P Importance Sampling and Stratification : Computation Issuer[ C] Proceeding of the 1998 Winter Simu2 lation Conference New York : IEEE Press [11 ]Fournie E Lasry J Numerical Methods in Finance[M] Cambridge : Cambridge University Press [ 12 ]Benedecte A Jean2Paul D A PDE approach to Asian options : Analytical and numerical evidence[j ] Journal of Banking & Fi2 nance : [13 ] Hull J White A The use of the control variate technique in option pricing[j ] Journal of Financial and Quantitative Analysis (4) : ( 79 )

7 4 : 79 [21 ]Schwartz A Bankruptcy workouts and debt contracts[j ] Journal of Law and Economics : [22 ]Moody Default and recovery rates of corporate bond issuers : A statistical review of Moodyπs Ratings performance [ ED/ OL ] http :/ / www moodys com/ 2002 [23 ]Lando D On Cox processes and credit2risky securities[j ] Review of Derivatives Research : [24 ]Vasicek O An equilibrium characterization of the term structure[j ] Journal of Financial Economics (2) : [25 ]Cox J C Ingersoll J E Ross S A An intertemporal general equilibrium model of asset prices[j ] Econometrica : [26 ]Cox J C Ingersoll J E Ross S A A theory of the term structure of interest rates[j ] Econometrica : 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 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 China ; 2 Tianjin University of Finance and Economics Tianjin China ; 3 Zhejiang University of Finance and Economics Hangzhou 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

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