ࡱ>  r5@ 0 bjbj22 5XXa&FFFFjjj~fSfSfS8SBU~JV.^"P^P^^7f.efyf $R>jkCe7fkk>FFP^^_{{{k FP^j^{k{{{:x,j^>V @WfS7u \Պ0 Iy~~FFFFjfdg{hifff>>~~/b5c{^~~b5 Application of the Moment Expansion Method to the Option Square Root Model Final Report for AMSC 663-664 Yun Zhou zhouyun@math.umd.edu Applied Mathematics and Scientific Computation University of Maryland, College Park, MD Advisor: Dr. Steven Heston  HYPERLINK "mailto:sheston@rhsmith.umd.edu" sheston@rhsmith.umd.edu Department of Finance, Robert H. Smith School of Business University of Maryland, College Park, MD May, 2009 Abstract The Options Square Root model or Heston model is the stochastic volatility model developed by Heston (1993). The governing equations consider not only the stochastic spot return but also the stochastic volatility, which has a correlation with the spot return. Heston (1993) gave a closed-form solution for the European Call option price based on the Fourier transform method. In this project we apply a moment expansion method to the Options Square Root model and compare the results with the closed form solution. The numerical results show good approximations up to the 4th order of moment. However, it also indicates divergence of the Gram-Charlier expansion. The convergence conditions for the Gram-Charlier expansion are discussed. 1 Introduction The Black-Scholes model has been widely and successfully used in explaining stock option prices. It is easy to calculate and explicitly model the relationships between all the variables. The Black-Scholes model assumes that stock returns are normally distributed and follow a constant variance diffusion process. However, empirical studies show that in reality, security prices do not follow a strict stationary log-normal process and the variances are stochastic. These facts make the stock return distribution skewned and kurtotic relative to a normal distribution. As a result, the Black-Scholes model misprices in-the-money and out-of-the-money options. Starting from this point, Hull and White (1987) propose a new stochastic volatility model. However, these types of models could not provide closed form solutions and involve more numerical techniques. Heston(1993) propose a new stochastic volatility model describing the evolution of volatility of the underlying asset with a closed-form solution. The basic Heston model assumes that St, the price of the asset at time t, is determined by a geometric Brownian motion:  EMBED Equation.DSMT4  (1) where t, the instantaneous variance, is a  HYPERLINK "http://en.wikipedia.org/wiki/CIR_process" \o "CIR process" CIR (Cox-Ingersoll-Ross) process:  EMBED Equation.DSMT4  (2)  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 are Wiener Processes with correlation .  is the average rate of return.  is the long volatility, or long run average volatility; as t goes to infinity, the expected value of t goes to .  is the mean reversion rate at which t reverts to .  EMBED Equation.DSMT4 is the volatility of the volatility. The Wiener Process Wt is characterized by three facts: W0 = 0 Wt is  HYPERLINK "http://en.wikipedia.org/wiki/Almost_surely" \o "Almost surely" almost surely continuous Wt has independent increments with distribution  EMBED Equation.DSMT4  ~ EMBED Equation.DSMT4  (for 0 d" s < t). N(, ) denotes a  HYPERLINK "http://en.wikipedia.org/wiki/Normal_distribution" \o "Normal distribution" normal distribution with  HYPERLINK "http://en.wikipedia.org/wiki/Expected_value" \o "Expected value" expected value  and  HYPERLINK "http://en.wikipedia.org/wiki/Variance" \o "Variance" variance  EMBED Equation.DSMT4 . If 0 d" s1 d" t1 d" s 2 d" t2, then Wt1"Ws1 and Wt2"Ws2 are independent random variables. The similar condition holds for n increments. An alternative characterization of the Wiener Process is an almost surely continuous  HYPERLINK "http://en.wikipedia.org/wiki/Martingale_(probability_theory)" \o "Martingale (probability theory)" martingale with W0 = 0 and  HYPERLINK "http://en.wikipedia.org/wiki/Quadratic_variation" \o "Quadratic variation" quadratic variation [Wt,Wt] = t (which means that Wt2-t is also a martingale). The CIR process is a  HYPERLINK "http://en.wikipedia.org/wiki/Markov_process" \o "Markov process" Markov process with continuous paths defined by the following  HYPERLINK "http://en.wikipedia.org/wiki/Stochastic_differential_equation" \o "Stochastic differential equation" stochastic differential equation:  EMBED Equation.DSMT4  where  and  EMBED Equation.DSMT4  are parameters. Value  EMBED Equation.DSMT4  follows a non-central Chi-Square distribution. The CIR process is widely used to model short term interest rates. There are some other ways to solve the Heston model, such as the Fast Fourier Transform (Carr and Madan, 1999) which needs the characteristic functions. In this project we use the moment expansion method to model the distribution of stock return driven by Heston model. Jarrow and Rudd (1982) use an Edgeworth expansion of the lognormal probability density function to model the distribution of the underlying asset and derive an option pricing formula. Corrado and Su (1996) use a Gram-Charlier series expansion of the normal probability density function to model the distribution of the underlying asset. This method captures the skewness and kurtosis deviations of the underlying asset distribution from a normal distribution. The option pricing formula is the sum of a Black-Scholes option price plus two approximate terms for non-normal skewness and kurtosis. The motivation for this project is to extend the moment expansion method to any order of moment in order to obtain highly accurate approximation of the asset return distributions. Thus, the approximate distributions depend on higher order of moment other than the first four orders. A cutoff can be determined to get a good enough truncated approximation. Moreover, this method can also be used to stochastic volatility models which do not have exact solutions or easily attainable characteristic functions. 2 Approach Let  EMBED Equation.DSMT4  be the spot return, according to equation (1), we have  EMBED Equation.DSMT4  For the log return distribution, the moment of order n is M. M is a function of the log return x, the initial variance  EMBED Equation.DSMT4 , the time to maturity  EMBED Equation.DSMT4  and n, the order of moment. We guess  EMBED Equation.DSMT4 , with initial conditions  EMBED Equation.DSMT4  where,  EMBED Equation.DSMT4 , the function of  EMBED Equation.DSMT4 , is the coefficient part of  EMBED Equation.DSMT4 . The moment M satisfies the Kolmogorov Backward equation:  EMBED Equation.DSMT4  (3). At the expiration time EMBED Equation.DSMT4 , EMBED Equation.DSMT4 . Considering n=1, then  EMBED Equation.DSMT4  (4) Substitute equation (4) into (3), we have  EMBED Equation.DSMT4  Then we get two ordinary differential equations,  EMBED Equation.DSMT4  (5)  (6) Solving equations (5) and (6) with the initial conditions, we get  EMBED Equation.DSMT4  Thus, the 1st moment is  EMBED Equation.DSMT4 . Similarly, using the backward equation (3) we generate a set of linear ordinary differential equations, from which moments can be obtained after solving these equations. There are two ways to solve these moments, one way is analytical. When n=1, one needs to solve for three coefficients and when n=4, one needs to solve for fifteen coefficients. The other way is numerical. I used a matrix exponential method which gives an exact solution. The set of ODEs can be written as  where matrix A is the coefficient matrix and y(t) is the unknown vector. Then the solution is (7) where b is the vector of initial value. For example, when n=1, the unknown vector  EMBED Equation.DSMT4  , coefficient matrix  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . According to equation (7), the unknown vector y(t) can be obtained. The results will be compared with the analytical ones in the validation section. After the moments are obtained, we need to normalize these raw moments into central moments with a zero mean and unit variance as follows:  EMBED Equation.DSMT4  where EMBED Equation.DSMT4  are raw moments,  EMBED Equation.DSMT4  are central moments,  EMBED Equation.DSMT4  is the mean value and  EMBED Equation.DSMT4 . Then we need to consider the way to implement moments into the Black-Scholes option pricing formula. European Call Option payoff at the expiration time T is the maximum of 0 and the stock price at the expiration time minus the strike price, i.e., EMBED Equation.DSMT4  at  EMBED Equation.DSMT4 . Thus, the call option price at the time to maturity  EMBED Equation.DSMT4  can be expressed as  EMBED Equation.DSMT4  (8) where n(z) is standard normal distribution,  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  is the volatility. We use the Gram-Charlier series expansion to seek an approximate distribution of the log return driven by the Heston model, g(z), then replace the normal distribution n(z) in equation (8) to get the approximate call option price. So we have:  EMBED Equation.DSMT4  (9) where  EMBED Equation.DSMT4 , the standard normal probability density function,  EMBED Equation.DSMT4  are the central moments of standard normal distribution of the order i,  EMBED Equation.DSMT4  are the Hermite polynomials,  EMBED Equation.DSMT4 . Then the call option price based on the Gram-Charlier expansion can be expressed as  EMBED Equation.DSMT4  (10)  EMBED Equation.DSMT4  is the call option price by the Black-Scholes model, and  EMBED Equation.DSMT4  are the coefficient parts which involve the integral of Hermite polynomials. Under a normal condition, we have  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , which makes these series approximation parts in equation (9) disappear. As a result, the approximate call option price is equal to the Black-Scholes option price in equation (10). The similar method has been used by Corrado and Su (1996). They use a truncated Gram-Charlier series expansion of the density function up to the 4th moment. The resulting truncated density gives an estimation of the non-normal skewness and kurtosis as follows:  EMBED Equation.DSMT4  (11) where EMBED Equation.DSMT4  are the 3rd and 4th moments respectively. Based on this truncated Gram-Charlier density expansion, they obtain the option pricing formula:  EMBED Equation.DSMT4  (12) where  is the Black-Scholes option pricing formula  EMBED Equation.DSMT4 . Using the computed central moments and the constant coefficient parts EMBED Equation.DSMT4 , the approximate call option price can be calculated. To determine the cutoff order of the moments, the Fourier transform solution is used as the truth in order to examine the relative error. Therefore, the moment expansion solutions can be used as an approximation of the Fourier transform solutions. Heston (1993) guess a solution to the Heston model, which involves two parts, one is the present value of the spot asset before optimal exercise, and the other is the present value of the strike-price payment. The solution has the following form:  EMBED Equation.DSMT4  Where EMBED Equation.DSMT4  is the price at time t of a unit discount bond that matures at time T. Both of these two terms satisfy equation (3).  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  satisfies the terminal condition,  EMBED Equation.DSMT4 . and have characteristic functions  EMBED Equation.DSMT4 respectively, which also satisfy equation (3).  EMBED Equation.DSMT4  can be obtained from the solutions of these characteristic functions, then the call option prices can be obtained. To check the accuracy, we compare the Fourier transform based solutions  EMBED Equation.DSMT4  with 1st to nth order moment expansion based solutions EMBED Equation.DSMT4 . The graph of  EMBED Equation.DSMT4  and n could help us to determine a cutoff and find a satisfactory estimation. 3 Validation 1) Moments The following table compares the numerical results from the matrix exponential method with the analytical results up to the 4th order. Only up to the 4th order of moment are obtained from the analytical method. The relative error indicates that results from the matrix exponential method give the exact solution as the analytical method. Parameters are in Table 2 Value(1). Order of MomentAnalyticalNumericalRelative Error11.09E-011.09E-01020.016820532270.01682053227030.00308719867910.0030871986791046.61E-046.61E-040Table 1. Compared Results on Moments 2) Reducing the Heston model to the Black-Scholes model The major difference between the Heston model and the Black-Scholes involve the nature of volatility. In the Heston model, it follows a stochastic dynamic motion while in the Black-Scholes it is a constant. For validation of the moment expansion method, we can set volatility in the Heston model as a constant and then compare the results with the Black-Scholes. In order to make volatility a constant, these two conditions should be satisfied: 1)  EMBED Equation.DSMT4 , 2)  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 . The computed option prices should be close to the Black-Scholes option prices. Fig. 1 shows the Root Means Square Error (RMSE) between the computed option price and the Black-Scholes up to the 20th order of moment used in a Gram-Charlier expansion. The RMSE is around 10-15. This implies that the moment expansion method is successfully implemented. ParametersValue (1)Value (2)Value (3)Mean reversion0.11.11.1Long-run variance0.010.010.011Initial variance0.010.010.013Correlation0.5-0.5-0.5Volatility of volatility 0.100.1Option maturity0.50.5 0.5Interest rate0.0230.0230.023Initial Stock Price707070-140Strike Price100100100Table 2. Parameters  Fig 1. Reduce the Heston model to the Black-Scholes 4 Numerical Results Fig. 2 gives the results on option pricing differences using the moment expansion method and the exact solution for the Heston model up to the 10th order of moment used in the Gram-Charlier expansion. Generally, we want the approximate results close to the exact solution which means, all the lines in the figure should be close to zero. When the order of moment increases in the Gram-Charlier expansion, the differences deviate from zero. When the order of moment is 10, the difference reaches the maximum. This figure indicates the divergence of the Gram-Charlier method in the Heston model.  Fig 2. Option pricing differences between Gram-Charlier and the exact solution of the Heston model Figure 3 shows the option price using the moment expansion method compared to the exact solution of the Heston model. The results imply the divergence of the Gram-Charlier method obviously. When the order of moment used in the Gram-Charlier expansion increases from 10th (red line) to 20th (black line), the approximate prices deviate from the truth.  Fig 3. Option pricing by the Gram-Charlier expansion and exact solution of the Heston model The RMSE on the log scale is represented in Fig.4. The figure gives the RMSE of the option pricing by the moment expansion method up to the 20th order of moment compared with the exact solution of the Heston model. RMSE is minimized when the order is 4, and then increases rapidly. For certain sets of parameters, up to the 4th order of moment used in the Gram-Charlier expansion might be a good approximation for the Heston model. However, the increasing RMSE also indicates that the method exhibits divergence for the Heston model. We need to consider the convergence conditions for the Gram-Charlier expansion method.  Fig 4. RMSE of Gram-Charlier on log scale 5 Discussion The poor convergence property of the Gram-Charlier expansion is mentioned by Cramer (1957). The source of divergence is the behavior of the probability density function at infinity. The probability density function must fall to zero faster than EMBED Equation.DSMT4 . This condition can be expressed as follows, which is called Cramers condition for convergence:  EMBED Equation.DSMT4  (13) Figure 5 and Figure 6 show examples of convergence and divergence when using the Gram-Charlier expansion to approximate distributions. In Fig. 5, the standard normal distribution  EMBED Equation.DSMT4  is used to approximate the normal distribution  EMBED Equation.DSMT4 . Cramers Condition of Convergence can be easily determined for EMBED Equation.DSMT4 , for which  EMBED Equation.DSMT4 satisfies the convergence conditions. As the order of moment increases, the approximate distribution approaches the desired distribution  EMBED Equation.DSMT4 .  Fig 5. Gram-Charlier Approximation for N(0,0.5)  Fig 6. Gram-Charlier Approximation for N(0,2) In Figure 6, the standard normal distribution is used to approximate the normal distribution  EMBED Equation.DSMT4 . Cramers Condition for Convergence does not hold in this case, as indicated by the divergence of the graph. These approximate distributions deviate from the desired distribution as the order of moment increases in the Gram-Charlier expansion. To determine the convergence condition for the Heston model, we need to analyze the probability density function of the asset return. The probability density function of Heston model has a complex structure. However, results (Dragulescu and Yakovenko, 2002) show that for large  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  is linear in x and quadratic for small  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  This property makes  EMBED Equation.DSMT4 , then Cramers Condition for Convergence fails to hold. This implies that using the Gram-Charlier method to approximate the Heston model gives divergence. 6 Conclusions and Future Work Moment expansion method is successfully applied to the Heston model. Numerical results show large divergence when using the Gram-Charlier expansion to approximate the distribution of asset returns in the Heston model. These approximate distributions deviate from the exact solution as the order of moment increases. Cramers Condition for Convergence for the Gram-Charlier expansion is discussed and two examples show the convergence and divergence cases when using the Gram-Charlier expansion to approximate the desired distributions. The probability density function of Heston model does not satisfy the Cramers Condition for Convergence. By comparing approximate results with the exact solution, we know that certain orders of moment may give a better approximation than the one given by the Black-Scholes. Another method, an Edgeworth expansion might also be a good choice for the Heston model since its convergence condition is not as strict as that of Gram-Charlier. Using the Edgeworth expansion instead of the Gram-Charlier expansion in the moment expansion method may be a worthy future endeavor. Other than the Heston model, the Gram-Charlier expansion can be applied to models which satisfy the Cramers Condition, such as the Binomial model. 7 References Blinnikov, S. and R. Moessner, 1998, Expansions for Nearly Gaussian Distribution, Astronomy & Astrophysics Supplement Series, 130, 193-205 Brown, D.M. Robinson, 2002, Skewness and kurtosis implied by option prices: a correction, Journal of Financial Research XXV, No.2, 279-282 Carr, P and D. Madan, 1999, Option Pricing and the Fast Fourier Transfer, Journal of Computational Finance, Vol. 2, No. 4, pp. 61-73 Corrado, 2007, The hidden martingale restriction in Gram-Charlier Option Prices, Journal of Futures Markets, 27, 6, 517-534 Corrado, C.J. and T. Su, 1996, Skewness and Kurtosis in S&P 500 index returns implied by option prices, Journal of Financial Research 19, 175-92. Cramer H., 1957, Mathematical Methods of Statistics. Princeton Univ. Press, Princeton Deagulescu, A and V. Yakovenko, 2002, Probability Distribution of Returns in the Heston Model with Stochastic Volatility, Quantitative Finance, Vol. 2, 443-453 Heston, 1993, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, The Review of Financial Studies 6, 2,327-343 Hull, J.C., and A. White, 1987, The Pricing of Options on Assets with Stochastic Volatilities, Journal of Finance, 42, 281-300 Jarrow, R. and A. Rudd, 1982, Approximate Option Valuation for Arbitrary Stochastic Process, Journal of Financial Economics, 10, 347-69 Jondeau, E. and M. Rockinger, 2001, Gram-Charlier Densities, Journal of Economic Dynamics & Control, 25, 1457-1483 Jurczenko, E., B. Maillet and B. Negrea, 2004, A note on skewness and kurtosis adjusted option pricing models under the Martingale restriction, Quantitative Finance, Vol. 4, 479-488 Shreve, 2004, Stochastic Calculus for Finance II continuous-time model Vahamaa, S., 2003, Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance, Finance Letter, 1(5), 6-12 8 Appendix 1) Derivation of Moments Starting from the Backward Equation:  EMBED Equation.DSMT4  (3) we can generate a group of linear ordinary differential equations as following,  (14) with initial conditions  EMBED Equation.DSMT4 . One method is to solve these ODEs recursively, the other one is to use matrix exponential. These ODEs can be written as where  EMBED Equation.DSMT4  is the unknown vector and A is coefficient matrix. 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In Brown and Robinson (2002) paper, they changed the sign of Q3 part to negative in equation (16)*, thus, I got the same result with them but keep the sign of Q3 part positive. n=4 (17)=  EMBED Equation.DSMT4  Q4 part is  EMBED Equation.DSMT4  (4) which is the same as Corrados paper.     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FMathType 5.0 Equation MathType EFEquation.DSMT49qM|DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HObjInfoDEquation Native E_1303511355FWWOle JA@AHA*_D_E_E_A  C i,j FMathType 5.0 Equation MathType EFEquation.DSMT49qXDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePagCompObjKiObjInfoMEquation Native N_1294743952FWWes!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  Q i FMathType 5.0 Equation MathType EFEquation.DSMT49q#;DSMT5WinAllBasicCodePagesOle SCompObjTiObjInfoVEquation Native WWTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  m==lnS 0 ++(r"-  s 2 2 )t,d 2 ==d 1 "-s t  ,Kn(d 2 )==S 0 e rt n(d 1 ) FMathType 5.0 Equation MathType EFEquation.DSMT49qDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_H_1294689240FWWOle aCompObjbiObjInfodEquation Native e_1294689395FWWOle pCompObjqiA@AHA*_D_E_E_A  C GC ==e 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EFEquation.DSMT49q#|DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  Y n (y,r)==qeEquation Native _1295727960;FWWOle CompObji "-  r22 ++rx n(x)H n (x)dx y" +" ==qe "-  r22 ++rx dn(x)H n"-1 (x) y" +" ==e "-  r22 ++rx n(x)H n"-1 (x)| y" "-qn(x)H n"-1 (x) y" +" de "-  r22 ++rx == 1 2p  e "-  12 (""-r) 2 H n"-1 (")"- 1 2p  e "-  12 (y"-r) 2 H n"-1 (y)"-qrn(x)H n"-1 (x) y" +" e "-  r22 ++rx dx=="-n(y"-r)H n"-1 (y)"-rY n"-1 (y,r) FMathType 5.0 Equation MathType EFEquation.DSMT49qO DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  YObjInfoEquation Native k _1294729689FWWOle  n ==q(e m++s t  z "-K)H n (z)n(z)dz "-d 2 " +" ==qe lnS 0 ++(r"-  s22 )t++s t  z H n (z)n(z)dz"-KqH n (z)n(z) "-d 2 " +" "-d 2 " +" dz==S 0 e rt ("-s t  )Y n"-1 ("-d 2 ,s t  )"-S 0 e rt n(d 1 )H n"-1 ("-d 2 )"-K[0"-n(d 2 )H n"-1 ("-d 2 )]==S 0 e rt ("-s t  )[("-s t  )Y n"-2 ("-d 2 ,s t  )"-n("-d 2 "-s t  )H n"-2 ("-d 2 )]==S 0 e rt ("-s t  ) 2 Y n"-2 ("-d 2 ,s t  )"-Kn(d 2 )H n"-2 ("-d 2 )("-s t  )......==S 0 e rt {"-p("-s t  ) jj==1n"-1 " H n"-1"-j ("-d 2 )n(d 1 )++("-s t  ) n N(d 1 )} FMathType 5.0 Equation MathTyCompObjiObjInfoEquation Native _1295076449FWWpe EFEquation.DSMT49q#´$DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  "-p("-s t  ) j H 3"-1"-j ("-d      !"#$%&),-./0123456789:;<=>?@ABCDEFGJMNOPQRSTUX[\]^_`cfghijklmnopqrstuvwxyz{|}~ 2 )n(d 1 )++("-s t  ) 3 N(d 1 ) j==13"-1 " =="-[("-s t  )H 3"-1"-1 ("-d 2 )++("-s t  ) 2 H 0 ("-d 2 )]n(d 1 )++("-s t  ) 3 N(d 1 )==s t  [(d 1 "-s t  "-s t  )n(d 1 )"-s 2 tN(d 1 )]=="-s t  [(2s t  "-d 1 )n(d 1 )++s 2 tN(d 1 )] FMathType 5.0 Equation MathType EFEquation.DSMT49qdXDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePagOle CompObjiObjInfoEquation Native es!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  "- 13!S 0 s t  [(2s t  "-d 1 )n(d 1 )++s 2 tN(d 1 )]_1295076543FWWOle CompObjiObjInfo FMathType 5.0 Equation MathType EFEquation.DSMT49qdXDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   13!S 0 s t Equation Native _1295076542FWWOle CompObji [(2s t  "-d 1 )n(d 1 )"-s 2 tN(d 1 )] FMathType 5.0 Equation MathType EFEquation.DSMT49qdDSMT5WinAllBasicCodePagesObjInfoEquation Native _1295727156FWWOle 'Times New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   13!S 0 s t  [(2s t  "-d 1 )n(d 1 )++s 2 tN(d 1 )]CompObj(iObjInfo*Equation Native +;_1295076574FWW FMathType 5.0 Equation MathType EFEquation.DSMT49qh@DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  "-p("-s t  ) j H 3"-1"-j ("-d 2 )n(d 1 )++("-s t  ) 4 N(d 1 ) j==14"-1 " =="-[("-s t  )H 4"-1"-1 ("-d 2 )++("-s t  ) 2 H 4"-1"-2 ("-d 2 )++("-s t  ) 3 H 0 ("-d 2 )]n(d 1 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S 0 s t  ((d 2 "-1"-3s t  (d"-s t  ))n(d)++s 3 t 3/2 N(d)) FMathType 5.0 Equation MathType EFEquation.DSMT49qOle CompObjiObjInfoOlePres000S L.  @&N: & MathType`-aK gOOb--$ \?hFbFbYYq^Wg- #"Times New Roman@wIw @wf- 2 0y{Times New Roman@wIw @wf- 2 ()()tl "Times New Roman@wIw @wf- 2 rtV#6"Times New Roman@wIw @wf- 2 BS.Times New Roman@wIw @wf-2 CSNdKeNdt] @Symbol @wIw @wf- 2 sS "Symbol` r@wIw @wf- 2 -SSymbol @wIw @wf- 2 =--r &|MathTypeUUpDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  C BS ==S 0 N(d)"-Ke "-rt N(d"-s t  ) & --"Systemf !-NANIQp|DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_EEquation Native _1303677827 F@W@WOle CompObj  i_A  C BS ==S 0 N(d)"-Ke "-rt N(d"-s t  ) FMathType 5.0 Equation MathType EFEquation.DSMT49qObjInfoOlePres000 Equation Native e_1303596078F@W@WDN S.  &N@ & MathTypeP `Times New RomanXwaw @w& f7- 2 ()D!"Times New RomanXwaw @w& f7- 2 A)`^Times New RomanXwaw @w& f7- 2 yeb"Symbol Xwaw @w& f7- 2 t)`xSymbol OXwaw @w& f7- 2 t)`4Symbol Xwaw @w& f7- 2 =)&UMathTypeUUIDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   y  (t)==e  A  t  b  & "System7& f7 !-NANI>ItDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   y  (t)==e  A  t  b  FMathType 5.0 Equation MathType EFEquation.DSMT49q 5h  4.   0&NOle CompObjiObjInfoOlePres000       !"#$%&'(*+,-./01234567:=>?@ABCDEFGHIJLMNOPSVWXYZ[\]^_`abcdeghijknqrstuvwxyz{|}~/ & MathType-AL A A'A|(J   N"Times New RomanXwaw @w fd- 2 '2(c "Times New RomanXwaw @w fd- 2 11w"Times New RomanXwaw @w fd-2 ,,2,11, x-Ogg "Times New RomanXwaw @w fd- 2 22a "Times New RomanXwaw @w fd- 2 12;"Times New RomanXwaw @w fd-2 ,11,10g-e "Times New RomanXwaw @w fd- 2 22% Times New RomanXwaw @w fd-"2 (2)(1)((1)(1))((1)L~"""~|!Times New RomanXwaw @w fd-2 (1))(1)|~")"Times New RomanXwaw @w fd-2 ijijijijp P"Times New RomanXwaw @w fd- 2 ijij.Times New RomanXwaw @w fd-2 CkjCiiCijiCjjl^TB>!:Times New RomanXwaw @w fd-2 kjCiCpHSymbolf Xwaw @w fd- 2 rsms_ ! 2 q2;"Symbol ,Xwaw @w fd- 2 +-+L 2 ++-Symbolf Xwaw @w fd-2 =-++++++++++,V+NN!d 2 +-+,%&MathTypeUU}DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  C i,j' =="-kjC i,j ++  12 (i++2)(i++1)C i++2,j"-1 ++(rs(i++1)j++m(i++1))C i++1,j ++(  12 s 2 (j++1)j++kq(j++1))C i,j++1 "-  12 (i++1)C i++1,j"-1 & -"System fd !-NANIM}lDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  C i,j' =="-kjC i,j ++  12 (i++2)(i++1)C iEquation Native )_1290651708F@W@WOle 8CompObj9i++2,j"-1 ++(rs(i++1)j++m(i++1))C i++1,j ++(  12 s 2 (j++1)j++kq(j++1))C i,j++1 "-  12 (i++1)C i++1,j"-1 FMathType 5.0 Equation MathType EFEquation.DSMT49qNl .  &N & MathTypeP `Times New Roman@wIw @wPf[-ObjInfo;OlePres000<Equation Native KH_1290651769F@W@W2 '()()6`^Times New Roman@wIw @wPf[-2 ytAytP88` Symbolv @wIw @wPf[- 2 =y&8MathTypeUU,DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   y  '(t)== A   y  (t) & "System[Pf[ !-NANIQ,DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   y  '(t)== A   y  (t) FMathType 5.0 Equation MathType EFEquation.DSMT49qN .   &N & MathTypeP `Ole QCompObjRiObjInfoTOlePres000 UTimes New Roman@wIw @wfR- 2 ()"Times New Roman@wIw @wfR- 2 At`^Times New Roman@wIw @wfR- 2 yteb8`Symbol ٘@wIw @wfR- 2 =t&:MathTypeUU.DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   y  (t)==e  A  t  b  & "SystemRfR !-NANIQ.DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APEquation Native fJ_1303677805 #F@W@WOle lCompObj"%miG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   y  (t)==e  A  t  b  FMathType 5.0 Equation MathType EFEquation.DSMT49qObjInfooOlePres000$&p*Equation Native c1TableՏ^ N .  &N@ & MathTypeP `Times New RomanXwaw @w fw-2 '()() 6DD`^Times New RomanXwaw @w fw- 2 yAy88`Symbolg Xwaw @w fw- 2 ttX`LSymbol~ Xwaw @w fw- 2 =t&SMathTypeUUGDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   y  '(t)== A   y  (t) & "Systemw fw !-NANI>GtDSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT ExtraWinAllCodePages!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   y  '(t)== A   y  (t)Oh+'0E-Ũaz&cdrᚔK98S;3Y4%N4s~Zn =vmi)}OF ݢZꐮi%pfsF ]34Q@55;k7 p/Ӽw\j PtδkEI|RDd |0 j # Aj2)%Mg`!)%Mg`Y0pxڥTOQnR0В( Q#v襤nUWPZj͠1Ƙńx1z`L]O@}ͷ߼GPC9fi[W8 ٺaM9Sg:?Dyʛ3,*Pր0xf_"ܴ?[dOLhTsF5|T̯`ÏGjឋîp4+5)XNRMPwF;n(_e2v]eѰl1>}R5-FbPgj "W یNtQy%]jo麶5#Hq׊J>?:\*w\5Mܔ9|<ķ|cy:|&g!N>v.,Qt=ӎZ4S37m ;Lo7_Dd b k c $Ak? 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