UnivPM_MathFinSolver1_readme.txt

We tested the routines of the library UnivPM_MathFinSolver1_readme with the following compilers:
Microsoft Visual C++ 6.0, Compaq Visual Fortran 6.

How to call the software library:

-C and C++ users:
    you must unzip the "definitivoC.zip" file which contains a simple main that calls the routine 
    UnivPM_MathFinSover1_1.dll. To initialize the parameters you must modify the "Cmain" file 
    and then run it using Microsoft Visual C++ 6.0 compiler. To use the routines UnivPM_MathFinSolver1_2.dll
    and UnivPM_MathFinSolver1_3.dll a main similar to the one provided  must be prepared by the user.

-Fortran 77 and Fortran 90 users:
    you must unzip the "definitivoFOR.zip" file which contains a simple main that calls the routine 
    UnivPM_MathFinSover1_1.dll. To initialize the parameters you must modify the "Fmain" file and 
    then run it using Compaq Visual Fortran 6 compiler. To use the routines UnivPM_MathFinSolver1_2.dll
    and UnivPM_MathFinSolver1_3.dll a main similar to the one provided  must be prepared by the user.


Note: the dll files have been placed in the same directory where the executable files (i.e. Fmain.exe
or Cmain.exe) are.


The routine UnivPM_MathFinSolver1_1.dll computes the first order approximation  to the transition probability 
density function p contained in the power series expansion in the correlation coefficients defined in Section 3.

C Parameters description:
int nv          : number of spatial variables (1<nv<102). The integer nv is the number of underlyings considered.
double x[]      : vector containing the initial point (see formula (3.2)). For i=0,1,...,nv-1 x[i] is the 
                  price of the (i+1)-th underlying at time to maturity t=0 (dimension nv).
                  Bound:  0.0005<x[i]<10, i=0,1,..,nv-1.
double sg[]     : vector containing the volatilities (dimension nv). For i=0,1,...,nv-1 sg[i] is the volatility 
                  associated to the (i+1)-th underlying.
                  Bound:  0.0005<sg[i]<0.6, i=0,1,..,nv-1.
double xxs[]    : vector containing the point where  the transition probability density at  time to maturity
                  t must be evaluated (dimension nv). For i=0,1,...,nv-1 xxs[i] is the price of the (i+1)-th 
                  underlying at time to maturity t.
                  Bound:  0.0005<xxs[i]<10, i=0,1,..,nv-1.
double rho[][]  : correlation matrix (dimension nv x nv). For i=0,1,...,nv-2, j=i+1,i+2,...,nv-1 the user must 
                  assign the elements rho[i][j], that is the correlation coefficient between the square roots of 
                  the (i+1)-th underlying and of the (j+1)-th underlying.
                  Bound: -1<rho[i][j]<1, i=0,1,...,nv-2, j=i+1,i+2,...,nv-1.
double t        : time to maturity where the transition probability density must be evaluated. Bound: 0<t<1.
int nt          : number of terms in the series expansion in time to maturity of the first order correction term 
                  that must be computed (see formula (7.1)). The first nt+1 terms of the series expansion in time
                  to maturity will be retained and summed. The largest value of nt allowable is 9. The strongly 
                  recommended value to use is nt=2.
double value    : it can be initialized to 0, since it must be ignored as an input parameter. The parameter value 
                  returns the first order approximation of the transition probability density. 




Fortran parameters description:
integer nv      : number of spatial variables (1<nv<102). The integer nv is the number of underlyings considered.
real*8 x()      : vector containing the initial point (see formula (3.2)).The components of the vector x() are the 
                  prices of the underlyings at time to maturity t=0 (dimension nv).
                  Bound:  0.0005<x(i)<10, i=1,2,..,nv.
real*8 sg()     : vector containing the volatilities (dimension nv). For i=1,2,...,nv sg(i) is the volatility 
                  associated to the i-th underlying.
                  Bound:  0.0005<sg(i)<0.6, i=1,2,..,nv.
real*8 xxs()    : vector containing the point where  the transition probability density at  time to maturity t 
                  must be evaluated (dimension nv). The components of xxs() are the prices of the underlyings at 
                  time to maturity t.
                  Bound:  0.0005<xxs(i)<10, i=1,2,..,nv.
real*8 rho(,)   : correlation matrix (dimension nv x nv). The user must assign the elements rho(i,j), 
                  i=1,2,...,nv-1, j=i+1,i+2,...,nv.
                  Bound: -1<rho(i,j)<1, i=1,2,...,nv-1, j=i+1,i+2,...,nv.
real*8 t        : time to maturity where the transition probability density must be evaluated. Bound: 0<t<1.
integer nt      : number of terms in the series expansion in time to maturity of the first order correction term 
                  that must be computed (see formula (7.1)). The first nt+1 terms of the series expansion in time 
                  to maturity will be retained and summed. The largest value of nt allowable is 9. The strongly 
                  recommended value is nt=2.
real*8 value    : do not initialize. The parameter value returns the first order approximation of the transition 
                  probability density. 





The routine UnivPM_MathFinSolver1_2.dll computes the zero-th order approximation to the transition probability density 
function p contained in the power series expansion in the correlation coefficients  defined in Section 3
appropriately rescaled as in formula (5.1).

C Parameters description:
int nv          : number of spatial variables (1<nv<102). The integer nv is the number of underlyings considered.
double x[]      : vector containing the initial point (see formula (3.2)). For i=0,1,...,nv-1 x[i] is the 
                  price of the (i+1)-th underlying at time to maturity t=0 (dimension nv).
                  Bound:  0.0005<x[i]<10, i=0,1,..,nv-1.
double sg[]     : vector containing the volatilities (dimension nv). For i=0,1,...,nv-1 sg[i] is the volatility 
                  associated to the (i+1)-th underlying.
                  Bound:  0.0005<sg[i]<0.6, i=0,1,..,nv-1.
double xxs[]    : vector containing the point where  the transition probability density at  time to maturity
                  t must be evaluated (dimension nv). For i=0,1,...,nv-1 xxs[i] is the price of the (i+1)-th 
                  underlying at time to maturity t.
                  Bound:  0.0005<xxs[i]<10, i=0,1,..,nv-1.
double t        : time to maturity where the transition probability density must be evaluated. Bound: 0<t<1.
double value    : it can be initialized  to 0, since it must be ignored as an input parameter. The parameter value 
                  returns the zero-th order  approximation of the transition probability density appropriately rescaled 
                  (see formula (5.1)).



Fortran parameters description:
integer nv      : number of spatial variables (1<nv<102). The integer nv is the number of underlyings considered.
real*8 x()      : vector containing the initial point (see formula (3.2)).The components of the vector x() are the 
                  prices of the underlyings at time to maturity t=0 (dimension nv).
                  Bound:  0.0005<x(i)<10, i=1,2,..,nv.
real*8 sg()     : vector containing the volatilities (dimension nv). For i=1,2,...,nv sg(i) is the volatility 
                  associated to the i-th underlying.
                  Bound:  0.0005<sg(i)<0.6, i=1,2,..,nv.
real*8 xxs()    : vector containing the point where  the transition probability density at  time to maturity t 
                  must be evaluated (dimension nv). The components of xxs() are the prices of the underlyings at 
                  time to maturity t.
                  Bound:  0.0005<xxs(i)<10, i=1,2,..,nv.
real*8 t        : time to maturity where the transition probability density must be evaluated. Bound: 0<t<1.
real*8 value    : do not initialize. The parameter value returns the zero-th order approximation 
                  of the transition probability density appropriately rescaled (see Section 5
                  formula (5.1)).




The routine UnivPM_MathFinSolver1_3.dll computes rho_{i,j} p_{1,i,j}, a first order correction in the power series 
expansion  in the correlation coefficients of the transition probability density function p contained in formula
(3.6) of Section 3 appropriately rescaled as in formula (5.2).

C Parameters description:
int nv          : number of spatial variables (1<nv<102). The integer nv is the number of underlyings considered.
double x[]      : vector containing the initial point (see formula (3.2)). For i=0,1,...,nv-1 x[i] is the 
                  price of the (i+1)-th underlying at time to maturity t=0 (dimension nv).
                  Bound:  0.0005<x[i]<10, i=0,1,..,nv-1.
double sg[]     : vector containing the volatilities (dimension nv). For i=0,1,...,nv-1 sg[i] is the volatility 
                  associated to the (i+1)-th underlying.
                  Bound:  0.0005<sg[i]<0.6, i=0,1,..,nv-1.
double xxs[]    : vector containing the point where  the transition probability density at  time to maturity
                  t must be evaluated (dimension nv). For i=0,1,...,nv-1 xxs[i] is the price of the (i+1)-th 
                  underlying at time to maturity t.
                  Bound:  0.0005<xxs[i]<10, i=0,1,..,nv-1.
double rhoij    : correlation coefficient rho[i-1][j-1] (i=1,2,...,nv-1, j=i+1,i+2,...,nv). The  indices i, j  
                  label the first order corrections of p appearing in (3.6).
                  Bound: -1<rhoij<1.
double t        : time to maturity where the transition probability density must be evaluated. Bound: 0<t<1.
int nt          : number of terms in the series expansion in time to maturity of the first order correction term 
                  that must be computed (see formula (7.1)). The first nt+1 terms of the series expansion in time
                  to maturity will be retained and summed. The largest value of nt allowable is 9. The strongly 
                  recommended value to use is nt=2.
i, j            : integers i and j (i=1,2,...,nv-1, j=i+1,i+2,...,nv). The indices i, j are the indices that 
                  label  the first order corrections of p appearing in (3.6).
double value    : it can be initialized to 0, since it must be ignored as an input parameter. The parameter value 
                  returns the first order approximation rho_{i,j}p_{1,i,j} of the transition probability density 
                  appropriately rescaled as in formula (5.2). 




Fortran parameters description:
integer nv      : number of spatial variables (1<nv<102). The integer nv is the number of underlyings considered.
real*8 x()      : vector containing the initial point (see formula (3.2)).The components of the vector x() are the 
                  prices of the underlyings at time to maturity t=0 (dimension nv).
real*8 sg()     : vector containing the volatilities (dimension nv). For i=1,2,...,nv sg(i) is the volatility 
                  associated to the i-th underlying.
                  Bound:  0.0005<sg(i)<0.6, i=1,2,..,nv.
real*8 xxs()    : vector containing the point where  the transition probability density at  time to maturity t 
                  must be evaluated (dimension nv). The components of xxs() are the prices of the underlyings at 
                  time to maturity t.
real*8 rhoij   : correlation coefficient rho(i,j) (i=1,2,...,nv-1, j=i+1,i+2,...,nv). The indices i, j label the 
                 first order corrections of p  appearing in (3.6). 
                  Bound: -1<rhoij<1.
real*8 t        : time to maturity where the transition probability density must be evaluated. Bound: 0<t<1.
integer nt      : number of the terms in the series expansion in time of the first order correction term that must 
                  be computed (see formula (7.1) Section 7). The largest value of nt allowable is 9. The strongly 
                  recommended value is nt=2.
i, j            : integers i and j (i=1,2,...,nv-1, j=i+1,i+2,...,nv). These are the indices that label the first order 
                  corrections of p appearing in (3.6).
real*8 value    : do not initialize. The parameter value returns the first order approximation rho_{i,j}p_{1,i,j} 
                  of the transition probability density appropriately rescaled as in formula (5.2).