Monte_Carlo_readme.txt

Two versions of a  Monte Carlo routine are given, the first one in C and the second one 
in Fortran 90. These routines recall the dynamic link routines UnivPM_MathFinSolver1_2.dll 
and UnivPM_MathFinSolver1_3.dll to compute the first order approximation to the transition 
probability density p contained in the power series expansion in the correlation coefficients 
defined in Section 3. The user must assign the payoff function of the financial product 
considered in the main code provided (C or Fortran) in order to allow the computation of the
corresponding price evaluating the integral (2.1) with the Monte Carlo routine provided here.




The user in the main program that contain the Monte Carlo routine must assign:

C main code
int ndim        : number of spatial variables (1<ndim<102). The integer ndim is the number of underlyings
                  considered. Note that ndim is the same as nv in the software library UnivPM_MathFinSolver1.
double xxs[]    : vector containing the point where  the transition probability density at time to maturity t
                  must be evaluated (dimension ndim). For i=0,1,...,ndim-1 xxs[i] is the price of the (i+1)-th
                  underlying  at time t.
                  Bound:  0.0005<xxs[i]<10, i=0,1,..,ndim-1.
double sg[]     : vector containing the volatilities (dimension ndim). For i=0,1,...,ndim-1 sg[i] is the volatility
                  associated to the (i+1)-th underlying.
                  Bound:  0.0005<sg[i]<0.6, i=0,1,..,ndim-1.
double rho[][]  : correlation matrix (dimension ndim x ndim). For  i=0,1,...,ndim-2, j=i+1,i+2,...,ndim-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,...,ndim-2, j=i+1,i+2,...,ndim-1.
double t        : time to maturity where the transition probability density must be evaluated. Bound: 0<t<1.
int nt          : defines the number of terms in the series expansion in time to maturity of the first order
                  correction terms 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.
int ncall       : number of function evaluations used by the Monte Carlo routine to evaluate the integral (2.1). 
                  Hint to choose ncall: let ep=max_{i=1,2,..,ndim} sg[i-1]^2*t  then choose ncall such that 
                  ncall>max(10000*ndim*ep, 50000).


The returned values are:

double tgral    : approximated value of the price of the financial product considered.
double sdev     : approximated value of the standard deviation of the random variable associated to tgral. This is a
                  probabilistic estimate of the error made approximating the integral in formula (2.1) with  tgral.




Fortran 90 main code:
integer ndim    : number of spatial variables (1<ndim<102). The integer ndim is the number of underlyings
                  considered. Note that ndim is the same as nv in the software library UnivPM_MathFinSolver1.
real*8 xxs()    : vector containing the point where  the transition probability density at time to maturity t must
                  be evaluated (dimension ndim). The components of xxs() are the prices of the underlyings at time
                  to maturity t.
                  Bound:  0.0005<xxs(i)<10, i=1,2,..,ndim.
real*8 sg()     : vector containing the volatilities (dimension ndim). For i=1,2,...,ndim sg(i) is the volatility
                  associated to the i-th underlying.
                  Bound:  0.0005<sg(i)<0.6, i=1,2,..,ndim.
real*8 rho(,)   : correlation matrix (dimension ndim x ndim). The user must assign the elements rho(i,j), 
                  i=1,2,...,ndim-1, j=i+1,i+2,...,ndim.
                  Bound: -1<rho(i,j)<1, i=1,2,...,ndim-1, j=i+1,i+2,...,ndim.
real*8 t        : time to maturity where the transition probability density must be evaluated. Bound: 0<t<1.
integer nt      : defines the number of terms in the series expansion in time to maturity of the first order correction
                  terms 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.
integer ncall   : number of function evaluations used by the Monte Carlo routine to evaluate the integral (2.1). 
                  Hint to choose ncall: let ep=max_{i=1,2,..,ndim} sg(i)^2*t  then choose ncall such that 
                  ncall>max(10000*ndim*ep, 50000).

The returned values are:

real*8 tgral    : approximated value of the price of the financial product considered.
real*8 sdev     : approximated value of the standard deviation of the random variable associated to tgral. This is a
                  probabilistic estimate of the error made approximating the integral in formula (2.1) with  tgral.



The user must assign  the payoff function of the (financial) product under evaluation. This 
assignement must be done in the Fortran function named "fxn" or in the C procedure named "fxn". Some comments
in the main code help the user to locate where the payoff function must be assigned.

The arguments of  "fxn" are:

C code
int ndim        : number of spatial variables (1<ndim<102). The integer ndim is the number of underlyings considered.
                  Note that ndim is the same as nv in the software library UnivPM_MathFinSolver1.
double x[]      : vector containing the point where  the payoff function must be evaluated (dimension ndim). The 
                  components of x[] are the prices of the underlyings at time to maturity t=0. For i=0,1,...,ndim-1
                  x[i] is the price of the (i+1)-th underlying.
               
Fortran code
integer ndim    : number of spatial variables (1<ndim<102). The integer ndim is the number of underlyings considered.
                  ndim is the same as nv in the software library UnivPM_MathFinSolver1.
real*8 x()      : vector containing the point where  the payoff function  must be evaluated (dimension ndim). 
                  The components of x() are the prices of the underlyings at time to maturity t=0. For i=1,2,...,ndim
                  x(i) is the price of the i-th underlying.


Finally we suggest to the user to test the validity of the choice of the parameter ncall made of running the main code 
with  payoff function identically equal to one. In this case the price obtained evaluating the integral (2.1) with the 
Monte Carlo routine must be one for every choice of the variable xxs[] and t. In fact in this case we are integrating 
over  (all) space a transition probability density. Note that if the integral mentioned above is not approximately one 
this may be due to the fact that ncall is too small or  to the fact that we are using the first order approximation of 
p outside of its region of validity. Some comments in the main code suggest how to  make this numerical test.



To help the reader we reproduce here the parameter description of UnivPM_MathFinSolver1_2.dll
and  UnivPM_MathFinSolver1_3.dll contained in UnivPM_MathFinSolver1_readme.txt.


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).