// Finds a minimum of a multidimensional function f in direction dir.
// Minimum is stored in min, its location in p :
static void min1dir(int dim, PnlVect * p, PnlVect * dir, double *min, double f(PnlVect *))
{
int j;
const double TOL=1.0e-10;
double xx,xmin,bx,ax;
// Initialise les variables globales
_n=dim;
_p = p;
_dir = dir;
func=f;
// [0,1] is the initial bracketing guess
ax=0.0;
xx=1.0;
minenc(&ax,&xx,&bx,f1dim); // Computes an acutal bracketing triplet
*min=min1dim(ax,xx,bx,f1dim,TOL,&xmin); // Computes the minimum of function f1dim
// (which is the minimum of function f in direction dir)
for(j=0;j<dim;j++){
double d = pnl_vect_get (dir , j);
pnl_vect_set (dir, j, d*xmin);
pnl_vect_set (p, j, pnl_vect_get (p, j) + pnl_vect_get (dir, j)); // Sets actual position of the minimum
}
}
// Conjugate gradient optimization of function f, given its gradient gradf.
// Minimum is stored in min, its location in p. Tolerance is asked :
static void optigc(int dim, PnlVect *p, double tol, double *min,
double f(PnlVect *),
void gradf(PnlVect *, PnlVect *))
{
int i, j;
/* Scalars used to define directions */
double gg,gam,fp,dgg;
PnlVect *g = pnl_vect_create (dim); /* Auxiliary direction :
gradient at the minimum */
PnlVect *h = pnl_vect_create (dim); /* Conjugate direction along
which to minimize */
PnlVect *grad = pnl_vect_create (dim); /* Gradient */
const int ITMAX = 20000;
const double EPS = 1.0e-18;
fp=f(p);
gradf(p,grad);
pnl_vect_clone (h, grad);
pnl_vect_clone (g, grad);
pnl_vect_mult_double (h ,-1.0);
pnl_vect_mult_double (g ,-1.0);
pnl_vect_mult_double (grad ,-1.0);
for(i=0; i<ITMAX; i++) {
min1dir(dim,p,h,min,f); // Minimizing along direction h
if(2.0*fabs((*min)-fp) <= tol*(fabs((*min))+fabs(fp)+EPS)) // Done : tolerance reached
{
pnl_vect_free (&g);
pnl_vect_free (&h);
pnl_vect_free (&grad);
return;
}
fp=(*min);
gradf(p,grad); // Computes gradient at point p, location of minimum
dgg=gg=0.0;
/* Computes coefficients applied to new direction for h */
gg = pnl_vect_scalar_prod (g, g); /* Denominator */
dgg = pnl_vect_scalar_prod (grad, grad) + pnl_vect_scalar_prod (g, grad); /* Numerator : Polak-Ribiere */
if(gg==0.0) // Gradient equals zero : done
{
pnl_vect_free (&g);
pnl_vect_free (&h);
pnl_vect_free (&grad);
return;
}
gam=dgg/gg;
for(j=0; j<dim; j++){ // Defining directions for next iteration
pnl_vect_set (g, j, - pnl_vect_get (grad, j));
pnl_vect_set (h, j, pnl_vect_get (g, j)+gam * pnl_vect_get (h, j));
}
}
perror ("Too many iterations in optigc\n");
}
static double f1dim( double x)
{
int j;
double val;
PnlVect * xt = pnl_vect_create (_n);
for(j=0; j<_n; j++){
pnl_vect_set (xt, j, pnl_vect_get (_p, j) + x * pnl_vect_get (_dir, j));
}
val=func(xt);
pnl_vect_free (&xt);
return val;
}
//------------------------Sampling the transition probability (v(0)=X_t, v(1)=int_0^t X_s ds)
//-------------------------dX_t = kappa(theta-X_t)dt + sigma sqrt(X_t)dW_t
//-------------------------------------In the case of the troncation serie
static void Sample_C( PnlVect* v,double t, double kappa, double sigma, double theta ,int generator)
{
//----------Declaration of variable
double gamma, lambda;
double tmp;
double tmp2;
int j,pss;
int order_tr; // Default value is equal to 20
//----------Initialization of parammter
tmp=0.;
j=0;
gamma = 4.*kappa/(sigma*sigma*(1.-exp(-kappa*t)));
lambda = pnl_vect_get(v,0)*gamma*exp(-kappa*t);
order_tr = 20;
//----------Begin operations
//----generate vt --> tmp
pss = pnl_rand_poisson(lambda*0.5,generator);
for(j=1;j<= pss;j++)
tmp = tmp + pnl_rand_gamma(1.,2., generator);
tmp = tmp +pnl_rand_gamma(2.*kappa*theta/(sigma*sigma),2.,generator);
tmp = tmp/gamma;
//----generate the variable Z
tmp2 =0.;
j=0;
pss = pnl_rand_bessel(2.*theta*kappa/(sigma*sigma)-1.,2.*kappa*sqrt(pnl_vect_get(v,0)*tmp)/(sigma*sigma*sinh(kappa*t*0.5)),generator);
for(j=1;j<=pss;j++)
{
tmp2=tmp2+X_3_sample( order_tr, t, kappa, sigma, generator);
}
//----generate int_0^t vs = X1 +X2 +X3 --> lambda
lambda=tmp2+X_2_sample( order_tr, t, kappa, sigma, theta , generator)+X_1_sample( order_tr, t, kappa, sigma, pnl_vect_get(v,0), tmp, generator);
//----set the new value
pnl_vect_set(v,0,tmp);
pnl_vect_set(v,1,lambda);
}
/**
* Solve a tridiagonal system defined by matrix whose three diagonals are
* defined by three values.
* @param low the double value of M(i,i-1)
* @param diag the double value of M(i,i)
* @param up the double value of M(i,i+1)
* @param rhs the right and side member of the system
* @param lhs contains the solution on exit
*/
void pnl_progonka(const double low,
const double diag, const double up,
const PnlVect * rhs,
PnlVect * lhs)
{
int i;
double *D,tmp;
D = malloc((rhs->size)*sizeof(double));
/* Do Down-Solve L y = rhs */
if((D[0]=diag)==0)
{
PNL_ERROR("division by zero","pnl_progonka");
}
pnl_vect_set(lhs,0,pnl_vect_get(rhs,0));
for(i=1; i<rhs->size; i++)
{
tmp=low/D[i-1];
if((D[i]=diag-tmp*up)==0)
{
PNL_ERROR("division by zero","pnl_progonka");
}
pnl_vect_set(lhs,i,pnl_vect_get(rhs,i)-tmp*pnl_vect_get(lhs,i-1));
}
/* Do Up-Solve L y = rhs */
pnl_vect_set(lhs,rhs->size-1,pnl_vect_get(lhs,rhs->size-1)/D[rhs->size-1]);
for(i=rhs->size-2;i>=0;i--)
{
pnl_vect_set(lhs,i,(pnl_vect_get(lhs,i)-up*pnl_vect_get(lhs,i+1))/D[i]);
}
free(D);
}
// Computes the price and the deltas of a claim using the lower bound of the
// price for an option
// that is paying a linear combination of assets :
static void lowlinearprice(int _dim, PnlVect *_eps, PnlVect *_x, PnlMat *_rac_C, PnlVect *_sigma, double _echeance, double *prix, PnlVect *deltas)
{
int i, j;
double arg;
double normv=0;
double tol=1e-15;
PnlVect *xopt = pnl_vect_create_from_double ( _dim+2, 1./sqrt(_dim+1.));
// Starting point for optimization : normalized vector
pnl_vect_set (xopt, _dim+1, 0.0);
// Initializing global variables to parameters of the problem
Dim=_dim;
Echeance=_echeance;
Sigma = _sigma;
Rac_C = _rac_C;
Eps = _eps;
X = _x;
optigc(Dim+2,xopt,tol,prix,Cost,Gradcost);
*prix = -1.0* (*prix); // Price is the maximum of function
for (i=0; i<Dim+1; i++)
{
double xopt_i = pnl_vect_get (xopt, i);
normv += xopt_i * xopt_i;
}
normv = sqrt (normv);
for(i=0; i<Dim; i++){
double tmp=0;
for(j=0; j<Dim+1; j++){
tmp+=pnl_mat_get (_rac_C, i+1, j) * pnl_vect_get (xopt, j);
}
arg=pnl_vect_get (xopt, Dim+1) + pnl_vect_get (Sigma, i+1) * tmp*sqrt(Echeance)/normv;
pnl_vect_set (deltas, i, pnl_vect_get (Eps, i+1) * cdf_nor(arg)); // Computing the deltas
}
pnl_vect_free (&xopt);
}
double call_by_density(double k, params *p)
{
int N ;
double m_dVarianceCurve;
double m_dGamma ;
double m_dOmega1 ;
double m_dOmega2 ;
double m_dCorrel;
int path,i ;
double sup ;
PnlVect *axis;
double strikeMin, res;
N = p->n;
m_dVarianceCurve= p->VarianceCurve;
m_dGamma = p->Gamma;
m_dOmega1 = p->Omega1;
m_dOmega2 = p->Omega2;
m_dCorrel = p->Correl;
path = 2000;
sup = 12;
axis = pnl_vect_create(path);
strikeMin = z(k * k,p);
res = 0;
for (i = 0; i < path; i++)
pnl_vect_set(axis,i,strikeMin + sup * i / (double)path);
for (i = 0; i < path - 1; i++)
res += (pnl_vect_get(axis,i+1) - pnl_vect_get(axis,i)) * MAX(sqrt(g(pnl_vect_get(axis,i),p)) - k, 0.0) *
exp(-pnl_vect_get(axis,i) * pnl_vect_get(axis,i) / 2.0) / sqrt(2.0 * M_PI);
return res;
}
double put_by_density(double k, params *p)
{
int N ;
double m_dVarianceCurve;
double m_dGamma ;
double m_dOmega1 ;
double m_dOmega2 ;
double m_dCorrel;
int path,i ;
double inf ;
PnlVect *axis;
double strikeMax, res;
N = p->n;
m_dVarianceCurve= p->VarianceCurve;
m_dGamma = p->Gamma;
m_dOmega1 = p->Omega1;
m_dOmega2 = p->Omega2;
m_dCorrel = p->Correl;
path = 2000;
inf = -12;
axis = pnl_vect_create(path);
strikeMax = z(k * k,p);
res = 0;
for (i = 0; i < path; i++)
pnl_vect_set(axis,i,inf + (i + 1) * (strikeMax - inf) / (double)path);
for (i = 0; i < path - 1; i++)
res += (pnl_vect_get(axis,i+1) - pnl_vect_get(axis,i)) * MAX(k - sqrt(g(pnl_vect_get(axis,i),p)), 0.0) *
exp(-pnl_vect_get(axis,i) * pnl_vect_get(axis,i) / 2.0) / sqrt(2.0 * M_PI);
return res;
}
double Barrier_l :: payoff (const PnlMat *path) {
double sum ;
PnlVect* final = pnl_vect_create(size_);
//On met dans final la dernière colonne de Path correspond à la valeur à maturité des sous-jacents.
pnl_mat_get_col(final, path, TimeSteps_);
sum = pnl_vect_scalar_prod(final, Coeff_) - Strike_;
//On vérifie que toutes les valeurs des sous-jacents soient au dessus de la barrière
//Si on en trouve une alors le prix de l'option est de 0
for (int i=0; i<TimeSteps_+1; i++){
for (int d=0; d<size_; d++){
if (pnl_mat_get(path,d,i) < pnl_vect_get(Bl_,d)){
pnl_vect_free(&final);
return 0;
}
}
}
double Barrier_u :: payoff (
double t,
const PnlMat* path) const
{
double sum ;
//Vecteur utilisé pour effectuer la somme de chaque actif à maturité
PnlVect* final = pnl_vect_create(size_);
//On met dans final la dernière colonne de Path correspond à la valeur à maturité des sous-jacents
pnl_mat_get_col(final, path, timeStep_);
sum = pnl_vect_scalar_prod(final, Coeff_) - strike_;
//On vérifie que toutes les valeurs des sous-jacents soient en dessous de la barrière
//Si on en trouve une alors le prix de l'option est de 0
for (int i=0; i<timeStep_+1; i++){
for (int d=0; d<size_; d++){
if (MGET(path,d,i) > pnl_vect_get(Bu_,d)){
pnl_vect_free(&final);
return 0;
}
}
}
double f_by_density(params *p)
{
int N ;
double m_dVarianceCurve;
double m_dGamma ;
double m_dOmega1 ;
double m_dOmega2 ;
double m_dCorrel;
int path,i ;
double sup ;
PnlVect *axis;
double res ;
N = p->n;
m_dVarianceCurve= p->VarianceCurve;
m_dGamma = p->Gamma;
m_dOmega1 = p->Omega1;
m_dOmega2 = p->Omega2;
m_dCorrel = p->Correl;
path = 2000;
sup = 12;
axis = pnl_vect_create(path);
res = 0;
for ( i = 0; i < path; i++)
pnl_vect_set(axis,i, -sup + 2 * sup * i / (double)path);
for ( i = 0; i < path - 1; i++)
res += (pnl_vect_get(axis,i + 1) - pnl_vect_get(axis,i)) *
sqrt(m_dVarianceCurve * (1 - m_dGamma) * exp(m_dOmega1 * pnl_vect_get(axis,i) - m_dOmega1 * m_dOmega1 / 2.0) +
m_dVarianceCurve * m_dGamma * exp(m_dOmega2 * pnl_vect_get(axis,i) - m_dOmega2 * m_dOmega2 / 2.0)) *
exp(-pnl_vect_get(axis,i) * pnl_vect_get(axis,i) / 2.0) / sqrt(2.0 *M_PI);
return res;
}
// Auxiliary cost function to minimize used in lowlinearprice :
static double Cost(PnlVect *ksi)
{
int i, j;
double p=0,arg=0;
double normv=0;
for (i=0; i<Dim+1; i++)
{
double ksi_i = pnl_vect_get (ksi, i);
normv += ksi_i * ksi_i;
}
normv = sqrt (normv);
for(i=0; i<Dim+1;i++){
double tmp=0;
for(j=0;j<Dim+1;j++){
tmp+=pnl_mat_get (Rac_C, i, j) * pnl_vect_get (ksi, j);
}
arg=pnl_vect_get (ksi, Dim+1) + pnl_vect_get (Sigma, i) * tmp*sqrt(Echeance)/normv;
p+=pnl_vect_get (Eps, i) * pnl_vect_get (X, i) * cdf_nor(arg);
}
return (-1.0*p); // The function is to be maximized
}
// Returning the price and the deltas of a basket option using its lower bound approximation :
static void lower_basket(int put_or_call,int dim, PnlVect *vol, PnlVect *poids, PnlVect *val_init, PnlVect *div, double cor, double tx_int, double strike, double echeance, double *prix, PnlVect* deltas)
{
int i,j;
// Initializing parameters
PnlVect *sigma = pnl_vect_create (dim+1);
PnlVect *x = pnl_vect_create (dim+1);
PnlVect *eps = pnl_vect_create (dim+1);
PnlMat *rac_C = pnl_mat_create (dim+1, dim+1);
pnl_vect_set (sigma, 0, 0);
for(i=1; i<dim+1;i++){
pnl_vect_set (sigma, i, pnl_vect_get (vol, i-1));
}
pnl_vect_set (x, 0, strike*exp(-tx_int*echeance));
for (i=1; i<dim+1; i++){
pnl_vect_set (x, i, fabs(pnl_vect_get (poids, i-1)) *
pnl_vect_get (val_init, i-1)*
exp(-pnl_vect_get (div, i-1)*echeance));
}
pnl_vect_set (eps, 0, -1);
for (i=1; i<dim+1; i++){
if(pnl_vect_get (poids, i-1)<0) pnl_vect_set (eps, i, -1);
else pnl_vect_set (eps, i, 1);
}
if(put_or_call==1)
{
pnl_vect_mult_double (eps, -1.0);
}
if(cor != 1){
PnlMat *C = pnl_mat_create (dim, dim);
// double *C=new double[dim*dim]; // Correlation matrix
for(i=0; i<dim; i++){
for(j=0; j<dim; j++){
if(i==j) pnl_mat_set (C, i, j, 1);
else pnl_mat_set (C, i, j, cor);
}
}
pnl_mat_chol (C);
for(i=0; i<dim+1; i++){
pnl_mat_set (rac_C, i, 0, 0);
pnl_mat_set (rac_C, 0, i, 0);
}
for(i=1; i<dim+1; i++){
for(j=1; j<=i;j++){
pnl_mat_set (rac_C, i, j, pnl_mat_get (C, i-1, j-1));
}
for(j=i+1;j<dim+1;j++){
pnl_mat_set (rac_C, i, j , 0);
}
}
/* Correlation was useful only to compute a square root of it */
pnl_mat_free (&C);
} else {
for(i=0; i<dim+1;i++){
pnl_mat_set (rac_C, i, 0, 0);
pnl_mat_set (rac_C, i, 1, 1);
for(j=2; j<dim+1;j++){
pnl_mat_set (rac_C, i, j, 0);
}
}
pnl_mat_set (rac_C, 0, 1, 0);
}
lowlinearprice(dim,eps,x,rac_C,sigma,echeance,prix,deltas); // Uses the general formula
/* In deltas are stored the derivatives along x[i], which differ from those
along val_init[i] */
for(i=0;i<dim;i++){
double d = pnl_vect_get (deltas, i);
pnl_vect_set (deltas, i, d * pnl_vect_get (x, i+1) / pnl_vect_get (val_init, i));
}
pnl_vect_free (&eps);
pnl_vect_free (&x);
pnl_vect_free (&sigma);
pnl_mat_free (&rac_C);
}
static int SingularPoints_Up(int am,double s,NumFunc_1 *p,double t,double r,double sigma,PnlVect *divid_dates,PnlVect *divid_amounts,int N,double h_up,double *ptprice,double *ptdelta)
{
int n;
double u,d,h,pu,pd,a1;
double K=p->Par[0].Val.V_DOUBLE;
double *v_min,*v_max,*new_vm,*new_vm1,*new_vm2,*new_vpm,*new_vpm1,*new_vpm2;
double *vect_t;
int * nb_critical;
int *divid_steps;
double value1,value2,value_price1,value_price2;
double dist1,dist2;
double TOL1=0.00000000001;
double TOL2=0.00000000001;
int i,j,k,l,jj,kk,Nb_div,new_nb_critical,n_k;
int max_critical=0;
int i1,index,nb_critical_put;
double m1,error,errp;
int old_nb_critical;
double x1,x2,y2,y11,a,b;
int n_max;
/*Number of Dividends Dates*/
Nb_div=divid_dates->size;
//Number maximum of singular points
n_max=50000;
/*Compute steps of the tree*/
n=N*Nb_div;
/*Memory allocations*/
divid_steps= malloc((n+1)*sizeof(int));
nb_critical=(int*)malloc(sizeof(int)*(n+1));
v_min=(double*)malloc(sizeof(double)*(n+1));
v_max=(double*)malloc(sizeof(double)*(n+1));
vm=(double*)malloc(sizeof(double)*(n_max));
vpm=(double*)malloc(sizeof(double)*(n_max));
new_vm=(double*)malloc(sizeof(double)*(n_max));
new_vm1=(double*)malloc(sizeof(double)*(n_max));
new_vm2=(double*)malloc(sizeof(double)*(n_max));
new_vpm=(double*)malloc(sizeof(double)*(n_max));
new_vpm1=(double*)malloc(sizeof(double)*(n_max));
new_vpm2=(double*)malloc(sizeof(double)*(n_max));
vect_t= malloc((n+1)*sizeof(double));
/*Up and Down factors*/
h=t/(double)n;
a1= exp(h*r);
u = exp(sigma*sqrt(h));
d= 1./u;
/*Risk-Neutral Probability*/
pu=(a1-d)/(u-d);
pd=1.-pu;
if ((pd>=1.) || (pd<=0.))
return NEGATIVE_PROBABILITY;
pu*=exp(-r*h);
pd*=exp(-r*h);
for(i=0;i<=n;i++)
vect_t[i]=h*(double)i;
//Compute steps related to the dividend dates
for(k=0;k<Nb_div;k++)
{
i=0;
while(vect_t[i]<pnl_vect_get(divid_dates,k)) i++;
if(fabs(pnl_vect_get(divid_dates,k)-vect_t[i])<1.e-10)
divid_steps[k]=i;
else
{
dist1=vect_t[i]-pnl_vect_get(divid_dates,k);
dist2=pnl_vect_get(divid_dates,k)-vect_t[i-1];
if (dist1<dist2)
divid_steps[k]=i;
else
divid_steps[k]=i-1;
}
}
/*Compute Minimum and Maximum of the stock at each step
taking in to account of the dividend payements*/
v_min[0]=s;
v_max[0]=s;
j=0;
for(i=1;i<=n;i++)
{
v_min[i]=v_min[i-1]*d;
v_max[i]=v_max[i-1]*u;
for(k=0;k<Nb_div;k++)
if(i==divid_steps[k])
{
v_min[i]=v_min[i]-pnl_vect_get(divid_amounts,Nb_div-k-1);
v_max[i]=v_max[i]-pnl_vect_get(divid_amounts,Nb_div-k-1);
}
}
/***Singular points at Maturity****/
if((v_min[n]<K)&&(v_max[n]>K))
{
nb_critical[n]=2;
//Abscissa
vm[0]=v_min[n];
vm[1]=K;
vm[2]=v_max[n];
//Ordinate
vpm[0]=(p->Compute)(p->Par,vm[0]);
vpm[1]=(p->Compute)(p->Par,vm[1]);
vpm[2]=(p->Compute)(p->Par,vm[2]);
}
else
{
nb_critical[n]=1;
//Abscissa
vm[0]=v_min[n];
vm[1]=v_max[n];
//Ordinate
vpm[0]=(p->Compute)(p->Par,vm[0]);
vpm[1]=(p->Compute)(p->Par,vm[1]);
}
/***Backward algorithm****/
for(i=1;i<=n;i++)
{
/*Min point*/
new_vm[0]=v_min[n-i];//Abscissa
value1=new_vm[0]*u;
value_price1=compute_pricesp(value1,nb_critical[n-i+1]);
value2=new_vm[0]*d;
value_price2=compute_pricesp(value2,nb_critical[n-i+1]);
new_vpm[0]=pu*value_price1+pd*value_price2;//Ordinate
/*Middle points*/
n_k=1;
for(j=1;j<nb_critical[n-i+1];j++)
{
for(jj=-1;jj<=1;jj=jj+2)
{
new_vm[n_k]=vm[j]*pow(u,(double)jj);//Abscissa
if((new_vm[n_k]>v_min[n-i])&&(new_vm[n_k]<v_max[n-i]))
{
value1=new_vm[n_k]*u;
value_price1=compute_pricesp(value1,nb_critical[n-i+1]);
value2=new_vm[n_k]*d;
value_price2=compute_pricesp(value2,nb_critical[n-i+1]);
new_vpm[n_k]=pu*value_price1+pd*value_price2;//Ordinate
n_k++;
}
}
}
/*Max point*/
new_vm[n_k]=v_max[n-i];//Abscissa
value1=new_vm[n_k]*u;
value_price1=compute_pricesp(value1,nb_critical[n-i+1]);
value2=new_vm[n_k]*d;
value_price2=compute_pricesp(value2,nb_critical[n-i+1]);
new_vpm[n_k]=pu*value_price1+pd*value_price2;//Ordinate
//Shift at the dividends dates
for(k=0;k<Nb_div;k++)
if(i==divid_steps[k])
{
for(j=0;j<=n_k;j++)
new_vm[j]+=pnl_vect_get(divid_amounts,Nb_div-k-1);
}
/*Sorting*/
nb_critical[n-i]=n_k;
Sort2Vect(nb_critical[n-i],new_vm,new_vpm);
//Remove singular points very close TOL1=e-10,TOL2=e-10
new_vm1[0]=new_vm[0];
new_vpm1[0]=new_vpm[0];
kk=0;
l=0;
do {
do {
l++;
}while((new_vm[l]<=new_vm1[kk]+TOL1)&&(l<nb_critical[n-i]));
kk++;
new_vm1[kk]=new_vm[l];
new_vpm1[kk]=new_vpm[l];
}while((l<nb_critical[n-i]));
new_nb_critical=kk;
nb_critical[n-i]=kk;
if(fabs(new_vm1[nb_critical[n-i]]-new_vm1[nb_critical[n-i]-1])<TOL2)
nb_critical[n-i]=new_nb_critical-1;
//Upper bound
i1=0;
index=0;
new_vm2[0]=new_vm1[0];
new_vpm2[0]=new_vpm1[0];
while(i1<nb_critical[n-i]-1)
{
l=1;
do
{
l++;
m1=(new_vpm1[i1+l]-new_vpm1[i1])/(new_vm1[i1+l]-new_vm1[i1]);
error=0.;
for(jj=1;jj<=l-1;jj++)
{
errp=m1*(new_vm1[i1+jj]-new_vm1[i1])+new_vpm1[i1]-new_vpm1[i1+jj];
if (errp<0) errp=-errp;
if (errp>error) error=errp;
}
}
while(!((error>h_up)||((i1+l)==nb_critical[n-i])));
index++;
new_vm2[index]=new_vm1[i1+l-1];
new_vpm2[index]=new_vpm1[i1+l-1];
i1=i1+l-1;
}
while(i1<nb_critical[n-i])
{
index++;
new_vm2[index]=new_vm1[i1+1];
new_vpm2[index]=new_vpm1[i1+1];
i1 =i1+1;
}
nb_critical[n-i]=index;
//American Call Case
if((am==1)&&((p->Compute)==&Call))
{
old_nb_critical=nb_critical[n-i];
if(MAX(0.,new_vm2[0]-K)<=new_vpm2[0])
{
l=1;
while((new_vpm2[l]>=MAX(0.,new_vm2[l]-K))&&(l<=nb_critical[n-i]))
{
l++;
if(l>nb_critical[n-i]) break;
}
if(l<=nb_critical[n-i])
{
nb_critical[n-i]=l+1;
x1=new_vm2[l-1];
x2=new_vm2[l];
y11=new_vpm2[l-1];
y2=new_vpm2[l];
a=(y2-y11)/(x2-x1);
b=(y11*x2-x1*y2)/(x2-x1);
new_vm2[l]=(K+b)/(1.-a);
new_vpm2[l]=MAX(0.,new_vm2[l]-K);
new_vm2[l+1]=new_vm2[old_nb_critical];
new_vpm2[l+1]=MAX(0.,new_vm2[l+1]-K);
}
}
}
/*
//Dynamic Programming:American Put Case
if((am==1)&&((p->Compute)==&Put))
for(j=0;j<=nb_critical[n-i];j++)
{
new_vpm2[j]=MAX(new_vpm2[j],K-new_vm2[j]);
}*/
//American Put Case
else
if((am==1)&&((p->Compute)==&Put))
{
if(MAX(0.,K-new_vm2[0])>=new_vpm2[0])
{
new_vm1[0]=new_vm2[0];
new_vpm1[0]=MAX(0.,K-new_vm2[0]);
l=1;
while((new_vpm2[l]<MAX(0.,K-new_vm2[l]))&&(l<=nb_critical[n-i]))
{
l++;
if(l>nb_critical[n-i]) break;
}
if(l>nb_critical[n-i])
{
new_vm1[1]=new_vm2[nb_critical[n-i]];
new_vpm1[1]=MAX(0.,K-new_vm1[1]);
nb_critical[n-i]=1;
}
else
if(l<=nb_critical[n-i])
{
x1=new_vm2[l-1];
x2=new_vm2[l];
y11=new_vpm2[l-1];
y2=new_vpm2[l];
a=(y2-y11)/(x2-x1);
b=(y11*x2-x1*y2)/(x2-x1);
new_vm1[1]=(K-b)/(1.+a);
new_vpm1[1]=MAX(0.,K-new_vm1[1]);
j=l;
nb_critical_put=1;
while((j<=nb_critical[n-i]))
{
nb_critical_put++;
new_vm1[nb_critical_put]=new_vm2[j];
new_vpm1[nb_critical_put]=new_vpm2[j];
j++;
};
nb_critical[n-i]=nb_critical_put;
}
for(j=0;j<=nb_critical[n-i];j++)
{
new_vm2[j]=new_vm1[j];
new_vpm2[j]=new_vpm1[j];
}
}
}
max_critical=MAX(nb_critical[n-i],max_critical);
//Copy
for(j=0;j<=nb_critical[n-i];j++)
{
vm[j]=new_vm2[j];
vpm[j]=new_vpm2[j];
}
/*Delta*/
if(i==(n-1))
*ptdelta=(vpm[nb_critical[n-i]]-vpm[0])/(vm[nb_critical[n-i]]-vm[0]);
}
/*Price*/
*ptprice=vpm[0];
//Memory desallocation
free(nb_critical);
free(divid_steps);
free(v_min);
free(v_max);
free(vpm);
free(vm);
free(new_vm);
free(new_vm1);
free(new_vm2);
free(new_vpm);
free(new_vpm1);
free(new_vpm2);
free(vect_t);
return OK;
}
static void pnl_basis_eval_test ()
{
PnlMat *X;
PnlVect *V, *x, *t, *D, *alpha, *lower, *upper;
PnlRng *rng;
PnlBasis *basis;
int j, deg, n;
double t0, x0, tol;
tol = 1E-8;
deg=5; //total degree
n=50;
D=pnl_vect_create(5);
x=pnl_vect_create(n);
t=pnl_vect_create(n);
t0=0.5;
x0=2.5;
rng = pnl_rng_create (PNL_RNG_MERSENNE);
pnl_rng_sseed (rng, 0);
/*
* Random points where the function will be evaluted
*/
pnl_vect_rng_uni(x,n,-5,4,rng);
pnl_vect_rng_uni(t,n,0,1,rng);
basis = pnl_basis_create_from_degree (PNL_BASIS_HERMITIAN, deg, 2);
alpha = pnl_vect_create (basis->nb_func);
X = pnl_mat_create (n, 2);
for(j=0;j<n;j++)
{
MLET (X, j, 0) = GET(t,j);
MLET (X, j, 1) = GET(x,j);
}
V=pnl_vect_create(n);
/*
* Vector of values for the function to recover
*/
for(j=0;j<n;j++)
{
LET(V,j)=fonction_a_retrouver(GET(t,j),GET(x,j));
}
pnl_basis_fit_ls (basis, alpha, X, V);
/*
* compute approximations of the derivatives (first order in time and
* second order in space )
*/
derive_approx_fonction(basis, D, alpha,t0,x0);
pnl_test_eq_abs (pnl_vect_get(D,0), fonction_a_retrouver(t0,x0), tol,
"deriv_approx_fonction", "derivative 0");
pnl_test_eq_abs (derive_x_approx_fonction(basis, alpha, t0, x0),
derive_x_fonction_a_retrouver(t0,x0), tol,
"deriv_approx_fonction", "derivative %% x");
pnl_test_eq_abs (pnl_vect_get(D,2), derive_xx_fonction_a_retrouver(t0,x0),
tol, "deriv_approx_fonction", "derivative %% xx");
pnl_test_eq_abs (pnl_vect_get(D,3), derive_t_fonction_a_retrouver(t0,x0),
tol, "deriv_approx_fonction", "derivative %% t");
pnl_test_eq_abs (pnl_vect_get(D,4), derive_xt_fonction_a_retrouver(t0,x0),
tol, "deriv_approx_fonction", "derivative %% tx");
pnl_basis_free (&basis);
/* reduced basis */
basis = pnl_basis_create_from_degree (PNL_BASIS_HERMITIAN, deg, 2);
lower = pnl_vect_create_from_list (2, 0., -5.);
upper = pnl_vect_create_from_list (2, 1., 4.);
pnl_basis_set_domain (basis, lower, upper);
pnl_basis_fit_ls (basis, alpha, X, V);
derive_approx_fonction(basis, D, alpha,t0,x0);
pnl_test_eq_abs (pnl_vect_get(D,0), fonction_a_retrouver(t0,x0), tol,
"deriv_approx_fonction (reduced)", "derivative 0");
pnl_test_eq_abs (derive_x_approx_fonction(basis, alpha, t0, x0),
derive_x_fonction_a_retrouver(t0,x0), tol,
"deriv_approx_fonction (reduced)", "derivative %% x");
pnl_test_eq_abs (pnl_vect_get(D,2), derive_xx_fonction_a_retrouver(t0,x0),
tol, "deriv_approx_fonction (reduced)", "derivative %% xx");
pnl_test_eq_abs (pnl_vect_get(D,3), derive_t_fonction_a_retrouver(t0,x0),
tol, "deriv_approx_fonction (reduced)", "derivative %% t");
pnl_test_eq_abs (pnl_vect_get(D,4), derive_xt_fonction_a_retrouver(t0,x0),
tol, "deriv_approx_fonction (reduced)", "derivative %% tx");
pnl_basis_free (&basis);
pnl_rng_free (&rng);
pnl_vect_free(&alpha);
pnl_vect_free(&x);
pnl_vect_free(&t);
pnl_vect_free(&V);
pnl_vect_free(&D);
pnl_vect_free(&lower);
pnl_vect_free(&upper);
pnl_mat_free(&X);
}
int MCGlassermanKim(double S0, NumFunc_1 *p, double T, double r, double q, double v0,double kappa,double theta,double sigma,double rho,int Nmc,int generator,double *ptprice, double *ptdelta,double *error_price)
{
//------------------Declaration of variable
int j,call_put;
PnlVect* vv;
double tmp1, tmp, tmp2, tmp3;
double tmpvar;
int init_mc;
double mt,sigmat;
double d1,d2;
double epsilon;
double K;
if ((p->Compute)==&Call)
call_put=0;
else
call_put=1;
K=p->Par[0].Val.V_PDOUBLE;
//-----------------Initialization of variable
vv = pnl_vect_create_from_double(2,0.);
pnl_vect_set(vv,0,v0);
epsilon = 0.01;
init_mc= pnl_rand_init(generator,1,(long)Nmc);
//-----------------Operation begins
tmp=0.;
tmp2=0.;
tmpvar=0.;
for(j=1;j<= Nmc;j++)
{
pnl_vect_set(vv,0,v0);
pnl_vect_set(vv,1,0.);
Sample_C( vv, T, kappa, sigma, theta , generator);
mt= (r-q -kappa*theta*rho/sigma) *T + rho*(pnl_vect_get(vv,0)-v0)/sigma+(kappa*rho/sigma-0.5)*pnl_vect_get(vv,1);
sigmat = sqrt((1-rho*rho)*pnl_vect_get(vv,1));
if(call_put==0)//call pricing
{
d1 = (log(S0/K)+mt)/sigmat+ sigmat;
d2 = d1 - sigmat;
tmp1 = S0*exp(mt-r*T+0.5*sigmat*sigmat)*cdf_nor(d1)-K*exp(-r*T)*cdf_nor(d2);
d1 = (log((S0+epsilon)/K)+mt)/sigmat+ sigmat;
d2 = d1 - sigmat;
tmp3 = (S0+epsilon)*exp(mt-r*T+0.5*sigmat*sigmat)*cdf_nor(d1)-K*exp(-r*T)*cdf_nor(d2);
}
else
{
d1 = (log(S0/K)+mt)/sigmat+ sigmat;
d2 = d1 - sigmat;
tmp1 = K*exp(-r*T)*(1.-cdf_nor(d2))-S0*(1.-exp(mt-r*T+0.5*sigmat*sigmat)*cdf_nor(d1));
d1 = (log((S0+epsilon)/K)+mt)/sigmat+ sigmat;
d2 = d1 - sigmat;
tmp3 = K*exp(-r*T)*(1.-cdf_nor(d2))-(S0+epsilon)*(1.-exp(mt-r*T+0.5*sigmat*sigmat)*cdf_nor(d1));
}
tmp = tmp1 +tmp;
tmp2 = tmp3+ tmp2;
//--------------confidence interval
tmpvar = tmp1*tmp1+tmpvar;
}
tmp = tmp /((double)Nmc);
tmp2 = tmp2/ ((double)Nmc);
tmpvar = tmpvar/((double) Nmc) - tmp*tmp;
*ptprice = tmp;
*ptdelta = (tmp2-tmp) /epsilon;
*error_price=sqrt(tmpvar/((double)Nmc));
//printf("the interval of confidence is +/- %f\n", 2.* sqrt(tmpvar/((double) Nmc)));
//-----------Free Memory
pnl_vect_free(&vv);
return init_mc;
}
/**
* compute the paths from vector which contains historical price and stock the result in a matrix with dimensions (T+1)*N for GARCH model
* @param[in] h vector which contain all values history of price
* @param[out] path matrix stock all path after compute
* @param[in] T Exercise time
* @param[in] alpha_zero garch parameter
* @param[in] alpha_one garch parameter
* @param[in] interest interest rate
* @param[in] lambda the constant unit risk premium
* @param[in] beta_one parameter garch
* @param[in] type_generator the type of generator for random number
* @return OK if ok otherwise return FAIL
*/
static int calculate_path(const PnlVect* h,PnlMat* path,double interest,int frequency,int N,int T,double alpha_zero,double alpha_one,double lambda,double beta_one,int type_generator)
{
double sigma,s,sigma_t;
int size;//size of vector
double s_t;//price at time t
double s_0;//price init
double presigma=alpha_zero/(1-beta_one-alpha_one*(1+pow(lambda,2)));
double preepsilon=0;
PnlMat* eps;//matrix contains all variables epsilon with value random
PnlMat* array_sigma;
int dimension=N;//colum of matrix
int samples;//row of matrix
int i,j;
interest=(frequency*interest)/252.;
size=h->size;
s_0=pnl_vect_get(h,0);
eps=pnl_mat_create(T-size+1,N);//matrix contains all variables epsilon with value random
array_sigma=pnl_mat_create(T-size+1,N);
samples=T-size+1;
pnl_rand_init(type_generator,dimension,samples);
pnl_mat_rand_normal(eps,samples,dimension,type_generator);
if(size==1)
{
s_t=s_0;
}
else
{
s_t=pnl_vect_get(h,size-1);
}
if(size>1)
{
for(i=0;i<size-1;i++)
{
presigma=sqrt(alpha_zero+alpha_one*pow((presigma*preepsilon-lambda*presigma),2)+beta_one*pow(presigma,2));
preepsilon=(1/presigma)*(log(pnl_vect_get(h,i+1)/pnl_vect_get(h,i))-interest+0.5*pow(presigma,2));
}
}
for(i=0;i<dimension;i++)
{
for(j=0;j<samples;j++)
{
if(j==0)
{
sigma=presigma;
pnl_mat_set(array_sigma,0,i,sigma);
if(size>1)
{
s=s_t*pow(M_E,(interest-0.5*pow(pnl_mat_get(array_sigma,0,i),2)+pnl_mat_get(array_sigma,0,i)*preepsilon));
}
else
{
s=s_0*pow(M_E,(interest-0.5*pow(pnl_mat_get(array_sigma,0,i),2)+pnl_mat_get(array_sigma,0,i)*pnl_mat_get(eps,0,i)));
}
pnl_mat_set(path,0,i,s);
}
else
{
sigma_t=pnl_mat_get(array_sigma,j-1,i);
sigma=sqrt(alpha_zero+alpha_one*pow((sigma_t*pnl_mat_get(eps,j-1,i)-lambda*sigma_t),2)+beta_one*pow(sigma_t,2));
pnl_mat_set(array_sigma,j,i,sigma);
s=pnl_mat_get(path,j-1,i)*pow(M_E,(interest-0.5*pow(sigma,2)+sigma*pnl_mat_get(eps,j,i)));
pnl_mat_set(path,j,i,s);
}
}
}
pnl_mat_free(&eps);
pnl_mat_free(&array_sigma);
return OK;
}
// Auxiliary gradient of function Cost, used in routine lowlinearprice :
static void Gradcost(PnlVect *ksi, PnlVect *g)
{
int i,j,k;
double normv=0, s;
for (i=0; i<Dim+1; i++)
{
double ksi_i = pnl_vect_get (ksi, i);
normv += ksi_i * ksi_i;
}
normv=sqrt(normv);
pnl_vect_set (g, Dim+1, 0);
for(j=0; j<Dim+1;j++){
pnl_vect_set (g, j, 0.);
for(i=0;i<Dim+1;i++){
double tmp=0;
for(k=0; k<Dim+1;k++){
tmp+=pnl_mat_get (Rac_C, i, k) * pnl_vect_get (ksi, k);
}
s=pnl_normal_density(pnl_vect_get (ksi, Dim+1) + pnl_vect_get (Sigma, i) *
tmp*sqrt(Echeance)/normv);
s*=pnl_vect_get (Eps, i) * pnl_vect_get (X, i);
if(j==Dim)
pnl_vect_set (g, Dim+1, pnl_vect_get (g, Dim+1) + s);
s*=pnl_vect_get (Sigma, i) * sqrt(Echeance)/normv;
s*=pnl_mat_get (Rac_C, i, j) - pnl_vect_get (ksi, j) *tmp/(normv*normv);
pnl_vect_set (g, j, pnl_vect_get (g, j) + s);
}
pnl_vect_set (g , j , -pnl_vect_get (g, j));
}
pnl_vect_set (g , Dim+1 , -pnl_vect_get (g, Dim+1));
}
void calibToVIXSmiles(double k1, double k2, double Theta , double RhoXY, params *p, FILE* MarketData, FILE* OUTOMEGA)
{
char chaine[TAILLE_MAX] = "";
int NumberMat,i,j;
PnlMat *Data;
PnlVect *aux;
PnlMat *OMEGA;
double varStateProces,T, Delta;
if(MarketData != NULL)
{
if(fgets(chaine, TAILLE_MAX, MarketData) != NULL)
i=0;
//LiborNumber = (int)atof ( chaine );
if(fgets(chaine, TAILLE_MAX, MarketData) != NULL)
NumberMat = (int) atof ( chaine );
Data = pnl_mat_create (NumberMat,7);
OMEGA = pnl_mat_create (2*NumberMat,3);
aux = pnl_vect_create (7);
for(j=0;j<NumberMat;j++)
{
if(fgets(chaine, TAILLE_MAX, MarketData) != NULL)
{
RowFromFile(chaine, NumberMat, aux);
for(i=0;i<7;i++)
{
pnl_mat_set(Data,j,i,pnl_vect_get(aux,i)/100.0);
}
}
}
fclose(MarketData);
}
for(i=0;i<NumberMat;i++)
{
p->VarianceCurve=pnl_mat_get(Data,i,1);
p->VIXFuture =pnl_mat_get(Data,i,2);
p->Strike =pnl_mat_get(Data,i,3);
p->PutVIX =pnl_mat_get(Data,i,4);
calibSkew(pnl_mat_get(Data,i,5), pnl_mat_get(Data,i,6), p);
if(i<NumberMat-1)
{
T = pnl_mat_get(Data,i+1,0);
Delta = pnl_mat_get(Data,i+1,0) - pnl_mat_get(Data,i,0);
}
else
{
Delta = pnl_mat_get(Data,i,0) - pnl_mat_get(Data,i-1,0);
T = pnl_mat_get(Data,i,0)+ Delta;
}
varStateProces = calcul_var(T, Delta, k1, k2, Theta , RhoXY);
pnl_mat_set(OMEGA,2*i,0, T-Delta);
pnl_mat_set(OMEGA,2*i,1, p->Omega1);
pnl_mat_set(OMEGA,2*i,2, p->VarianceCurve);
pnl_mat_set(OMEGA,2*i+1,0, T-p->Gamma*Delta);
pnl_mat_set(OMEGA,2*i+1,1, p->Omega2);
pnl_mat_set(OMEGA,2*i+1,2, p->VarianceCurve);
}
if (OUTOMEGA != NULL)
{
fprintf( OUTOMEGA, "%d \n", 2*NumberMat ); // Ecriture du caractère A
for(i=0;i<2*NumberMat;i++)
fprintf( OUTOMEGA, "%f \t %f \t %f \n", pnl_mat_get(OMEGA,i,0), pnl_mat_get(OMEGA,i,1), pnl_mat_get(OMEGA,i,2)); // Ecriture du caractère A
fclose(OUTOMEGA);
}
//pnl_mat_print(OMEGA);
}
static void prix_en_0_ls(double *res_prix, PnlMat *asset, int M, int N, int N_trading, double spot, double T, param *P, PnlBasis *basis)
{
PnlMat *V, *res_beta, *res_no_call;
PnlMatInt *H, *mod_H;
PnlVectInt *res_zeta, *res_tau,*res_theta;
PnlVect *tmp_prix;
int j, i, zeta_j, tau_j, theta_j;
double sprix,s;
double h=T/N;
//initialisation
V=pnl_mat_new();
res_no_call=pnl_mat_new();
res_beta=pnl_mat_new();
res_zeta=pnl_vect_int_new();
res_theta=pnl_vect_int_new();
res_tau=pnl_vect_int_new();
tmp_prix=pnl_vect_create(M);
H=pnl_mat_int_new();
mod_H=pnl_mat_int_new();
//calcul du vecteur H
calcul_H(H,asset,M,N,N_trading,spot,P);
calcul_mod_H(mod_H,H,M,N,N_trading,P);
//calcul du vecteur theta
theta(res_theta,mod_H,M,N,N_trading,P);
//calcul du prix_no_call protection
prix_no_call(res_no_call,M,N,asset,spot,T,P,basis);
//calcul du prix standard protection
prix(V,res_no_call,M,N,asset,res_theta,spot,T,P,basis);
//calcul de tau, zeta et beta
tau(res_tau,M,N,V,asset,res_theta,P);
zeta(res_zeta,res_tau,res_theta);
beta(res_beta,spot,asset,T,N,P);
//calcul de la somme Monte Carlo
for(j=0;j<M;j++)
{
s=0;
tau_j=pnl_vect_int_get(res_tau,j);
theta_j=pnl_vect_int_get(res_theta,j);
zeta_j=pnl_vect_int_get(res_zeta,j);
if(tau_j<theta_j)
{
pnl_vect_set(tmp_prix,j,pnl_mat_get(res_beta,zeta_j,j)*low(pnl_mat_get(asset,tau_j,j),P));
}
else
{
pnl_vect_set(tmp_prix,j,pnl_mat_get(res_beta,zeta_j,j)*pnl_mat_get(res_no_call,theta_j,j));
}
for(i=1;i<=zeta_j;i++) s=s+h*pnl_mat_get(res_beta,i,j)*c(pnl_mat_get(asset,i,j),spot,P);
pnl_vect_set(tmp_prix,j,pnl_vect_get(tmp_prix,j)+s);
}
sprix=pnl_vect_sum(tmp_prix);
pnl_mat_free(&V);
pnl_mat_free(&res_beta);
pnl_mat_int_free(&H);
pnl_mat_int_free(&mod_H);
pnl_vect_int_free(&res_zeta);
pnl_vect_int_free(&res_tau);
pnl_vect_int_free(&res_theta);
pnl_vect_free(&tmp_prix);
*res_prix=sprix/M;
}