int main(void)
{
double r1, r2, r3, x;
int i;
x = 10.0*10.0*10.0;
printf("10^i\tsh1\t\tsh2\t\tsh3\t\terr(sh2)\terr(sh3)\n");
for(i = 2; i > -N; i--)
{
x /= 10.0;
r1 = sh1(x);
r2 = sh2(x);
r3 = sh3(x);
printf("10^%d\t%e\t%e\t%e\t%e\t%e\n", i, r1, r2, r3, (r1-r2)/r1, (r1-r3)/r1);
}
return 0;
}
void KahaleSmileSection::compute() {
std::pair<Size,Size> afIdx = ssutils_->arbitragefreeIndices();
leftIndex_ = afIdx.first;
rightIndex_ = afIdx.second;
if(deleteArbitragePoints_) {
while(leftIndex_>1 || rightIndex_<k_.size()-1) {
ssutils_ = boost::shared_ptr<SmileSectionUtils>(new SmileSectionUtils(*source_,moneynessGrid_,f_));
std::pair<Size,Size> afIdx = ssutils_->arbitragefreeIndices();
leftIndex_ = afIdx.first;
rightIndex_ = afIdx.second;
QL_REQUIRE(rightIndex_>leftIndex_,
"arbitrage free region must at least contain two points (only index is " << leftIndex_ << ")");
if(leftIndex_>1) {
moneynessGrid_.erase(moneynessGrid_.begin()+leftIndex_-1);
k_.erase(k_.begin()+leftIndex_-1);
c_.erase(c_.begin()+leftIndex_-1);
leftIndex_--;
rightIndex_--;
}
if(rightIndex_<k_.size()-1) {
moneynessGrid_.erase(moneynessGrid_.begin()+rightIndex_+1);
k_.erase(k_.begin()+rightIndex_+1);
c_.erase(c_.begin()+rightIndex_+1);
rightIndex_--;
}
}
}
cFunctions_ = std::vector<boost::shared_ptr<cFunction> >(rightIndex_-leftIndex_+2);
// extrapolation in the leftmost interval
Brent brent;
bool success;
Real secl = 0.0;
do {
success=true;
try {
Real k1 = k_[leftIndex_];
Real c1 = c_[leftIndex_];
Real c0 = c_[0];
secl = (c_[leftIndex_]-c_[0]) / (k_[leftIndex_]-k_[0]);
Real sec = (c_[leftIndex_+1]-c_[leftIndex_]) / (k_[leftIndex_+1]-k_[leftIndex_]);
Real c1p;
if(interpolate_) c1p=(secl+sec)/2;
else {
c1p=(blackFormula(Option::Call, k1+gap_, f_, sqrt(source_->variance(k1+gap_)))-
blackFormula(Option::Call, k1, f_, sqrt(source_->variance(k1))))/gap_;
QL_REQUIRE(secl < c1p && c1p <= 0.0,"dummy");
// can not extrapolate so throw exception which is caught below
}
sHelper1 sh1(k1,c0,c1,c1p);
Real s = brent.solve(sh1,QL_KAHALE_ACC,0.20,0.00,QL_KAHALE_SMAX); // numerical parameters hardcoded here
sh1(s);
boost::shared_ptr<cFunction> cFct1(new cFunction(sh1.f_,s,0.0,sh1.b_));
cFunctions_[0]=cFct1;
} catch(...) {
leftIndex_++;
success=false;
}
} while(!success && leftIndex_ < rightIndex_);
QL_REQUIRE(leftIndex_ < rightIndex_, "can not extrapolate to left, right index of af region reached ("
<< rightIndex_ << ")");
// interpolation
Real cp0 = 0.0, cp1 = 0.0;
if(interpolate_) {
for(Size i = leftIndex_; i<rightIndex_; i++) {
Real k0 = k_[i];
Real k1 = k_[i+1];
Real c0 = c_[i];
Real c1 = c_[i+1];
Real sec = (c_[i+1]-c_[i]) / (k_[i+1]-k_[i]);
if(i==leftIndex_) cp0 = leftIndex_ > 0 ? (secl + sec) / 2.0 : sec;
Real secr;
if(i==rightIndex_-1) secr=0.0;
else secr = (c_[i+2]-c_[i+1]) / (k_[i+2]-k_[i+1]);
cp1 = (sec+secr) / 2.0;
aHelper ah(k0,k1,c0,c1,cp0,cp1);
Real a;
try {
a = brent.solve(ah,QL_KAHALE_ACC,0.5*(cp1+(1.0+cp0)),cp1+QL_KAHALE_EPS,1.0+cp0-QL_KAHALE_EPS);
// numerical parameters hardcoded here
} catch(...) {
// from theory there must exist a zero. if the solver does not find it, it most
// probably lies close one of the interval bounds. Just choose the better bound
// and relax the accuracy. This does not matter in practice usually.
Real la = std::fabs(ah(cp1+QL_KAHALE_EPS));
Real ra = std::fabs(ah(1.0+cp0-QL_KAHALE_EPS));
if( la < QL_KAHALE_ACC_RELAX || ra < QL_KAHALE_ACC_RELAX) { // tolerance hardcoded here
a = la < ra ? cp1+QL_KAHALE_EPS : 1.0+cp0-QL_KAHALE_EPS;
}
else
QL_FAIL("can not interpolate at index " << i);
}
ah(a);
boost::shared_ptr<cFunction> cFct(new cFunction(ah.f_,ah.s_,a,ah.b_));
cFunctions_[leftIndex_ > 0 ? i-leftIndex_+1 : 0]=cFct;
cp0=cp1;
}
}
// extrapolation of right wing
do {
success=true;
try {
Real k0 = k_[rightIndex_];
Real c0 = c_[rightIndex_];
Real cp0;
if(interpolate_) cp0 = 0.5*(c_[rightIndex_]-c_[rightIndex_-1])/(k_[rightIndex_]-k_[rightIndex_-1]);
else {
cp0=(blackFormula(Option::Call, k0, f_, sqrt(source_->variance(k0)))-
blackFormula(Option::Call, k0-gap_, f_, sqrt(source_->variance(k0-gap_))))/gap_;
}
boost::shared_ptr<cFunction> cFct;
if(exponentialExtrapolation_) {
cFct = boost::shared_ptr<cFunction>(new cFunction(-cp0/c0 ,std::log(c0)-cp0/c0*k0));
}
else {
sHelper sh(k0,c0,cp0);
Real s;
s = brent.solve(sh,QL_KAHALE_ACC,0.20,0.0,QL_KAHALE_SMAX); // numerical parameters hardcoded here
sh(s);
cFct = boost::shared_ptr<cFunction>(new cFunction(sh.f_,s,0.0,0.0));
}
cFunctions_[rightIndex_-leftIndex_+1]=cFct;
} catch (...) {
rightIndex_--;
success=false;
}
} while(!success && rightIndex_ > leftIndex_);
QL_REQUIRE(leftIndex_ < rightIndex_, "can not extrapolate to right, left index of af region reached (" <<
leftIndex_ << ")");
}
void KahaleSmileSection::compute() {
std::pair<Size, Size> afIdx = ssutils_->arbitragefreeIndices();
leftIndex_ = afIdx.first;
rightIndex_ = afIdx.second;
cFunctions_ = std::vector<boost::shared_ptr<cFunction> >(
rightIndex_ - leftIndex_ + 2);
// extrapolation in the leftmost interval
Brent brent;
bool success;
Real secl = 0.0;
do {
success = true;
try {
Real k1 = k_[leftIndex_];
Real c1 = c_[leftIndex_];
Real c0 = c_[0];
secl = (c_[leftIndex_] - c_[0]) / (k_[leftIndex_] - k_[0]);
Real sec = (c_[leftIndex_ + 1] - c_[leftIndex_]) /
(k_[leftIndex_ + 1] - k_[leftIndex_]);
Real c1p;
if (interpolate_)
c1p = (secl + sec) / 2;
else {
c1p = (blackFormula(Option::Call, k1 + gap_, f_,
sqrt(source_->variance(k1 + gap_))) -
blackFormula(Option::Call, k1, f_,
sqrt(source_->variance(k1)))) /
gap_;
QL_REQUIRE(secl < c1p && c1p <= 0.0, "dummy");
// can not extrapolate so throw exception which is caught
// below
}
sHelper1 sh1(k1, c0, c1, c1p);
Real s = brent.solve(sh1, QL_KAHALE_ACC, 0.20, 0.00,
QL_KAHALE_SMAX); // numerical parameters
// hardcoded here
sh1(s);
boost::shared_ptr<cFunction> cFct1(
new cFunction(sh1.f_, s, 0.0, sh1.b_));
cFunctions_[0] = cFct1;
// sanity check - in rare cases we can get digitials
// which are not monotonic or greater than 1.0
// due to numerical effects. Move to the next index in
// these cases.
Real dig = digitalOptionPrice(k1 / 2.0);
QL_REQUIRE(dig >= -c1p && dig <= 1.0, "dummy");
}
catch (...) {
leftIndex_++;
success = false;
}
} while (!success && leftIndex_ < rightIndex_);
QL_REQUIRE(
leftIndex_ < rightIndex_,
"can not extrapolate to left, right index of af region reached ("
<< rightIndex_ << ")");
// interpolation
Real cp0 = 0.0, cp1 = 0.0;
if (interpolate_) {
for (Size i = leftIndex_; i < rightIndex_; i++) {
Real k0 = k_[i];
Real k1 = k_[i + 1];
Real c0 = c_[i];
Real c1 = c_[i + 1];
Real sec = (c_[i + 1] - c_[i]) / (k_[i + 1] - k_[i]);
if (i == leftIndex_)
cp0 = leftIndex_ > 0 ? (secl + sec) / 2.0 : sec;
Real secr;
if (i == rightIndex_ - 1)
secr = 0.0;
else
secr = (c_[i + 2] - c_[i + 1]) / (k_[i + 2] - k_[i + 1]);
cp1 = (sec + secr) / 2.0;
aHelper ah(k0, k1, c0, c1, cp0, cp1);
Real a;
bool valid = false;
try {
a = brent.solve(
ah, QL_KAHALE_ACC, 0.5 * (cp1 + (1.0 + cp0)),
cp1 + QL_KAHALE_EPS, 1.0 + cp0 - QL_KAHALE_EPS);
// numerical parameters hardcoded here
valid = true;
}
catch (...) {
// delete the right point of the interval where we try to
// interpolate
moneynessGrid_.erase(moneynessGrid_.begin() + (i + 1));
k_.erase(k_.begin() + (i + 1));
c_.erase(c_.begin() + (i + 1));
cFunctions_.erase(cFunctions_.begin() + (i + 1));
rightIndex_--;
i--;
}
if (valid) {
ah(a);
boost::shared_ptr<cFunction> cFct(
new cFunction(ah.f_, ah.s_, a, ah.b_));
cFunctions_[leftIndex_ > 0 ? i - leftIndex_ + 1 : 0] = cFct;
cp0 = cp1;
}
}
}
// extrapolation of right wing
do {
success = true;
try {
Real k0 = k_[rightIndex_];
Real c0 = c_[rightIndex_];
Real cp0;
if (interpolate_)
cp0 = 0.5 * (c_[rightIndex_] - c_[rightIndex_ - 1]) /
(k_[rightIndex_] - k_[rightIndex_ - 1]);
else {
cp0 = (blackFormula(Option::Call, k0, f_,
sqrt(source_->variance(k0))) -
blackFormula(Option::Call, k0 - gap_, f_,
sqrt(source_->variance(k0 - gap_)))) /
gap_;
}
boost::shared_ptr<cFunction> cFct;
if (exponentialExtrapolation_) {
QL_REQUIRE(-cp0 / c0 > 0.0, "dummy"); // this is caught
// below
cFct = boost::shared_ptr<cFunction>(
new cFunction(-cp0 / c0, std::log(c0) - cp0 / c0 * k0));
} else {
sHelper sh(k0, c0, cp0);
Real s;
s = brent.solve(sh, QL_KAHALE_ACC, 0.20, 0.0,
QL_KAHALE_SMAX); // numerical parameters
// hardcoded here
sh(s);
cFct = boost::shared_ptr<cFunction>(
new cFunction(sh.f_, s, 0.0, 0.0));
}
cFunctions_[rightIndex_ - leftIndex_ + 1] = cFct;
}
catch (...) {
rightIndex_--;
success = false;
}
} while (!success && rightIndex_ > leftIndex_);
QL_REQUIRE(
leftIndex_ < rightIndex_,
"can not extrapolate to right, left index of af region reached ("
<< leftIndex_ << ")");
}
