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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2006 Warren Chou
Copyright (C) 2007, 2008 StatPro Italia srl
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<https://www.quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
/*! \file replicatingvarianceswapengine.hpp
\brief Replicating engine for variance swaps
*/
#ifndef quantlib_replicating_varianceswap_engine_hpp
#define quantlib_replicating_varianceswap_engine_hpp
#include <ql/exercise.hpp>
#include <ql/instruments/europeanoption.hpp>
#include <ql/instruments/varianceswap.hpp>
#include <ql/pricingengines/vanilla/analyticeuropeanengine.hpp>
#include <utility>
namespace QuantLib {
//! Variance-swap pricing engine using replicating cost,
/*! as described in Demeterfi, Derman, Kamal & Zou,
"A Guide to Volatility and Variance Swaps", 1999
\ingroup forwardengines
\test returned variances verified against results from literature
*/
class ReplicatingVarianceSwapEngine : public VarianceSwap::engine {
public:
typedef std::vector<std::pair<
ext::shared_ptr<StrikedTypePayoff>, Real> > weights_type;
// constructor
ReplicatingVarianceSwapEngine(ext::shared_ptr<GeneralizedBlackScholesProcess> process,
Real dk = 5.0,
const std::vector<Real>& callStrikes = std::vector<Real>(),
const std::vector<Real>& putStrikes = std::vector<Real>());
void calculate() const override;
protected:
// helper methods
void computeOptionWeights(const std::vector<Real>&,
Option::Type,
weights_type& optionWeights) const;
Real computeLogPayoff(Real, Real) const;
Real computeReplicatingPortfolio(
const weights_type& optionWeights) const;
Rate riskFreeRate() const;
DiscountFactor riskFreeDiscount() const;
Real underlying() const;
Time residualTime() const;
private:
ext::shared_ptr<GeneralizedBlackScholesProcess> process_;
Real dk_;
std::vector<Real> callStrikes_, putStrikes_;
};
// inline definitions
inline ReplicatingVarianceSwapEngine::ReplicatingVarianceSwapEngine(
ext::shared_ptr<GeneralizedBlackScholesProcess> process,
Real dk,
const std::vector<Real>& callStrikes,
const std::vector<Real>& putStrikes)
: process_(std::move(process)), dk_(dk), callStrikes_(callStrikes), putStrikes_(putStrikes) {
QL_REQUIRE(process_, "no process given");
QL_REQUIRE(!callStrikes.empty() && !putStrikes.empty(),
"no strike(s) given");
QL_REQUIRE(*std::min_element(putStrikes.begin(),putStrikes.end())>0.0,
"min put strike must be positive");
QL_REQUIRE(*std::min_element(callStrikes.begin(), callStrikes.end())==
*std::max_element(putStrikes.begin(), putStrikes.end()),
"min call and max put strikes differ");
}
inline void ReplicatingVarianceSwapEngine::computeOptionWeights(
const std::vector<Real>& availStrikes,
const Option::Type type,
weights_type& optionWeights) const {
if (availStrikes.empty())
return;
std::vector<Real> strikes = availStrikes;
// add end-strike for piecewise approximation
switch (type) {
case Option::Call:
std::sort(strikes.begin(), strikes.end());
strikes.push_back(strikes.back() + dk_);
break;
case Option::Put:
std::sort(strikes.begin(), strikes.end(), std::greater<>());
strikes.push_back(std::max(strikes.back() - dk_, 0.0));
break;
default:
QL_FAIL("invalid option type");
}
// remove duplicate strikes
auto last = std::unique(strikes.begin(), strikes.end());
strikes.erase(last, strikes.end());
// compute weights
Real f = strikes.front();
Real slope, prevSlope = 0.0;
for (auto k=strikes.begin();
// added end-strike discarded
k<strikes.end()-1;
++k) {
slope = std::fabs((computeLogPayoff(*(k+1), f) -
computeLogPayoff(*k, f))/
(*(k+1) - *k));
ext::shared_ptr<StrikedTypePayoff> payoff(
new PlainVanillaPayoff(type, *k));
if ( k == strikes.begin() )
optionWeights.emplace_back(payoff,slope);
else
optionWeights.emplace_back(payoff, slope - prevSlope);
prevSlope = slope;
}
}
inline Real ReplicatingVarianceSwapEngine::computeLogPayoff(
const Real strike,
const Real callPutStrikeBoundary) const {
Real f = callPutStrikeBoundary;
return (2.0/residualTime()) * (((strike - f)/f) - std::log(strike/f));
}
inline
Real ReplicatingVarianceSwapEngine::computeReplicatingPortfolio(
const weights_type& optionWeights) const {
ext::shared_ptr<Exercise> exercise(
new EuropeanExercise(arguments_.maturityDate));
ext::shared_ptr<PricingEngine> optionEngine(
new AnalyticEuropeanEngine(process_));
Real optionsValue = 0.0;
for (auto i = optionWeights.begin(); i < optionWeights.end(); ++i) {
ext::shared_ptr<StrikedTypePayoff> payoff = i->first;
EuropeanOption option(payoff, exercise);
option.setPricingEngine(optionEngine);
Real weight = i->second;
optionsValue += option.NPV() * weight;
}
Real f = optionWeights.front().first->strike();
return 2.0 * riskFreeRate() -
2.0/residualTime() *
(((underlying()/riskFreeDiscount() - f)/f) +
std::log(f/underlying())) +
optionsValue/riskFreeDiscount();
}
// calculate variance via replicating portfolio
inline void ReplicatingVarianceSwapEngine::calculate() const {
weights_type optionWeigths;
computeOptionWeights(callStrikes_, Option::Call, optionWeigths);
computeOptionWeights(putStrikes_, Option::Put, optionWeigths);
results_.variance = computeReplicatingPortfolio(optionWeigths);
DiscountFactor riskFreeDiscount =
process_->riskFreeRate()->discount(arguments_.maturityDate);
Real multiplier;
switch (arguments_.position) {
case Position::Long:
multiplier = 1.0;
break;
case Position::Short:
multiplier = -1.0;
break;
default:
QL_FAIL("Unknown position");
}
results_.value = multiplier * riskFreeDiscount * arguments_.notional *
(results_.variance - arguments_.strike);
results_.additionalResults["optionWeights"] = optionWeigths;
}
inline Real ReplicatingVarianceSwapEngine::underlying() const {
return process_->x0();
}
inline Time ReplicatingVarianceSwapEngine::residualTime() const {
return process_->time(arguments_.maturityDate);
}
inline Rate ReplicatingVarianceSwapEngine::riskFreeRate() const {
return process_->riskFreeRate()->zeroRate(residualTime(), Continuous,
NoFrequency, true);
}
inline
DiscountFactor ReplicatingVarianceSwapEngine::riskFreeDiscount() const {
return process_->riskFreeRate()->discount(residualTime());
}
}
#endif
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