Implied Volatility difference between VanillaOption and DividendVanillaOption

[George Cowie]

Hello,

  I recently came across a discrepancy between the implied volatilities returned by a VanillaOption and a DividendVanillaOption.  I'm using identical inputs for both except for empty date/amount vectors being passed in for the latter.  I would expect my code (attached below) to return the same result in both cases, but I'm seeing differences (albeit small). 

 Here's the output:

'VanillaOption' Implied Volatility: 0.336308

'DividendVanillaOption' Implied Volatility: 0.335391

  In both cases, the impliedVolatility call is going to use the Crank Nicolson pricing model, but the VanillaOption instrument uses the FDAmericanEngine, whereas the DividendVanillaOption is using the FDDividendAmericanEngine.  

  - Has this topic been covered before?  If so, can someone point me to that discussion (didn't see it in the archive)

  - Shouldn't a VanillaOption have the same implied volatility as a DividendVanillaOption with no dividends and otherwise identical inputs?

Thanks,

George

  Here's the code:

//Built on Debian Wheezy, g++ 4.7.2, Boost 104900

#include <ql/quantlib.hpp>

int main(int argc, char** argv)

{

  QuantLib::Option::Type optionType = QuantLib::Option::Call ;

  double value = 1.00 ;

  double underlying = 100.0 ;

  double strike = 100.0 ;

  QuantLib::Spread dividendYield = 0.00 ;

  QuantLib::Rate riskFreeRate = 0.00 ;

  int length = 2 ;

  double volguess = 0.4 ;

  

  QuantLib::Date today = QuantLib::Date::todaysDate();

  QuantLib::Settings::instance().evaluationDate() = today;

  

  QuantLib::DayCounter dc = QuantLib::Actual360();

  boost::shared_ptr<QuantLib::SimpleQuote> spot(new QuantLib::SimpleQuote(underlying));

  boost::shared_ptr<QuantLib::SimpleQuote> vol(new QuantLib::SimpleQuote(volguess));

  

  boost::shared_ptr<QuantLib::BlackVolTermStructure> volTS = 

    boost::shared_ptr<QuantLib::BlackVolTermStructure>(

      new QuantLib::BlackConstantVol(today, 

     QuantLib::NullCalendar(), 

     QuantLib::Handle<QuantLib::Quote>(vol), dc));

  boost::shared_ptr<QuantLib::SimpleQuote> qRate(new QuantLib::SimpleQuote(dividendYield));

  boost::shared_ptr<QuantLib::YieldTermStructure> qTS = boost::shared_ptr<QuantLib::YieldTermStructure>(new QuantLib::FlatForward(today, QuantLib::Handle<QuantLib::Quote>(qRate), dc));

  boost::shared_ptr<QuantLib::SimpleQuote> rRate(new QuantLib::SimpleQuote(riskFreeRate));

  boost::shared_ptr<QuantLib::YieldTermStructure> rTS = boost::shared_ptr<QuantLib::YieldTermStructure>(new QuantLib::FlatForward(today, QuantLib::Handle<QuantLib::Quote>(rRate), dc));

  QuantLib::Date exDate = today + length;

  QuantLib::Settings::instance().evaluationDate() = today;

  

  boost::shared_ptr<QuantLib::Exercise> exercise(new QuantLib::AmericanExercise(today, exDate));

  boost::shared_ptr<QuantLib::StrikedTypePayoff> payoff(new QuantLib::PlainVanillaPayoff(optionType, strike));

  typedef QuantLib::BlackScholesMertonProcess BSMProcess; 

  boost::shared_ptr<QuantLib::GeneralizedBlackScholesProcess> process = 

    boost::shared_ptr<BSMProcess>(new BSMProcess(QuantLib::Handle<QuantLib::Quote>(spot),

QuantLib::Handle<QuantLib::YieldTermStructure>(qTS),

QuantLib::Handle<QuantLib::YieldTermStructure>(rTS),

QuantLib::Handle<QuantLib::BlackVolTermStructure>(volTS)));

  boost::shared_ptr<QuantLib::VanillaOption> vanillaOption(new QuantLib::VanillaOption(payoff, exercise)) ;

  double vanillaImplied = vanillaOption->impliedVolatility(value, process) ; 

  std::cout << "'VanillaOption' Implied Volatility: " << vanillaImplied << std::endl ;

  std::vector< QuantLib::Date > myDividendDates ;

  std::vector< QuantLib::Real > myDividendAmounts ;  

  boost::shared_ptr<QuantLib::DividendVanillaOption> dividendVanillaOption(new QuantLib::DividendVanillaOption(payoff, exercise,myDividendDates,myDividendAmounts)) ;

  double dividendVanillaImplied = dividendVanillaOption->impliedVolatility(value, process) ; 

  std::cout << "'DividendVanillaOption' Implied Volatility: " << dividendVanillaImplied << std::endl ;

  

  return 0 ;

}

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[Luigi Ballabio]

Hello,

    it's a bug. and an old one at that: see <https://sourceforge.net/p/quantlib/mailman/message/28749403/>. The FDDividendAmericanEngine gives the wrong price in your case, and of course this affects the implied-volatility calculation. Do you think you may have some time to look into it?

Later,

    Luigi

On Sat, Apr 26, 2014 at 10:10 PM, George Cowie <[hidden email]> wrote:

Hello,

  I recently came across a discrepancy between the implied volatilities returned by a VanillaOption and a DividendVanillaOption.  I'm using identical inputs for both except for empty date/amount vectors being passed in for the latter.  I would expect my code (attached below) to return the same result in both cases, but I'm seeing differences (albeit small). 

 Here's the output:

'VanillaOption' Implied Volatility: 0.336308

'DividendVanillaOption' Implied Volatility: 0.335391

  In both cases, the impliedVolatility call is going to use the Crank Nicolson pricing model, but the VanillaOption instrument uses the FDAmericanEngine, whereas the DividendVanillaOption is using the FDDividendAmericanEngine.  

  - Has this topic been covered before?  If so, can someone point me to that discussion (didn't see it in the archive)

  - Shouldn't a VanillaOption have the same implied volatility as a DividendVanillaOption with no dividends and otherwise identical inputs?

Thanks,

George

  Here's the code:

//Built on Debian Wheezy, g++ 4.7.2, Boost 104900

#include <ql/quantlib.hpp>

int main(int argc, char** argv)

{

  QuantLib::Option::Type optionType = QuantLib::Option::Call ;

  double value = 1.00 ;

  double underlying = 100.0 ;

  double strike = 100.0 ;

  QuantLib::Spread dividendYield = 0.00 ;

  QuantLib::Rate riskFreeRate = 0.00 ;

  int length = 2 ;

  double volguess = 0.4 ;

  

  QuantLib::Date today = QuantLib::Date::todaysDate();

  QuantLib::Settings::instance().evaluationDate() = today;

  

  QuantLib::DayCounter dc = QuantLib::Actual360();

  boost::shared_ptr<QuantLib::SimpleQuote> spot(new QuantLib::SimpleQuote(underlying));

  boost::shared_ptr<QuantLib::SimpleQuote> vol(new QuantLib::SimpleQuote(volguess));

  

  boost::shared_ptr<QuantLib::BlackVolTermStructure> volTS = 

    boost::shared_ptr<QuantLib::BlackVolTermStructure>(

      new QuantLib::BlackConstantVol(today, 

     QuantLib::NullCalendar(), 

     QuantLib::Handle<QuantLib::Quote>(vol), dc));

  boost::shared_ptr<QuantLib::SimpleQuote> qRate(new QuantLib::SimpleQuote(dividendYield));

  boost::shared_ptr<QuantLib::YieldTermStructure> qTS = boost::shared_ptr<QuantLib::YieldTermStructure>(new QuantLib::FlatForward(today, QuantLib::Handle<QuantLib::Quote>(qRate), dc));

  boost::shared_ptr<QuantLib::SimpleQuote> rRate(new QuantLib::SimpleQuote(riskFreeRate));

  boost::shared_ptr<QuantLib::YieldTermStructure> rTS = boost::shared_ptr<QuantLib::YieldTermStructure>(new QuantLib::FlatForward(today, QuantLib::Handle<QuantLib::Quote>(rRate), dc));

  QuantLib::Date exDate = today + length;

  QuantLib::Settings::instance().evaluationDate() = today;

  

  boost::shared_ptr<QuantLib::Exercise> exercise(new QuantLib::AmericanExercise(today, exDate));

  boost::shared_ptr<QuantLib::StrikedTypePayoff> payoff(new QuantLib::PlainVanillaPayoff(optionType, strike));

  typedef QuantLib::BlackScholesMertonProcess BSMProcess; 

  boost::shared_ptr<QuantLib::GeneralizedBlackScholesProcess> process = 

    boost::shared_ptr<BSMProcess>(new BSMProcess(QuantLib::Handle<QuantLib::Quote>(spot),

QuantLib::Handle<QuantLib::YieldTermStructure>(qTS),

QuantLib::Handle<QuantLib::YieldTermStructure>(rTS),

QuantLib::Handle<QuantLib::BlackVolTermStructure>(volTS)));

  boost::shared_ptr<QuantLib::VanillaOption> vanillaOption(new QuantLib::VanillaOption(payoff, exercise)) ;

  double vanillaImplied = vanillaOption->impliedVolatility(value, process) ; 

  std::cout << "'VanillaOption' Implied Volatility: " << vanillaImplied << std::endl ;

  std::vector< QuantLib::Date > myDividendDates ;

  std::vector< QuantLib::Real > myDividendAmounts ;  

  boost::shared_ptr<QuantLib::DividendVanillaOption> dividendVanillaOption(new QuantLib::DividendVanillaOption(payoff, exercise,myDividendDates,myDividendAmounts)) ;

  double dividendVanillaImplied = dividendVanillaOption->impliedVolatility(value, process) ; 

  std::cout << "'DividendVanillaOption' Implied Volatility: " << dividendVanillaImplied << std::endl ;

  

  return 0 ;

}

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[George Cowie]

Thanks for the info Luigi, will have a look this week and see if I can sort it out

On Apr 28, 2014, at 8:55 AM, Luigi Ballabio <[hidden email]> wrote:

Hello,

    it's a bug. and an old one at that: see <https://sourceforge.net/p/quantlib/mailman/message/28749403/>. The FDDividendAmericanEngine gives the wrong price in your case, and of course this affects the implied-volatility calculation. Do you think you may have some time to look into it?

Later,

    Luigi

On Sat, Apr 26, 2014 at 10:10 PM, George Cowie <[hidden email]> wrote:

Hello,

  I recently came across a discrepancy between the implied volatilities returned by a VanillaOption and a DividendVanillaOption.  I'm using identical inputs for both except for empty date/amount vectors being passed in for the latter.  I would expect my code (attached below) to return the same result in both cases, but I'm seeing differences (albeit small). 

 Here's the output:

'VanillaOption' Implied Volatility: 0.336308

'DividendVanillaOption' Implied Volatility: 0.335391

  In both cases, the impliedVolatility call is going to use the Crank Nicolson pricing model, but the VanillaOption instrument uses the FDAmericanEngine, whereas the DividendVanillaOption is using the FDDividendAmericanEngine.  

  - Has this topic been covered before?  If so, can someone point me to that discussion (didn't see it in the archive)

  - Shouldn't a VanillaOption have the same implied volatility as a DividendVanillaOption with no dividends and otherwise identical inputs?

Thanks,

George

  Here's the code:

//Built on Debian Wheezy, g++ 4.7.2, Boost 104900

#include <ql/quantlib.hpp>

int main(int argc, char** argv)

{

  QuantLib::Option::Type optionType = QuantLib::Option::Call ;

  double value = 1.00 ;

  double underlying = 100.0 ;

  double strike = 100.0 ;

  QuantLib::Spread dividendYield = 0.00 ;

  QuantLib::Rate riskFreeRate = 0.00 ;

  int length = 2 ;

  double volguess = 0.4 ;

  

  QuantLib::Date today = QuantLib::Date::todaysDate();

  QuantLib::Settings::instance().evaluationDate() = today;

  

  QuantLib::DayCounter dc = QuantLib::Actual360();

  boost::shared_ptr<QuantLib::SimpleQuote> spot(new QuantLib::SimpleQuote(underlying));

  boost::shared_ptr<QuantLib::SimpleQuote> vol(new QuantLib::SimpleQuote(volguess));

  

  boost::shared_ptr<QuantLib::BlackVolTermStructure> volTS = 

    boost::shared_ptr<QuantLib::BlackVolTermStructure>(

      new QuantLib::BlackConstantVol(today, 

     QuantLib::NullCalendar(), 

     QuantLib::Handle<QuantLib::Quote>(vol), dc));

  boost::shared_ptr<QuantLib::SimpleQuote> qRate(new QuantLib::SimpleQuote(dividendYield));

  boost::shared_ptr<QuantLib::YieldTermStructure> qTS = boost::shared_ptr<QuantLib::YieldTermStructure>(new QuantLib::FlatForward(today, QuantLib::Handle<QuantLib::Quote>(qRate), dc));

  boost::shared_ptr<QuantLib::SimpleQuote> rRate(new QuantLib::SimpleQuote(riskFreeRate));

  boost::shared_ptr<QuantLib::YieldTermStructure> rTS = boost::shared_ptr<QuantLib::YieldTermStructure>(new QuantLib::FlatForward(today, QuantLib::Handle<QuantLib::Quote>(rRate), dc));

  QuantLib::Date exDate = today + length;

  QuantLib::Settings::instance().evaluationDate() = today;

  

  boost::shared_ptr<QuantLib::Exercise> exercise(new QuantLib::AmericanExercise(today, exDate));

  boost::shared_ptr<QuantLib::StrikedTypePayoff> payoff(new QuantLib::PlainVanillaPayoff(optionType, strike));

  typedef QuantLib::BlackScholesMertonProcess BSMProcess; 

  boost::shared_ptr<QuantLib::GeneralizedBlackScholesProcess> process = 

    boost::shared_ptr<BSMProcess>(new BSMProcess(QuantLib::Handle<QuantLib::Quote>(spot),

QuantLib::Handle<QuantLib::YieldTermStructure>(qTS),

QuantLib::Handle<QuantLib::YieldTermStructure>(rTS),

QuantLib::Handle<QuantLib::BlackVolTermStructure>(volTS)));

  boost::shared_ptr<QuantLib::VanillaOption> vanillaOption(new QuantLib::VanillaOption(payoff, exercise)) ;

  double vanillaImplied = vanillaOption->impliedVolatility(value, process) ; 

  std::cout << "'VanillaOption' Implied Volatility: " << vanillaImplied << std::endl ;

  std::vector< QuantLib::Date > myDividendDates ;

  std::vector< QuantLib::Real > myDividendAmounts ;  

  boost::shared_ptr<QuantLib::DividendVanillaOption> dividendVanillaOption(new QuantLib::DividendVanillaOption(payoff, exercise,myDividendDates,myDividendAmounts)) ;

  double dividendVanillaImplied = dividendVanillaOption->impliedVolatility(value, process) ; 

  std::cout << "'DividendVanillaOption' Implied Volatility: " << dividendVanillaImplied << std::endl ;

  

  return 0 ;

}

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