[Quantlib-dev] re "BS call option price lower than intrinsic value - bug #3417114"

[R Y <fan...@gm...>]

The bug report was invalidated and closed too soon that I haven't been able
to put a comment on that one, so let me post it here...
The bug was that the result of BS() formula, c=BS(F,K,sigma,"call") for a
call option doesn't satisfy c>=F-K for certain K<<F (i.e. deep in the money
strike).

The issue is caused by floating point precision. That is true. But I was
hoping the dev team could dig a bit further for the cause but it seems I
have to do that myself and lay out the answer here...

*The root cause is that there is a flaw in the implementation of the
cumulative normal CDF function such that the function f(x) is not
monotonically increasing in x.*

I'm not an expert in numerical analysis so I don't know what's wrong exactly
with the implementation. But if we use the cdf function in boost::math, we
fix the problem. A sketch of code below (assuming discount factor 1.0):

  // Compute call value manually.
  double d1 = std::log(F/K)/(sigma*sqrt(T)) + 0.5*sigma*sqrt(T);
  double d2 = d1 - sigma*sqrt(T);
  boost::math::normal N;
  double nd1 = boost::math::cdf(N, d1);
  double nd2 = boost::math::cdf(N, d2);
  double c = F * nd1 - K * nd2;