re "BS call option price lower than intrinsic value - bug #3417114"

Posted by R Yan on Oct 03, 2011; 12:31pm
URL: http://quantlib.414.s1.nabble.com/re-BS-call-option-price-lower-than-intrinsic-value-bug-3417114-tp9130.html

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;

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